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0704.1102

Spectral analysis for convolution operators on locally compact groups

arXiv:0704.1102v1 [math-ph] 9 Apr 2007 Spectral analysis for convolution operators on locally compact groups M. Măntoiu1 and R. Tiedra de Aldecoa2 1 Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P. O. Box 1-764, 014700 Bucharest, Romania 2 Département de mathématiques, Université de Cergy-Pontoise, 2, Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France E-mails: [email protected], [email protected] Abstract We consider operators Hµ of convolution with measures µ on locally compact groups. We characterize the spectrum of Hµ by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of Hµ, respectively. The proofs rely on commutator methods. Key words and phrases: locally compact group, convolution operator, positive commutator, point spec- trum, singular spectrum. 2000 Mathematics Subject Classification: 34L05, 81Q10, 44A35, 22D05 1 Introduction Any selfadjoint operator H in a Hilbert space H, with spectral measure EH and spectrum σ(H), is reduced by an orthogonal decomposition H = Hac(H)⊕Hp(H)⊕Hsc(H), that we briefly recall (cf. [30, Sec. 7.4]). Denote by Bor(R) the family of Borel subsets of R. Then, for any f ∈ H, one has the positive Borel measure H : Bor(R) → [0,∞), A 7→ ν H(A) := ‖EH(A)f‖ = 〈f, EH(A)f〉 . We say that f belongs to the spectral subspace Hp(H) if ν H is pure point, f belongs to the spectral subspace Hac(H) if ν H is absolutely continuous with respect to the Lebesgue measure, and f belongs to the spectral subspace Hsc(H) if ν H is singularly continuous with respect to the Lebesgue measure. One also uses the nota- tions Hc(H) := Hac(H)⊕Hsc(H) for the continuous subspace of H and Hs(H) := Hp(H)⊕Hsc(H) for the singular subspace of H . The sets σp(H) := σ H |Hp(H) , σac(H) := σ H |Hac(H) , σsc(H) := σ H |Hsc(H) σc(H) := σ(H |Hc(H)) and σs(H) := σ(H |Hs(H)) are called pure point spectrum, absolutely continuous spec- trum, continuous spectrum, and singular spectrum of H , respectively. An important issue in spectral theory consists in determining the above spectral subspaces or subsets for concrete selfadjoint operators. Under various assumptions this has been performed for important classes of op- erators: Schrödinger and more general partial differential operators, Toeplitz operators, Wiener-Hopf operators, and many others. Since the mathematical literature on this subject is considerable, it seems pointless to try to indicate references. In the present article we consider locally compact groups X , abelian or not, and convolution operators Hµ, acting on L 2(X), defined by suitable measures µ belonging to M(X), the Banach ∗-algebra of complex Radon measures on X . The case µ = χS , the characteristic function of a compact generating subset, leads to Hecke operators associated to the left regular representation (notice that our groups need not to be discrete). The precise definitions and statements are gathered in the next section. Essentially, our result consists in determining http://arxiv.org/abs/0704.1102v1 subspaces K1µ and K µ of L 2(X), explicitly defined in terms of µ and the family Hom(X,R) of continuous group morphisms Φ : X → R, such that Hp(Hµ) ⊂ K µ and Hs(Hµ) ⊂ K µ. The cases K µ = {0} or K µ = {0} are interesting; in the first case Hµ has no eigenvalues, and in the second case Hµ is purely absolutely continuous. The subspaces K1µ and K µ can be calculated explicitly only in very convenient situations. Rather often we are only able to show that they differ from H. In Section 3 we prove the results stated and discussed in Section 2. The proofs rely on a modification of a positive commutator technique called the method of the weakly conjugate operator. This method, an unbounded version of the Kato-Putnam theorem [26, Thm. XIII.28], developed and used in various situations [5, 6, 19, 23, 24, 25, 27], is recalled in Section 3.1. The last section is devoted to examples. We refer to [2, 3, 4, 7, 8, 9, 11, 15, 17, 18, 20, 21, 22, 29] for some related works on the spectral theory of operators on groups and graphs. Some of these articles put into evidence (Hecke-type) operators with large singular or singular continuous components. In [25], where analogous technics are used, one gets restrictions on the singular spectrum for adjacency operators on certain classes of graphs (which could be of non-Cayley type). 2 The main result We give in this section the statement of our main result for convolution operators on arbitrary locally compact groups (LCG). The reader is referred to [10, 16] for general information on the theory of LCG. 2.1 Statement of the main result Let X be a LCG with identity e, center Z(X) and modular function ∆. Let us fix a left Haar measure λ on X , using the notation dx := dλ(x). The associated right Haar measure ρ is defined by ρ(E) := λ(E−1) for each Borel subset E of X . Whenever X is compact, λ is normalized, i.e. λ(X) = 1. On discrete groups the counting measure (assigning mass 1 to every point) is considered. The notation a.e. stands for “almost everywhere” and refers to the Haar measure λ. The Lebesgue space Lp(X) ≡ Lp(X, dλ), 1 ≤ p ≤ ∞, of X with respect to λ is endowed with the usual norm ‖f‖p := dx |f(x)|p We are interested in convolution of functions by measures. Namely, we consider for every measure µ ∈ M(X) and every function f ∈ Lp(X), 1 ≤ p < ∞, the convolution of µ and f given (essentially) by (µ ∗ f)(x) := dµ(y) f(y−1x) for a.e. x ∈ X. It is known [16, Thm. 20.12] that µ ∗ f ∈ Lp(X) and that ‖µ ∗ f‖p ≤ ‖µ‖ ‖f‖p, where ‖µ‖ := |µ|(X) is the norm of the measure µ. Since we are mainly concerned with the hilbertian theory, we consider in the sequel the convolution operator Hµ, µ ∈ M(X), acting in the Hilbert space H := L 2(X): Hµf := µ ∗ f, f ∈ H. The operator Hµ is bounded with norm ‖Hµ‖ ≤ ‖µ‖, and it admits an adjoint operator H µ equal to Hµ∗ , the convolution operator by µ∗ ∈ M(X) defined by µ∗(E) = µ(E−1). If the measure µ is absolutely continuous w.r.t. the left Haar measure λ, so that dµ = a dλ with a ∈ L1(X), then µ∗ is also absolutely continuous w.r.t. λ and dµ∗ = a∗dλ, where a∗(x) := a(x−1)∆(x−1) for a.e. x ∈ X . In such a case we simply write Ha for Hadλ. We shall always assume that Hµ is selfadjoint, i.e. that µ = µ Let U(H) stands for the group of unitary operators in H and let L : X → U(H) be the left regular representation of X . Then Hµ is equal to the strong operator integral dµ(y)L(y), and µ 7→ Hµ is the integrated form of L. We recall that given two measures µ, ν ∈ M(X), their convolution µ∗ν ∈ M(X) is defined by the relation [10, Eq. 2.34] generalizing the usual convolution of L1-functions: d(µ ∗ ν)(x) g(x) := dµ(x)dν(y) g(xy) ∀g ∈ C0(X), where C0(X) denotes the C ∗-algebra of continuous complex functions on X decaying at infinity. The inequality ‖µ ∗ ν‖ ≤ ‖µ‖ ‖ν‖ holds. Given µ ∈ M(X), let ϕ : X → R be such that the linear functional F : C0(X) → C, g 7→ dµ(x)ϕ(x)g(x) is bounded. Then there exists a unique measure in M(X) associated to F , due to the Riesz-Markov representa- tion theorem. We write ϕµ for this measure, and we simply say that ϕ is such that ϕµ ∈ M(X). Let us call real character any continuous group morphism Φ : X → R. Their set forms a real vector space Hom(X,R), which can be infinite dimensional. Definition 2.1. Let µ = µ∗ ∈ M(X). (a) A real character Φ is semi-adapted to µ if Φµ,Φ2µ ∈ M(X), and (Φµ) ∗ µ = µ ∗ (Φµ). The set of real characters that are semi-adapted to µ is denoted by Hom1µ(X,R). (b) A real character Φ is adapted to µ if Φ is semi-adapted to µ,Φ3µ ∈ M(X), and (Φµ) ∗ (Φ2µ) = (Φ2µ) ∗ (Φµ). The set of real characters that are adapted to µ is denoted by Hom2µ(X,R). Let Kjµ := Φ∈Hom µ(X,R) ker(HΦµ), for j = 1, 2; then our main result is the following. Theorem 2.2. Let X be a LCG and let µ = µ∗ ∈ M(X). Then Hp(Hµ) ⊂ K µ and Hs(Hµ) ⊂ K A more precise result is obtained in a particular situation. Corollary 2.3. Let X be a LCG and let µ = µ∗ ∈ M(X). Assume that there exists a real character Φ adapted to µ such that Φ2 is equal to an nonzero constant on supp(µ). Then Hµ has a purely absolutely continuous spec- trum, with the possible exception of an eigenvalue located at the origin, with eigenspace ker(Hµ) = ker(HΦµ). Corollary 2.3 specially applies to adjacency operators on certain classes of Cayley graphs, which are Hecke-type operators in the regular representation, thus convolution operators on discrete groups. Remark 2.4. Using the method of the weakly conjugate operator, some extra results (as a Limiting Absorp- tion Principle, global smooth operators, perturbations of Hµ) can also be obtained. For simplicity we do not include them here, even if they can be inferred quite straightforwardly from [5] and [6]. Improvements in the assumptions are also possible, but with the cost of more complicated statements and proofs. Proposition 2.1 in [6] shows the generality of the method. 2.2 Comments and remarks (A) One obstacle in applying Theorem 2.2 is the fact that certain locally compact groups admit few nonzero real characters, maybe none. We say that x ∈ X is compact, and we write x ∈ B(X), if the closed subgroup generated by x is compact. If the order of x ∈ B(X) is finite, then x is clearly a compact element (but in non-discrete groups there could be others). Although B(X) is the union of all the compact subgroups of X , it is in general neither a subgroup, nor a closed set in X . We write B(X) for the closed subgroup generated by B(X). A continuous group morphism sends compact subgroups to compact subgroups. But the unique compact subgroup of R is {0}. Thus a real character on X annihilates B(X). It is not clear in general that the “smallness” of the vector space Hom(X,R) is related to a tendancy for convolution operators on X to have a substantial singular subspace, but for certain classes of groups this is indeed the case. For example, if X is compact, then X = B(X), Hom(X,R) = {0} and all operators Hµ, µ ∈ M(X), are pure point (see (B) below). (B) It is not at all exceptional for a convolution operator to have eigenvalues. For example, if a type I repre- sentation U is contained in the left regular representation L, then any function a ∈ L1(X) which is transformed by (the integrated form of) U into a compact operator will lead to a convolution operator Ha having eigenvalues. To consider just an exteme case, let us assume that X is a CCR group and that L is completely reducible. Then Ha can be written as a direct sum of compact operators, thus it has pure point spectrum. These conditions are fulfilled in the very particular case of compact groups. Actually, in this case, the irreducible representations are all finite-dimensional, so even convolution operators by elements of M(X) are pure point. (C) The occurrence in Theorem 2.2 of the subspaces Kjµ is not as mysterious as it could seem at first sight. For example, if µ = δe, then Φµ = 0 for any Φ ∈ Hom(X,R), so that K µ = H. Accordingly Hµ = 1, with spectrum composed of the single eigenvalue 1 with corresponding eigenspace H. Another simple example is obtained by considering X compact. On one hand the single real character is Φ = 0, with associated subspaces Kjµ = H for any µ = µ ∗ ∈ M(X). On the other hand we know from (B) that Hp(Hµ) is also equal to H. If the support of µ is contained in a subgroup Y of X with 0 < λ(Y ) < ∞, then a direct calculation shows that the associated characteristic function χY is an eigenvector of Hµ with eigenvalue µ(Y ). Actually, since (Φµ)(Y ) = 0 for any Φ ∈ Hom(X,R) with Φµ ∈ M(X), CχY is contained in ker(HΦµ). 3 Proof of the main result The proof of Theorem 2.2 relies on an abstract method, that we briefly recall in a simple form. 3.1 The method of the weakly conjugate operator The method of the weakly conjugate operator works for unbounded operators, but for our purposes it will be enough to assume H bounded. It also produces estimations on the boundary values of the resolvent and information on wave operators, but we shall only concentrate on spectral results. We start by introducing some notations. The symbol H stands for a Hilbert space with scalar product 〈·, ·〉 and norm ‖ · ‖. Given two Hilbert spaces H1 and H2, we denote by B(H1,H2) the set of bounded operators from H1 to H2, and put B(H) := B(H,H). We assume that H is endowed with a strongly continuous unitary group {Wt}t∈R. Its selfadjoint generator is denoted by A and has domain D(A). In most of the applications A is unbounded. Definition 3.1. A bounded selfadjoint operator H in H belongs to C1(A;H) if one of the following equivalent condition is satisfied: (i) the map R ∋ t 7→ W−tHWt ∈ B(H) is strongly differentiable, (ii) the sesquilinear form D(A)×D(A) ∋ (f, g) 7→ i 〈Hf,Ag〉 − i 〈Af,Hg〉 ∈ C is continuous when D(A) is endowed with the topology of H. We denote by B the strong derivative in (i) calculated at t = 0, or equivalently the bounded selfadjoint operator associated with the extension of the form in (ii). The operator B provides a rigorous meaning to the commutator i[H,A]. We shall write B > 0 if B is positive and injective, namely if 〈f,Bf〉 > 0 for all f ∈ H \ {0}. Definition 3.2. The operator A is weakly conjugate to the bounded selfadjoint operator H if H ∈ C1(A;H) and B ≡ i[H,A] > 0. For B > 0 let us consider the completionB of H with respect to the norm ‖f‖B := 〈f,Bf〉 . The adjoint space B∗ of B can be identified with the completion of BH with respect to the norm ‖g‖B∗ := g,B−1g One has then the continuous dense embeddings B∗ →֒ H →֒ B, and B extends to an isometric operator from B to B∗. Due to these embeddings it makes sense to assume that {Wt}t∈R restricts to a C0-group in B∗, or equivalently that it extends to a C0-group in B. Under this assumption (tacitly assumed in the sequel) we keep the same notation for these C0-groups. The domain of the generator of the C0-group in B (resp. B endowed with the graph norm is denoted by D(A,B) (resp. D(A,B∗)). In analogy with Definition 3.1 the requirement B ∈ C1(A;B,B∗) means that the map R ∋ t 7→ W−tBWt ∈ B(B,B ∗) is strongly differentiable, or equivalently that the sesquilinear form D(A,B)×D(A,B) ∋ (f, g) 7→ i 〈f,BAg〉 − i 〈Af,Bg〉 ∈ C is continuous when D(A,B) is endowed with the topology of B. Here, 〈·, ·〉 denotes the duality between B and Theorem 3.3. Assume that A is weakly conjugate to H and that B ≡ i[H,A] belongs to C1(A;B,B∗). Then the spectrum of H is purely absolutely continuous. Note that the method should be conveniently adapted when absolute continuity is expected only in a sub- space of the Hilbert space. This is the case considered in the sequel. 3.2 Proof of Theorem 2.2 In this section we construct suitable weakly conjugate operators in the framework of section 2, and we prove our main result. For that purpose, let us fix a real character Φ ∈ Hom(X,R) and a measure µ = µ∗ ∈ M(X). We shall keep writing Φ for the associated operator of multiplication in H. In most of the applications this operator is unbounded; its domain is equal to D(Φ) ≡ {f ∈ H | Φf ∈ H}. One ingredient of our approach is the fact that multiplication by morphisms behaves like a derivation with respect to the convolution product: for suitable functions or measures f, g : X → C, one has Φ(f ∗ g) = (Φf)∗g+f∗(Φg). Using this observation we show in the next lemma that the commutator i[Hµ,Φ] (constructed as in Definition 3.1) is related to the operator HΦµ. This provides a partial explanation of our choice of the “semi-adapted” and “adapted” conditions. Lemma 3.4. (a) If Φ is semi-adapted to µ, then Hµ ∈ C1(Φ;H), and i[Hµ,Φ] = −iHΦµ ∈ B(H). Simi- larly, −iHΦµ ∈ C 1(Φ,H), and i[−iHΦµ,Φ] = −HΦ2µ ∈ B(H). Moreover, the equality [Hµ, HΦµ] = 0 holds. (b) If Φ is adapted to µ, then −HΦ2µ ∈ C 1(Φ,H), and the equality [HΦµ, HΦ2µ] = 0 holds. Proof. (a) Let Φ be semi-adapted to µ and let f ∈ D(Φ). Then one has µ,Φµ ∈ M(X) and f,Φf ∈ H. Thus µ ∗ f ∈ D(Φ), and the equality Φ(Hµf) = (Φµ) ∗ f + µ ∗ (Φf) holds in H. It follows that i(HµΦ−ΦHµ) is well-defined on D(Φ) and is equal to −iHΦµ. Hence Condition (ii) of Definition 3.1 is fulfilled. The proof that −iHΦµ belongs toC 1(Φ,H) and that i[−iHΦµ,Φ] = −HΦ2µ is similar. Finally the equality [Hµ, HΦµ] = 0 is clearly equivalent to the requirement (Φµ) ∗ µ = µ ∗ (Φµ). (b) The proof is completely analogous to that of point (a). If Φ is semi-adapted to µ, we set K := i[Hµ,Φ] = −iHΦµ and L := i[K,Φ] = i[−iHΦµ,Φ] = −HΦ2µ (for the sake of simplicity, we omit to write the dependence of these operators in Φ and µ). The first part of the previous lemma states that Hµ and K belongs to C 1(Φ;H). In particular, it follows that K leaves invariant the domain D(Φ), and the operator A := 1 (ΦK +KΦ) is well-defined and symmetric on D(Φ). Similarly, if Φ is adapted to µ, the second part of the lemma states that L belongs to C1(Φ;H). Therefore the operator L leaves D(Φ) invariant, and the operator A′ := 1 (ΦL+ LΦ) is well-defined and symmetric on D(Φ). Lemma 3.5. (a) If Φ is semi-adapted to µ, then the operator A is essentially selfadjoint on D(Φ). The domain of its closure A is D(A) = D(ΦK) = {f ∈ H | ΦKf ∈ H} and A acts on D(A) as the operator ΦK − i (b) If Φ is adapted to µ, then the operator A′ is essentially selfadjoint on D(Φ). The domain of its closure A′ is D(A′) = D(ΦL) = {f ∈ H | ΦLf ∈ H}. Proof. One just has to reproduce the proof of [11, Lemma 3.1], replacing their couple (N,S) by (Φ,K) for the point (a) and by (Φ, L) for the point (b). In the next lemma we collect some results on commutators with A or A′. The commutation relations exhibited in Lemma 3.4, i.e. [Hµ,K] = 0 if Φ is semi-adapted to µ and [K,L] = 0 if Φ is adapted to µ, are essential. Lemma 3.6. If Φ is semi-adapted to µ, then (a) The quadratic form D(A) ∋ f 7→ i 〈Hµf,Af〉 − i 〈Af,Hµf〉 extends uniquely to the bounded form defined by the operator K2, (b) The quadratic form D(A) ∋ f 7→ i K2f,Af Af,K2f extends uniquely to the bounded form defined by the operator KLK + 1 K2L+ LK2 (which reduces to 2KLK if Φ is adapted to µ), (c) If Φ is adapted to µ, then the quadratic form D(A′) ∋ f 7→ i 〈Kf,A′f〉 − i 〈A′f,Kf〉 extends uniquely to the bounded form defined by the operator L2. The proof is straightforward. Computations may be performed on the core D(Φ). These results imply that Hµ ∈ C 1(A;H), K2 ∈ C1(A;H) and (when Φ is adapted) K ∈ C1(A′;H). Using these results we now establish a relation between the kernels of the operators Hµ, K and L. Lemma 3.7. If Φ is semi-adapted to µ, then one has ker(Hµ) ⊂ Hp(Hµ) ⊂ ker(K) ⊂ Hp(K). If Φ is adapted to µ, one also has Hp(K) ⊂ ker(L) ⊂ Hp(L). Proof. Let f be an eigenvector of Hµ. Due to the Virial Theorem [1, Proposition 7.2.10] and the fact that Hµ belongs to C1(A;H), one has 〈f, i[Hµ, A]f〉 = 0. It follows by Lemma 3.6.(a) that 0 = f,K2f = ‖Kf‖2, i.e. f ∈ ker(K). The inclusion Hp(Hµ) ⊂ ker(K) follows. Similarly, by using A ′ instead of A and Lemma 3.6.(c) one gets (when Φ is adapted) the inclusion Hp(K) ⊂ ker(L), and the lemma is proved. Assume now that Φ is semi-adapted to µ. Then we can decompose the Hilbert space H into the direct sum H = K ⊕ G, where K := ker(K) and G is the closure of the range KH. It is easy to see that Hµ and K are reduced by this decomposition and that their restrictions to the Hilbert space G are bounded selfadjoint operators. In the next lemma we prove that this decomposition of H also reduces the operator A if Φ is adapted to µ. Lemma 3.8. If Φ is adapted to µ, then the decomposition H = K ⊕ G reduces the operator A. The restriction of A to G defines a selfadjoint operator denoted by A0. Proof. We already know that on D(A) = D(ΦK) one has A = ΦK − i L. By using Lemma 3.7 it follows that K ⊂ ker(A) ⊂ D(A). Then one trivially checks that (i) A [K ∩ D(A)] ⊂ K, (ii) A [G ∩ D(A)] ⊂ G and (iii) D(A) = [K ∩ D(A)] + [G ∩ D(A)], which means that A is reduced by the decomposition H = K⊕G. Thus by [30, Theorem 7.28] the restriction of A to D(A0) ≡ D(A) ∩ G is selfadjoint in G. Proof of Theorem 2.2. We know from Lemma 3.7 that Hp(Hµ) ⊂ ker(−iHΦµ) for each Φ ∈ Hom µ(X,R). This obviously implies the first inclusion of the theorem. Let us denote by H0 and K0 the restrictions to G of the operators Hµ and K . We shall prove in points (i)-(iii) below that if Φ is adapted to µ, then the method of the weakly conjugate operator, presented in Section 3.1, applies to the operators H0 and A0 in the Hilbert space G. It follows then that G ⊂ Hac(Hµ), and a fortiori that Hsc(Hµ) ⊂ K = ker(−iHΦµ). Since this result holds for each Φ ∈ Hom µ(X,R), the second inclusion of the theorem follows straightforwardly. (i) Lemma 3.6.(a) implies that i(H0A0 −A0H0) is equal in the form sense to K 0 on D(A0) ≡ D(A) ∩ G. Therefore the corresponding quadratic form extends uniquely to the bounded form defined by the operator K20 . This implies that H0 belongs to C 1(A0;G). (ii) Since B0 := i[H0, A0] ≡ K 0 > 0 in G, the operator A0 is weakly conjugate to H0. So we define the space B as the completion of G with respect to the norm ‖f‖B := 〈f,B0f〉 . The adjoint space of B is denoted by B∗ and can be identified with the completion of B0G with respect to the norm ‖f‖B∗ := f,B−10 f can also be expressed as the closure of the subspace KH = K0G with respect to the same norm ‖f‖B∗ =∥∥|K0|−1f ∥∥. Due to Lemma 3.6.(b) the quadratic form D(A0) ∋ f 7→ i 〈B0fA0f〉 − i 〈A0f,B0f〉 extends uniquely to the bounded form defined by the operator 2K0L0K0, where L0 is the restriction of L to G. We write i[B0, A0] for this extension, which clearly defines an element of B(B,B (iii) Let {Wt}t∈R be the unitary group in G generated by A0. We check now that this group extends to a C0-group in B. This easily reduces to proving that for any t ∈ R there exists a constant C(t) ≥ 0 such that ‖Wtf‖B ≤ C(t)‖f‖B for all f ∈ D(A0). Due to point (ii) one has for each f ∈ D(A0) ‖Wtf‖ B = 〈f,B0f〉+ dτ 〈Wτf, i[B0, A0]Wτf〉 ≤ ‖f‖ B + 2‖L0‖ ∫ |t| dτ ‖Wτf‖ Since G →֒ B, the function (0, |t|) ∋ τ 7→ ‖Wτf‖ B ∈ R is bounded. Thus we get the inequality ‖Wtf‖B ≤ e|t|‖L0‖ ‖f‖B by using a simple form of the Gronwall Lemma. Therefore {Wt}t∈R extends to a C0-group in B, and by duality {Wt}t∈R also defines a C0-group in B ∗. This concludes the proof of the fact that B0 extends to an element of C1(A0;B,B ∗). Thus all hypotheses of Theorem 3.3 are satisfied and this gives the result. Proof of Corollary 2.3. Since L = −HΦ2µ is proportional with Hµ, one has ker(Hµ) = Hp(Hµ) ⊂ Hs(Hµ) ⊂ ker(HΦµ) due to Lemma 3.7 and Theorem 2.2. Using this with µ replaced by iΦµ, one easily gets the identity ker(HΦµ) = ker(Hµ). Therefore ker(Hµ) = Hp(Hµ) = Hs(Hµ) = ker(HΦµ), and the claim is proved. 4 Examples 4.1 Perturbations of central measures In this section, we exploit commutativity in a non-commutative setting by using central measures. The group X is assumed to be unimodular. By definition, the central measures are the elements of the center Z[M(X)] of the convolution Banach ∗-algebra M(X). They can be characterized by the condition µ(yEy−1) = µ(E) for any y ∈ X and any Borel set E ⊂ X . The central (or class) functions are the elements of Z[M(X)] ∩ L1(X) = Z[L1(X)]. Thus a characteristic function χE is central iff λ(E) < ∞ and E is invariant under all inner automorphisms. The relevant simple facts are the following: if µ is central, Φ ∈ Hom(X,R) and Φµ ∈ M(X), then Φµ is also central (this follows from the identity Φ(yxy−1) = Φ(y)+Φ(x)−Φ(y) = Φ(x), ∀x, y ∈ X). On the other hand, if µ is arbitrary but supported on B(X), then Φµ = 0 for any real character Φ. Thus all the commutation relations in Definition 2.1 are satisfied, and one gets from Theorem 2.2 the following result: Corollary 4.1. Let X be a unimodular LCG, let µ0 = µ∗0 ∈ M(X) be a central measure, and let µ1 = µ M(X) with supp(µ1) ⊂ B(X). Then Hp(Hµ0+µ1) ⊂ Φ∈Hom(X,R) Φµ0,Φ 2µ0∈M(X) ker(HΦµ0) Hs(Hµ0+µ1) ⊂ Φ∈Hom(X,R) Φµ0,Φ 2µ0,Φ 3µ0∈M(X) ker(HΦµ0 ). In order to get more explicit results, we restrict ourselves in the next section to a convenient class of LCG, generalizing both abelian and compact groups. 4.2 Convolution operators on central groups Following [12], we say that X is central (or of class [Z]) if the quotient X/Z(X) is compact. Central groups possess a specific structure [12, Thm. 4.4]: If X is central, then X isomorphic to a direct productRd×H , where H contains a compact open normal subgroup. Proposition 4.2. Let X be a central group and µ0 = µ∗0 ∈ M(X) a central measure such that supp(µ0) is compact and not included in B(X). Let µ1 = µ 1 ∈ M(X) with supp(µ1) ⊂ B(X) and set µ := µ0 + µ1. Then Hac(Hµ) 6= {0}. Proof. Central groups are unimodular [13, Prop. p. 366], and Φµ,Φ2µ,Φ3µ ∈ M(X) for any Φ ∈ Hom(X,R), due to the hypotheses. Furthermore we know by [12, Thm. 5.7] that B(X) = B(X) is a closed normal subgroup of X and that X/B(X) is isomorphic to the direct product Rd ×D, where D is a discrete torsion-free abelian group. But the groups Rd ×D are exactly those for which the real characters separate points [12, Cor. p. 335]. Therefore for any x ∈ supp(µ0) \ B(X) there exists Φ ∈ Hom(X,R) such that Φ(x) 6= 0. Thus HΦµ is a nonzero convolution operator, and the claim follows by Corollary 4.1. In a central groupX there exists plenty of central compactly supported measures. For instance there always exists in X [12, Thm. 4.2] a neighbourhood base of e composed of compact sets S = S−1 which are invariant (under the inner automorphisms), i.e. central groups belong to the class [SIN]. Therefore the measures µ0 = χS dλ satisfy µ0 = µ 0 and are subject to Proposition 4.2. Actually this also applies to central characteristic functions χS with “large” S, since in X any compact set is contained in a compact invariant neighbourhood of the identity [13, Lemma p. 365]. One can also exihibit central measures satisfying Proposition 4.2 defined by continuous functions. Indeed we know by [13, Thm. 1.3] that for any neighbourhood U of the identity e of a central group X there exists a non-negative continuous central function aU , with supp(aU ) ⊂ U and aU (e) > 0. A simple way to construct examples is as follows. Let X := K×Y , where K is a compact group with Haar measure λK and Y is an abelian LCG with Haar measure λY . Clearly X is central and B(X) = K × B(Y ). Let E be a finite family of invariant subsets of K such that each E ∈ E satisfies λK(E) > 0 and E −1 ∈ E . For each E ∈ E , let IE be a compact subset of Y such that λY (IE) > 0 and (IE) −1 = IE−1 . Suppose also that IE0 is not a subset of B(Y ) for some E0 ∈ E . Then one easily shows that the set S := E∈E E × IE satisfies the following properties: S is compact, S = S−1, S is invariant, and S not included in B(X). Thus Hac(HχS+µ1) 6= {0} for any µ1 = µ 1 ∈ M(X) with supp(µ1) ⊂ K × B(Y ), due to Proposition 4.2. The following two examples are applications of the preceding construction. Example 4.3. Let X := S3 ×Z, where S3 is the symmetric group of degree 3. The group S3 has a presentation〈 a, b | a2, b2, (ab)3 , and its conjugacy classes are E1 = E 1 = {e}, E2 = E 2 = {a, b, aba} and E3 = E−13 = {ab, ba}. Set E := {E2, E3} and choose IE1 , IE2 two finite symetric subsets of Z, each of them containing at least two elements. Clearly these sets satisfy all the requirements of the above construction. Thus Hac(HχS ) 6= {0} if S := E∈E E × IE . Example 4.4. Let X = SU(2)×R, where SU(2) is the group (with Haar measure λ2) of 2×2 unitary matrices of determinant +1. For each ϑ ∈ [0, π] let C(ϑ) be the conjugacy class of the matrix diag(eiϑ, e−iϑ) in SU(2). A direct calculation (using for instance Euler angles) shows that λ2 ϑ∈J C(ϑ) > 0 for each J ⊂ [0, π] with nonzero Lebesgue measure. Set E1 := ϑ∈(0,1) C(ϑ), E2 := ϑ∈(2,π)C(ϑ), E := {E1, E2}, IE1 := (−1, 1), and IE2 := (−3,−2) ∪ (2, 3). Clearly these sets and many others satisfy all the requirements of the above construction. Thus Hac(HχS ) 6= {0} if S := E∈E E × IE . A nice example of a central group which is not the product of a compact and an abelian group can be found in [14, Ex. 4.7]. In a simple situation one even gets purely absolutely continuous operators; this should be compared with the discussion in Section 2.2: Example 4.5. Let X be a central group, let z ∈ Z(X) \ B(X), and set µ := δz + δz−1 + µ1 for some µ1 = µ 1 ∈ M(X) with supp(µ1) ⊂ B(X). Then µ satisfies the hypotheses of Proposition 4.2, and we can choose Φ ∈ Hom(X,R) such that Φ(z) = 1 Φ(z2) 6= 0 (note in particular that z /∈ B(X) iff z2 /∈ B(X) and that Φµ1 = 0). Thus Hs(Hµ) ⊂ ker(HΦµ). But f ∈ H belongs to ker(HΦµ) = ker HΦ(δz+δz−1) f(z−1x) = f(zx) for a.e. x ∈ X . This periodicity w.r.t. the non-compact element z2 easily implies that the 2-function f should vanish a.e. and thus that Hac(Hµ) = H. 4.3 Abelian groups We consider in this section the case of locally compact abelian groups (LCAG), whose theory can be found in the monograph [16]. LCAG are particular cases of central groups. Their convolution algebra M(X) is abelian, so spectral results on convolution operators can be deduced from the preceding section. We shall not repete them here, but rather invoke duality to obtain properties of a class of multiplication operators on the dual group Let X stands for a LCAG with elements x, y, z, . . ., and let X̂ be the dual group of X , i.e. the set of characters of X endowed with the topology of compact convergence on X . The elements of X̂ are denoted by ξ, η, ζ, . . . and we shall use the notation 〈x, ξ〉 for the expression ξ(x). The Fourier transform m of a measure µ ∈ M(X) is given by m(ξ) ≡ [F (µ)](ξ) := dµ(x) 〈x, ξ〉, ξ ∈ X̂. We recall from [16, Thm. 23.10] that m belongs to the C∗-algebra BC(X̂) of bounded continuous complex functions on X̂ , and that ‖m‖∞ ≤ ‖µ‖ (showing that the bound ‖Hµ‖ ≤ ‖µ‖ is not optimal in general). Actually the subspace F (M(X)) is dense in BC(X̂), and the subspace F (L1(X)) is densely contained in C0(X̂), the ideal of BC(X̂) composed of continuous complex functions on X̂ vanishing at infinity. For a suitably normalized Haar measure on X̂ , the Fourier transform also defines a unitary isomorphism from H onto 2(X̂), which we denote by the same symbol. It maps unitarily Hµ on the operator Mm of multiplication with m = F (µ). Moreover µ = µ∗, iff m is real, and σ(Hµ) = σ(Mm) = m(X̂), σp(Hµ) = σp(Mm) = {s ∈ R | λ∧ (m−1(s)) > 0}, where λ∧ is any Haar measure on X̂ . This does not solve the problem of determining the nature of the spectrum, at least for three reasons. First, simple or natural conditions on µ could be obscured when using the Fourier transform; the function m = F (µ) could be difficult to compute or to evaluate. Second, the dual group X̂ can be complicated. We are not aware of general results on the nature of the spectrum of multiplication operators on LCAG. Third, even for X̂ = Rd, the spectral theory of multiplication operators is quite subtle. For the particular case X̂ = Rd, one finds in [1, Sec. 7.1.4 & 7.6.2] refined results both on the absolute continuity and on the occurence of singular continuous spectrum for multiplication operators. Let us recall that there is an almost canonical identification ofHom(X,R)with the vector spaceHom(R, X̂) of all continuous one-parameter subgroups of X̂ . For a given real character Φ, we denote by ϕ ∈ Hom(R, X̂) the unique element satisfying 〈x, ϕ(t)〉 = eitΦ(x), ∀t ∈ R, x ∈ X. Definition 4.6. The function m : X̂ → C is differentiable at ξ ∈ X̂ along the one-parameter subgroup ϕ ∈ Hom(R, X̂) if the function R ∋ t 7→ m(ξ + ϕ(t)) ∈ C is differentiable at t = 0. In such a case we write (dϕm) (ξ) for m(ξ + ϕ(t)) . Higher order derivatives, when existing, are denoted by dkϕm, k ∈ N. This definition triggers a formalism which has some of the properties of the differential calculus on Rd. However a differentiable function might not be continuous. Moreover, if X̂ is totally disconnected, then the theory is trivial: Every complex function defined on X̂ is differentiable with respect to the single trivial element of Hom(R, X̂), and the derivative is always zero. If µ ∈ M(X) is such that Φµ ∈ M(X), then [28, p. 68] m = F (µ) is differentiable at any point ξ along the one-parameter subgroup ϕ and −iF (Φµ) = dϕm. Let us fix a bounded continuous function m : X̂ → R such that F−1(m) ∈ M(X). We say that the one-parameter subgroup ϕ : R → X̂ is in Hom1m(R, X̂) if m is twice differentiable w.r.t. ϕ and dϕm, d F (M(X)). If, in addition, m is thrice differentiable w.r.t. ϕ and d3ϕm ∈ F (M(X)) too, we say that ϕ belongs to Hom2m(R, X̂). Then next result follows directly from Corollary 4.1. Corollary 4.7. Let X be a LCAG and let m0,m1 be real functions with F−1(m0),F−1(m1) ∈ M(X) and supp(F−1(m1)) ⊂ B(X). Then Hp(Mm0+m1) ⊂ ϕ∈Hom1m0 (R, bX) ker(Mdϕm0) Hs(Mm0+m1) ⊂ ϕ∈Hom2m0 (R, bX) ker(Mdϕm0). It is worth noting that for X abelian, the following assertions are equivalent [16, Thm. 24.34 & Cor. 24.35]: (i) B(X) = {e}, (ii) the real characters separate points, (iii) the dual group X̂ is connected, (iv)X is isomorphic to Rd ×D, where D is a discrete torsion-free abelian group. Up to our knowledge, Corollary 4.7 is not known in the present generality. It is a by-product of a theory working in a non-commutative framework and it is obviously far from being optimal. We hope to treat the spec- tral analysis of (unbounded) multiplication operators on LCAG in greater detail in a forthcoming publication. One may interpret our use of Hom(X,R) in Theorem 2.2 as an attempt to involve “smoothness” and “derivatives” in spectral theory for groups which might not be abelian or might not have a given Lie structure. 4.4 Semidirect products Let N,G be two discrete groups with G abelian (for which we use additive notations), and let τ : G → Aut(N) be a group morphism. Let X := N ×τ G be the τ -semidirect produt of N by G. The multiplication in X is defined by (n, g)(m,h) := (nτg(m), g + h), so that (n, g)−1 = (τ−g(n −1),−g). In the sequel we only consider real characters Φ ∈ Hom(X,R) of the form Φ = φ ◦ π, where φ ∈ Hom(G,R) and π : X → G is the canonical morphism given by π(n, g) := g. Proposition 4.8. Let a0 = a∗0 : X → C have a finite support and satisfy n1,n2∈N m=n1τg1 (n2) a0(n1, g1)a0(n2, g2) = n1,n2∈N m=n1τg2(n2) a0(n1, g2)a0(n2, g1), ∀g1, g2 ∈ G, ∀m ∈ N. (4.1) Let a1 = a 1 : X → C have a finite support contained in B(X) and be such that a1 ∗ a0 = a0 ∗ a1. Then Hs(Ha0+a1) ⊂ φ∈Hom(G,R) H(φ◦π)a0 Proof. Since a := a0 + a1 has a finite support, we only have to check that Φa ∗ a− a ∗ Φa = Φa ∗ Φ2a− Φ2a ∗ Φa = 0 for any Φ = φ ◦ π, φ ∈ Hom(G,R). (4.2) Since a1 ∗ a0 = a0 ∗ a1 and Φa1 = 0 for each Φ ∈ Hom(X,R), we are easily reduced to check (4.2) only for a0. Let (m,h) ∈ X ; then a direct calculation gives (Φa0 ∗ a0 − a0 ∗ Φa0)(m,h) = φ(2g − h) n1,n2∈N m=n1τg(n2) a0(n1, g)a0(n2, h− g). This leads to the identity (Φa0 ∗ a0 − a0 ∗ Φa0)(m,h) = −(Φa0 ∗ a0 − a0 ∗ Φa0)(m,h) by using Condition (4.1) and the change of variable g′ := h− g. Thus Φa0 ∗ a0 − a0 ∗ Φa0 = 0. By a similar argument one obtains that Φa0 ∗ Φ 2a0 − Φ 2a0 ∗ Φa0 = 0 (the extra factor involved in this computation is symmetric with respect to the change of variables). The proposition tells us that Ha has a non-trivial absolutely continuous component if there exists a real character φ ∈ Hom(G,R) such that (φ ◦ π)a 6= 0. Consequently, as soon as supp(a) is not included in N × B(G), we are done. For instance if G = Zd, we simply have to ask for the existence of an element (n, g) ∈ supp(a) with g 6= 0. In the remaining part we indicate several situations to which Proposition 4.8 applies; the perturbation a1 is left apart for simplicity. Procedure 4.9. Let G0 be a finite subset of G such that G0 = −G0. For each g ∈ G0 let Ng be a finite subset of N such that τg = N−1g for each g ∈ G0. Set Ng × {g}. This is a convenient description of the most general finite subset of X satisfying S−1 = S (which is equivalent to χS = χ S). Condition (4.1) amounts to #{(n1, n2) ∈ Ng1 ×Ng2 | m = n1τg1(n2)} = #{(n1, n2) ∈ Ng2 ×Ng1 | m = n1τg2(n2))} (4.3) for each g1, g2 ∈ G0 and m ∈ N . Under these assumptions Proposition 4.8 applies and HχS has a non-trivial absolutely continuous component if G0 ∩ [G \ B(G)] 6= ∅. One can ensure in various situations that S is a system of generators (which is needed to assign a connected Cayley graph to (X,S)). This happens, for instance, if G0 generates G, ∪g∈G0Ng generates N and the unit e of N belongs to Ng for each g ∈ G0. Example 4.10. Let N := S3 = a, b | a2, b2, (ab)3 , G := Z, G0 := {−1, 1},N−1 := {a, aba},N1 := {a, b}, and τg(n) := a gna−g for each g ∈ Z and n ∈ S3. Then direct calculations show that all the assumptions in Pro- cedure 4.9 are verified. Hence Hac(HχS ) 6= {0} for X := S3 ×τ Z if S = {(a,−1), (aba,−1), (a, 1), (b, 1)}. Actually, by applying Corollary 2.3 with Φ(n, g) := g, one finds that the single possible component of the sin- gular spectrum of HχS is an eigenvalue located at the origin. However a careful inspection shows us that HχS is injective, and thus that Hac(HχS ) = H. A rather simple (but not trivial) possibility consists in takingNg ≡ N0 independent on g ∈ G0 in Procedure 4.9. In such a case S = N0 ×G0 and χS = χN0 ⊗ χG0 (which does not implies in general that HχS is a tensor product of operators). If we also assume that N0 is invariant under the set {τg | g ∈ G0}, then all the necessary assumptions are satisfied. For instance, the invariance of N0 permits to define a bijection between the two sets in Formula (4.3). A particular case would be to choose G := Zd, with G0 := {(±1, . . . , 0), . . . , (0, . . . ,±1)}. By using the real morphism φ : Zd → R defined by φ(±1, . . . , 0) =: ±1, . . . , φ(0, . . . ,±1) =: ±1, one can apply Corollary 2.3 to conclude that, except the possible eigenvalue 0, HχS is purely absolutely continuous. The following two examples are applications of the preceding construction. Example 4.11. We consider a simple type of wreath product. Take G a discrete abelian group and put N := RJ , where R is an arbitrary discrete group and J is a finite set on which G acts by (g, j) 7→ g(j). Then {rj}j∈J := {rg(j)}j∈J defines an action of G on R J , thus we can construct the semidirect product RJ ×τ G. If G0 = −G0 ⊂ G and R0 = R 0 ⊂ R are finite subsets with G0 ∩ [G \ B(G)] 6= ∅, then N0 := R satisfies all the required conditions. Thus Hac(HχS ) 6= {0} if S := N0 ×G0. Example 4.12. Let G be a discrete abelian group, let N be the free group generated by the family {a1, . . . , an}, and set N0 := a±11 , . . . , a . Choose a finite set G0 = −G0 ⊂ G with G0 ∩ [G \B(G)] 6= ∅ and an action τ on N such that the conditions on S := N0×G0 are satisfied (for instance τg , g ∈ G0, may act by permutation on the generators). Then HχS has a non-trivial absolutely continuous part. Virtually the methods of this article could also be applied to non-split group extensions. Acknowledgements M.M. acknowledges support from the contract 2-CEx 06-11-34. R.T.d.A. thanks the Swiss National Science Foundation for financial support. Part of this work was done while M.M. was visiting the University of Paris XI. He would like to thank Professor B. Helffer for his kind hospitality. Both authors are grateful to V. Georgescu and S. Richard for useful discussions. References [1] W. O. Amrein, A. Boutet de Monvel, and V. Georgescu. C0-groups, commutator methods and spectral theory of N -body Hamiltonians, volume 135 of Progress in Math. Birkhäuser, Basel, 1996. [2] L. Bartholdi and R. I. Grigorchuk. On the spectrum of Hecke type operators related to some fractal groups. Tr. Mat. Inst. Steklova, 231(Din. Sist., Avtom. i Beskon. Gruppy):5–45, 2000. [3] L. Bartholdi and W. Woess. Spectral computations on lamplighter groups and Diestel-Leader graphs. J. Fourier Anal. Appl., 11(2):175–202, 2005. [4] C. Béguin, A. Valette, and A. Zuk. On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper’s operator. J. Geom. Phys., 21(4):337–356, 1997. [5] A. Boutet de Monvel, G. Kazantseva, and M. Măntoiu. Some anisotropic Schrödinger operators without singular spectrum. Helv. Phys. Acta, 69(1):13–25, 1996. [6] A. Boutet de Monvel and M. Măntoiu. The method of the weakly conjugate operator. In Inverse and algebraic quantum scattering theory (Lake Balaton, 1996), volume 488 of Lecture Notes in Phys., pages 204–226, Berlin, 1997. Springer. [7] J. Breuer. Singular continuous spectrum for the Laplacian on a certain sparse tree. preprint on math.SP/0604552. [8] J. Breuer. Singular continuous and dense point spectrum for sparse tree with finite dimensions. preprint on math.SP/0608159. [9] W. Dicks and T. Schick. The spectral measure of certain elements of the complex group ring of a wreath product. Geom. Dedicata, 93:121–137, 2002. [10] G. B. Folland. A course in abstract harmonic analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1995. [11] G. Georgescu and S. Golénia. Isometries, Fock spaces, and spectral analysis of Schrödinger operators on trees. J. Funct. Anal., 227(2):389–429, 2005. [12] S. Grosser and M. Moskowitz. On central topological groups. Trans. Amer. Math. Soc., 127:317–340, 1967. [13] S. Grosser and M. Moskowitz. Representation theory of central topological groups. Trans. Amer. Math. Soc., 129:361–390, 1967. [14] S. Grosser and M. Moskowitz. Compactness conditions in topological groups. J. Reine Angew. Math., 246:1–40, 1971. [15] R. I. Grigorchuk and A. Żuk. The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata, 87(1-3):209–244, 2001. [16] E. Hewitt and K. A. Ross. Abstract harmonic analysis. Vol. I: Structure of topological groups. Integra- tion theory, group representations, volume 115 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. [17] P. de la Harpe, A. Guyan Robertson, and A. Valette. On the spectrum of the sum of generators for a finitely generated group. Israel J. Math., 81(1-2):65–96, 1993. [18] P. de la Harpe, A. Guyan Robertson, and A. Valette. On the spectrum of the sum of generators of a finitely generated group. II. Colloq. Math., 65(1):87–102, 1993. [19] A. Iftimovici and M. Măntoiu. Limiting absorption principle at critical values for the Dirac operator. Lett. Math. Phys., 49(3):235–243, 1999. [20] M. Kambites, P. V. Silva, and B. Steinberg. The spectra of lamplighter groups and Cayley machines. Geom. Dedicata, 120:193–227, 2006. [21] H. Kesten. Symmetric random walks on groups. Trans. Amer. Math. Soc., 92:336–354, 1959. [22] L. Malozemov and A. Teplyaev. Pure point spectrum of the Laplacians on fractal graphs. Pure point spectrum of the Laplacians on fractal graphs. J. Funct. Anal., 129:390–405, 1995. [23] M. Măntoiu and M. Pascu. Global resolvent estimates for multiplication operators. J. Operator Th., 36(2):283–294, 1996. [24] M. Măntoiu and S. Richard. Absence of singular spectrum for Schrödinger operators with anisotropic potentials and magnetic fields. J. Math. Phys., 41(5):2732–2740, 2000. [25] M. Măntoiu, S. Richard, and R. Tiedra de Aldecoa. Spectral analysis for adjacency operators on graphs. preprint on math-ph/0603020 and to appear in Ann. H. Poincaré. [26] M. Reed and B. Simon. Methods of modern mathematical physics, IV: Analysis of Operators. Academic Press, New York, 1978. [27] S. Richard. Some improvements in the method of the weakly conjugate operator. Lett. Math. Phys., 76(1):27–36, 2006. [28] J. Riss. Eléments de calcul différentiel et théorie des distributions sur les groupes abéliens localement compacts. Acta Math., 89:45–105, 1953. [29] B. Simon. Operators with singular continuous spectrum VI: Graph Laplaciens and Laplace-Beltrami operators. Proc. Amer. Math. Soc., 124:1177–1182, 1996. [30] J. Weidmann. Linear operators in Hilbert spaces. Springer-Verlag, New York, 1980.

0704.1103

Spin Dynamics Of $qqq$ Wave Function On Light Front In High Momentum Limit Of QCD : Role Of $qqq$ Force

Spin Dynamics Of qqq Wave Function On Light Front In High Momentum Limit Of QCD : Role Of qqq Force A.N.Mitra ∗ 244 Tagore Park, Delhi-110009, India Abstract The contribution of a spin-rich qqq force ( in conjunction with pairwise qq forces) to the analytical structure of the qqq wave func- tion is worked out in the high momentum regime of QCD where the confining interaction may be ignored, so that the dominant effect is Coulombic. A distinctive feature of this study is that the spin-rich qqq force is generated by a ggg vertex ( a genuine part of the QCD Lagrangian ) wherein the 3 radiating gluon lines end on as many quark lines, giving rise to a (Mercedes-Benz type) Y -shaped diagram. The dynamics is that of a Salpeter-like equation ( 3D support for the ker- nel) formulated covariantly on the light front, a la Markov-Yukawa Transversality Principle (MYTP) which warrants a 2-way intercon- nection between the 3D and 4D Bethe-Salpeter (BSE) forms for 2 as well as 3 fermion quarks. With these ingredients, the differential equation for the 3D wave function φ receives well-defined contribu- tions from the qq and qqq forces. In particular a negative eigenvalue of the spin operator iσ1.σ2 × σ3 which is an integral part of the qqq force, causes a characteristic singularity in the differential equation, signalling the dynamical effect of a spin-rich qqq force not yet consid- ered in the literature. The potentially crucial role of this interesting effect vis-a-vis the so-called ‘spin anomaly’ of the proton, is a subject of considerable physical interest. ∗Email: (1)[email protected]; (2)[email protected] http://arxiv.org/abs/0704.1103v1 1 Introduction The concept of a fundamental 3-body force (on par with a 2-body force) is hard to realize in physics, leaving aside certain ad hoc representations of higher order effects, for example those of ∆, N∗ resonances in hadron physics. At the deeper quark-gluon level on the other hand, a truly 3-body qqq force shows up as a folding of a ggg vertex ( a genuine part of the gluon Lagrangian in QCD) with 3 distinct q̄gq vertices, so as to form a Y -shaped diagram (see fig 1 below). Indeed a 3-body qqq force of this type, albeit for ‘scalar’ gluons, was first suggested by Ernest Ma [1], when QCD was still in its infancy. [ A similar representation is also possible for NNN interaction via ρρρ or σσσ vertices, but was never in fashion in the literature [2]]. We note in passing that a Y -shaped (Mercedes-Benz type) picture [3] was once considered in the context of a preon model for quarks and leptons. In the context of QCD as a Yang-Mills field, a ggg vertex has a momentum representation of the form [4] Wggg = −igsfabc[(k1 − k2)λδµν + (k2 − k3)µδνλ + (k3 − k1)νδλµ] (1.1) where the 4-momenta emanating from the ggg vertex satisfy k1+k2+k3 = 0, and fabc is the color factor. When this vertex is folded into 3 q̄gq vertices of the respective forms gsū(p 1)iγµ{λa1/2}u(p1), and two similar terms, the resultant qqq interaction matrix (suppressing the Dirac spinors for the 3 quarks) becomes Vqqq = [iγ(1).(k2 − k3)γ(2).γ(3) + {2}+ {3}]{λ1λ2λ3}/{k21k22k23} (1.2) where ki = pi − p′i; λi are the color matrices which get contracted into the corresponding scalar triple products in an obvious notation. [Note that the flavour indices are absent here since the quark gluon interaction is flavour blind]. This interaction will be considered in conjunction with 3 pairs of qq forces within the framework of a Bethe-Salpeter type dynamics to be specified be- low. Before proceeding in this direction, it is in order to explain a possi- ble motivation behind the use of a direct qqq force with such a rich spin dependence. Apart from the intrinsic beauty of this term, the immediate provocation for its use comes from the issue of ”proton spin” which, after making headlines about two decades ago, has come to the fore once again, thanks to the progress of experimental techniques in polarized deep inelastic scattering off polarized protons, and their variations thereof, which allow for an experimental determination of certain key QCD parameters by relating them to certain observable quantities emanating from external probes ; ( see a recent review [5] for references and other details). On the other hand it is also of considerable theoretical interest to determine these very quantities directly from the intrinsic premises of QCD provided one has a ” good ” qqq wave function to play with. Such a plea would have sounded rather utopian in the early days of QCD when phenomenology was the order of the day. Today however many aspects of QCD are understood well enough to make such studies worthwhile by hindsight, with possible ramifications beyond their educational value. The role of the direct 3-body force may be seen in this light, while working for simplicity in the experimentally accessi- ble regime of valence quarks. To that end this paper is specifically concerned with the effect of the 3-body force (1.2) on the analytical structure of the qqq wave function, while the formalism dealing with the contributions of the various operators ( iγµγ5, 2-gluon effects, etc) to the proton spin is reserved for a subsequent paper. 1.1 Theoretical Ingredients In the valence quark regime, we need to consider a qqq system governed by pairwise qq forces as well as a direct 3-quark force of the type (1.2). A further simplification occurs in the high momentum regime where the effect of confin- ing forces may be neglected, so that only coulombic forces need be kept track of. We shall take the dynamics of a qqq hadron in the high − momentum regime to be governed by a Bethe-Salpeter Equation (BSE) whose kernel is a sum of three pairs of (coulombic) qq forces plus a single 3-body term (1.2). Unfortunately the 4D form of the BSE is too general to be of prac- tical value for qqq dynamics, so a better option is the Salpeter Equation [6] representing its instantaneous form. And, except for its lack of covariance, the Salpeter equation has the remarkable property of 3D-4D interlinkage, a feature that had been present all along in the original formulation itself [6], but had somehow remained hidden from view in the literature, until clarified [7] in the context of a comprehensive two − tier BSE formalism developed independently in a covariant manner [8]. (A practical significance of the ‘two-tier’ formalism is that the reduced 3D form is ideal for the determi- nation of hadron mass spectra of both qq̄ and qqq types [9, 10], while the reconstructed 4D form is convenient for the evaluation of transition ampli- tudes [9] via Feynman techniques for loop diagrams; see also Munz et al ref.(9) ). A covariant formulation of the BSE is centered around the hadron 4-momentum Pµ in accordance with the Markov-Yukawa Transversality Prin- ciple (MYTP) [11, 12], which was shown to be a ‘gauge principle’ in disguise [13]. It ensures that the interactions among the constituents be transverse to the direction of Pµ. In the high momentum regime to be considered here, the confining interaction has been ignored for simplicity, which leaves the 3D form of the BS dynamics inadequate for mass spectral determination, yet its dynamical on the spin-structure of the wave function should be realistic enough for dealing with the hadron spin in the high momentum limit. Now to another vital element of the theory: Although the Salpeter Equa- tion, as the instant form of BSE, admits of a covariant formulation in the rest frame of the total hadron 4-momentum Pµ, it suffers from certain ill- defined 4D loop integrals due to a ‘Lorentz-mismatch’ among the rest-frames of the participating hadronic composites, resulting in time-like momentum components in the exponential/gaussian factors associated with their vertex functions. This is especially true of triangle loops, such as applicable to the pion form factor and ρ − ππ coupling [14] where this disease causes unwar- ranted ”complexities” in the amplitudes, while two-quark loops just escape this pathology. For a possible remedy against this disease, without losing the benefit of an MYTP - based 3D-4D interconnection, a promising candidate is the light-front approach of Dirac [15] by virtue of its bigger (7) stabil- ity group compared with 6 for the instant form theory. Its basic simplicity was first noticed by Weinberg [16] who formulated the infinite momentum frame towards the same end. A covariant generalization of MYTP on the light front, requiring the use of two null 4-vectors nµ and ñµ that satisfy n2 = ñ2 = 0 and n.ñ = 1, cures the ‘Lorentz mismatch’ disease noted above [17]. The remaining problem of n- dependence of transition amplitudes is solved through a simple ansatz of ‘Lorentz completion’ [17, 18], so as to yield a Lorentz-invariant pion e.m. form factor in accord with experiment . A sim- ilar approach had been considered by Carbonell et al [19] in the context of the Kadychevsky-Karmanov formalism [20, 21], except for their missing out on the second (dual) null-vector ñ which happens to be crucial for recovering a Lorentz - invariant structure in a more natural way. 1.2 Plan of the Paper The plan of the paper is based on an interlinked 3D-4D BSE formalism char- acterized by a Lorentz-covariant 3D support for its kernel a la MYTP [11, 12], adapted to the light front (LF) [17]. The full structure of the kernel is a sum of 3 pairs of coulombic qq forces plus a qqq force, Eq.(1.2), whose 3D support is implied by the fact that all internal momenta q be transverse to Pµ , viz., q̂µ = qµ − q.PPµ/P 2. However the propagators are left untouched in their standard 4D forms. [The light front formulation requires a collinear frame [17] which is further specified in Section 2 ]. The strategy now lies in a step-wise reduction of the (Salpeter-like) 4D Master Equation involving the actual (4D) fermionic wave function Ψ. Step A consists in expressing Ψ in terms of an auxiliary (bosonic) 4D quantity Φ satisfying an equivalent (bosonic) 4D (Salpeter-like) equation. Step B involves a ” Gordon reduction ” to eliminate the γµ matrices in favour of σµν matrices. Finally Step C con- sists in a 3D reduction of this ‘bosonized’ Master Equation, and a subsequent reconstruction of the original Ψ in 4D form by a suitable reversal of steps. In the process a 3D scalar wave function φ is introduced which not only fa- cilitates an explicit solution of the 3D (albeit fully covariant) Salpeter-like equation but is also a key component of the (reconstructed) 4D fermionic wave function Ψ. To facilitate the process of 3D-4D interlinkage, a Green’s function approach is employed a la [22], from which it is straightforward to derive the corresponding wave functions via the appropriate ‘pole’ limits. The entire exercise involves a close correspondence between the (earlier) in- stantaneous form [22] and the (later) LF form [17], so as to project only the latter by making free use of the results of the former. A new element will be a generalization of the earlier formalism [22] so as to include the effect of the 3-body force, Eq (1.2), on the structure of the qqq Green’s function which will require a more elaborate strategy, keeping in view the relative strengths of qq and qqq forces. After a short account of the correspondence between the instant and LF forms of the dynamics, Section 2 is devoted to Steps A and B for converting the Master Equation for the actual 4D qqq wave function Ψ (fermionic) to an equivalent 4D (bosonic) form involving Φ, with all γ-matrices eliminated in favour of Pauli matrices, by exploiting the effective 3D support for the qq and qqq kernels on the light-front, as described above. Section 3 de- scribes the first part of Step C, viz., the use of Green′s- function method for the 3D reduction of the Master Equation for the 4D Green’s function G corresponding to Φ, resulting in an integral equation for the 3D Green’s function Ĝ corresponding to phi, specialized to the large momentum regime on the light-front. The second part of Step C, namely its reconstruction back to the 4D quantity G a la [22], leading to a formal connection between Φ and φ, is the subject of Section 4, now with the added complexity behind the inclusion of the 3-body qqq force along with the (dominant) pairwise qq forces. [ The subsequent reconstruction of the 4D fermionic form Ψ, follows from the results of Section 2]. As a further aid to the understanding of the 3D-4D interconnection, Appendix A describes the complete procedure for a prototype qq subsystem, whose results are freely used for justifying several steps in Sections 3 and 4. In Section 5, the full 3D BSE for φ is set up in a simplified form designed to check for the dynamical effect of spin present in both the qq and qqq forces. on the structure of the differential equation for φ (the dominant effect being of the latter ! ). Section 6 is devoted to a short critique of the role of the Vqqq term on the analytic srtucture of the qqq wave function, while the derivation of the requisite QCD parameters for the baryon from the forward scattering amplitude off the iγµγ5 operator, via the quark constituents, as well as a more elaborate 2-gluon contribution to the proton spin, is reserved for a subsequent paper. 2 Salpeter Eq on LF : From Ψ To Φ 2.1 Master Eq : Instant Vs LF Forms of Dynamics Since the first step in the formulation of a covariant Salpeter-like equation on the light front (LF) is to establish a correspondence between the instant and LF forms of the dynamics, we first recall some definitions [17] for the LF quantities p± = p0±p3 defined covariantly as p+ = n.p 2 and p− = −ñ.p while the perpendicular components continue to be denoted by p⊥ in both notations. Now for a typical internal momentum qµ, the parallel component P.qPµ/P 2 of the instant form translates in the LF form as q3µ = zPnnµ, where Pn = P.ñ, and z = n.q/n.P . As a check, q̂ 2 = q2 + z2M2 which shows that zM plays the role of the third component of q̂ on LF. Next, we collect some of the more important definitions / results of the LF formalism [17] q⊥ = q − qnn; q̂ = q⊥ + zPnn; z = q.n/P.n; qn = q.ñ; (2.1) Pn = P.ñ;P.q = Pnq.n + P.nqn; q̂.ñ = P⊥.q⊥ = 0; P.q̂ = Pnq.n; q̂ 2 = q2 +M2z2;P 2 = −M2 For a qqq baryon, there are two internal momenta, each separately satisfying the relations (2.1). Note that for any 4-vector A, A.n and −An correspond to 1/ 2 times the usual light front quantities A± = A0 ± Az respectively. But since a physical amplitude must not depend on the orientation n, a simple device termed Lorentz−completion via the collinear trick [17] yields a Lorentz-invariant amplitude for a transition process with three external lines P, P ′, P ′′(= P+P ′) as follows. Since collinearity implies P⊥.P = 0, the 4-scalar product P.P ′ = P.nP ′n+PnP ′.n+P⊥.P simplifies to P.nP ′n+PnP Then ‘ Lorentz completion ’ simply amounts to reversing the last step via the ‘zero’ quantity P⊥.P , so as to recover the Lorentz invariant quantity P.P ′ at the end ! And as a practical simplification, one does not even have to use the n, ñ symbols; it suffices to use the more familiar light-front components A± for the covariant quantities 2[A.n,−An] respectively. For ready reference, the precise correspondence between the instant and LF definitions of the ‘parallel (z)’ and ‘time-like (0) ’ components of the various 4- momenta for a qqq baryon ( i = 1,2,3) [22, 23] : piz; pi0 = ; p̂i ≡ {pi⊥, piz} (2.2) The last part of Eq.(2.2) defines a covariant 3-vector on the LF that will frequently appear in the reduced 3D BSE for the qqq proton. Our goal is to write down ( and solve) the Master equation for three fermion quarks complete with all internal d.o.f.’s, in the presence of both qq and direct qqq forces which reads [8] : Ψ(p1p2p3) = SF (p1)SF (p2)g d4q′12 (2π)4 γ(1)µ γ ν Dµν(k12)Ψ(p 2, p3) +SF (p1)SF (p2)SF (p3) d4q′12d (2π)8 VqqqΨ(p 3) (2.3) where the definitions for the various momenta, and the phase conventions for the quark propagators are those of [8], while the direct 3-quark interaction Vqqq in the last term is new, and given by (1.2). The central problem is now the reduction of this Master equation (2.3) through the three steps (A, B, C) vide Sect (1.2), as outlined below. 2.2 Reduction of Master Eq from Ψ to Φ Since permutation symmetry plays a crucial role for a qqq hadron for fermion quarks, we define at the outset a pair of internal variables (ξ; η) with, say, the index #3 as basis, as [22] 2ξ3 = p1 − p2; 6η3 = −2p3 + p1 + p2; P = p1 + p2 + p3 (2.4) where the time-like and space- like parts of each are given by (2.2), and the corresponding 3-vector defined as p̂i ≡ {pi⊥, piz}. Two identical sets of momentum pairs ξ1, η1 and ξ2, η2 are similarly defined, but can be expressed in terms of the set (2.4) via permutation symmetry. Now Step A, which is designed to dispose of the fermion d.o.f.’s for a qqq system, consists in defining an auxiliary scalar function Φ related to the actual BS wave function Ψ by Ψ = Π123S F i (−pi)Φ(pip2p3)W (P ) (2.5) Here the quantity W (P ) is independent of the internal momenta but includes the spin-cum-flavour wave functions χ, φ of the 3 quarks involved (see [?] : W (P ) = [χ′φ′ + χ′′φ′′]/ 2 (2.6) where φ′, φ′′ are the standard flavour functions of mixed symmetry [25] [not to be confused with the 3D wave function φ !], and χ′ ,χ′′ are the corresponding relativistic spin functions. The latter may be defined either in terms of the quark # indices as in Eqs (1.2) or (2.3), or sometimes more conveniently in a common Dirac matrix space as [24, 26] |χ′ >; |χ′′ >= [ M − iγ.P [iγ5; iγ̂µ/ 2]βγ ⊗ [[1; γ5γ̂µ]u(P )]α (2.7) where the first factor is the βγ-element of a 4 x 4 matrix in the joint spin space of the quark #s 1, 2 [26], and the second factor the α element of a 4 x 1 spinor in the spin space of quark # 3; C is a charge conjugation matrix with the properties [27] −γ̃µ = C−1γµC; γ̃5 = C−1γ5C; and γ̂µ is the component of γµ orthogonal to Pµ. Finally, the representations of the flavour functions φ′, φ′′ satisfy the following relations in the ”3” basis < φ′′|1;~τ (3)|φ′′ >=< φ′|1;− ~τ (3)|φ′ > 2.3 Gordon Reduction on Vqq3 & Vqqq At this stage we indicate the effect of Gordon reduction on the pairwise kernels V (ξ̂iη̂i) and the 3-body kernel Vqqq, following the original treatment of [29], (also quoted in [8] ). The strategy lies in a close scrutiny of Eqs.(2.3-2.5) with a view to eliminate the Dirac matrices in favour of the Pauli matrices σµν . To that end, we express the Dirac propagators SF (pi) as iSF (pi) = (mq − iγ(i).pi)/∆i; ∆i = m2q + p2i Now employing the notation V (i)µ = (mq − iγ(i).pi)iγ(i)µ (2.8) the result of Gordon reduction is expressed by the formula [29, 8] V (1).V (2) = −M212−(q̂+q̂′)2+i(q̂+q̂′).(σ̂1+σ̂2)×(q̂−q̂′)−σ ik (q−q′)j(q−q′)k (2.9) where we have used a mixed vector-tensor notation for the various 3-vectors on the light-front in the sense of (2.2) and a short-hand notation q for q12. It is now possible to identify the pairwise kernel arising from the first group of terms on the rhs of Eq.(2. 3) after expressing it in LF notation of Eq.(2.2) as follows: Vqq3 ≡ V (ξ̂3, ξ̂′′3 ) = g2sV (1).V (2) × (q̂12 − q̂′′12)2 (2.10) where we have included a net color factor of (−4/3) arising from a folding of the factor F12 for each qq pair in a 3 ∗ state, into λ3/2 for the spectator quark # 3 in a 3 state; the corresponding gluon propagator has been taken on the light-front, in the notation of (2.2). Similarly the 3-body kernel Vqqq in Eq.(2.3) may be identified as a sum of 3 terms arising from a folding of (1.2) with the 3 Dirac factors (mq − iγ(i).pi) in the second term of (2.3). Vqqq = (−4/3)g4s [V (1).(k̂2 − k̂3)× V (2).V (3) + {2}+ {3}]/{k̂21k̂22k̂23} (2.11) using the notation V (i)µ of (2.8). The net effect of the (antisymmetric) com- bination of the color matrices λ is represented by the factor (−4/3) sitting in front; and ki = pi − p′i. The ‘bosonized’ form of Eq. (2.3) is now Φ(p1p2p3) = d4q′12 (2π)4 V (ξ̂3, ξ̂ 3 )Φ(p 2, p3) ∆1∆2∆3 d4q′12d (2π)8 VqqqΦ(p 3) (2.12) Figure 1: (a) Pictorial view of 2- & 3- body interactions; (b) ‘Mercedez-Benz’ diagram for qqq-force in terms of the structures (2.10) and (2.11) for the 2- and 3-body kernels respectively. And the Gordon reduction formula (2.9 ) may be repeatedly used to used to further simplify the structures of Vqq[Eq (2.10)] and Vqqq[ Eq. (2.12)], which are collected as follows. 2.3.1 Further Reduction of Vqq3 & Vqqq The 2-body and 3-body interactions are pictorially represented by fig 1(a) where the vertices of the triangleABC stand for the position vectors (r1, r2, r3) of the 3 quarks in configuration space, while the centroid O has the position vector R = ((r1 + r2 + r3)/3. The corresponding momenta of the gluon lines connecting them for pairwise qq interactions are precisely the quanti- ties qij − q′ij that appear as arguments in Eq. (2.10), while the momenta associated with the gluon lines connecting all the 3 quarks together, a la the Y -shaped fig.(1b), are the quantities k̂i corresponding to the vectors (ri −R). In the notation of Eq.(2.4), the former momenta are precisely ξ̂i, while the latter are the dual variables η̂i. [This interpretation will prove useful in the subsequent analysis]. The pairwise terms Vqqi were earlier considered in detail [8]. The ‘con- vective’ terms are of the form (p1 + p2).(p 1 + p 2) which simplifies to −M212 ≈ −4M2/9, plus smaller terms. The spin-orbit terms are unimportant for the ground state of the qqq proton, but the spin-spin terms survive for its ground state S = 1/2; their angular average works out as : σ̂i.σ̂j = [Σ2 − 9] where we have introduced the total spin operator Σ for the proton state, with Σ2 = 3. Collecting these results the sum of the 3 pairwise interactions (2.10) work out as Vqq3 = g ∑ 4M2p3z (q̂12 − q̂′′12)2 − 3p3z(Σ2 − 9)] (2.13) We now consider the more interesting term Vqqq, Eq.(2.12). The numerator can be broken up into 3 distinct groups : a) convective; b) spin-orbit; c) spin- sp[in-spin. Using the notation 2p̄i = pi+p i, the convective part vanishes (due to antisymmetry with too few d.o.f.’s) : 4[p̄1.p̄2][2p̄3.(k1 − k2)] ⇒ 0 The spin-orbit part comes from two groups of terms, one of which vanishes: 2p̄1.(k1 − k2)[σ̂3.(¯̂p3 × k̂3) ⇒ 0 where a mixed notation is used for the Pauli matrices, light-faced tensor for the 4D and hatted vector for the 3D forms. The other group gives rise to a non-zero contribution as 4p̄1.p̄2[iσ µν k3ν(k1µ − k2µ)] ≈ (−4iM2/9) σ̂i.ξ̂3 × η̂3 (2.14) in the notation of Eq.(2.4). Note the overall antisymmetry of the last factor (independent of the index 3). Finally the (totally antisymmetric) spin-spin-spin contribution SSS is defined as : SSS = [iσ(1)µν k̂1ν(k̂2µ − k̂3µ)iσ λρ k̂2ρiσ λσ k̂3σ (2.15) The simplification of this expression (which is elementary) is helped by the identity σ(i)µν = ǫµνaσ̂ a . The next step is angular averaging over each of the independent 3-vectors involved in each of the three terms above, namely, < k̂ik̂j >= k̂ 2δij/3. The resulting expression for SSS is SSS = {k̂21k̂22 k̂23}σ̂1.σ̂2 × σ̂3 Substituting for SSS in Eq (2.11) and ignoring the spin-orbit terms (for the ground state of the proton) the net 3-body force is Vqqq = −8ig4s (σ̂1.σ̂2 × σ̂3) (2.16) 3 3D Reduction By Green’s FunctionMethod We are now in a position to implement Step C of our program ( establishing a 3D-4D interconnection between the corresponding wave functions) , for which it is convenient to employ the Green’s function approach [22]. Calling the 4D Green’s functions associated with Ψ and Φ by GF and GS respectively, the connection between them analogously to Eq.(2.5) may be written as GF (ξη; ξ ′η′) = W (P )⊗Π123S−1F i (−pi)GS(ξη; ξ′η′)Π123S−1F i (−p′i)W̄ (P ′) (3.1) where we have indicated the 4-momentum arguments of the Green’s functions involved, in a common S3 basis (ξ, η), and expressed the spin-flavour depen- dence of GF as a matrix product implied by the notation W (P )⊗ W̄ (P ′). Now to show the 3D-4D interconnection it is enough to work at the level of the ‘scalar’ Green’s function GS ( relabelled as G for simplicity), since it is trivial to include the spin-flavour d.o.f.’s in a matrix notation via (2.5) later. The main steps for the scalar qqq Green’s function are shown next, while Appendix A sketches the corresponding process for a typical qq subsystem, from 3D reduction to 4D reconstruction, so as to serve as a simpler prototype for the actual qqq case. 3.1 Reduction from 4D G(ξη; ξ′η′) to 3D Ĝ(ξ̂η̂; ξ̂′η̂′) We shall now outline a Green’s function method to establish a 3D-4D inter- linkage between the 4D Φ and the 3D φ, following a generalization of the procedure developed some time ago [22], so as to include the effect of the direct 3-body force Vqqq. To that end we shall rederive the qq denominator functions Dij of [22] arising from single integrations over dξi0, to show that they are exactly proportional to the 3D denominator function D123 arising from a double integration over dξ0dη0 in the direct 3-body term to be consid- ered here. [This is a big improvement over the earlier derivation [22] where this property was missing, and helps pave the way for putting together the effects of both 2- and 3-body forces within a common dynamical framework The full 4D Green’s function for ‘scalar’ quarks is G(ξη; ξ′η′) (taking out the c.m δ-fn), while its 3D counterpart Ĝ is [22] Ĝ(ξ̂η̂; ξ̂′η̂′) = dξ0dη0dξ 0G(ξη; ξ ′η′) (3.2) where the time-like subscript ‘0’ should be read as the corresponding LF component in the sense of Eq.(2.2). Note that both G and Ĝ are S3 sym- metric, since the combination dξ0dη0 has this property. But the two hybrid Green’s functions defined as [22] G̃3ξ(ξ̂3η3; ξ̂ dξ30dξ 30G(ξη; ξ ′η′); G̃3η(ξ3η̂3; ξ dη30dη 30G(ξη; ξ ′η′) (3.3) are not S3 symmetric (hence they are indexed), since the integration now involves only one of the two ξ,η variables. Next, the 4D G(ξη; ξ′η′) satisfies a BSE of the form i(2π)4G(ξη; ξ′η′) = d4ξ′′ 4∆1∆2 V (ξ̂3, ξ̂ 3 )G(ξ 3η3; ξ d4ξ′′d4η′′ 9(2π)4∆1∆2∆3 VqqqG(ξ 3η3; ξ 3) (3.4) which is analogous to Eq. (2.12) for the corresponding wave functions, except for a change of variables wherein the factors 1/4 and 1/9 in the first and second groups of terms stem from the relations (2.4) among the corresponding variables. It is understood that both these types of interaction have covariant 3D support, and they incorporate the effect of ‘Gordon reduction’ as outlined in Sect 2.3 above. The 3D reduction of (3.4) is now achieved by integrating it a la (3.2), leading to a structure of the form : (2π)3Ĝ(ξ̂η̂; ξ̂′η̂′) = d3ξ̂′′3V (ξ̂3, ξ̂ 3 )Ĝ(ξ̂ 3 ; ξ̂ 3(2π)3D123 d3ξ′′d3η′′VqqqĜ(ξ 3 ; ξ 3) (3.5) To explain the structure of certain factors, Eq (3.5) shows that we have two types of 3D denominator functions Dij and D123, associated with pairwise 2-body and direct 3-body forces respectively. It is easily shown that they are simply related to each other. To that end, Appendix A already shows the structure of Dij : D12 = MD12+ ; D12+ = 2M [ω 1⊥p2+ + ω 2⊥p1+ − P12−p1+p2+] (3.6) where ω2i⊥ equals m q + p i⊥. Using the on-shellness (∆3 = 0) of the spectator (#3) then gives P12− = P−− p3− which reduces further to P−−ω23⊥/p3+. Its substitution back in (3.6) leads to the S3 symmetric result: p3+D12+ ≡ D123 = 2 {p2+p3+ω21⊥} − 2p1+p2+p3+P− (3.7) Here we have anticipated the structure of D123 associated with the Vqqq term whose formal definition is P 2+dq12−dp3− 4M2(2iπ)2∆1∆2∆3 (3.8) where the (double) contour integrations in the indicated variables leads to the desired result. The resultant 3D BSE for Ĝ is now (2π)3D123Ĝ(ξ̂η̂; ξ̂ ′η̂′) = d3ξ̂′′3V (ξ̂3, ξ̂ 3)Ĝ(ξ̂ 3 ; ξ̂ 3(2π)3 d3ξ′′d3η′′VqqqĜ(ξ 3 ; ξ 3) (3.9) The structure of this equation reveals some interesting symmetries, when (2.13) and (2.16) are substituted for Vqq3 and Vqqq respectively. Namely, in the first group of terms the η variable is the spectator since the integration is only over the ξ variable, while in the second group of terms, their relative roles are interchanged with the ξ variable effectively a spectator ! This fact is not immediately apparent because of the double integration involved, but a little reflection shows that the absence of the ξ variable in (2.16) effectively ensures its spectator status in the argument of the corresponding Green’s function. We shall see this feature more clearly in the coordinate space representation of the φ equation (corresponding to Eq (3.9) for Ĝ, to be considered in Section 5. But first we turn our attention to the reconstruction of the 4D Green’s function from the 3D form Ĝ satisfying (3.9). To that end we note that the qq and qqq group of terms in (3.9) need different strategies and are best handled separately, one at a time, (temporarily) ignoring the presence of the other. 4 Reconstruction of 4D Wave Function 4.1 Reconstruction of G with qq Forces Only Considering first the qq group, we proceed exactly as in (A.10) of Appendix A for the qq system. Namely, first express G̃3η, eq.(3.3), in terms of the 3D Ĝ, using the notation of (2.2) and the result of (3.7): G̃3η(ξ3η̂3; ξ 2iπp3z∆1∆2 Ĝ(ξ̂η̂; ξ̂′η̂′) D′123 2iπp′3z∆ (4.1) In a similar way the fully 4D G function is expressible in terms of the hybrid function G̃3ξ as G(ξη; ξ′η′) = 2iπp3z∆1∆2 G̃3ξ(ξ̂3η3; ξ̂ D′123 2iπp′3z∆ (4.2) Now since the G̃3ξ function is not determined from qqq dynamics alone, we invoke an ansatz similar to, but more symmetrical than, [22] : G̃3ξ(ξ̂3η3; ξ̂ 3) = Ĝ(ξ̂η̂; ξ̂ ′η̂′)F (p3, p 3) (4.3) where the balance of the p3 (spectator) dependence is in the (as yet unknown) F function, subject to an explicit self-consistency check. To that end, try the ( symmetrical) Lorentz-invariant form F (p3, p δ(∆3 −∆′3) (4.4) and integrate both sides of (4.4) w.r.t. dp30dp 30, in the notation of (2.2), to show that the consistency check is met with the mass shell value of the spectator momentum : 4p3zp Substitution from A3 in (4.4) then gives the symmetrical form F (p3, p 3) = 4p3zp δ(∆3 −∆′3) 2πi∆3 whence the 4D G-fn in terms of Ĝ via the sequence (4.3) and (4.2), works out as G(ξη; ξ′η′) = Ĝ(ξ̂η̂; ξ̂′η̂′) D′123 δ(∆3 −∆′3) (2πi)5 (4.5) 4.2 Reconstruction of G with qqq Forces Only Next we consider only the qqq group of terms for a reconstruction from Ĝ ( Eq (3.5)) to G (Eq (3.4)). This case is more akin to the qq case of Appendix A in the sense that if we use the collective indices (ξ, η) ≡ ρ and (ξ′, η′) ≡ ρ′ for the initial and final state arguments of the G-function we can define two kinds of G̃ as follows. G̃(ρ̂; ρ′) = dρ0G(ρ; ρ ′); G̃(ρ; ρ̂′) = dρ′0G(ρ; ρ ′) (4.6) where the integrals on the RHS are each of the double integral type, so that the definitions of the denominator functions involved are given by Eq.(3.8). Thus, following (A. 9), G̃ and Ĝ are connected as follows G̃(ρ, ρ̂′) = D123(ρ̂) (2iπ)2∆1∆2∆3 Ĝ(ρ̂; ρ̂′) (4.7) together with a second one with the roles of ρ, ρ′ interchanged. Continuing exactly as in Appendix A, G of Eq (3.4) gets expressed in terms of G̃ on the RHS, since the interaction Vqqq does not involve the variables ρ 0. Thence another application of Eq.(4.7) for expressing G̃(ρ; ρ̂′) in terms of Ĝ finally yields the desired connection: G(ρ; ρ′) = D123(ρ̂) (2iπ)2∆1∆2∆3 Ĝ(ρ̂; ρ̂′) D123(ρ̂ (2iπ)2∆′1∆ (4.8) where the symbol ρ stands collectively for (ξ, η), as does ρ′. Note that in this case the reconstruction of G in terms of of (the fully reduced) Ĝ is unique, and does not require any extra ansatz like (4.4). 4.3 Putting Both Vqq & Vqqq Together We have now two distinct types of 3D-4D interconnections valid for pure qq and pure qqq forces respectively. But when both types of forces are present, an exact interconnection is very difficult to derive. We shall therefore strive for an approximate but sufficiently realistic solution based on the relative strengths of the two forces, namely, an overwhelming preponderance of qq over qqq forces. To give effect to this strategy, we first note that the 3D level does not involve any approximation since Eq. (3.9) treats both types of interaction on par. It is only at the 4D level of reconstruction that an approximation is necessary for putting the two forces together. Now taking account of the dominance of qq over qqq forces, the simplest choice is to prefer the structure of Eq (4.5) over that of Eq. (4.8). And although the ratio of the two 3D-4D interconversion jackets in (4.5) over (4.8) is a singular quantity δ(∆3 −∆′3)× (2πi) it causes no harm for physical amplitudes since such singular quantities get ironed out on integration over the various momenta [23]. With this un- derstanding, Eq.(4.5) represents the reconstruction of the full 4D Green’s function G(ξη; ξ′η′) in terms of the 3D quantity Ĝ(ξ̂η̂; ξ̂′η̂′) where the latter now satisfies Eq (3.9) which includes the effect of both qq and qqq forces. 4.4 3D-4D Interconnection for Wave Functions Finally the spectral representations of G(ξη; ξ′η′) and Ĝ(ξ̂η̂; ξ̂′η̂′) for the qqq system, exactly on the lines of (A.12-A.13) for a qq subsystem near a bound state pole P 2 = −M2 , are G(ξη; ξ′η′) = Φn(ξ, η)Φ ′, η′)/(P 2 +M2); (4.9) and a similar equation for Ĝ vis-a-vis φ. These give the connection between the 4D wave function Φ which satisfies Eq (2.12), and the 3D wave function φ corresponding to the Green’s function Ĝ which satisfies (3.9). For purposes of evaluating transition amplitudes via Feynman diagrams, it is convenient to index Φ as Φ1 +Φ2 +Φ3, and a corresponding indexing for the associated vertices as V = V1 + V2 + V3 , as in ref [22], so as to keep track of which quark is involved in which vertex. Φ3(ξ, η) ≡ ∆1∆2∆3 φ(ξ̂, η̂) δ(∆3) (2πi)5/2 (4.10) The δ-function in eq.(4.7) has nothing to do with any connectedness problem; see ref.[22] for detailed reasons. Finally, the use of Eq.(2.5) with (4.10) yields an explicit structure for the actual (fermionic) wave function Ψ as Ψ(ξ, η) = Π123SF (pi)D123 [φ(ξ̂, η̂) δ(∆3)∆3 (2πi)5/2 ]×W (P ) (4.11) 5 Complete φ Equation In Coordinate Space Our final task is to set up (and solve) the 3D BSE for φ which may be inferred from (3.9) by making use of a spectral representation similar to (4.9), and going to the pole P 2 = −M2 : (2π)3D123φ(ξ̂, η̂) = d3ξ̂′′3Vqq3φ(ξ̂ 3 , η̂3) 3(2π)3 d3ξ′′d3η′′Vqqqφ(ξ̂ 3 , η̂ 3) (5.1) where Vqq3 is given by (2.13); Vqqq by (2.16); and D123 by (3.7). To transform this equation in coordinate space, define the combinations analogous to (2.4) 2s3 = r1 − r2; 6t3 = −2r3 + r1 + r2 (5.2) Then a Fourier transform to the coordinate− space representation gives for the pairwise terms Vqq3 the coulombic structure V (2)(s) = d3ξ̂′3 (2π)3 exp [iξ̂′3.s3]× [ ∑ 8M2g2sp3z 27(ξ̂3 − ξ̂′3)2 (Σ2 − 9)p3z] (5.3) which multiplies the coordinate space wave function φ(s, t) ( S3 symmetric in its arguments). Similarly, the (S3 symmetric) double Fourier transform of Vqqq is defined as V (3)(s, t) = −4ig4s d3ξ̂′d3η̂′ 3(2π)6 exp [iξ̂′.s+ iη̂′.t]× (η̂3 − η̂′3)2 [σ̂1.(σ̂2× σ̂3)] (5.4) The integration in (5.3) involves a 3D δ-function δ3(ŝ3) ; and the double integration in (5.4) additionally involves the factot 1/|t3| due to the η in- tegration. The δ function being singular may be ‘regularized’ by using a differential representation (acting on the scalar function φ(s, t)), with |ŝ3| = |s3| : δ3(ŝ3) (2π)3) φ(s, t) =⇒ 4π|s3| [∂2s3]φ(s, t) (5.5) The result of integration in (5.3) is then expressed as V (2)(s) = 8M2αsp3z 27|s3| 2 − 9) 27|s3| ] (5.6) Next the 3-body term term (5.4) integrates, with the help of (5.5), to [iσ̂1.(σ̂2 × σ̂3)] Now the product 1/s3t3 can be approximately replaced by the S3 symmetric expression 2/[s23 + t 3] which can be taken out of the summation sign (with index ‘3’ dropped) to give ∂2s3 which in turn equals (3/2)(∂ s + ∂ t ), an S3 symmetric sum with index ‘3’ dropped again. This finally leads from (5.4) to the S3 symmetric form of the 3-body term in coordinate space: V (3)(s, t) = 4α2s [iσ̂1.(σ̂2 × σ̂3)] 3(s2 + t2) [∂2s + ∂ t ] (5.7) For further manipulations it is convenient to replace the antisymmetric spin operator A = [iσ̂1.(σ̂2 × σ̂3)] by one of its possible eigenvalues as follows: Squaring this quantity yields in a simple way A2 = −A− 15 + Σ2; Σ ≡ σ̂1 + σ̂2 + σ̂3 from which one obtains the successive results A3 + A2 + 15A = Σ2A; A2(A+ 1)2 = [Σ2 − 15]2 The last equation yields the four solutions A = −1 15− Σ2 − 1/4; A = −1 Σ2 − 15− 1/4 (5.8) whose substitution in Eqs (5.4) and (5.7) summarises the full content of the total spin effect of the Vqqq term. Finally, the denominator term D123 is a differential operator in coordinate space : D123/4 = [(−m2q + ∂23⊥)∂1z∂2z]− iM∂1z∂2z∂3z (5.9) Thus φ satisfies the following equation in coordinate space : D123φ(s, t) = [V (2)(s) + V (3)(s, t)]φ(s, t) (5.10) where the various operators are given by (5.6) - (5.9). 5.1 Simplified Form of the φ Equation (5.10) Since this paper is intended as a preliminary mathematical framework for the dynamical effect of the spin terms, especially the spin-rich Vqqq term on the spatial structure of the proton wave function, pending a full-fledged study of the proton spin anomaly, we shall at this stage merely outline a qualitative procedure to determine the nature of the solution of Eq. (5.10), with particular reference to the possible role of the eigenvalues (5.8) of A, on the solution of this differential equation. To that end, we shall make some drastic simplifications, starting with the differential operator (5.9) whose principal terms (up to second order ), in momentum space, work out as D123 ≈ ) + [ − 2m2q](ξ2z + η2z) + (m2q −M2/9) Further simplification arises with the ‘special’ value M = 3mq which then yields a simple yet transparent expression in coordinate space : D123 ≈ [−∂2s − ∂2t ] (5.11) an operator with full rotational invariance and S3 symmetry in the 3D space ( on light-front) defined by Eq. (2.2). In a similar vein, the 2-body terms (5.6) can be simplified by the replacements piz ≈ M/3, so as to yield V (2)(s, t) = 2M3αs s2 + t2 Mαs(Σ 2 − 9) s2 + t2 [∂2s + ∂ t ] (5.12) where we have made a further simplification based on certain standard in- equalities [31] s2 + t2 And the 3-body term (5.7) may be written more compactly in terms of of the operator A with eigenvalues (5.8) as V (3)(s, t) = α2s 3(s2 + t2) [∂2s + ∂ t ] (5.13) One has now a differential equation for (5.10), with the simplified operators (5.11-13) which exhibit a 6D symmetry. Taking R2 = s2 + t2, the 6D Lapla- cian for the ground state of the proton (with all angular d.o.f.’s dropped) takes the simpler form [4M2/9 + Mαs(Σ 2 − 9) 4Aα2s ](∂2R + ∂R)φ(R) + 8M3αs φ(R) = 0 which on rescaling with the dimensionless variable X = MR leads to the simpler form 2 − 9) Aα2s√ ](∂2X + ∂X)φ+ φ = 0 (5.14) 5.2 Spin Effects of Vqq & Vqqq Terms on φ Singularity The first thing to notice from the φ Equation, (5.14), is that the spin- dependent parts of both Vqq and Vqqq appear in the multiplying factor with the 6D Laplacian acting on φ, and are proportional to αs and α s respectively, with the Vqq term having a milder singularity (∼ X−1) than the Vqqq term (∼ X−2). Further information on the singularity of the differential equa- tion (at points other than R = 0) hinges on the nature of the eigenvalues of the spin operator A which appears in above multiplying factor. Now these eigenvalues are given by (5.8), two of which are complex for the state of the proton (Σ2 = 3), but the other two are real. Of the real solutions the positive eigenvalue does not yield a zero in this multiplying factor but the negative eigenvalue (≈ −4) does give a zero for a real value of R at a point X20 between X2 = 0 and X2 = ∞ : X20 − Aα2s√ = 0;X0 = + [α2s/32− Aα2s√ ] (5.15) where the value of Σ2 = 3 for the proton state has been substituted. Indeed this zero ( for A ≈ −4) is a key element of the dynamical effect of the spin - rich 3-body force term, although the spin effect of the 2-body Vqq term is marginal. To study this effect more quantitatively, we seek an approximate solution of Eq. (5.14) in the neighbourhood of the zero at (5.15) for which a crude nu- merical estimate suggests the following location. Taking a 3-flavour structure αs = 2π/[9 lnM/Λ], with Λ ≈ 150MeV , one finds αs ≈ 0.39, whence X0 ≈ 0.0689±X1; X1 ≡ 0.5841 This shows that the qq term gives ∼ 10% shift around the central value of X1, which corresponds to R ≈ 0.12fm, and is almost entirely due to the spin effect of the Vqqq term. Therefore in the spirit of this qualitative investigation, it makes sense to drop this 10% effect, in which case Eq (5.14) simplifies to ](∂2X + ∂X)φ+ φ = 0; X1 = 0.584 (5.16) To bring it nearer to a standard form, transform to the independent variable z according to zX20 = X 0 −X2, which yields z(1 − z)∂2zφ− 3z∂zφ− β 1− zφ = 0; β ≡ αs X1 ≈ 0.058 (5.17) This equation is almost (not quite) of the hypergeometric form but it can be reduced to a standard one (with singularities located at z = 0, 1,∞) [32] by exploiting the smallness ( β = 0.058 ) of the last term to replace it with a constant ( with value corresponding to z = 1/2) : z(1 − z)∂2zφ− 3z∂zφ− φ = 0 (5.18) An equivalent equation may be obtained with the transformation z = 1−x : x(1 − x)∂2xφ+ 3(1− x)∂xφ− φ = 0 (5.19) where x = 1 corresponds to the location X = X1 of the singularity, as indicated in Eq. (5.16). [ Note that this singularity corresponds to the negative eigenvalue of A, thus reflecting the dynamical effect of the spin-rich Vqqq term, while a positive or complex eigenvalue of A would result in the disappearance of this singularity ]. The solution of Eq.(5.19) is then given by [32] φ = F (a, b|3|x); a + b = 2; ab = (5.20) with the entire machinery of hypergeometric functions available [32] for ex- ploring its properties according to need. The scaled variable x is related to the 6D distance R by MR = X1x where X1 ≈ 1.5αs is directly related to the location of the singularity induced by the spin structure of the Y-shaped 3-body force . And the reconstruction of the full Bethe Salpeter wave func- tion Ψ as a sum of 3 pieces Ψi is now only a matter of substituting (5.20) in (4.11), whence the corresponding vertex functions Vi are immediately iden- tified [22, 23] for specific transition amplitudes a la Feynman diagrams. 6 Retrospect : Critique Of The 3-Body Force In retrospect, we have considered, in conjunction with pairwise qq forces, the effect of a new type of 3-body force Vqqq, rich in spin content, on the ana- lytical structure of the qqq wave function in the high momentum regime of QCD where the confining interaction is unimportant, rendering the dominant force Coulombic. As to the anatomy of this (spin-rich) Vqqq , we have taken it to be generated by a ggg vertex ( a genuine part of the QCD Lagrangian ) wherein the 3 radiating gluon lines end on as many quark lines, giving rise to a (Mercedes-Benz type) Y -shaped diagram, a la fig 1. From a physical point of view it is natural to expect that such a spin-rich structure should play a potentially crucial role in the so-called ‘spin anomaly’ of the proton, a subject that seems once again to have raised its head in the context of new polarized beam techniques now available [5] for resolving the issue experi- mentally. With that end in view, our strategy has been to determine the dynamical effect of the spin dependence of Vqq and Vqqq forces on the ana- lytical structure of the internal 3D wave function φ. [It is emphasized that this effect must be carefully distinguished from the ” kinematical ” effect of spin, which manifests in several other ways, namely through the presence of various γ matrices that appear in equations like (4.11) connecting φ to the full 4D wave function Ψ]. Indeed we found in Section 5 that while the spin- dynamical effect of 2-body forces is marginal, that of the spin-rich 3-body force is quite pronounced, and shows up through the possible ” eigenvalues ”, Eq (5.7), of the spin operator (iσ1.σ2 × σ3) which is a part of Vqqq : only a negative eigenvalue (there is only one !), induces a singularity in the dif- ferential equation for φ, but not others. The resulting dynamical effect is expressed by a hypergeometric function, Eq.(5.20), which gets folded into the 4D BS amplitude Ψ via Eq.(4.11), thus fulfilling the original (limited) motivation behind this study. Unfortunately, the dynamical framework underlying the contents of this paper has had a long history born out of the author’s long involvement with the so-called Bethe-Salpeter Equation (BSE), often with extended periods of stalemate (and frustration !), but mostly centred around the quest for a sat- isfactory definition of probability within the BSE framework. Eventually it became possible to settle for a toned down version of BSE, that of a Salpeter- like equation ( 3D support for the kernel), which is amenable to a probability interpretation at the 3D level. And its 4D features, although present in the original formulation itself [6], had for decades remained hidden from view, but were finally dug out [7] in the context of an independent approach designed to explore a 3D-4D interconnection between the corresponding BS amplitudes when the kernel has a 3D support [8]. Further, the lack of covariance in the original formulation [6] was subsequently remedied via a special (instanta- neous) frame of reference in which the composite hadron of 4-momentum Pµ is at rest [33]. This result turned out to be in conformity with the Markov- Yukawa Transversality Principle (MYTP) [11, 12], which happens to be a ‘gauge principle’ in disguise [13] : It ensures that the interactions among the constituents be transverse to the direction of Pµ. Subsequent refinements have been mostly technical , especially the use of Dirac’s light-front formula- tion so as to extend the dynamical range of validity of the BS framework by overcoming certain practical problems like ‘Lorentz mismatch disease’ [17] associated with different vertices of a given Feynman diagram. It therefore looked worthwhile to employ this old-fashioned formalism in the present context of a new kind of 3-body force Vqqq, with enough de- tails put in for a reasonably self-contained description without having to dig frequently into the original sources. This has also given an opportunity to make several refinements, especially the derivation of a common denomina- tor function associated with both the qq and qqq forces and extending the earlier formalism to accommodate the Vqqq force, for which this theoretical framework seems to be well suited. The next part of this programme involves the derivation of the requi- site QCD parameters for the baryon spin, for which some key ingredients are i) the forward scattering amplitude off the iγµγ5 operator, inserted at the individual quark lines, and ii) a more elaborate 2-gluon contribution to the proton spin, which is reserved for a subsequent paper within the same formalism. 7 Acknowledgements Several colleagues have contributed to the evolution of the BS formalism, with and without active authorships, and the present paper also shares the general acknowledgement. Yet two items stand out in the specific context of this paper : First, the analysis in Section 5 owes its origin to a simple- minded methodology due to his late Father, Jatindranath Mitra. Second, the pedagogical techniques of qqq symmetry with several d.o.f.,s, which have played a key role in this paper, are due to the late Mario Verde as described in Hand Buch der Physik Vol 24, 1957. The author also acknowledges the comments of Aalok Mishra which led to the use of the eigenvalues of the spin operator A in Section 5. He is also grateful to Vineet Ghildyal for help with figure 1. Appendix A : qq Subsystem Formalism Using the correspondence (2.2) of the text, most of the covariant instant form results of [22] may be taken over to the present light-front situation [7]. In this Appendix, we outline the structure of the 4D / 3D interconnection between the corresponding Green’s functions for the qq sub-system as a prototype for the actual qqq system considered in Section 2 of the text. The qq Green’s functions satisfy the respective equations [22]: (2π)4iG(q, q′;P ) = d4q′′V (q̂, q̂′′)G(q′′, q′;P ); (A.1) Ĝ(q̂, q̂′) = dq0dq 0G(q, q ′;P ) (A.2) Integrating both sides of (A.1) gives via (A.2), the 3D BSE for a bound state (no inhomogeneous term !): (2π)3D(q̂)Ĝ(q̂, q̂′) = d3q̂′′V (q̂, q̂′′)Ĝ(q̂′′, q̂′) (A.3) where the 3D denominator function D(q̂) is defined as D(q̂) (A.4) leading (for general unequal mass kinematics) to [8] D(q̂) = D+(q̂); D+(q̂) = 2P+[q̂ λ(M2, m21, m ] (A.5) where λ is the triangle function of its arguments. Now define the hybrid Green’s functions [22]: G̃(q̂, q′) = dq0G(q, q ′;P ); G̃(q, q̂′) = dq′0G(q, q ′;P ) (A.6) Using (A.6) on the RHS of (A.1) gives (2π)4iG(q, q′;P ) = d3q̂′′V (q̂, q̂′′)G̃(q̂′′, q′) (A.7) Integrating (A.1) w.r.t. dq′0 only, and using (2.7) again, gives (2π)4iG̃(q, q̂′) = d3q̂′′V (q̂, q̂′′)Ĝ(q̂′′, q̂′) (A.8) From (A.8) and (A.3), G̃ and Ĝ are connected as: G̃(q, q̂′) = D(q̂) 2iπ∆1∆2 Ĝ(q̂, q̂′) (A.9) Interchanging q and q′ in (A.9) gives the dual result G̃(q̂, q′) = D(q̂′) 2iπ∆′1∆ Ĝ(q̂, q̂′) (A.10) Substitution in (A.7) gives the desired 3D-4D interconnection G(q, q′;P ) = D(q̂) 2iπ∆1∆2 Ĝ(q̂, q̂′;P ) D(q̂′) 2iπ∆′1∆ (A.11) Next, spectral representations for the 4D and 3D G-fns in (A.11) give [22] G(q, q′;P ) = Φn(q;P )Φ ′;P )/(P 2 +M2); (A.12) Ĝ(q̂, q̂′) = φn(q̂)φ ′)/(P 2 +M2) (A.13) where Φn and φn are 4D and 3D wave functions, so that their interconnection (valid for any bound state n), is expressed by: Γ(q̂) ≡ ∆1∆2Φ(q;P ) = D(q̂)φ(q̂) (A.14) which tells us that the covariant vertex function Γ on the light-front is a function of q̂ only [22] via its definition (2.2). This derivation is a prototype for the qqq case of text. References [1] Ernest Ma, Phys. Rev.D12, 2105 (1975). [2] B. McKellar, private communication ; 1994. [3] A. N. Mitra, Phys Rev. D28, 1745 (1983). [4] J.C. Taylor, Gauge Theories of Weak Interactions, Cambridge Univ Press, 1978. [5] S.D. Bass Rev. Mod. Phys. 77, 1257-1302 (2005). [6] E.E. Salpeter, Phys.Rev. 87 (1952) 328. [7] A. N. Mitra and B. M. Sodermark, Nucl. Phys. A 695, 328 (2001). [8] S. Chakrabarty et al, Prog. Part. Nucl. Phys. 22, 43-180 (1989). [9] N. N. Singh and A.N. Mitra, Phys. Rev.D38, 1454 (1988); C.R. Munz et al, Nucl. Phys. A578, 397 (1994). [10] A. Sharma et al, Phys.Rev.D50 (1994) 454. [11] M.A. Markov, Sov. J. Phys. 3 (1940) 452. [12] H. Yukawa, Phys.Rev. 771950) 219. [13] J. Lukierski and M. Oziwicz, Phys. Lett.B69, 339 (1977). [14] I. Santhanam et al, Intl J Mod Phys E2, 219 (1993) [15] P.A.M. Dirac, Rev. Mod. Phys. 21 (1949) 392. [16] S. Weinberg, Phys. Rev. 150 (1966) 1313 [17] A.N. Mitra, Phys. Lett.B463 (1999) 293. [18] A.N. Mitra, Nucl. Phys. A684, 313c (2000) [19] J. Carbonell et al, Phys. Rep.400C, 215 (1998) [20] V. Kadychevsky Nucl. Phys. B6, 125 (1968) [21] V.A. Karmanov, Nucl.Phys. B166, 378 (1980). [22] A.N. Mitra Int J Mod Phys.A14, 4781 (1999). [23] A.N. Mitra and B.M. Sodermark, lanl ark hep-ph/0104219. [24] A.N. Mitra and A. Mittal, Phys.Rev.D29, 1399 (1984). [25] R.P. Feynman et al, Phys.Rev. D3, 2706 (1971) [26] R. Blankenbecler et al, Nucl.Phys.12, 629 (1959) [27] A.S. Davidov Quantum Mechanics, Pergamon Press Oxford, 1965. [28] A.N. Mitra and M.H. Ross, Phys. Rev.158, 1630 (1967) [29] A.N. Mitra, Zeits. f. Phys. Particles & Fields, C8, 25 (1981). [30] A.N. Mitra and B.M. Sodermark, hep-ph/0104219 [31] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Nat. Bur. of Standards 1965 [32] Morse-Feshbach, Methods of Theoretical Physics, McGraw Hill, New York, 1953. [33] A.N. Mitra & S. Bhatnagar, Intl. J Mod Phys A 7, 121 (1991) http://arxiv.org/abs/hep-ph/0104219 http://arxiv.org/abs/hep-ph/0104219 Introduction Theoretical Ingredients Plan of the Paper Salpeter Eq on LF : From To Master Eq : Instant Vs LF Forms of Dynamics Reduction of Master Eq from to Gordon Reduction on Vqq3 & Vqqq Further Reduction of Vqq3 & Vqqq 3D Reduction By Green's Function Method Reduction from 4D G(; ' ' ) to 3D "705EG("705E"705E;"705E'"705E') Reconstruction of 4D Wave Function Reconstruction of G with qq Forces Only Reconstruction of G with qqq Forces Only Putting Both Vqq & Vqqq Together 3D-4D Interconnection for Wave Functions Complete Equation In Coordinate Space Simplified Form of the Equation (5.10) Spin Effects of Vqq & Vqqq Terms on Singularity Retrospect : Critique Of The 3-Body Force Acknowledgements

0704.1104

On the induction of the four-dimensional Lorentz-breaking non-Abelian Chern-Simons action

On the induction of the four-dimensional Lorentz-breaking non-Abelian Chern-Simons action M. Gomes,1 J. R. Nascimento,1, 2 E. Passos,2 A. Yu. Petrov,2 and A. J. da Silva1 1Instituto de F́ısica, Universidade de São Paulo Caixa Postal 66318, 05315-970, São Paulo, SP, Brazil∗ 2Departamento de F́ısica, Universidade Federal da Paráıba Caixa Postal 5008, 58051-970, João Pessoa, Paráıba, Brazil† Abstract A four-dimensional Lorentz-breaking non-Abelian Chern-Simons like action is generated as a one-loop per- turbative correction via an appropriate Lorentz-breaking coupling of the non-Abelian gauge field to a spinor field. This term is shown to be regularization dependent but nevertheless it can be found unambiguously in different regularization schemes at zero and at finite temperature. Electronic address: mgomes,jroberto,[email protected] Electronic address: jroberto,passos,[email protected] http://arxiv.org/abs/0704.1104v2 mailto:mgomes,jroberto,[email protected] mailto:jroberto,passos,[email protected] During the last years, different aspects of the Lorentz symmetry breaking have been intensively studied [1]. One of the theoretical consequences of this effect is the birefringence of light in vacuum. After the formulation of the concept of noncommutativity of the space-time [2], which implies in Lorentz symmetry breaking (see the discussion in [3]), The interest in this subject has greatly increased . One of the implications of the Lorentz symmetry breaking is the possibility of introducing a lot of new couplings in the Standard Model [4]. These terms may arise from radiative corrections to some Lorentz-breaking field theories at zero [5, 6, 7, 8, 9, 10] and at finite temperature [11, 12, 13, 14]. Alternatively, they may be induced from the deformation of the canonical commutation relation through the use of the noncommutative fields method [15, 16]. Recently, the renormalizability of the Yang Mills (YM) theory with a four-dimensional non- Abelian Lorentz-breaking Chern-Simons (CS) term was studied in [17]. The induction of such Lorentz-breaking CS term starting from a pure YM was investigated within the noncommuta- tive fields method in [16]. In the present work we show how the same CS term can be induced through radiative corrections starting from a YM theory coupled with fermions in the presence of an interaction of the fermions with a constant external field at zero and at finite temperture. We start with the following model which represents a non-Abelian generalization of the spinor electrodynamics with the Lorentz-breaking coupling Lf = ψ (i∂/−m− γ5b/)δij − gγµAaµ(Ω ψj . (1) Here bρ is a constant four-vector. The Aµ = A a is a Yang-Mills field coupled to spinors ψ which carry the isotopic indices, ψ = (ψi), with i taking values from 1 to N with N being the dimension of the chosen representation of the Lie algebra. The Ωa = (Ωa)ij are the Lie group generators in this representation satisfying the relations: [Ωa, Ωb] = ifabcΩc and Tr[ΩaΩb] = δab. The one-loop effective action of the gauge field Aaµ is SYM+Sf [b,A] where SYM is the Yang-Mills action and Sf [b,A] can be expressed in the form of the following functional trace: Sf [b,A] = −iTr ln(i∂/−m− γ5b/− gγµAaµΩ a). (2) This functional trace can be rewritten as Sf [b,A] = Sf [b] + S f [b,A], with the first term being Sf [b] = −iTr ln(i∂/ − m − γ5b/). The nontrivial dynamics is concentrated in the second term S ′f [b,A], which is given by the power series: S ′f [b,A] = iTr i∂/−m− γ5b/ gγµAaµΩ . (3) To make explicit the non-Abelian Chern-Simons term we should expand this expression up to the third order in the gauge field: S ′f [b,A] = S f [b,A] + S f [b,A] + . . . (4) where f [b,A] = i∂/−m− γ5b/ γµAaµΩ i∂/−m− γ5b/ γνAbνΩ b], (5) f [b,A] = i∂/−m− γ5b/ γµAaµΩ i∂/−m− γ5b/ γλAbλΩ i∂/−m− γ5b/ γνAcνΩ c]. (6) and the elipsis stands for higher order terms in the gauge field. Using the above expressions, it is easy now to verify that the one loop effective action expanded up to first order in bµ may be written as S ′f [b,A] = d4x kρǫ ρµνλ(∂λA igAaµA abc) (7) where kρ is kρ = 2ig (2π)4 2 + 3m2)− 4pρ(b · p) (p2 −m2)3 . (8) This result exactly reproduces the structure of the non-Abelian Lorentz-breaking Chern-Simons term described in [17]. One can observe that the expressions (7,8), after reduction to the Abelian case, coincide with the known Abelian results [14, 18]. Apparently, there is a relation between the induced Lorentz-breaking Chern-Simons term and Adler-Bell-Jackiw anomaly as both situation are observed for the well known triangle graph. This issue has been discursed in Ref.[6]. Also, the interesting discussion of the problem of ambiguities in the Lorentz-breaking theories is presented in [7]. By power counting, the momentum integral in expression (8) involves terms with logarithmic divergence so that different regularization prescriptions will produce diverse outcomes. Lorentz preserving regularizations, more precisely any regularization in which we can make: pµpν → will produce finite results. By adopting the method of dimensional regularization [19], the above integral is promoted to D dimensions and a straightforward calculation yields kρ = 2ig (2π)D (p2 −m2)3 )p2 + 3m2] 4g2 (4−D) Γ((4−D)/2) Γ(3)(4π)D/2 bρ, (9) which coincides with the result found in [18] for the Abelian situation. If, instead of dimensional regularization, the integral in Eq. (8) is kept in four dimensions the regularization enforced the replacement kρ = 6ig 2m2bρ (2π)4 (p2 −m2)3 bρ, (10) which now agrees with the Abelian result obtained in [20]. To develop calculations in the finite temperature case, let us now assume that the system is in the state of thermal equilibrium at a temperature T = 1/β. In this case, we can use the Matsubara formalism for fermions, which consists in taking p0 ≡ iωn = (n + 1/2)2πiβ and replacing the integration over the zeroth component of the momentum by a discrete sum (1/2π) dp0 → iβ Thus, the Eq. (7) can be written as S ′f [b,A] = d4x kρ(β)ǫ ρµνλ(∂λA igAaµA abc). (11) Hereafter all expressions are in the Euclidean space (all greek indices run from 1 to 4). The vector kρ(β) is given by kρ(β) = (2π)3 bρ(3m 2 − p2) + 4pρ(b · p) (p2 +m2)3 . (12) By extending the ~p integration to d dimensions it follows that the time-like component of kρ(β) is k4(β) = (2π)d 3M2n − ~p 2 (~p 2 +M2n) , (13) where M2n = ω 2. Using the prescription of dimensional regularization [19], we have k4(β) = − (4π)d/2 [dΓ(2 − d/2) − 6Γ(3− d/2)] (M2n) 2−d/2 λ−1/2 (3− d) Γ(λ) [(n+ 1/2)2 + a2]λ , (14) where a = mβ/2π and λ = 2− d/2. At this point the following identity [21]: [(n + b)2 + a2]λ πΓ(λ− 1/2) Γ(λ)(a2)λ−1/2 + 4 sin(πλ) (z2 − a2)λ exp 2π(z + ib)− 1 , (15) valid for 1/2 < λ < 1 can be used to get k4(β) = +4 (3− d)(a2) 2 Γ(λ) sin(πλ) (z2 − a2)λ exp 2π(z + ib)− 1 . (16) In d = 3 this gives k4(β) = b4, (17) i.e., the same result (9) without any dependence on the temperature, which agrees with the one obtained in [22] for the Abelian situation. If instead of (12) we use (10) as the starting point for the computation of finite temperature effects we get k4(β) = b4( F (a)), (18) where F (a) = dz(z2 − a2)1/2 tanh(πz) cosh2(πz) , (19) has the following asymptotics: F (a → ∞) → 0 (T → 0) and F (a → 0) → 1/2π2 (T → ∞), see Fig.1. 0.0035 0.004 0.0045 0.005 0.0055 0.006 0 0.2 0.4 0.6 0.8 1 1.2 FIG. 1: The function F (a) is different from zero everywhere. At zero temperature (β → ∞), the function tends to a nonzero value 1 Let us now consider the space part, ki(β), of the vector kρ(β). In this case, the expression (12) can be rewritten as: ki(β) = (2π)3 bi(3m 2 − p2) + 4pi(b · p) (p2 +m2)3 , (20) Then, considering this expression formally in d space dimensions, we can replace pipj by δij , hence we get ki(β) = (2π)d 4m2 − (d−4 )~p2 −M2n (~p2 +M2n) , (21) which now furnishes ki(β) = 4m2g2 Γ(3− d (4π)d/2 (m2 + ω2n) (a2)λ−1/2Γ(λ) [(n+ 1 )2 + a2]λ , (22) where we have introduced λ = 3 − d . We cannot apply the relation (15) for d = 3, because the integral in that expression does not converge. Thus, let us perform the analytic continuation of that relation; we obtain [13] (z2 − a2)λ exp 2π(z + ib)− 1 3− 2λ (z2 − a2)λ−1 × (23) exp 2π(z + ib)− 1 (2− λ)(1 − λ) (z2 − a2)λ−2 exp 2π(z + ib)− 1 Thus for d = 3 the Eq. (22) takes the form ki(β) = bi( F (a)), (24) where F (a) was defined in (19). Thus, we see that at high temperature the Chern-Simons coef- ficient is twice its value at zero temperature, i.e., ki(β → 0) = 12π2 . On the other hand, at zero temperature, one recovers the result ki(β → ∞) = 14π2 . We have generated the non-Abelian Lorentz-breaking Chern-Simons term via the Lorentz- breaking coupling of the Yang-Mills field with the spinor field at zero and at finite temperature. The essential property of the result is that within the framework of dimensional regularization this term turns out to be finite. We note that the derivative expansion approach naturally allows to preserve the gauge invariance for the quantum corrections. It is natural to expect that at least some of other Lorentz-breaking terms which existence was predicted in [4] also can be generated via appropriate couplings of the gauge or gravity fields with some matter fields. We have also obtained the coefficient kρ for the non-Abelian Lorentz-breaking Chern-Simons term at the finite temperature. We found that the results for this term turn out to be dependent on the regularization scheme both at zero and at finite temperature (in a particular regularization scheme the time-like component was found to be temperature independent). Considering the dependence on the regularization scheme, one should note that the momentum integral determining the value of the vector kρ is formally superficially divergent, thus dependence of its finite part on the renormalization procedure is very natural. However, in the regularization schemes suggested in the paper the divergent part identically disappears as a consequence of the rotational invariance of the relevant integrands. Acknowledgements. This work was partially supported by Conselho Nacional de Desenvolvi- mento Cient́ıfico e Tecnológico (CNPq) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP). The work by A. Yu. P. has been supported by CNPq-FAPESQ DCR program, CNPq project No. 350400/2005-9. [1] V. Kostelecky, S. Samuel, Phys. Rev. D 39, 683 (1989); S. Coleman, S. Glashow, Phys. Rev. D 59, 116008 (1999); J. Magueijo, L. Smolin, Phys. Rev. D 67, 044017 (2003); Class. Quant. Grav. 21, 1725 (2003); O. Bertolami, C. Carvalho, Phys. Rev. D 74, 083020 (2006). [2] N. Seiberg, E. Witten, JHEP 09, 032 (1999). [3] S. Carroll, J. Harvey, V. Kostelecky, C. Lane, T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001); C. E. Carlson, C. D. Carone, and R. F. Lebed, Phys. Lett. B 518, 201 (2001); C. E. Carlson, C. D. Carone, and R. F. Lebed, Phys. Lett. B 549, 337 (2002); O. Bertolami, L. Guisado, JHEP 12, 013 (2003); J. Collins, A. Perez, D. Sudarsky, L. Urrutia, and H. Vucetich, Phys. Rev. Lett. 93, 191301 (2004); A. F. Ferrari, H. O. Girotti, M. Gomes, Phys. Rev. D 73 047703, 2006. [4] V. A. Kostelecky, Phys. Rev. D 69, 105009 (2003); ”Lorentz violation and gravity”, hep-ph/0412406. [5] J. M. Chung and P. Oh, Phys. Rev. D 60, 067702 (1999). [6] M. Perez-Victoria, Phys. Rev. Lett. 83, 2518 (1999). [7] R. Jackiw, Int. J. Mod. Phys. B14, 2011 (2000). [8] A. A. Andrianov, P. Giacconi and R. Soldati, JHEP 0202, 030 (2002). [9] M. Chaichian, W. F. Chen and R. Gonzalez Felipe, Phys. Lett. B 503, 215 (2001). [10] B. Altschul, Phys. Rev. D 69, 125009 (2004). B. Altschul, Phys. Rev. D 70, 101701 (2004). [11] D. Bazeia, T. Mariz, J. R. Nascimento, E. Passos, R. F. Ribeiro, J. Phys. A 36, 4937 (2003). [12] T. Mariz, J. R. Nascimento, E. Passos, R. F. Ribeiro, Phys. Rev. D 70, 024014 (2004). [13] L. Cervi, L. Griguolo, D. Seminara, Phys. Rev. D 64, 105003 (2001). [14] T. Mariz, J. R. Nascimento, E. Passos and R. F. Ribeiro, JHEP 10, 019 (2005). [15] J. M. Carmona, J. L. Cortes, J. Gamboa, F. Mendez, Phys. Lett. B 565, 222 (2003); JHEP 03, 058 (2003); J. Gamboa, J. Lopez-Sarrion, Phys. Rev. D 71, 067702 (2005); H. Falomir, J. Gamboa, J. Lopez-Sarrion, F. Mendez, A. J. da Silva, Phys. Rev. D 74, 047701 (2006). [16] H. Falomir, J. Gamboa, J. Lopez-Sarrion, F. Mendez, A. J. da Silva, Phys. Lett. B 632, 740 (2006). [17] D. Colladay and P. McDonald, ”One-loop renormalization of pure Yang-Mills with Lorentz violation”, hep-ph/0609084. [18] A. A. Andrianov, P. Giacconi and R. Soldati, JHEP 02, 030 (2002). [19] G. ‘t Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972). http://arxiv.org/abs/hep-ph/0412406 http://arxiv.org/abs/hep-ph/0609084 [20] R. Jackiw and V. A. Kostelecky, Phys. Rev. Lett. 82, 3572 (1999). [21] L. H. Ford, Phys. Rev. D 21, 933 (1980). [22] D. Ebert, V. Ch. Zhukovsky and A. S. Razumovsky, Phys. Rev. D 70, 025003 (2004). References

0704.1105

Neutrino Astronomy with High Spatial Resolution is Already Existing

2007: Neutrino Astronomy with High Spatial Resolution is Already Existing NEUTRINO ASTRONOMY WITH HIGH SPATIAL RESOLUTION IS ALREADY EXISTING V.A. Rantsev-Kartinovξ RRC “Kurchatov Institute”, Kurchatov Sq. 1, 123182, Moscow, Russia ξ email:[email protected] Abstract By basing on observations of skeletal structures of the Sun and assuming that some of them are located inside of star, and also that a filamentary (linear) matter (whose a model earlier was put forward by B.U. Rodionov) is in basis of these internal structures the author consider pos- sible processes of images formation of these structures inside the Sun and theirs coming out into space and also gives an elementary estimations of its parameters, which allow: i) to form their images in a flux of electronic neu- trinos; ii) to carry out these images from within of the Sun into space; iii) to develop these images in form of a con- comitant flux of soft x-ray, which next is recorded by telescope of soft x-ray. It is supposed the processes con- sidered here, actually, can be accepted as future base of neutrino astronomy with high spatial resolution. I. INTRODUCTION The analysis (by means of the method of multilevel dy- namical contrasting (MMDC) developed and described by the author earlier [1a, 1b]) of images of the Sun in range of waves lengths of a soft x-ray has resulted in revealing of fractal skeletal structures (FSS) of the Sun [2]. The to- pology of these structures appeared identical to the same that was revealed and described earlier in a wide range of spatial scales, the phenomena and environments [3]. MMDC analysis of images of solar spots (SS) in soft x- ray has shown they have three-dimensional structure of the same topology (see Fig. 1). Figure 1. The two fragments of image of active zone of the Sun surface and below its schematic presentations are presented here. It is visible the SS presents coaxial tubular structures of the Sun of filamentary matter which are pushed out from bowels of our star to its surface. The to- pology of these structures is the same that in observable before in dust deposit of tokomak T-10. Tridimentionality of the structures, their inter-lacings and connections are precisely traced here. Diameters of these structures are ~ 2 109 cm and their upper butt-ends are towered above the Sun surface up to height ~ 1010 cm. Hence from this in [2] the hypothesis has been put for- ward - the Sun has internal FSS which can be by devel- opment of one of possible forms of filamentary matter (FM) (for example, which was suggested by B.U.Rodionov [4]), and SS are part of FSS of the Sun which have been squeezed out during its activity from within to its surface. The revealed fragments of FSS of the Sun which have radius of rotation around of a star axis less radius of its disk on breadth of their locations (see Fig. 2) can be good acknowledgement of this hypothesis. Figure 2. Fragment of the Sun image, = 195 Å (SOHO). The edge of a solar disk is seen overhead. Complexly weaved structure of the FM, which has coaxial tubular structures (CTS) collected of similar blocks but of the smaller size is visible. Diameter of CTS ~ 3.3 1010 cm, di- 1-4244-0914-4/07/$25.00 ©2007 IEEE. 1589 ameters of blocks (of which they are collected) are ~ 109 cm, diameter of the FM (going out from the center of its butt-end) is ~ 8 109 cm. This CTS has radius of its rotation around of the Sun axis of some smaller than radius of so- lar disk on breadth of its location. As soft x-ray cannot take out the image of "interior" of the Sun into a space the author considers the following general scheme of its formation: 1) neutrinos by passing through FSS of the Sun FM are coding and take out into space their image in their own flux; 2) this image trans- forms by any mechanism into an accompanying flux of x- ray quantums which develops on the screen of a tele- scope. For realization of such scheme it is necessary to describe the prospective nature and properties of FM of the Sun and to find mechanisms of formation and devel- oping of its image in x-ray. II. PARAMETERS OF MODIFIED MODEL OF FILAMENTARY MATTER Suggested the FM model not up to the end still is de- veloped and described, but it is suitable for primary esti- mations of basic parameters of such matter. Filaments of such FM represents infinite chains of quarks in the Bose- condensate state (BCS) strung onto quantified a magnetic field flux which minimal value is equal ϕ0 = hc/e. Here, h - the Plank constant, c - velocity of light in vacuum, e - a charge of electron. Let's consider that all quarks are in BCS, they are relativistic and have identical energy ~ 5 MeV. Then the quark filament radius (rq ~ 5·10 -13 cm) is defined by size of the de Broil wave, a magnetic field in- tensity (flux of which into one quantum connects every- thing quarks of such infinite filament into one whole) in- side of such filament is Hq ~ 3·10 17 Gs and energy of quarks connection is ~ 5 GeV. According to the model, such the quark filament can be surrounded by an electron cloud in BCS. The minimal radius of such shell of relativ- istic electrons, which is covering a back magnetic flux ( 0φ ) is re ~ 3·10 -12 cm, and a field inside this shell is He ~ 7·1015 Gs. The most important result of the analysis of revealed structures in the universe (with participation of the author) is its fractality. Above described model of FМ does not allow building structures of generations. Therefore the author modified it, assuming, the FM was formed at very hot universe. The quark filaments of final length in those days could have a shell (intercepting the back flux of a magnetic field) too of quarks. Internal cords and external shells of extended filaments of such matter can present consecutive chains of quarks in BCS with various combi- nations of charge composition. In particular, it can exist filaments which are consisting of identical sequence of the quarks which are in BCS (2×(-1/3)+2×(+2/3)+2×(- 1/3)+· ·). The charge agreeing with unit of length of any- one filament is equal to zero. Extended filaments of FМ can be assembled of separate cylindrical blocks the length of which is connected with constant thin structure α=1/137. These blocks of the minimal length can be ac- cepted as "linear atoms" (LA) of such FM. From here for estimated calculations it is possible to believe, that quan- tity of nucleons in such LA of FM NN ~ 6·10 2, linear den- sity of nucleons in it nN ~ 6·10 13 cm-1, its mass maq ~ 1.2·10-21 g, length la ~ 10 -11 cm, the relation of its length to diameter (which it is equal ~ 4rq ~ 2·10 -12 cm) ~ 5. The density of substance of FM is ~ 4·1013 cm-3, i.e. only in 5 times there is less than density of a nuclear matter. The LA of FM can build FSS, [3], showing their universality. As average density of substance of Sun ρС ~ 10 g/сm 3 the volume fraction of FМ inside a star should be small. The LA radius of FМ in 104 times is less than radius of hydro- gen atom it is neutral and does not interact with usual at- oms. Interaction of this matter with usual nuclei must be extremely weak. Such FM can exist neutral and inside the Sun as the temperature even of its most bowels is much lower than energy of connection even of electrons in ex- ternal shell of its filaments inasmuch as energy of connec- tion of electrons in FM is ~ 5 МэВ. For such LA a linear density of electrons in them is ne ~ 3·10 13 cm-1, mass mae ~ 6·10-22 g ~ 3·102 mp, where mp - mass of a proton, the lin- ear density of substance in them ~ 6·10-11 g·cm-1, the rela- tion of length to its diameter is ~ 2. III. FRACTAL SKELETAL STRUCTURES OF FILAMENTARY MATTER Inasmuch as the topology of revealed FSS of the Sun was found identical to one which was before observable in the dust carbon deposits taken from chambers of plasma installations, the model of FSS construction of the FM of LA has been chosen similar to construction FSS of carbon nanotubes [3]. The mass, number of electrons, length and diameter of tubes of generations with number n is in this case completely determined and the latest can be estimated along formulas (1): .51062~2~.;5106~ 112112 cmdlcmd nnn −−−− ⋅⋅⋅⋅⋅ , (1) From here blocks of various generations of such FSS have scale factor K = 5. As tubular filament of 2-nd generations with radial connections consists of ~ 80 LA, that number of LA in such filament of generation with number n is Nl ~ (80)n-1=23(n-1)·10n-1, full number of electrons in it is N1e ~ NN·Nl n = 3·102·23(n-1)·10n-1 = 3·102·23(n-1)·10n+1, its vol- ume is Vl n ~ 0.25·π·dn 2 ~ 4.3·10-34·53(n-1) cm3 and average electrons density in it is nle n ~ Nle n ≈ 24n+32/52n-38 cm-3. Observable filaments of FM in images of the Sun have the characteristic size in diameter ~ 3 109 cm. From here, ac- cording to expression (1), we finds number of generation of blocks of this structure, n ~ 30 and then according to above we calculates average density of electrons in these filaments, nle n ~ 2.5·1030 cm-3. IV. FORMATION AND DEVELOPMENT OF IMAGES OF THE FRACTAL SKELETAL STRUCTURES OF FILAMENTARY MATTER OF THE SUN Average density of the Sun matter is ~ 10·g·cm-3, av- erage electrons density of its plasmas is ne C ~ 5·1024 cm-3 ~5·105·nle n cm-3. On sharp gradients of electron density, occurs amplified oscillations of electronic neutrinos and turning of their some part into muonic ones [5], it is espe- cial if a condition of a resonance for this process is satis- fied. Thus, the image of FSS of the Sun FM can form in a neutrino flux, which is coming through a body of the Sun in a direction of the observer, inasmuch in areas with rather homogeneous electron density neutrino do not co- operate almost with substance. For those regions for which the condition of the resonant oscillations is satis- fied the image contrast in the neutrinos flow can make up value almost 100 %. Neutrino dispersion on filaments of FM, due to a high tension of a magnetic field in them, can promote also for formation of FSS image. Thus, the neu- trino flow is able to forming and freely to carry over the image of internal structure of the star from within into outside. ] Further there is a problem connected with displaying of such image. The fact is that we do not have the corresponding screen, capable to show such image. Because of very small value of interaction cross-section, even thickness of sphere of the Earth is not capable to play a role of such screen. Here the author puts forward a hypothesis, that in space between the Earth and the Sun there are same structures of FM, but rarer as they belong to considerably higher numbers of generations. The electronic neutrinos passing across fibers of FM, and cooperating with their magnetic field (according to conclusions of paper [6]) are able to generate the quantums extending in a direction of the observer. It is necessary to note here, that only at first sight, the setting of problem about of electromagnetic interactions of neutrinos can appear absurd inasmuch initially Pauli postulated neutrino as electrically neutral particles, i.e., as a noninteracting particles with electromagnetic fields. However it turned out, that electromagnetic properties at the massive neutrino arise at the account of its interaction with vacuum of Standard model. According to theory Weinberg – Glashow - Salam, the neutrino which is moving inside an external electromagnetic field, at the time moment of t in a point with coordinate r with some probability breaks up onto a virtual electron and +W - бозон, and at the time moment of t′ in a point with coordinate r' virtual the electron and +W - бозон mutually are absorbing each other, and converting again into the real neutrino. Quantum-mechanical indeterminancy principle, «time - energy», 2/≥∆∆ tE allows existence of such particles during of small time intervals of the order Et ∆∆ /~ . For the virtual +W - boson the estimation of this time gives: .102~ 272 ccm −⋅≈∆ Inasmuch a virtual particles taking place in the given process are a charged particles, then their interaction with an external magnetic field changes their state. In such case we have amendments to movement of the massive neutrinos. This amendment can be expressed as the amendment to value of the neutrino mass provided that the impulse neutrino does not vary: mEmcE ∆=∆ )/( 4 . One of amendments to energy can be the energy of interaction of magnetic moment of neutrino, νµ , with an external magnetic field, H , ( )( HUH νµ−= ). Differently, the Dirac massive neutrino at interaction with vacuum obtains a magnetic moment. At that it is accepted definition: the magnetic moment of neutrino is directed lengthways of neutrino spin, and for antineutrino - against At calculations of radiation amendments to the neutrino mass it is possible to take account influence of an external electromagnetic field precisely. It is allowing revealing a dynamic nature of magnetic moment of the massive neutrino and mass. It turned out, that they are complex nonlinear functions of H and E . If electromagnetic fields are weak, i.e., 0, BHE << , where 0 104.4 ⋅== e cmB e Gs, (here em - electronic mass) that the magnetic moment of Dirac neutrino (in frames of the Standard model) accepts its the static value which (in system of units where 1== c ) is equal [6]: mmeGF 0 ννν µ ⋅≈⋅≈= − eV/Gs (2). Here 251003.1 −−⋅= mGF - constant of Fermi pm - mass of proton, 9 0 108.52 −⋅== cm µ eV/Gs – electronic magneton of Bor, А=1.74 10-27. From here, it follows - 00 →νµ , at 0→νm . According to [6], for a case of a weak enough of the constant magnetic field ( λ0BH << , where 11 mλ ) and in presence of large value of a transverse impulse of neutrino ( cmp W>>⊥ ) the neutrino magnetic moment demonstrates dependence, as from intensity of a magnetic field, so from energy of neutrino. The theoretical substantiation of this is given in [6]. For our case it is possible to consider 0νν µµ ≈ . At linear along field - ( H0νµ ) of approximation [6], and homogeneous magnetic field - ( H0νµ ), the energy neutrino - ( HE ) will be written down as: −= ⊥2 ννν ζµ E HEE H , (3) where: 222222 , ⊥⊥ +=+= νννννν pmEpmE , (4) and spinal a number 1±=ζ sets orientation of particle spin lengthways or against direction of magnetic field. From laws of conservation of energy and impulse if photons are emitted it is following, the frequency of emitted photons are determined by expression (5): +=′−= casesother allin 0 Z , (5) where, Z =β – longitudinal (in relation to direction of field) component of velocity neutrino, Ω – corner between neutrino impulse p and photon k . From (5) it is visible, only such neutrino can emit photon the spin of which is directed against magnetic field (ζ = −1), and radiation is accompanied by change of projection value onto direction of magnetic field: ζ = −1 → ζ = +1. For construction of image which was formed in the neutrino flux (by means of a flux of quantums obtained from them) it is necessary to put 0=Ω , i.e., these quantums should have the same direction of spreading, as neutrinos. Precisely the same formula has been obtained for radiation of neutron moving across of constant magnetic field earlier [7], theirs identity is evident if instead of magnetic moment of neutrino 0νµ , to take magnetic moment of neutron nµ . Now, on the basis of obtained image in a range of lengths of waves of soft x-ray (100-400) Å, if energy of solar neutrinos is 0.4~νE (MeV), we will able to estimate, what rest mass of neutrino should be in order to the formula (5) was satisfied. This waves range corresponds to interval of frequencies 1710)5,02(~ ⋅−ω s-1. Let's consider a case of perpendicular distribution of neutrino flux concerning direction of magnetic field, and 1710~ω s-1 which suits to experimental observations. Now, on the basis of expressions (2) and (5), for the given case it is possible to write down expression which gives the top limit for definition of rest mass of neutrino: )(107.1~ m −⋅≤ . (6) In papers [6] and [7] expressions of probability of quantums radiation of light by the neutron and neutrino at movement of theirs in the constant magnetic field are listed, accordingly. On the basis of the expressions which are given in these papers, it is possible to show, that the full probability of quantums radiation at movement neutrino across a direction of homogeneous magnetic field ( H ) with taking into account (2) and (6) is proportional to 8H a value: W , (7) i.e., quickly grows with increase of the magnetic field value. This value quickly grows with increase of power of homogeneous magnetic field, and also at 1→β . From the analysis of expression (7) it follows, that the estimation of probability of the given process needs to be carried out or along fields for which scale of lengths of radiated waves have maximal power of the magnetic field or when 1→β . Here, it is needed to consider a three cases when the flux of the Dirac neutrino generates quantums at its interaction with: a) a dipole of magnetic field of the Sun (which at the highest activity can reach value ~ 102 Gs; b) with a magnetic flux of base filaments of FM, which are taking place in cosmos between the Earth and the Sun and connected with fractal structure of FM inside a star (i.e. located near to a surface of a star); c) with a magnetic flux of base filaments of FM, which is taking place in space between the Earth and the Sun and connected with fractal structure of FM of the Earth (i.e. taking place near to a surface of the Earth). Inasmuch near to the Sun surface, a power of magnetic dipole field, density of basic filaments of FM and flux neutrino have the greatest values therefore the basic contribution to formation of soft x-ray quantums (which have predominant direction - to the Earth) will give areas closely adjoining to the Sun surface. Expres- sion (6) for our case (a) gives 15108.1 −⋅≤amν eV, and in the case (b) - 1.5≤bmν eV, that practically coincides with value obtained in experiments. As the neutrino mag- netic moment is proportional to its mass (see (2)) and very small, that (according to (7)) the probability of process for the case (a) (although, 412 102~1 −⋅− β ) is very small value too. Estimations of probability of emission of quan- tums by the neutrino flux for a case (c) are showing the probability of emission of quantums in this case in 300 times is less than in a case (b). So, absolutely clearly, the basic contribution to quantums radiation by flux neutrinos can give born quantums only in the case (b). At the same time there is no necessity in averaging of magnetic field over volume of superficial layer occupied with FM struc- ture of such generation a diameter of which is little biger than the Sun disk diameter. However and in this case our estimations have shown, that at taking into account of de- scribed above model of construction of skeletal structures of FM and gathering of quantums during of several tens of minutes by entrance surface of telescope ~ 104 cm2), it is impossible to obtain dense enough flux of attendant quantums (which are generated due to interaction of flux solar neutrinos with magnetic field of FM located near to the Sun surface) for construction of image of solar in- sides. If neutrinos are driving in very strong and variable magnetic field they also are able to radiate attendant quan- tums. Inasmuch as LA of FM inside itself carry direct and back the magnetic fluxes (which are quantized), the neu- trino movement across of FM with oriented blocks actu- ally means their movement in strongly variable magnetic field which frequency of change is determined by average of linear density of LA of FM along a trajectory. The layer of such oriented LAs of FM can be formed of free LAs which are not included in rotating FSS of the Sun which form original halo on the distance determined by equality of gravitation and centrifugal force acting on them. The estimation gives such halo will be on distance ~ 108 km from a star. As length of wave in which have been obtained images of the Sun (and which were dis- cussed here) is ~ 10-6 cm, then the average linear density of LAs of FM in halo should be ~ 106 cm-1. As the possi- ble mechanism of development of the latent image in the neutrino flux, it would be possible to consider also their interaction with probable halo from particles of "a dark matter» (DM), but for this purpose it is necessary to know, first of all, its physical nature which remains a rid- dle for us in present time, and its interaction with FSS of FM and neutrino. Here it is necessary to note onto that fact, the LAs of FM may be considered also as the DM particles. If the conclusion of the author about observation by him of FSS of FM appears precisely confirmed, then we know the answer of a problem about the mechanism of conversion of the image of interiors of the Sun coded in a neutrino flux, and therefore it will be necessary to find this mechanism only. At the given moment the estima- tions, obtained from the analysis of a database of images of the Sun shows that at the taking into account of the model of construction of FSS of FM described above and gathering of quantums during several tens of minutes by the area of an entrance lens of a telescope by square ~ 104 cm2, the obtained flux of accompanying x-ray quantums (generated for the account not for a while yet of the un- known mechanism) is found by sufficient for develop- ment of the image of insides of the Sun. At the same time the spatial resolution can reach value up to 5 107 cm that almost in 104 times higher than it was been obtained in up-to-the-minute the Superkamioknade project [8] (see item 5). More detailed consideration of the given problem is complicated because an all available calculations of probability of radiation of quantums are carried out for movement neutrino in a homogeneous magnetic field while evidently, the neutrino pass across fibers of FM inside of which there is high power of mag- netic field that can lead to essential increase of probability of radiation and which can take place here. V. SUPERKAMIOKNADE NEUTRINO TELESCOPE Abundantly clearly, the neutrino telescope can be created and on the basis of direct neutrino detecting. It has been carried out at experiments SUPERKAMIOKANDE (Japan). The principle of action of this telescope is based on the fact the vigorous neutrino, passing through envi- ronment and cooperating with electrons gives to them the significant impulses which is directed along a neutrino trajectory. Such electrons emit a cone of light of Cer- enkov radiation in a direction of their movement. This ra- diation is registered by specially created sensitive pho- tomultipliers. The cylindrical tank filled highly with cleared water is the base of a telescope. Its height is 36 m and diameter - 34 m. The full weight of a tank is 50 kilo- ton. Walls and a bottom of this tank are covered with as- sembly with 11146 specially designed and very sensitive photomultipliers with low-noise photo-cathodes in diame- ter 50 cm. The arrangement of these photomultipliers was such that all sites of volume of this tank were looked through by them with big reliability. Due to creation of this telescope the image of the Sun almost for 3 years of an exposition has been obtained. The remarkable result of this huge, laborious work is submitted in a Fig. 3. Figure 3. This is a neutrino profile of the Sun which was been obtained with help Super-Kamiokande of the detec- tor [8]. One pixel of the image corresponds to one degree which should be compared with a half of degree of a solar seen disk. Distribution of contrast is determined by a cor- ner of neutrino dispersion on electrons. VI. NEUTRINO ASTRONOMY OF THE HIGH SPATIAL RESOLUTION The modern model of the Sun guarantees thermonu- clear reactions only in its central part. Direct detecting of a solar neutrino flux should correspond to scale of the im- age of one third of its visual disk. Modern the neutrino telescope, [8], is not capable to register inside spatial structures even of the nearest star, especially of far galax- ies. The size of the image of a solar disk (see Fig. 3), ob- tained with its help for 2.5 years in a neutrino flux, ap- peared almost in 25 times more of visual one and 75 times more the size corresponding to neutrino-active area of the star. How it was been shown above if solar neutrino are Dirac-neutrino then they possess by not the zero magnetic moment and at interaction with a magnetic field they are able to radiate quantums of light. Inasmuch as value of magnetic moment of the neutrino is very small, in order the probability of such process was not zero, very big in- tensity of a magnetic field is necessary. It can take place only inside filaments of the matter which was been con- sidered above. If suggested FM is a reality then there can be and an neutrino astronomy of the high spatial resolu- tion. The fact is that the image of "insides" of the star is coded in a neutrino flux given birth inside of the same star. Further, this flux carry out image to outside and (by means of interaction with FSS of FM of the Sun and halo of oriented, free LAs or DM particles) translates this im- age into coded (but already in a flux of attendant quan- tums) the image, decoding of which occurs on the screen of a telescope. Presence of halo can give an explanation to occurrence of a horizontal strip in the image of a solar disk in a neutrino flux (see Fig. 3), because of neutrino dispersion on it. The vertical width of this strip, with tak- ing into account of geometrical optics, corresponds to dis- tance of this halo from the observer. Construction of images in an optical or x-ray range has very high spatial resolution. Therefore the described above neutrino astronomy can take place only under con- dition of existence of FM, similar which has been de- scribed in item 2. The examples presented in paper shows that the hypotheses suggested by the author can have a reality and are the facts of development of FSS of FM in- side the Sun and its nearest space environment. VI. SUMMARY The fact of observation of the images displaying inter- nal structure of the star, can solve many problems con- nected to neutrino physics: a) to prove existence of neu- trino oscillations, i.e., that solar neutrinos are Dirac- neutrinos; b) to study of FSS of FM and their dynamics inside stars/galaxies through the analysis of their images in various ranges of lengths of waves; c) to study dynamic character of the moment and neutrino mass and their in- teractions with LA of FM or DM particles. Thus, already now we have a neutrino astronomy which allows us to look into bowels of stars and galaxies, to observe their internal structure and to study processes taking place inside them, to reveal FSS of FM and to study their properties, as inside stars, as in their environ- ment. All this can give a new push into researching of space objects, understanding of processes of their forma- tion, and searching a new energy sources taking place in the universe. VII. ACKNOWLEDGMENTS The author is deeply grateful to V.I. Kogan for invariable support and interest to these researches. VIII. REFERENCES 1. A.B.Kukushkin, V.A.Rantsev-Kartinov, “Self- similarity of plasma networking in a broad range of length scales: from laboratory to cosmic plasmas”, RSI, vol. 70, pp. 1387-1391, (1999). 2. Rantsev-Kartinov V.A., “Revelation of the Sun Self- Similarity Skeletal Structures”, in Proc. 32nd EPS Conference on Plasma Phys. Tarragona, 27 June - 1 July 2005 ECA Vol.29C, P-2.155 (2005), http://eps2005.ciemat.es/papers/pdf/P2_155.pdf. 3. A.B.Kukushkin, V.A.Rantsev-Kartinov: a) “Similarity of skeletal objects in the range 10-5 cm to 1023 cm”, Phys.Lett. A, vol. 306, pp. 175-183, (2002); b) “Skeletal structures in high-current electric discharges and laser-produced plasmas: observations and hypotheses”, Ed. F. Gerard, Nova Science Publishers, New York, Advances in Plasma Phys. Research, vol. 2, pp. 1-22, (2002). 4. Rodionov B.U., “Thready (linear) dark matter possible displays”, Gravitation and Cosmology, vol. 8, Sup- plement I, pp. 214-215, (2002). 5. Gonzalez-Garcia M.C., C.N. Yang and Yosef Nir, “Neutrino Masses and Mixing: Evidence and Implica- tions”, Rev. Mod. Phys., vol. 75, p. 345, (2003). 6. Borisov A.V., Zukovskiy V.Ch., Ternov A.I., “Elec- tromagnetic properties of the Dirac massive neutrino at an external electromagnetic field”, Izvestiya vuzov Ser. Fiz. 3, pp. 64-70, (1988), (in russian). 7. Borisov A.V., Vshivcev A.S., Zukovskiy V.Ch., Emi- nov P.A., Uspekhi Fizicheskikh Nauk, 167, № 3, (1997) 241-267, (in russian). 8. McDonald, Art; Spiering, Christian; Schönert, Stefan; Kearns, Edward T.; Kajita, Takaaki, “Astrophysical Neutrino Telescopes”, RSI, vol. 75, № 2, p. 293, (2004).

0704.1106

Analysis of $\Omega_c^*(css)$ and $\Omega_b^*(bss)$ with QCD sum rules

Analysis of Ω∗ (css) and Ω∗ (bss) with QCD sum rules Zhi-Gang Wang 1 Department of Physics, North China Electric Power University, Baoding 071003, P. R. China Abstract In this article, we calculate the masses and residues of the heavy baryons Ω∗c(css) and Ω (bss) with spin-parity 3 with the QCD sum rules. The nu- merical values are compatible with experimental data and other theoretical estimations. PACS number: 14.20.Lq, 14.20.Mr Key words: Ω∗c(css), Ω b(bss), QCD sum rules 1 Introduction Several new excited charmed baryon states have been observed by BaBar, Belle and CLEO Collaborations, such as Λc(2765) +, Λ+c (2880), Λ c (2940), Σ c (2800), Ξ c (2980), Ξ+c (3077), Ξ c(2980) , Ξ c(3077) [1, 2]. The charmed baryons provide a rich source of states, including possible candidates for the orbital excitations. They serve as an excellent ground for testing predictions of the constituent quark models and heavy quark symmetry [3]. The charmed and bottomed baryons, which contain a heavy quark and two light quarks, provides an ideal tool for studying dynamics of the light quarks in the presence of a heavy quark. The u, d and s quarks form an SU(3) flavor triplet, 3× 3 = 3̄ + 6, two light quarks can form diquarks with a symmetric sextet and an antisymmetric antitriplet. For the S-wave baryons, the sextet contains both spin-1 and spin-3 states, while the antitriplet contains only spin-1 states. By now, the 1 antitriplet states (Λ+c , Ξ c), and the and 3 sextet states (Ωc,Σc,Ξ and (Ω∗c ,Σ c ) have been established. The baryon Ω∗c , a css candidate for the partner of the strange baryon Ω(sss), was observed by BaBar collaboration in the radiative decay Ω∗c → Ωcγ [4]. The baryon Ωc(css) was reconstructed in decays to the final states Ω −π+, Ω−π+π0, Ω−π+π−π+ and Ξ−K−π+π+. It lies about 70.8±1.0±1.1MeV above the Ωc, and it is the last singly-charmed baryon with zero orbital momentum observed experimentally In this article, we calculate the mass and residue of the Ω∗c (and Ω b as byproduct, the Ω∗b has not been observed experimentally yet) with the QCD sum rules [6, 7]. In the QCD sum rules, operator product expansion is used to expand the time-ordered currents into a series of quark and gluon condensates which parameterize the long distance properties of the QCD vacuum. Based on current-hadron duality, we can 1E-mail,[email protected];[email protected]. http://arxiv.org/abs/0704.1106v3 obtain copious information about the hadronic parameters at the phenomenological side. The article is arranged as follows: we derive the QCD sum rules for the masses and residues of the Ω∗c and Ω b in section 2; in section 3, numerical results and discussions; section 4 is reserved for conclusion. 2 QCD sum rules for the Ω∗c and Ω In the following, we write down the two-point correlation functions Πaµν(p 2) in the QCD sum rules approach, Πaµν(p 2) = i d4xeip·x〈0|T Jaµ(x)J̄ ν (0) |0〉 , (1) Jaµ(x) = ǫijks i (x)Cγµsj(x)Q k(x) , (2) λaNµ(p, s) = 〈0|Jaµ(0)|Ω∗a(p, s)〉 , (3) where the upper index a represents the c and b quarks respectively; the Nµ(p, s) and λa stand for the Rarita-Schwinger spin vector and residue of the baryon Ω respectively. i, j and k are color indexes, C is charge conjunction matrix, and µ and ν are Lorentz indexes. The correlation functions Πaµν(p) can be decomposed as follows: Πaµν(p) = −gµν 6pΠa1(p2) + Πa2(p2) + · · · , (4) due to Lorentz covariance. The first structure gµν 6p has an odd number of γ-matrices and conserves chirality, the second structure gµν has an even number of γ-matrices and violate chirality. In the original QCD sum rules analysis of the nucleon masses and magnetic moments [8], the interval of dimensions (of the condensates) for the odd structure is larger than the interval of dimensions for the even structure, one may expect a better accuracy of results obtained from the sum rules with the odd structure. In this article, we choose the two tensor structures to study the masses and residues of the heavy baryons Ω∗c and Ω b , as the masses of the heavy quarks break the chiral symmetry explicitly. According to basic assumption of current-hadron duality in the QCD sum rules approach [6], we insert a complete series of intermediate states satisfying unitarity principle with the same quantum numbers as the current operator Jaµ(x) into the correlation functions in Eq.(1) to obtain the hadronic representation. After isolating the pole terms of the lowest states Ω∗a, we obtain the following result: Πaµν(p 2) = −gµνλ2a MΩ∗a+ 6p + · · · , (5) where we have used the relation to sum over the Rarita-Schwinger spin vector, Nµ(p, s)N̄ν(p, s) = −(6p +MΩ∗a) gµν − − 2pµpν pµγν − pνγµ 3MΩ∗a . (6) In the following, we briefly outline operator product expansion for the correlation functions Πaµν(p) in perturbative QCD theory. The calculations are performed at large space-like momentum region p2 ≪ 0, which corresponds to small distance x ≈ 0 required by validity of operator product expansion. We write down the ”full” propagators Sij(x) and S (x) of a massive quark in the presence of the vacuum condensates firstly [6]2, Sij(x) = iδij 6x 2π2x4 − δijms 4π2x2 − δij 〈s̄s〉+ iδij ms〈s̄s〉 6x− 〈s̄gsσGs〉 iδijx ms〈s̄gsσGs〉 6x− 32π2x2 Gijµν(6xσµν + σµν 6x) + · · · , Q (x) = (2π)4 d4ke−ik·x 6k −mQ σαβ(6k +mQ) + (6k +mQ)σαβ (k2 −m2Q)2 〈αsGG 〉δijmQ k2 +mQ 6k (k2 −m2Q)4 + · · · , (7) where 〈s̄gsσGs〉 = 〈s̄gsσαβGαβs〉 and 〈αsGGπ 〉 = 〈 αsGαβG 〉, then contract the quark fields in the correlation functions Πaµν(p) with Wick theorem, and obtain the result: Πaµν(p) = 2iǫijkǫi′j′k′ d4x eip·xTr γµSii′(x)γνCS jj′(x)C Q (x) . (8) Substitute the full s, c and b quark propagators into above correlation functions and complete the integral in coordinate space, then integrate over the variable k, we can obtain the correlation functions Πai (p 2) at the level of quark-gluon degree of freedom: 2One can consult the last article of Ref.[6] for technical details in deriving the full propagator. Πa1(p 2) = − dxx(1 − x)2(x+ 2) m̃2a − p2 m̃2a − p2 −ms〈s̄s〉 dxx(x− 2) log m̃2a − p2 192π2 〈αsGG dxx(x − 2) log m̃2a − p2 ms〈s̄gsσGs〉 m̃2a − p2 576π2 (1− x)2(x+ 2) x2(m̃2a − p2) 〈s̄s〉2 m2a − p2 ms〈s̄gsσGs〉 m2a − p2 + · · · , (9) Πa2(p 2) = − dx(1− x)2(x+ 2) m̃2a − p2 m̃2a − p2 mams〈s̄s〉 dx(x− 2) log m̃2a − p2 576π2 〈αsGG − 3x2 + 2x+ 9) log m̃2a − p2 576π2 〈αsGG x4 − x3 − 3x2 + 5x− 2 x(1− x) m̃2a − p2 mams〈s̄gsσGs〉 m̃2a − p2 ma〈s̄s〉2 m2a − p2 mams〈s̄gsσGs〉 m2a − p2 + · · · , (10) where m̃2a = We carry out operator product expansion to the vacuum condensates adding up to dimension-6. In calculation, we take assumption of vacuum saturation for high dimension vacuum condensates, they are always factorized to lower condensates with vacuum saturation in the QCD sum rules, factorization works well in large Nc limit. In this article, we take into account the contributions from the quark condensate 〈s̄s〉, mixed condensate 〈s̄gsσGs〉, gluon condensate 〈αsGGπ 〉, and neglect the contributions from other high dimension condensates, which are suppressed by large denominators and would not play significant roles. Once analytical results are obtained, then we can take current-hadron duality below the threshold s0a and perform Borel transformation with respect to the variable P 2 = −p2, finally we obtain the following sum rules: λ2a exp ∫ s0a dxx(1− x)2(x+ 2) m̃2a − s ms〈s̄s〉 ∫ s0a dxx(x− 2) exp 192π2 ∫ s0a dxx(x− 2) exp ms〈s̄gsσGs〉 dxx exp 576π2 (1− x)2(2 + x) 〈s̄s〉2 ms〈s̄gsσGs〉 , (11) MΩ∗aλ a exp ∫ s0a dx(1− x)2(x+ 2) m̃2a − s mams〈s̄s〉 ∫ s0a dx(x− 2) exp 576π2 ∫ s0a − 3x2 + 2x+ 9) exp 576π2 〈αsGG x4 − x3 − 3x2 + 5x− 2 x(1− x) m̃2a exp mams〈s̄gsσGs〉 dx exp ma〈s̄s〉2 mams〈s̄gsσGs〉 , (12) where th = (ma + 2ms) 2 and ∆a = Differentiate the above sum rules with respect to the variable 1 , then eliminate the quantity λΩ∗a , we obtain two QCD sum rules for the masses MΩ∗a : ∫ s0a dxx(1 − x)2(x+ 2) m̃2a − s s exp ms〈s̄s〉 ∫ s0a dxx(x− 2)s exp 192π2 ∫ s0a dxx(x− 2)s exp a〈s̄gsσGs〉 dx exp 576π2 (1− x)2(2 + x) m2a〈s̄s〉2 a〈s̄gsσGs〉 ∫ s0a dxx(1 − x)2(x+ 2) m̃2a − s ms〈s̄s〉 ∫ s0a dxx(x− 2) exp 192π2 ∫ s0a dxx(x− 2) exp ms〈s̄gsσGs〉 dxx exp 576π2 〈αsGG (1− x)2(x+ 2) 〈s̄s〉2 ms〈s̄gsσGs〉 , (13) M2Ω∗a = ∫ s0a dx(1− x)2(x+ 2) m̃2a − s s exp mams〈s̄s〉 ∫ s0a dx(x− 2)s exp 576π2 〈αsGG ∫ s0a − 3x2 + 2x+ 9)s exp 576π2 x4 − x3 − 3x2 + 5x− 2 x(1 − x) m̃4a exp m3ams〈s̄gsσGs〉 m3a〈s̄s〉2 m3ams〈s̄gsσGs〉 ∫ s0a dx(1− x)2(x+ 2) m̃2a − s mams〈s̄s〉 ∫ s0a dx(x− 2) exp 576π2 ∫ s0a − 3x2 + 2x+ 9) exp 576π2 x4 − x3 − 3x2 + 5x− 2 x(1 − x) m̃2a exp mams〈s̄gsσGs〉 dx exp ma〈s̄s〉2 mams〈s̄gsσGs〉 . (14) 3 Numerical results and discussions The input parameters are taken to be the standard values 〈q̄q〉 = −(0.24±0.01GeV)3, 〈s̄s〉 = (0.8 ± 0.2)〈q̄q〉, 〈s̄gsσGs〉 = m20〈s̄s〉, m20 = (0.8 ± 0.2)GeV2, 〈αsGGπ 〉 = (0.33GeV)4, ms = (0.14±0.01)GeV, mc = (1.4±0.1)GeV and mb = (4.8±0.1)GeV [6, 7, 9]. The contribution from the gluon condensate 〈αsGG 〉 is less than 4%, and the uncertainty is neglected here. For the octet baryons with I(JP ) = 1 ), the mass of the proton (the ground state) is Mp = 938MeV, and the mass of the first radial excited state N(1440) (the Roper resonance) is M1440 = (1420 − 1470)MeV ≈ 1440MeV [10]. For the decuplet baryons with I(JP ) = 3 ) , the mass of the ∆(1232) (the ground state) is M1232 = (1231 − 1233)MeV ≈ 1232MeV, and the mass of the first radial excited Eq.(11) Eq.(12) perturbative term +80% +83% 〈s̄s〉 +12% +10% 〈s̄gsσGs〉 −4% −2% 〈s̄s〉2 +12% +7% 〈αsGG 〉 +1% +2% Table 1: The contributions from different terms in the sum rules for the Ω∗c with the central values of the input parameters. Eq.(11) Eq.(12) perturbative term +78% +80% 〈s̄s〉 +10% +10% 〈s̄gsσGs〉 −4% −3% 〈s̄s〉2 +15% +12% 〈αsGG 〉 +1% +1% Table 2: The contributions from different terms in the sum rules for the Ω∗b with the central values of the input parameters. state ∆(1600) is M1600 = (1550 − 1700)MeV ≈ 1600MeV [10]. The separation between the ground states and first radial excited states is about 0.5GeV. So in the QCD sum rules for the baryons with the light quarks, the threshold parameters s0 are always chosen to be s0 = Mgr + 0.5GeV [8, 11], here gr stands for the ground states. The threshold parameters for the heavy baryons Ω∗c and Ω b can be chosen to be s0 = (2.8 + 0.5)2GeV2 and s0 = (6.1 + 0.5)2GeV2, respectively. The mass of the bottomed baryon Ω∗b with spin-parity is about MΩ∗ = (6.04 − 6.09)GeV, which is predicted by the quark models and lattice QCD [12, 13]. In this article, the threshold parameters and Borel parameters are taken as = 11.0GeV2 and M2 = (2.5 − 3.5)GeV2 for the charmed baryon Ω∗c , and = 45.0GeV2 and M2 = (5.0 − 6.0)GeV2 for the bottomed baryon Ω∗b . The contributions from different terms for the central values of the input parameters are presented in Table.1 and Table.2, respectively. From the two tables, we can expect convergence of the operator product expansion. In the two sum rules in Eqs.(11-12), the contributions from the terms proportional to the quark conden- sate 〈s̄s〉 and mixed condensate 〈s̄gsσGs〉 are suppressed due to the small mass ms comparing with the terms proportional to the 〈s̄s〉2. Furthermore, from the ’full’ propagator of the s quark, we can see that the mixed condensate 〈s̄gsσGs〉 is com- panied with additional large denominators, its contribution is even smaller. In the right-hand side of Eqs.(11-12), the terms proportional to the 〈s̄s〉2 are suppressed 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 M2(GeV2) Central value; Upper bound; lower bound. 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 M2(GeV2) Central value; Upper bound; Lower bound. Figure 1: MΩ∗c with Borel parameter M 2, A from Eq.(13) and B from Eq.(14). by the exponents exp[−m2a/M2], which is balanced by the factor exp[−M2Ω∗a/M in the left-hand side. Although the masses of the c quark and Ω∗c baryon are much smaller than the corresponding ones of the b quark and Ω∗b baryon, the Borel parameters M2 are different, for the central values of the Borel parameters M2, exp[M2 /M2−m2b/M2] > exp[M2Ω∗c/M 2−m2c/M2]. It is not unexpected, the contri- butions from the 〈s̄s〉2 are larger in the sum rules for the Ω∗b baryon than the ones for the Ω∗c baryon. If we approximate the phenomenological spectral density with the perturbative term, the contribution from the pole term is as large as (28− 54)% for the charmed baryon Ω∗c and (33 − 50)% for the bottomed baryon Ω∗b . We can choose smaller Borel parameter M2 or larger threshold parameters s0a to enhance the contributions from the ground states. However, if we take larger threshold parameter s0a, the contribution from the first radial excited state maybe included in; on the other hand, for smaller Borel parameterM2, the sum rules are not stable enough, the uncertainty with variation of the Borel parameter is large. In the case of the multiquark states, the standard criterion of the lowest pole dominance cannot be satisfied, we have to resort to new criterion to overcome the problem, for detailed discussions about this subject, one can consult Ref.[14]. Taking into account all uncertainties of the input parameters, finally we obtain the values of the masses and residues of the heavy baryons Ω∗c and Ω b , which are shown in Figs.1-4 respectively, MΩ∗c = (2.72± 0.12)GeV , = (6.04± 0.13)GeV , λΩ∗c = (0.047± 0.008)GeV , = (0.057± 0.011)GeV , (15) 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 M2(GeV2) Central value; Upper bound; Lower bound. 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 M2(GeV2) Central value; Upper bound; Lower bound. Figure 2: λΩ∗c with Borel parameter M 2, A from Eq.(11) and Eq.(13), and B from Eq.(12) and Eq.(14). 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 M2(GeV2) Central value; Upper bound; Lower bound. 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 M2(GeV2) Central value; Upper bound; Lower bound. Figure 3: MΩ∗ with Borel parameter M2, A from Eq.(13) and B from Eq.(14). 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 M2(GeV2) Central value; Upper bound; Lower bound. 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 M2(GeV2) Central value; Upper bound; Lower bound. Figure 4: λΩ∗ with Borel parameter M2, A from Eq.(11) and Eq.(13), and B from Eq.(12) and Eq.(14). from A from Eq.(11) and Eq.(13), and MΩ∗c = (2.80± 0.08)GeV , = (6.08± 0.12)GeV , λΩ∗c = (0.046± 0.007)GeV , = (0.060± 0.011)GeV , (16) from Eq.(12) and Eq.(14). The average values are about MΩ∗c = (2.76± 0.10)GeV , = (6.06± 0.13)GeV , λΩ∗c = (0.047± 0.008)GeV , = (0.058± 0.011)GeV . (17) The value of the mass MΩ∗c is compatible with the experimental data MΩ∗c = (2.768±0.003)GeV [10], the interpolating current Jcµ(x) can couple with the charmed baryon Ω∗c and give reasonable mass. The value of the mass MΩ∗b for the bottomed baryon Ω∗b with is compatible with other theoretical calculations, MΩ∗ = (6.04− 6.09)GeV, such as the quark models and lattice QCD [12, 13]. Once reasonable values of the residues λΩ∗c and λΩ∗b are obtained, we can take them as basic input parameters and study the hadronic processes [15], for example, the radiative decay Ω∗c → Ωcγ, with the light-cone QCD sum rules or the QCD sum rules in external field. 4 Conclusion In this article, we calculate the masses and residues of the heavy baryons Ω∗c(css) and Ω∗b(bss) with the QCD sum rules. The numerical values are compatible with the experimental data and other theoretical estimations. Once reasonable values of the residues λΩ∗c and λΩ∗b are obtained, we can take them as basic parameters and study the hadronic processes, for example, the radiative decay Ω∗c → Ωcγ, with the light-cone QCD sum rules or the QCD sum rules in external field. Acknowledgments This work is supported by National Natural Science Foundation, Grant Number 10405009, 10775051, and Program for New Century Excellent Talents in University, Grant Number NCET-07-0282, and Key Program Foundation of NCEPU. References [1] B. Aubert, et al, Phys. Rev. Lett. 98 (2007) 012001; R. Mizuk, et al, Phys. Rev. Lett. 98 (2007) 262001; B. Aubert, et al, arXiv:hep-ex/0607042; R. Chistov, et al, Phys. Rev. Lett. 97 (2006) 162001; M. Artuso, et al, Phys. Rev. Lett. 86 (2001) 4479. [2] T. Lesiak, arXiv:hep-ex/0612042; J. L. Rosner, arXiv:hep-ph/0612332. [3] J. G. Koerner, D. Pirjol, M. Kraemer, Prog. Part. Nucl. Phys. 33 (1994) 787; F. Hussain, G. Thompson, J. G. Koerner, arXiv:hep-ph/9311309. [4] B. Aubert, et al, Phys. Rev. Lett. 97 (2006) 232001. [5] J. L. Rosner, Phys. Rev. D52 (1995) 6461. [6] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147 (1979) 385, 448; L. J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rept. 127 (1985) [7] S. Narison, QCD Spectral Sum Rules, World Scientific Lecture Notes in Physics 26 (1989) 1; P. Colangelo and A. Khodjamirian, hep-ph/0010175. [8] B. L. Ioffe, Nucl. Phys. B188 (1981) 317; B. L. Ioffe, A. V. Smilga, Nucl. Phys. B232 (1984) 109; V. M. Belyaev, B. L. Ioffe, Sov. Phys. JETP 56 (1982) 493; V. M. Belyaev, B. L. Ioffe, Sov. Phys. JETP 57 (1983) 716. [9] B. L. Ioffe, Prog. Part. Nucl. Phys. 56 (2006) 232; and references therein. [10] W.-M. Yao, et al, J. Phys. G33 (2006) 1. [11] C. B. Chiu, J. Pasupathy, S. L. Wilson, Phys. Rev. D32 (1985) 1786; S. L. Wilson, J. Pasupathy, C. B. Chiu, Phys. Rev. D36 (1987) 1451; C. B. Chiu, S. L. Wilson, J. Pasupathy, J. P. Singh, Phys. Rev. D36 (1987) 1553. http://arxiv.org/abs/hep-ex/0607042 http://arxiv.org/abs/hep-ex/0612042 http://arxiv.org/abs/hep-ph/0612332 http://arxiv.org/abs/hep-ph/9311309 http://arxiv.org/abs/hep-ph/0010175 [12] D. Ebert, R. N. Faustov, V. O. Galkin, Phys. Rev. D72 (2005) 034026; E. Jenkins, Phys. Rev. D55 (1997) 10. [13] N. Mathur, R. Lewis, R. M. Woloshyn, Phys. Rev. D66 (2002) 014502. [14] Z. G. Wang, S. L. Wan, Nucl. Phys. A778 (2006) 22; Z. G. Wang, S. L. Wan, J. Phys. G34 (2007) 505; Z. G. Wang, S. L. Wan, Chin. Phys. Lett. 23 (2006) 3208; Z. G. Wang, Nucl. Phys. A791 (2007) 106. [15] H. Y. Cheng, C. K. Chua, Phys. Rev. D75 (2007) 014006; C. Chen, X. L. Chen, X. Liu, W. Z. Deng, S. L. Zhu, arXiv:0704.0075[hep-ph]. http://arxiv.org/abs/0704.0075 Introduction QCD sum rules for the c* and b* Numerical results and discussions Conclusion

0704.1107

The dependence of the estimated luminosities of ULX on spectral models

Draft version August 8, 2021 Preprint typeset using LATEX style emulateapj v. 6/22/04 THE DEPENDENCE OF THE ESTIMATED LUMINOSITIES OF ULX ON SPECTRAL MODELS A. Senorita Devi1, R. Misra2, V. K. Agrawal3 and K. Y. Singh1 Draft version August 8, 2021 ABSTRACT Data from Chandra observations of thirty nearby galaxies were analyzed and 365 X-ray point sources were chosen whose spectra were not contaminated by excessive diffuse emission and not affected by photon pile up. The spectra of these sources were fitted using two spectral models (an absorbed power-law and a disk blackbody) to ascertain the dependence of estimated parameters on the spectral model used. It was found that the cumulative luminosity function depends on the choice of the spectral model, especially for luminosities > 1040 ergs/s. In accordance with previous results, a large number (∼ 80) of the sources have luminosities > 1039 ergs/s (Ultra-Luminous X-ray sources) with indistinguishable average spectral parameters (inner disk temperature ∼ 1 keV and/or photon index Γ ∼ 2) with those of the lower luminosities ones. After considering foreground stars and known background AGN, we identify four sources whose minimum luminosity exceed 1040 ergs/s, and call them Extremely Luminous X-ray sources (ELX). The spectra of these sources are in general better represented by the disk black body model than the power-law one. These ELX can be grouped into two distinct spectral classes. Two of them have an inner disk temperature of < 0.5 keV and hence are called “supersoft” ELX, while the other two have temperatures & 1.3 keV and are called “hard” ELX. The estimated inner disk temperatures of the supersoft ELX are compatible with the hypothesis that they harbor intermediate size black holes, which are accreting at ∼ 0.5 times their Eddington Luminosity. The radiative mechanism for hard ELX, seems to be Inverse Comptonization, which in contrast to standard black holes systems, is probably saturated. Extensive variability analysis of these ELX, will be able to distinguish whether these two spectral class represent different systems or they are spectral states of the same kind of source. Subject headings: Galaxies: general - X-rays: binaries 1. INTRODUCTION In the last few years, Chandra observations of nearby galaxies have detected many non-nuclear X-ray point sources (Kaaret et al. 2001 ; Matsumoto et al. 2001; Zezas & Fabbiano 2002), some of which have isotropic lu- minosities> 1039 ergs/s and are called Ultra luminous X- ray Sources (ULX). While some of these ULX are super- nova remnants (e.g. Ryder et al. 1993; Fox et al. 2000), it is believed that the majority of them are compact ac- creting systems. Indeed, ASCA X-ray spectral studies of many ULX have revealed that they display the charac- teristics of accreting black holes (Makishima et al. 2000; Mizuno et al. 2001). ULX have also been called Super- Eddington Sources (Fabbiano 1989, 2004) and Intermedi- ate luminosity X-ray objects (Roberts & Warwick 2000; Colbert & Mushotzky 1999). Since these ULX sources emit radiation at a rate larger than the Eddington luminosity for a ten-solar mass black hole, they are believed to harbor a black hole of mass 10M⊙<M<10 5M⊙ (Colbert & Mushotzky 1999; Makishima et al. 2000) where the upper limit is con- strained by the fact that a more massive black hole would have settled into the nucleus due to dynamical friction (Kaaret et al. 2001 ). Black holes in this mass range are called Intermediate Mass Black Holes (IMBH), since they 1 Department Of Physics, Manipur University, Canchipur, Imphal-795003, Manipur, India; [email protected] 2 Inter-University Center for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune-411007, India; [email protected] 3 Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400 005,In- seem to represent the missing component of the black hole mass spectrum with masses prevailing in the gap be- tween those of stellar mass black holes found in Galactic X-ray binaries and those associated with Active Galac- tic Nuclei, M ∼ 106 − 109 M⊙ (Richstone et al. 1998). Miller & Colbert (2004) and Miller (2005) review the present evidence for IMBH in ULX and Liu & Mirabel (2005) have compiled a catalogue of some ULX and prop- erties. Alternate models for ULX are that their luminosi- ties are super-Eddington (Begelman 2002) or that their emission is beamed from a geometrically thick accre- tion disk (King et al. 2001). However, it has been ar- gued that in the latter case, such thick ”funnel” shaped disks enhance the observed flux by just a factor of few (Misra & Sriram 2003). For all of these models, the creation of such sources (Portegies, Zwart & McMillian 2002; Taniguchi et al. 2000; Madau & Rees 2001) and process by which they sustain high accretion rates (King et al. 2001), are largely unknown. Investigations on the nature of ULX have been un- dertaken by studying the spectra and variability of in- dividual sources. For example analysis of the spectra of NGC 1313 X-1, X-2 (Miller et al. 2003) and M81 X- 9 (Miller, Fabian & Miller 2004), revealed the presence of a cool accretion disk component (kTin ∼ 0.1 − 0.5 keV), suggesting that ULX indeed harbor IMBH. Tran- sitions between two spectral states, similar to those seen in Galactic black hole systems, have been reported in NGC 1313 X-1(Colbert & Mushotzky 1999) and two sources in IC342 (Kubota et al. 2001). Spectral transi- tions have also been reported in two sources in NGC 1313 http://arxiv.org/abs/0704.1107v1 2 Senorita Devi et al. (Feng & Kaaret 2006). The large collecting area of XMM-Newton, allows for detailed spectral fits to ULX, which often comprise of two components (Wang et al. 2004; Feng & Kaaret 2005). However, Gonçalves & Soria (2006), have argued that such soft spectral components depend on the complex- ity of the fitting model. An interesting object is the brightest X-ray point source in M 82, whose intrinsic luminosity has been measured to be as high as 1.6 × 1041 ergs/s (Ptak and Griffiths 1999). The detection of a 54 mHz quasi-periodic oscillation in its X-ray light curve suggests that the source is a compact object and not a background AGN (Strohmayer & Mushotzky 2003; Dewangan et al. 2006a). The spectra of this source can be fitted by a power-law with photon index, Γ ∼ 2 (Fiorito & Titarchuk 2004), but is more consistently fit- ted with a flatter power-law with an high energy cut- off around ∼ 6 keV, which can be interpreted as opti- cally thick, saturated Comptonization (Agrawal & Misra 2006). A quasi-periodic oscillation has also been discov- ered in the bright X-ray source of Holmberg IX, which is similar to the source in M82 in having a flat spec- trum (Γ ∼ 1) with a ∼ 9 keV cutoff (Dewangan et al. 2006b). Recently, Stobbart et al. (2006) found that the XMM-Newton spectra of eleven of the eighteen ULX studied by them, showed such high energy curvature. Chandra observations of NGC 5204 X-1, also reveals the presence of an optically thick Comptonized component (Roberts et al. 2006). While these results of individual ULX are intriguing, there does not seem to be any signif- icantly distinguishable spectral property of ULX, and in general their spectra can be described either by steep or flat power-law indices, with and without soft components (e.g. Dewangan et al. 2005). Another line of investigation is to construct the cu- mulative luminosity function and histograms of spec- tral parameters of a large sample of X-ray sources (e.g. Colbert & Ptak 2002). The hope here is that, in case ULX are a distinct class of sources and/or they can be classified into distinct subgroups, the luminosity func- tion should exhibit a break and their spectral parame- ters should show clustering. Swartz et al. (2004) ana- lyzed data from 82 galaxies and estimated the luminos- ity function and spectral parameters of the X-ray point sources in these galaxies. They found that the average photon index (as well as the distribution) of the ULX and the less luminous sources is nearly same. Moreover, their spatial and variability distributions are also similar. While, their analysis revealed that the luminosity func- tions of ULX depend on the host galaxy type and star formation rate, they did not find any significant evidence for breaks in them. As emphasized by Swartz et al. (2004), the power-law model they used to fit the data was chosen as an empir- ical one. They attempted to fit all the sources with the power-law model and only for those sources that did not provide a reasonable fit, they used other models like disk black body. They note that for many sources that are well fitted by the power-law model, other spectral models can also represent the spectra. In this work, we consider a smaller sample of 30 galaxies but fit the spectra of the points sources, with both a power-law and a disk black- body model. In principle, the intrinsic (i.e. the absorp- tion corrected) luminosity inferred for a source may be different for the two spectral models. Our first motiva- tion here is to make a qualitative estimate of this differ- ence by noting the dependency of the luminosity function on the spectral model used. A second motivation for this work is based on the expectation that estimations of a different spectral parameter like the inner disk tempera- ture (as compared to the photon index), maybe better in distinguishing ULX from other sources or they may re- veal dependencies (like correlations between luminosity and temperature) which could shed light on the nature of these sources. As mentioned earlier, the spectra of several ULX are complex requiring more than one com- ponents. Thus, the two spectral models, an absorbed power-law and and an absorbed disk blackbody, should be considered as empirical ones, which can adequately represent low count data and these models need not be the correct physical representation of the actual source spectrum. It is clear that ULX are fairly common in nearby galax- ies. For example, Swartz et al. (2004) identified 154 of them in 82 galaxies. If these sources are Eddington lim- ited, then black holes & 10M⊙ may indeed be quite com- mon. However, only a few of these sources have luminosi- ties greater than 1040 ergs/s. It is these sources, (the best example being the bright source in M82 X-1), that require black hole with masses, M ∼ 102−4M⊙. The de- velopment of a self-consistent theory which explains the process by which such black holes are created and un- dergo high accretion, is theoretically challenging. Hence, it is important to estimate the number of sources whose minimum intrinsic luminosity exceeds 1040 ergs/s. Here the minimum value of the luminosity should not only in- clude the statistical spectral fitting error, but also the variations in the luminosity estimation that may occur upon using different viable spectral models. The third motivation of this work is to identify such sources which we call Extremely Luminous X-ray sources (ELX). To avoid possible ambiguities in determining the luminos- ity, we have chosen only those sources whose spectra are not contaminated by excessive diffuse emission and which are not affected by photon count pile up. Identification of such relatively “clean” systems would allow for more detailed studies of their properties to be undertaken and would be the first step toward understanding their na- ture. A similar analysis has been undertaken recently by Winter et al. (2006) for thirty galaxies observed by XMM-Newton. Since their motivation was to check if ULX also exhibit soft and hard spectral states like black hole binaries, they limited their analysis to bright sources where detailed spectral analysis could be done. They found 16 sources as possible low-state ULX and 26 as high state ones. For the high state ULX, the observed range for the black body temperature was 0.1− 1.0 keV, with the more luminous sources having lower temper- ature. The Chandra analysis undertaken here, can be compared with this contemporary work and as we discuss later, the analysis are more or less consistent with each other. This consistency is important, since although the larger collecting area of XMM-Newton, allows for more complex spectral analysis, the higher angular resolution of Chandra ensures that a source spectrum is not contam- inated by diffuse emission and flux from nearby sources. Extremely luminous X-ray point sources 3 2. OBSERVATIONS AND DATA REDUCTION The names of the thirty host galaxies and the details of the Chandra ACIS observations are tabulated in Ta- ble 1. Distances to the galaxies were obtained from Swartz et al. (2004) and references therein. This sample of galaxies is a subset of those analyzed by Swartz et al. (2004). No particular criterion was imposed on the selec- tion, since our motivation is limited to obtaining enough sources and not to study dependency on galaxy type. The data reduction and analysis were done using CIAO3.2 and HEASOFT6.0.2. Using the CIAO source detection tool wavdetect, X-ray point sources were ex- tracted from the level 2 event list. It was found that at least 60 counts are required to fit the spectral data with a two parameter model and hence only those sources with net counts ≥ 60 were chosen for the spectral analysis. Choosing a lower threshold of 50 counts resulted in a large number of sources for which spectral parameters could not be constrained. To avoid photon pile up ef- fects, a conservative threshold of the count rate being > 0.05 counts/s was imposed which led to the rejec- tion of fifteen sources which have been listed in Table 3 of the appendix. For some sources, typically near the nucleus, it was difficult to find nearby source free back- ground regions and hence these sources were also not included in the analysis. Sources embedded in exces- sive diffuse emission (i.e. when the background flux was larger than 2 counts/arcsec2 ) were also rejected. Typi- cally, this amounted to considering only those sources for which the estimated background counts were less than 20. For each data set, observation-specific bad pixel lists were set in the ardlib parameter file. Using a combina- tion of CIAO tools and calibration data, the source and background spectra were extracted. These selection criteria makes the sample incomplete both in low and high luminosity ends. Thus the results obtained should not be used for quantitative analysis of the luminosity functions. The motivation here is evaluate dependency on spectral models and to identify sources which have high intrinsic luminosity. Thus, care has been taken to avoid possibly contaminated data, even if such criteria result in a loss of sources. Spectral analysis was done using XSPEC version 12.2 and the data was fitted in the energy range 0.3-8.0 keV. All sources were fitted with two spectral models, the ab- sorbed power-law and an absorbed disk blackbody. Ab- sorption was taken into account using the XSPEC model phabs. Since the number of counts in each spectrum was typically low, the C-statistics was used for the analysis. Technically, the C-statistics is not appropriate for high counts and/or for background subtracted data. However, it was ascertained that the model parameters obtained ei- ther by C-statistics or χ2 statistics, were consistent with each other for high count rate sources. This is reassuring, because if the results depended on the statistics used, it would be imperative to use the correct statistics for high count sources (which need not be χ2) taking into account the correct (possibly non-Gaussian) error profiles. An important problem, when fitting low count data with a two parameter (plus normalization) model, is the possibility of many local minima in the discerning statistic (in this case C-statistic) space. Hence, we take a cautious approach and do not fit the data using the Fig. 1.— The intrinsic luminosity (in 0.3-8.0 keV energy range) estimated by fitting the power-law model versus the bolometric intrinsic luminosity estimated using the disk black body model. XSPEC minimization routine. Instead we compute the C-statistic for a range of parameter values (using the XSPEC command steppar) and find the global minimum. Such a technique is numerically expensive, but it ensures that the global minimum has been found and the correct errors are obtained for the best fit parameters. 3. RESULTS The 365 sources considered in this analysis were classi- fied into three categories, depending on whether the data was better fit by the disk black body model (23 sources) or the power-law one (67 sources) or both (275 sources). The criterion chosen to determine a better fit to the data was that C-statistic difference between the models should be larger than 2.7. If the difference was less, than both model fits were considered to equally represent the data. Although such a criterion is ad hoc (considering the un- certainties in the actual error statistics) and count rate dependent, it does serve as an qualitative guideline to differentiate between those systems which can be repre- sented by a power-law emission and/or a black body one. The spectral parameters for all sources for the power-law and disk blackbody models are tabulated in Tables 4 and 5 of the appendix. For ULX, the physically relevant parameter is the in- trinsic bolometric luminosity which should be used to de- fine and identify them. However, given the limitations of an instrument’s energy sensitivity range, the bolometric luminosity is spectral model dependent. For a power-law the bolometric luminosity cannot be estimated and only a lower limit can be obtained using the observed energy range. Since our motivation is to show how the bolomet- ric luminosity is affected by the use of different spectral models, we have plotted in Figure 1, the bolometric lu- minosity for the disk black body model versus the lower 4 Senorita Devi et al. TABLE 1 Sample Galaxy properties Galaxy Distance (Mpc) ObsID Texp(ks) N(≥ 60cts) NGC0253 2.6 969 13.98 13 NGC0628 9.7 2058 46.16 7 NGC0891 10.0 794 50.90 14 NGC1291 8.9 795 39.16 14 NGC1316 17.0 2022 29.85 9 NGC1399 18.3 319 55.94 36 NGC1569 2.2 782 96.75 16 NGC2403 3.1 2014 35.59 4 NGC3034 3.9 361 33.25 5 NGC3079 15.6 2038 26.57 5 NGC3379 11.1 1587 31.52 7 NGC3556 14.1 2025 59.36 15 NGC3628 10.0 2039 57.96 14 NGC4125 24.2 2071 64.23 8 NGC4365 20.9 2015 40.42 9 NGC4374 17.4 803 28.47 4 NGC4449 3.7 2031 26.59 12 NGC4485/90 7.8 1579 19.52 9 NGC4559 10.3 2027 10.70 1 NGC4579 21.0 807 33.90 3 NGC4594 9.6 1586 18.51 18 NGC4631 7.6 797 59.21 12 NGC4649 16.6 785 36.87 23 NGC4697 11.8 784 39.25 19 NGC5055 9.2 2197 27.99 16 NGC5128 4.0 962 36.50 22 NGC5194/5 8.4 1622 26.80 18 NGC5457 7.0 2065 9.63 4 NGC5775 26.7 2940 58.21 15 NGC6946 5.5 1043 58.28 16 Note. — (Texp)the exposure time in ks; (N) the number of point sources with total counts from the source≥ 60 as detected by wavdetect with fluxscale= 1 limit to the luminosity using the power-law model. The figure represents only those sources which can be rep- resented by both models. The figure shows that while for most sources the difference in luminosities is not sub- stantial, there are sources with estimated luminosities & 1039 erg/s, where the disk black body luminosity es- timation is significantly smaller than the power-law one. This happens for sources for which the spectral index is large. To show the dependence of the luminosity function on the fitting model, we compute the cumulative luminos- ity function in two ways. For the disk black body cu- mulative luminosity function (DBCLF), the luminosity obtained form fitting a black body is used, except for those sources that are fitted better with a power-law for which the power-law model estimated luminosity is considered. Similarly, for the power-law cumulative lu- minosity function, (PLCLF) the luminosity corresponds to the power-law fit, except for those sources which are better represented by a disk black body spectrum. In Figure 2, the solid line represents the DBCLF while the PLCLF is plotted as a dotted one. There are less num- ber of sources with L > 1040 ergs/s for disk black body preferred representation. Moreover, there is a significant difference in the slope of the two luminosity function and the presence of a faint break at L ∼ 1040 ergs/s for the PLCLF, is not evident in the DBCLF. Of particular in- terest is the number of sources whose minimum luminos- ity (i.e. the minimum of the two lower limits obtained by fitting the two spectral models) exceeds a certain value. In Figure 2, the dashed line represents such a minimum cumulative luminosity function, which reveals that there are eight sources with minimum luminosity greater 1040 ergs/s (ELX) and ∼ 80 sources with minimum luminosity greater 1039 ergs/s (ULX). Figures 3 (a) shows the variation of the luminosity ver- sus the disk black body temperature, While most of the sources have an inner disk temperature ∼ 1 keV, as is evident from the distribution (Figure 3 b), there is a pop- ulation of high luminosity source with low (∼ 0.1 keV) temperature. Although the number of sources is low, there seems to be some evidence, that ELX (i.e. sources with luminosities > 1040 ergs/s ) can be divided in two groups, a “super-soft” group with temperature less than 0.2 keV and an harder group with temperature ∼ 2 keV. Figure 3 may be compared with the results obtained by Winter et al. (2006) using XMM-Newton data for a dif- ferent sample and selection criteria. They also find that sources with luminosities > 5 × 1039 ergs/s have have a similar bimodal distribution in temperature as shown in Figure 3. This supports the hypothesis that ELX can be divided into two groups and this is not an artifact of sample selection bias. Figure 4 (a) shows the variation of luminosity with power-law photon index for those sources which can be fitted by a power-law model. There is a clear correla- tion between the two. This correlation does not seem to be due to overestimation of column density, since no such correlation is seen in the luminosity versus NH plot (Figure 5 b). Similar to the analysis using the Extremely luminous X-ray point sources 5 Fig. 2.— The cumulative luminosity function using the disk black body model (solid line), the power-law model (dotted line) and the minimum cumulative luminosity function (dashed line). 0.1 1 10 Fig. 3.— (a) The luminosity versus inner disk temperature and (b) the distribution of the inner disk temperature for sources whose spectra can be modeled as disk black body emission. The triangles represent sources which are better fitted by the disk black body model as compared to the power-law one. The two solid lines represent the expected luminosity versus maximum temperature relations for accretion disks radiating at one and one-tenth of the the Eddington Luminosity. Two sources which were identified as foreground stars (see text) are not included in this plot. 0 2 4 6 8 10 Fig. 4.— (a) The luminosity versus power-law index and (b) the distribution of the power-law index for sources whose spectra can be modeled as a power-law emission. The triangles represent sources which are better fitted by the power-law model as compared to the disk black body one. Two sources which were identified as background AGN (see text) are not included in this plot. disk black body model, there is a group of “super-soft” sources (i.e. photon spectral index > 3) which are also highly luminous (L > 1040 ergs/s). The column den- sity versus luminosity plots for both the power-law and disk black body models (Figure 5), reveal an absence of correlation, which is indicative that there may not be a bias in the analysis, i.e. the luminosities are not being over-estimated because of a NH overestimation. In this analysis, there are eight sources which have an apparent minimum luminosity greater than 1040 ergs/s. However, two of these sources (NGC 5055, R.A: 13 15 30.18, Dec: +42 03 13.5 and NGC 4594, R.A: 12 39 45.22, Dec: -11 38 49.8) are foreground stars based on the optical images of the galaxies. Optical spec- troscopy of a source in NGC 5775 ( R.A: 14 53 55.8, Dec: +3 33 28.02) reveals that it is a background AGN (Gutiérrez & López-Corredoira 2005), while a source in NGC 1399 ( R.A: 14 53 55.8, Dec: +3 33 28.02) is a BLAGN (Green et al. 2004). The spectral properties of the other four sources, which we call Extremely Lumi- nous sources (ELX) are tabulated in Table 2. The NGC 0628 source reported in Table 2, is a different source than the well studied ULX, CXOU J013651.1+154547. The luminosity of this highly variable source (Krauss et al. 2005) is ∼ 1039 ergs/s and its spectral properties are listed in Table 4. The source in NGC 6946 is a well known variable source (Liu & Mirabel 2005) and has been called X7 with L ∼ 1039.22 ergs/sec(Lira et al. 2000), IXO 85 (Colbert & Ptak 2002) and source no. 56 (Holt et al. 2003). Although, the best fit parameters for C-statistics are shown, these results have been checked using χ2 statistics and by C-statistics fit for unbinned and back- 6 Senorita Devi et al. 36 38 40 42 Fig. 5.— The column density versus luminosity for sources fitted with (a) the disk black body model and (b) the power-law model. ground not subtracted data and it was found that the parameters values are consistent within errors and the errors on the estimated luminosities do not vary by more than a factor two. In general, these sources are better represented by disk black body emission than a power- law model, except for the source in NGC 6946, which however requires an exceptionally large power-law pho- ton spectral index, (Γ > 5). The spectral properties of these bright sources suggest that they may be divided into two groups. The first group of four sources (Table 2), are represented by low inner disk temperatures (< 0.5 keV), and hence may be called “supersoft” sources. In contrast the second group of three sources, have harder spectra with inner disk temperatures & 1 keV or with power-law photon index (Γ ∼ 2) and hence may be called hard sources. For this group, the spectra are marginally fitted better with a disk blackbody emission, although considering the uncertainties in the spectral fitting, a power-law representation may also be acceptable. 4. SUMMARY AND DISCUSSION Chandra observations of thirty galaxies were analyzed and the spectra of their points sources were fitted us- ing both a power-law and a disk black body emission model. Only those sources were chosen, which were bright enough to allow a meaningful spectral fit, but whose data was not contaminated by excessive diffuse emission and/or effected by pile-up. It was found that the shape of the luminosity function especially at the high luminosity end, depends on the choice of the spec- tral model. In accordance with earlier results, a large number of the sources (∼ 80 ) have a luminosity which exceeds 1039 ergs/s and hence satisfy the standard definition of be- ing Ultra Luminous X-ray sources (ULX) and do not seem to have any spectral distinction when compared with sources having lower luminosity. In this sample of 365 sources, there are four source which we refer to as Extremely luminous X-ray sources (ELX) since their lu- minosities were estimated to exceed 1040 ergs/s. These sources are in general better described by disk black- body emission and can be distinctively grouped into two classes. This is consistent with the results of an indepen- dent analysis using XMM-Newton data (Winter et al. 2006). The members of the first class have soft spec- tra with best fit inner disk temperature < 0.5 keV, while for the other class the spectra is harder with inner disk temperature & 1.3 keV. If disk black body emission is indeed the correct radia- tive process for the supersoft class then the inner disk temperature should correspond to the maximum color temperature of a disk, which can be estimated to be Tcol ∼ 0.3 keV L )−1/2 (1) where L40 is the luminosity in 10 40 ergs/s, f is the color factor and LEdd is the Eddington luminosity. In Figure 3 (a), the two solid lines represent this relationship for L/LEdd = 1 and 0.1. Thus within the uncertainties, the supersoft sources are compatible with having pure black body disk emission, and have L ∼ 0.5LEdd. ELX which are members of the hard class have inner disk temperatures which are higher than that expected from a Eddington limited black body accretion disks. Hence, for these source the radiative mechanism is prob- ably inverse Comptonization of soft photons. Detailed spectral analysis, which included XMM observations, of the bright X-ray source in M82 X-1 has revealed that its spectrum is better fitted by a saturated Comptonization model (Agrawal & Misra 2006), which is also the case for the the bright X-ray source in Holmberg IX. Holmberg IX is not part of the sample studied here and M82 X-1 has been excluded because of pile-up effects and excess dif- fuse emission. Thus, these sources, with estimated lumi- nosities ∼ 1041 erg/s, could also be members of the hard class of ELX. Thus it seems that like the the hard state of standard black hole binaries, the hard class ELX also have spectra which is due to thermal Comptonization, however unlike black hole binaries, in ELX the Comp- tonization seems to be saturated. Thus, it is tempting to draw by analogy, that the two spectral classes of ELX are actually two spectral states of the same kind of ob- ject. This can be verified if spectral transition between the two classes is observed. With the identification of these ELX and other sources from the literature, it is now possible to undertake a more extensive study of their properties. Temporal variability of these sources will shed more light on the nature of these enigmatic sources. The authors thank the referee for useful comments and suggestions which have significantly improved the paper. ASD thanks CSIR and IUCAA for support. REFERENCES Agrawal, V. K., & Misra, R. 2006, ApJ, 638, L83 Begelman, M. C., 2002, ApJ, 568, L97. Extremely luminous X-ray point sources 7 TABLE 2 Spectral properties of sources with minimum luminosity > 1040ergs/s galaxy R.A. Dec. Counts Bins kTin (keV) log LDbb C-statDbb Γ log LPow C-statPow NGC0628 01 36 47.45 +15 47 45.01 200 7 0.09 +0.04 −0.03 41.65 +2.49 −1.60 0.7 9.56 +0.44 −2.42 42.18 +0.36 −1.49 NGC6946 20 35 0.13 +60 9 7.97 1936 66 0.32 +0.02 −0.03 39.46 +0.08 −0.06 162.2 5.14 +0.34 −0.27 40.41 +0.21 −0.18 134.8 NGC4579 12 37 40.30 +11 47 27.48 1696 66 1.35 +0.10 −0.09 40.56 +0.02 −0.02 65.0 1.75 +0.11 −0.11 40.37 +0.00 −0.02 NGC5775 14 53 58.90 + 3 32 16.78 1358 60 1.97 +0.29 −0.22 41.11 +0.02 −0.02 55.8 1.86 +0.22 −0.11 40.95 +0.10 −0.06 Note. — Host Galaxy name; Right Ascension; Declination; Total counts; Number of energy bins after rebinning; Best fit inner disk temperature; Bolometric luminosity estimate using disk black body model; C-statistic for disk black body model; Photon index Γ for the power-law model; Luminosity estimate (0.3-8.0 keV) using the power-law model; C-statistic for power-law model; Colbert, E. J. M., & Mushotzky, R. F. 1999, ApJ, 519, 89 Colbert, E. J. M., & Ptak, A. F. 2002, ApJS, 143, 25 Dewangan, G. C., Griffiths, R. E., & Rao, A. R. 2006, ApJ, 641, Dewangan, G. C., Titarchuk, L., & Griffiths, R. E. 2006, ApJ, 637, Dewangan, G. C., Griffiths, R. E., Choudhury, M., Miyaji, T., & Schurch, N. J. 2005, ApJ, 635, 198 Feng, H., & Kaaret, P. 2006, ApJ, 650, L75 Feng, H., & Kaaret, P. 2005, ApJ, 633, 1052 Fabbiano, G. 1989, ARA&A, 27, 87 Fabbiano, G. 2004, Revista Mexicana de Astronomia y Astrofisica Conference Series, 20, 46 Fiorito, R. & Titarchuk, L., 2004, ApJ, 614, L113. Fox, D. W., et al. 2000, MNRAS, 319, 1154 Green, P. J., et al. 2004, ApJS, 150, 43 Gonçalves, A. C., & Soria, R. 2006, MNRAS, 371, 673 Gutiérrez, C. M., & López-Corredoira, M. 2005, ApJ, 622, L89 Holt, S. S., Schlegel, E. M., Hwang, U., & Petre, R. 2003, ApJ, 588, 792 Kaaret, P., et al. 2001, MNRAS, 321, L29. King, A. R. et al. 2001, ApJ, 552, L109. Krauss, M. I., Kilgard, R. E., Garcia, M. R., Roberts, T. P., & Prestwich, A. H. 2005, ApJ, 630, 228 Kubota, A., et al., 2001, ApJ, 547, L119. Lira, P., Lawrence, A., & Johnson, R. A. 2000, MNRAS, 319, 17 Liu, Q. Z., & Mirabel, I. F. 2005, A&A, 429, 1125 Madau, P., & Rees, M. J., 2001, ApJ, 551, L27. Makishima, K., et al. 2000, ApJ, 535, 632 Matsumoto, H., et al. 2001, ApJ, 547, L25. Miller, J. M., Fabbiano, G., Miller, M. C., & Fabian, A. C., 2003, ApJ, 585, L37. Miller, M. C., & Colbert, E. J. M. 2004, International Journal of Modern Physics D, 13, 1 Miller, J. M., Fabian, A. C., & Miller, M. C., 2004, ApJ, 607, 931. Miller, J. M. 2005, Ap&SS, 300, 227 Misra, R., & Sriram, K., 2003, ApJ, 584, 981. Mizuno, T., Kubota, A., & Makishima, K. 2001, ApJ, 554, 1282 Portegies Zwart, S. F., & McMillian, S. L. W., 2002, ApJ, 576, 899. Ptak A., & Griffiths R., 1999, ApJ, 517, L85 Richstone, D., et al. 1998, Nature, 395, A14 Roberts, T. P., & Warwick, R. S. 2000, MNRAS, 315, 98 Roberts, T. P., Kilgard, R. E., Warwick, R. S., Goad, M. R., & Ward, M. J. 2006, MNRAS, 371, 1877 Ryder, S., Staveley-Smith, L., Dopita, M., Petre, R., Colbert, E., Malin, D., & Schlegel, E. 1993, ApJ, 416, 167 Stobbart, A.-M., Roberts, T. P., & Wilms, J. 2006, MNRAS, 368, Strohmayer, T. E., & Mushotzky, R. F., 2003, ApJ, 586, L61. Swartz, D. A., Ghosh, K. K., Tennant, A. F., Wu K. 2004, ApJS, 154, 519. Taniguchi, Y., Shioya, Y., Tsuru, T. G. & Ikeuchi, S., 2000, PASJ, 52, 533. Wang, Q. D., Yao, Y., Fukui, W., Zhang, S. N., & Williams, R. 2004, ApJ, 609, 113 Winter, L. M., Mushotzky, R. F., & Reynolds, C. S. 2006, ApJ, 649, 730 Zezas, A., & Fabbiano, G., ApJ, 577, 726. APPENDIX 8 Senorita Devi et al. TABLE 3 List of sources with count rate greater than 0.05 counts/s Galaxy R.A. Dec. Count rate s−1 NGC0253 0 47 32.97 -25 17 48.80 0.0845164 NGC0253 0 47 22.59 -25 20 50.87 0.0657110 NGC0253 0 47 17.55 -25 18 11.18 0.0569807 NGC1569 4 31 16.85 +64 49 50.13 0.0567468 NGC2403 7 36 55.61 +65 35 40.85 0.0746877 NGC2403 7 36 25.53 +65 35 40.02 0.1521207 NGC3628 11 20 15.75 +13 35 13.70 0.0530390 NGC4374 12 25 11.92 +12 51 53.53 0.0563400 NGC4449 12 28 17.83 +44 6 33.86 0.0525013 NGC4485 12 30 43.26 +41 38 18.36 0.0509909 NGC4485 12 30 30.56 +41 41 42.33 0.0758634 NGC4559 12 35 58.56 +27 57 41.91 0.1233310 NGC4559 12 35 51.71 +27 56 4.05 0.1984267 NGC4631 12 41 55.56 +32 32 16.90 0.0554584 NGC6946 20 35 0.74 +60 11 30.74 0.1448118 Note. — The spectra of these sources would be affected by pile-up and hence have not been included in the sample TABLE 4 Spectral Properties of point sources fitted with the Power-Law model Galaxy R.A. Decl. nH(10 −2) Γ log(L) ergs/s Cstat d. o. f. NGC0253 0 47 43.07 -25 15 29.28 0.15 +0.53 −0.15 +2.31 −1.21 37.70 +1.20 −0.18 9.43 4 NGC0253 0 47 42.80 -25 15 2.02 0.77 +0.51 −0.49 +0.66 −0.77 38.26 +0.27 −0.12 3.36 4 NGC0253 0 47 35.25 -25 15 11.53 0.48 +0.05 −0.13 +0.22 −0.33 38.65 +0.05 −0.08 38.94 19 NGC0253 0 47 34.28 -25 17 3.32 5.66 +5.98 −5.66 +4.62 −4.95 40.82 +3.28 −2.90 3.21 2 NGC0253 0 47 34.00 -25 16 36.51 1.04 +0.11 −0.10 +0.22 −0.22 39.01 +0.08 −0.07 10.93 27 NGC0253 0 47 33.55 -25 18 16.51 0.00 +0.18 −0.00 +0.99 −0.44 37.33 +0.15 −0.10 2.37 2 NGC0253 0 47 32.05 -25 17 21.43 3.17 +2.20 −2.46 +1.10 −1.10 38.61 +0.72 −0.32 1.63 3 NGC0253 0 47 30.98 -25 18 26.23 1.55 +1.46 −1.55 +1.21 −1.21 38.24 +0.65 −0.21 1.76 3 NGC0253 0 47 28.01 -25 18 20.21 0.93 +1.13 −0.92 +1.76 −1.54 38.02 +1.03 −0.23 2.13 1 NGC0253 0 47 25.20 -25 19 45.22 0.00 +0.15 −0.00 +0.66 −0.22 37.96 +0.09 −0.09 5.10 4 NGC0253 0 47 18.50 -25 19 13.94 0.00 +0.15 −0.00 +0.44 −0.22 37.97 +0.07 −0.05 3.94 5 NGC0253 0 47 40.66 -25 14 11.71 0.69 +0.72 −0.56 +1.98 −1.32 37.96 +1.11 −0.27 1.07 3 NGC0253 0 47 17.65 -25 18 26.45 0.11 +0.49 −0.11 +1.54 −0.77 38.12 +0.27 −0.12 3.13 3 NGC0628 1 36 51.06 +15 45 46.86 0.03 +0.05 −0.03 +0.22 −0.11 39.22 +0.04 −0.02 50.58 36 NGC0628 1 36 47.45 +15 47 45.01 0.89 +0.11 −0.33 +0.44 −2.42 42.18 +0.36 −1.49 8.56 4 . . . . . . . . . Note. — Host galaxy name; Right Ascension; Declination; nH , equivalent hydrogen column density; Γ, photon power-law index; L, X-ray luminosity in the energy range: 0.3-8.0 keV; C-statistics; degree of freedom. The complete version of this table is in the electronic edition of the Journal. The printed edition contains only a sample. Extremely luminous X-ray point sources 9 TABLE 5 Spectral Properties of point sources fitted with the disk black body model Galaxy R.A. Decl. nH(10 −2) kTin (keV) log(L) ergs/s Cstat d. o. f. NGC0253 0 47 43.07 -25 15 29.28 0.00 +0.32 −0.00 +0.42 −0.25 37.69 +0.37 −0.00 9.41 4 NGC0253 0 47 42.80 -25 15 2.02 0.44 +0.36 −0.32 +1.95 −0.52 38.40 +0.26 −0.08 3.95 4 NGC0253 0 47 35.25 -25 15 11.53 0.24 +0.07 −0.07 +0.21 −0.15 38.71 +0.04 −0.03 35.79 19 NGC0253 0 47 34.28 -25 17 3.32 3.08 +5.36 −3.07 +9.38 −0.36 38.69 +2.04 −0.60 3.37 2 NGC0253 0 47 34.00 -25 16 36.51 0.64 +0.12 −0.10 +0.19 −0.16 39.06 +0.03 −0.03 8.60 27 NGC0253 0 47 33.55 -25 18 16.51 0.00 +0.08 −0.00 +0.69 −0.29 37.49 +0.20 −0.06 3.85 2 NGC0253 0 47 32.05 -25 17 21.43 2.04 +1.82 −1.62 +4.71 −0.80 38.68 +0.40 −0.09 1.37 3 NGC0253 0 47 30.98 -25 18 26.23 0.96 +0.96 −0.86 +8.00 −0.91 38.41 +0.73 −0.12 1.84 3 NGC0253 0 47 28.01 -25 18 20.21 0.62 +0.77 −0.56 +8.80 −0.55 38.08 +0.90 −0.12 2.51 1 NGC0253 0 47 25.20 -25 19 45.22 0.00 +0.06 −0.00 +0.86 −0.33 38.12 +0.21 −0.09 7.71 4 NGC0253 0 47 18.50 -25 19 13.94 0.00 +0.05 −0.00 +0.73 −0.41 38.22 +0.15 −0.09 6.36 5 NGC0253 0 47 40.66 -25 14 11.71 0.42 +0.47 −0.38 +2.76 −0.40 37.94 +0.38 −0.18 1.64 3 NGC0253 0 47 17.65 -25 18 26.45 0.08 +0.32 −0.07 +8.76 −0.65 38.23 +1.04 −0.13 2.85 3 NGC0628 1 36 51.06 +15 45 46.86 0.00 +0.00 −0.00 +0.08 −0.11 39.42 +0.02 −0.04 119.44 36 NGC0628 1 36 47.45 +15 47 45.01 0.75 +0.58 −0.39 +0.04 −0.03 41.65 +2.49 −1.60 0.66 4 . . . . . . . . . Note. — Host galaxy name; Right Ascension; Declination; nH , equivalent hydrogen column density; kTin, inner disk temperature; L, Bolometric X-ray luminosity; C-statistics; degree of freedom. The complete version of this table is in the electronic edition of the Journal. The printed edition contains only a sample.

0704.1108

Evidence for Symplectic Symmetry in Ab Initio No-Core Shell Model Results for Light Nuclei

Evidence for Symplectic Symmetry in Ab Initio No-Core Shell Model Results for Light Nuclei Tomáš Dytrych, Kristina D. Sviratcheva, Chairul Bahri, and Jerry P. Draayer Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA James P. Vary Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA Lawrence Livermore National Laboratory, L-414, 7000 East Avenue, Livermore, California, 94551, USA and Stanford Linear Accelerator Center, MS81, 2575 Sand Hill Road, Menlo Park, California, 94025, USA Clear evidence for symplectic symmetry in low-lying states of 12C and 16O is reported. Eigenstates of 12C and 16O, determined within the framework of the no-core shell model using the JISP16 NN realistic interaction, typically project at the 85-90% level onto a few of the most deformed symplectic basis states that span only a small fraction of the full model space. The results are nearly independent of whether the bare or renormalized effective interactions are used in the analysis. The outcome confirms Elliott’s SU(3) model which underpins the symplectic scheme, and above all, points to the relevance of a symplectic no-core shell model that can reproduce experimental B(E2) values without effective charges as well as deformed spatial modes associated with clustering phenomena in nuclei. Recently developed realistic interactions, such as J- matrix inverse scattering potentials [1] and modern two- and three-nucleon potentials derived from meson ex- change theory [2] or by using chiral effective field theory [3], succeed in modeling the essence of the strong inter- action for the purpose of input into microscopic shell- model calculations that target reproducing characteris- tic features of light nuclei. The ab initio No-Core Shell Model (NCSM) [4] which employs such modern realistic interactions, yields a good description of the low-lying states in few-nucleon systems [5] as well as in more com- plex nuclei like 12C [4, 6]. In addition to advancing our understanding of the propagation of the nucleon-nucleon force in nuclear matter and clustering phenomena [7, 8], modeling the structure of 12C, 16O and similar nuclei is also important for gaining a better understanding of other physical processes such as parity-violating electron scattering from light nuclei [9] and results gained through neutrino studies [10] as well as for making better predic- tions for capture reaction rates that figure prominently, for example, in the burning of He in massive stars [11]. In this letter we report on investigations that show that realistic eigenstates for low-lying states determined in NCSM calculations for light nuclei with the JISP16 realis- tic interaction [1], predominantly project onto few of the most deformed Sp(3,R)-symmetric basis states that are free of spurious center-of-mass motion. This reflects the presence of an underlying symplectic sp(3,R) ⊃ su(3) ⊃ so(3) algebraic structure [22], which is not a priori im- posed on the interaction and furthermore is found to re- main unaltered after a Lee-Suzuki similarity transforma- tion used to accommodate the truncation of the infinite Hilbert space by renormalization of the bare interaction. This in turn provides insight into the physics of a nucleon system and its geometry. Specifically, nuclear collective states with well-developed quadrupole and monopole vi- brational modes and rotational modes are described nat- urally by irreducible representations (irreps) of Sp(3,R). The present study points to the possibility of achieving convergence of higher-lying collective modes and reach- ing heavier nuclei by expanding the NCSM basis space beyond its current limits through Sp(3,R) basis states that span a dramatically smaller subspace of the full space. In this way, the symplectic no-core shell-model (Sp-NCSM) with realistic interactions and with a mixed Sp(3,R) irrep extension will allow one to account for even higher ~Ω configurations required to realize experimen- tally measured B(E2) values without an effective charge, and to accommodate highly deformed spatial configura- tions [12] that are required to reproduce α-cluster modes, which may be responsible for shaping, e.g., the second 0+ state in 12C and 16O [8]. We focus on the 0+gs ground state and the lowest 2+(≡2+ ) and 4+(≡4+ ) states in the oblate 12C nucleus as well as the 0+gs in the ‘closed-shell’ 16O nucleus. The NCSM eigenstates for these states are reasonably well converged in the Nmax = 6 (or 6~Ω) model space with an effective interaction based on the JISP16 realistic interac- tion [1], which typically leads to rapid convergence in the NCSM evaluations, describes NN data to high accuracy and is consistent with, but not constrained by, meson ex- change theory, QCD or locality. In addition, calculated binding energies as well as other observables for 12C such as B(E2;2+ →0+gs), B(M1;1 →0+gs), ground-state proton rms radii and the 2+ quadrupole moment all lie reason- ably close to the measured values. While symplectic al- gebraic approaches have achieved a very good reproduc- tion of low-lying energies and B(E2) values in light nuclei [13, 14] and specifically in 12C using phenomenological interactions [15] or truncated symplectic basis with sim- http://arxiv.org/abs/0704.1108v1 plistic (semi-) microscopic interactions [16, 17], here, for the first time, we establish, the dominance of the sym- plectic Sp(3,R) symmetry in light nuclei, and hence their propensity towards development of collective motion, as unveiled through ab initio calculations of the NCSM type starting with realistic two-nucleon interactions. The symplectic shell model [18, 19] is based on the noncompact symplectic sp(3,R) algebra. The classical realization of this symmetry underpins the dynamics of rotating bodies and has been used, for example, to de- scribe the rotation of deformed stars and galaxies [20]. In its quantal realization it is known to underpin the successful Bohr-Mottelson collective model and has also been shown to be a multiple oscillator shell generaliza- tion of Elliott’s SU(3) model. Consequently, symplectic basis states bring forward important information about nuclear shapes and deformation in terms of (λ, µ), which serve to label the SU(3) irreps within a given Sp(3,R) irrep, for example, (0, 0), (λ, 0) and (0, µ) describe spher- ical, prolate and oblate shapes, respectively. The significance of the symplectic symmetry for a mi- croscopic description of a quantum many-body system emerges from the physical relevance of its 21 generators constructed as bilinear products of the momentum (pα) and coordinate (qβ) operators, e.g. pαpβ , pαqβ , and qαqβ with α, β = x, y, and z for the 3 spatial directions. Hence, the many-particle kinetic energy, the mass quadrupole moment operator, and the angular momentum are all elements of the sp(3,R) ⊃ su(3) ⊃ so(3) algebraic struc- ture. It also includes monopole and quadrupole collective vibrations reaching beyond a single shell to higher-lying and core configurations, as well as vorticity degrees of freedom for a description of the continuum from irrota- tional to rigid rotor flows. Alternatively, the elements of the sp(3,R) algebra can be represented as bilinear prod- ucts in harmonic oscillator (HO) raising and lowering op- erators, which means the basis states of a Sp(3,R) irrep can be expanded in a 3-D HO (m-scheme) basis which is the same basis used in the NCSM, thereby facilitating calculations and symmetry identification. The basis states within a Sp(3,R) irrep are built by applying symplectic raising operators to a np-nh (n- particle-n-hole, n = 0, 2, 4, ...) lowest-weight Sp(3,R) state (symplectic bandhead), which is defined by the usual requirement that the symplectic lowering operator annihilates it. The raising operator induces a 2~Ω 1p-1h monopole or quadrupole excitation (one particle raised by two shells) together with a smaller 2~Ω 2p-2h correc- tion for eliminating the spurious center-of-mass motion. If one were to include all possible lowest-weight np-nh starting state configurations (n ≤ Nmax), and allowed all multiples thereof, one would span the full NCSM space. The lowest-lying eigenstates of 12C and 16O were calcu- lated using the NCSM as implemented through the Many Fermion Dynamics (MFD) code [21] with an effective interaction derived from the realistic JISP16 NN poten- 0 2 4 6 8 10 12 3 0p-0h all 0p-0h dominant 0p-0h+2p-2h Sp(3,R) 0 2 4 6 8 10 12M Sp(3,R) FIG. 1: NCSM space dimension as a function of the maxi- mum ~Ω excitations, Nmax, compared to that of the Sp(3,R) subspace: (a) J = 0, 2, and 4 for 12C, and (b) J = 0 for 16O. tial [1] for different ~Ω oscillator strengths. For both nuclei we constructed all of the 0p-0h and 2~Ω 2p-2h (2 particles raised by one shell each) symplectic band- heads and generated their Sp(3,R) irreps up to Nmax = 6 (6~Ω model space). Analysis of overlaps of the symplec- tic states with the NCSM eigenstates for 2~Ω, 4~Ω, and 6~Ω model spaces (Nmax = 2, 4, 6) reveals the dominance of the 0p-0h Sp(3,R) irreps. For the 0+gs and the lowest 2+ and 4+ states in 12C there are nonnegligible overlaps for only 3 of the 13 0p-0h Sp(3,R) irreps, namely, the leading (most deformed) representation specified by the shape deformation of its symplectic bandhead, (0 4), and carrying spin S = 0 together with two (1 2) S = 1 ir- reps with different bandhead constructions for protons and neutrons. For the ground state of 16O there is only one possible 0p-0h Sp(3,R) irrep, (0 0) S = 0. In addi- tion, among the 2~Ω 2p-2h Sp(3,R) irreps only a small fraction contributes significantly to the overlaps and it in- cludes the most deformed configurations that correspond to oblate shapes in 12C and prolate ones in 16O. The typical dimension of a symplectic irrep in the Nmax = 6 space is on the order of 10 2 as compared to 107 for the full NCSM m-scheme basis space. As Nmax is increased the dimension of the J = 0, 2, and 4 symplec- tic space built on the 0p-0h Sp(3,R) irreps for 12C grows very slowly compared to the NCSM space dimension (Fig. 1a). The dominance of only three irreps additionally re- duces the dimensionality of the symplectic model space, which remains a small fraction of the NCSM basis space even when the most dominant 2~Ω 2p-2h Sp(3,R) irreps are included. The space reduction is even more dramatic in the case of 16O (Fig. 1b). This means that a space spanned by a set of symplectic basis states is computa- tionally manageable even when high-~Ω configurations are included. The overlaps of the most dominant symplectic states with investigated NCSM eigenstates for the 12C and the 16O in the 0, 2, 4 and 6~Ω subspaces are given in Table I and II. In order to speed up the calculations, we retained only the largest amplitudes of the NCSM states, those sufficient to account for at least 98% of the norm which is quoted also in the table. The results show that approx- imately 85% of the NCSM eigenstates for 12C (16O) fall within a subspace spanned by the few most significant 0p-0h and 2~Ω 2p-2h Sp(3,R) irreps, with the 2~Ω 2p- 2h Sp(3,R) irreps accounting for 5% (10%) and with the leading irrep, (0 4) for 12C and (0 0) for 16O, carrying close to 70% (75%) of the NCSM wavefunction. Furthermore, the S = 0 part of all three NCSM eigen- states for 12C is almost entirely projected (95%) onto only six S = 0 symplectic irreps included in Table I, with as much as 90% of the spin-zero NCSM states accounted for solely by the leading (0 4) irrep. The S = 1 part is also remarkably well described by merely two Sp(3,R) irreps. Similar results are observed for the ground state of 16O. Another striking property of the low-lying eigenstates is revealed when the spin projections of the converged NCSM states are examined. Specifically, as shown in Fig. 2, their Sp(3,R) symmetry and hence the geometry of the nucleon system being described is nearly indepen- dent of the ~Ω oscillator strength. The symplectic sym- metry is present with equal strength in the spin parts of the NCSM wavefunctions for 12C as well as 16O regard- less of whether the bare or the effective interactions are used. This suggests that the Lee-Suzuki transformation, which effectively compensates for the finite space trun- cation by renormalization of the bare interaction, does not affect the Sp(3,R) symmetry structure of the spa- tial wavefunctions. Hence, the symplectic structure de- TABLE I: Probability distribution of NCSM eigenstates for 12C across the dominant 0p-0h and 2~Ω 2p-2h Sp(3,R) irreps, ~Ω=15 MeV. 0~Ω 2~Ω 4~Ω 6~Ω Total J = 0 Sp(3,R) (0 4)S = 0 46.26 12.58 4.76 1.24 64.84 (1 2)S = 1 4.80 2.02 0.92 0.38 8.12 (1 2)S = 1 4.72 1.99 0.91 0.37 7.99 2~Ω 2p-2h 3.46 1.02 0.35 4.83 Total 55.78 20.05 7.61 2.34 85.78 NCSM 56.18 22.40 12.81 7.00 98.38 J = 2 Sp(3,R) (0 4)S = 0 46.80 12.41 4.55 1.19 64.95 (1 2)S = 1 4.84 1.77 0.78 0.30 7.69 (1 2)S = 1 4.69 1.72 0.76 0.30 7.47 2~Ω 2p-2h 3.28 1.04 0.38 4.70 Total 56.33 19.18 7.13 2.17 84.81 NCSM 56.18 21.79 12.73 7.28 98.43 J = 4 Sp(3,R) (0 4)S = 0 51.45 12.11 4.18 1.04 68.78 (1 2)S = 1 3.04 0.95 0.40 0.15 4.54 (1 2)S = 1 3.01 0.94 0.39 0.15 4.49 2~Ω 2p-2h 3.23 1.16 0.39 4.78 Total 57.50 17.23 6.13 1.73 82.59 NCSM 57.64 20.34 12.59 7.66 98.23 tected in the present analysis for 6~Ωmodel space is what would emerge in NSCM evaluations with a sufficiently large model space to justify use of the bare interaction. 11 12 13 14 15 16 17 18 Bare (a) J=0 h�̄Ω ——(MeV) 11 12 13 14 15 16 17 18 Bare (b) J=2 h�̄Ω ——(MeV) 11 12 13 14 15 16 17 18 Bare (c) J=4 h�̄Ω ——(MeV) 12 13 14 15 16 Bare (d) J=0 h�̄Ω ——(MeV) FIG. 2: Projection of the S = 0 (blue, left) [and S = 1 (red, right)] Sp(3,R) irreps onto the corresponding significant spin components of the NSCM wavefunctions for (a) 0+gs, (b) 2 and (c) 4+ in 12C and (d) 0+gs in 16O, for effective interaction for different ~Ω oscillator strengths and bare interaction. In addition, as one varies the oscillator strength ~Ω, the projection of the NCSM wavefunctions onto the sym- plectic subspace changes only slightly (see, e.g., Fig. 3 for the 0+gs state of 12C and 16O). The symplectic structure is preserved, only the Sp(3,R) irrep contributions change because the S = 0 (S = 1) part of the NCSM eigen- states decrease (increase) towards higher ~Ω frequen- cies. Clearly, the largest contribution comes from the leading Sp(3,R) irrep (black diamonds), growing to 80% of the NCSM wavefunctions for the lowest ~Ω. These results can be interpreted as a strong confirmation of El- liott’s SU(3) model since the projection of the NCSM states onto the 0~Ω space [Fig. 3, blue (lowest) bars] is a projection of the NCSM results onto the SU(3) shell model. The outcome is consistent with what has been shown to be a dominance of the leading SU(3) symmetry for SU(3)-based shell-model studies with realistic inter- actions in 0~Ω model spaces. It seems the simplest of TABLE II: Probability distribution of the NCSM eigenstate for the J = 0 ground state in 16O across the 0p-0h and dom- inant 2~Ω 2p-2h Sp(3,R) irreps, ~Ω=15 MeV. 0~Ω 2~Ω 4~Ω 6~Ω Total Sp(3,R) (0 0)S = 0 50.53 15.87 6.32 2.30 75.02 2~Ω 2p-2h 5.99 2.52 1.32 9.83 Total 50.53 21.86 8.84 3.62 84.85 NCSM 50.53 22.58 14.91 10.81 98.83 11 12 13 14 15 16 17 18 h�̄Ω ——(MeV) 12 13 14 15 16 h�̄Ω ——(MeV) FIG. 3: Ground 0+ state probability distribution over 0~Ω (blue, lowest) to 6~Ω (green, highest) subspaces for the most dominant 0p-0h + 2~Ω 2p-2h Sp(3,R) irrep case (left) and NCSM (right) together with the leading irrep con- tribution (black diamonds), (0 4) for 12C (a) and (0 0) for 16O (b), as a function of the ~Ω oscillator strength, Nmax = 6. Elliott’s collective states can be regarded as a good first- order approximation in the presence of realistic interac- tions, whether the latter is restricted to a 0~Ω model space or richer multi-~Ω NCSM model spaces. The 0+gs and 2 states in 12C, constructed in terms of the three Sp(3,R) irreps with probability amplitudes defined by the overlaps with the NCSM wavefunctions for Nmax = 6 case, were also used to determine B(E2 : → 0+gs) transition rates. The latter, increasing from 101% to 107% of the corresponding NCSM numbers with increasing ~Ω, clearly reproduce the NCSM results. In summary, we have shown that ab initio NCSM cal- culations with the JISP16 nucleon-nucleon interaction display a very clear symplectic structure, which is un- altered whether the bare or effective interactions for var- ious ~Ω strengths are used. Specifically, NCSM wave- functions for the lowest 0+gs, 2 and 4+ states in 12C and the ground state in 16O project at the 85-90% level onto a few 0p-0h and 2~Ω 2p-2h spurious center-of-mass free symplectic irreps. Furthermore, while the dimensionality of the latter is only ≈ 10−3% that of the NCSM space, they closely reproduce the NCSM B(E2) estimates. The wavefunctions for 12C are strongly dominated by the three leading 0p-0h symplectic irreps, with a clear dom- inance of the most deformed (0 4)S = 0 collective con- figuration. The ground state of 16O is dominated by the single 0p-0h irrep (0 0)S = 0. The results confirm for the first time the validity of the Sp(3,R) approach when realistic interactions are invoked in a NCSM space. This demonstrates the importance of the Sp(3,R) symmetry in light nuclei while reaffirming the value of the simpler SU(3) model upon which it is based. The results further suggest that a Sp-NCSM extension of the NCSM may be a practical scheme for achieving convergence to mea- sured B(E2) values without the need for introducing an effective charge. In short, the NCSM with a modern re- alistic interaction supports the development of collective motion in nuclei which is realized through the Sp-NCSM and as is apparent in its 0~Ω Elliott model limit. Discussions with many colleagues, but especially Bruce R. Barrett, are gratefully acknowledged. This work was supported by the US National Science Foundation, Grant Nos 0140300 & 0500291, and the Southeastern Universi- ties Research Association, as well as, in part, by the US Department of Energy Grant Nos. DE-AC02-76SF00515 and DE-FG02-87ER40371 and at the University of Cal- ifornia, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. Tomáš Dytrych acknowl- edges supplemental support from the Graduate School of Louisiana State University. [1] A. M. Shirokov, A. I. Mazur, S. A. Zaytsev, J. P. Vary, and T. A. Weber, Phys. Rev. C 70, 044005 (2004); A. M. Shirokov, J. P. Vary, A. I. Mazur, S. A. Zaytsev, and T. A. Weber, Phys. Letts. B 621, 96(2005); A. M. Shirokov, J. P. Vary, A. I. Mazur, and T. A. Weber, Phys. Letts. B 644, 33 (2007). [2] R. Machleidt, F. Sammarruca, and Y. Song, Phys. Rev. C 53, R1483 (1996); R. Machleidt, Phys. Rev. C 63, 024001 (2001). [3] D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001 (2003). [4] P. Navrátil, J. P. Vary, and B. R. Barrett, Phys. Rev. Lett. 84, 5728 (2000); Phys. Rev. C 62, 054311 (2000). [5] P. Navrátil and B. R. Barrett, Phys. Rev. C 57, 562 (1998); 59, 1906 (1999); P. Navrátil, G. P. Ka- muntavičius, and B. R. Barrett, Phys. Rev. C 61, 044001 (2000). [6] P. Navrátil and W. E. Ormand, Phys. Rev. C 68, 034305 (2003). [7] S. Karataglidis et al., Phys. Rev. C 52, 861 (1995). [8] Y. Funaki et al., Phys. Rev. C 67, R051306 (2003). [9] M. J. Musolf and T. W. Donnelly, Nucl. Phys. A546, 509 (1992). [10] A. C. Hayes, P. Navrátil, and J. P. Vary, Phys. Rev. Lett. 91, 012502 (2003). [11] C. R. Brune et al., Phys. Rev. Lett. 83, 4025 (1999). [12] K. T. Hecht and D. Braunschweig, Nucl. Phys. A295, 34 (1978); Y. Suzuki, Nucl. Phys. A448, 395 (1986). [13] G. Rosensteel and D. J. Rowe, Ann. Phys. N.Y. 126, 343 (1980). [14] J. P. Draayer, K. J. Weeks and G. Rosensteel, Nucl. Phys. A413, 215 (1984). [15] J. Escher and A. Leviatan, Phys. Rev. C 65, 054309 (2002). [16] F. Arickx, J. Broeckhove, and E. Deumens, Nucl. Phys. A377, 121 (1982). [17] S. S. Avancini and E. J. V. de Passos, J. Phys. G 19, 125 (1993). [18] G. Rosensteel and D. J. Rowe, Phys. Rev. Lett. 38, 10 (1977). [19] D. J. Rowe, Reports on Progr. in Phys. 48, 1419 (1985). [20] G. Rosensteel, Astrophys. J. 416 291, (1993). [21] J. P. Vary, “The Many-Fermion-Dynamics Shell-Model Code,” Iowa State University, 1992 (unpublished); J. P. Vary and D. C. Zheng, ibid 1994 (unpublished). [22] We use lowercase (capital) letters for algebras (groups).

0704.1109

Spinning Strings, Black Holes and Stable Closed Timelike Geodesics

Spinning Strings, Black Holes and Stable Closed Timelike Geodesics. Valéria M. Rosa∗ Departamento de Matemática, Universidade Federal de Viçosa, 36570-000 Viçosa, M.G., Brazil Patricio S. Letelier† Departamento de Matemática Aplicada-IMECC, Universidade Estadual de Campinas, 13081-970 Campinas, S.P., Brazil The existence and stability under linear perturbation of closed timelike curves in the spacetime associated to Schwarzschild black hole pierced by a spinning string are studied. Due to the super- position of the black hole, we find that the spinning string spacetime is deformed in such a way to allow the existence of closed timelike geodesics. PACS numbers: 04.20.Gz, 04.20.Dg, 04.20.Jb Keywords: Closed Timelike Geodesics, Linear Stability, Time Machines, Black Holes, Cosmic Strings,Torsion Lines The existance of closed timelike curves (CTCs) in the Gödel universe and other apacetimes is a worrying fact since these curves show a clear violation of causality. In some cases these CTCs can be disregarded by en- ergy considerations. Their existance requires an external force acting along the whole CTC, process that may con- sume a great amount of energy. The energy needed to travel along a CTC in Gödel’s universe is computed in [1]. When the external force is null the energy needed to travel is also null. Therefore, in principle, the existence of closed timelike geodesics (CTGs) presents a bigger prob- lem of breakdown of causality. The classical problem of the existence of closed geodesics in Riemannian geometry was solved by Hadamard [2] in two dimensions and by Cartan [3] in an arbitrary number of dimensions. As a topological prob- lem, the existence of CTGs was proved by Tipler [4] in a class of four-dimensional compact Lorentz manifolds with covering space containing a compact Cauchy sur- face. In a compact pseudo-Riemaniann manifold with Lorentzian signature (Lorentzian manifold) Galloway [5] found sufficient conditions to have CTGs, see also [6]. To the best of our knowledge there are four solution to the Einstein equations generated by matter with positive mass density that contain CTGs: a) Soares [7] found a class of cosmological models, solutions of Einstein- Maxwell equations, with a subclass where the timelike paths of matter are closed. For these models the exis- tence of CTGs is demonstrated and explicit examples are given. These CTGs are not linearly stable [8]. b) Stead- man [9] described the behavior of CTGs in a vacuum exterior for the van Stockum solution that represents an infinite rotating dust cylinder. For this solution explicit examples of CTCs and CTGs are shown. There are stable CTGs in this spacetime [8]. c) Bonnor and Steadman [10] studied the existence of CTGs in a spacetime with two spinning particles each one with magnetic moment equal ∗Electronic address: e-mail: [email protected] †Electronic address: e-mail: [email protected] to angular moment and mass equal to charge (Perjeons), in particular, they present a explicit CTG. This partic- ular CTG is not stable, but there exist many other that are stable [11]. d) There are linearly stable CTGs [8] in one of the Gödel-type metrics with not flat background studied by Gürses et al. [12][13]. For CTGs in a space- time associated to a cloud of strings with negative mass density see [14]. These CTGs are not stable [8]. The existence of CTCs in a spacetime whose source is a spinning string has been investigated by many authors (see for instance [15]-[19]). The interpretation of these strings as torsion line defects can be found in [20], [21], see also [22][23]. These torsion line defects appear when one tries to stabilize two rotating black holes kept apart by spin repulsion [24]. Also, the black hole thermody- namics associated to a static black hole pierced by a non rotating string was studied some time ago by Aryal et al. [25]. In the present work we study the existence and stabil- ity of CTCs under linear perturbations in the spacetime associated to Schwarzschild black hole (BH) pierced by a spinning string. Even though this spacetime is more a mathematical curiosity than an example of a real space- time we believe that the study of stability of CTCs and CTGs can shed some light into the existence of this rather pathological curves. In particular, we study sufficient conditions to have linearly stable CTGs. We find that these conditions are not very restrictive and can be eas- ily satisfied. Furthermore, we compared them with the same conditions studied by Galloway [5] for a compact Lorentzian manifold. Let us consider the spacetime with metric, ds2 = (1− )(dt−αdϕ)2− 1− 2m −r2(dθ2+β2 sin2 θdϕ2), where α = 4S and S is the string’s spin angular momen- tum per unit of length, β = 1 − 4λ and λ is the string’s linear mass density that is equal to its tension (λ ≤ 1/4). In the particular case, α = 0 and β = 1, the metric (1) reduces to the Schwarzschild solution. When m = 0, Eq. (1) represents a spinning string, with the further special- http://arxiv.org/abs/0704.1109v2 mailto:e-mail: [email protected] mailto:e-mail: [email protected] ization β = 1 (not deficit angle) we have a pure massless torsion line defect [20] [21]. Therefore the metric (1) can be considered as representing the spacetime associated to a Schwarzschild black hole pierced by a spinning string. Let us denote by γ a closed curve given in its paramet- ric form by, t = t0, r = r0, ϕ ∈ [0, 2π], θ = , (2) where t0 and r0 are constants. When γ is parametrized with an arbitrary parameter σ, we have a timelike curve when dx > 0. This condition reduces to gϕϕ > 0, i.e., (1− 2m/r0)α2 − r20β2 > 0. (3) A generic CTC γ satisfies the system of equations given ẍµ + Γ αβ ẋ αẋβ = Fµ(x), (4) where the overdot indicates derivation with respect to αβ are the Christoffel symbols and F µ is a specific external force (aµ = Fµ). The nonzero component of the four-acceleration of γ is (r0 − 2m)(α2m− r30β2)ϕ̇2. (5) Our goal is to study the behavior of closed timelike geodesics. Therefore taking α as one of the two solutions α2m− r3 β2 = 0, (6) we have ar = 0. Under this condition (3) is satisfied when r0 > 3m, that put the CTG outside the black hole. Let γ̃ be the curve obtained from γ after a small per- turbation ξ, i.e., x̃µ = xµ + ξµ. From equations (4) one finds that the system of differential equations satisfied by the perturbation ξ is [26], + 2Γαβµ uµ + Γαβµ,λξ λuβuµ = Fα,λξ λ. (7) For the above mentioned closed timelike geodesic the system (7) reduces to ξ̈0 + k1ξ̇ 1 = 0, (8) ξ̈1 + k2ξ̇ 0 + k3ξ 1 = 0, (9) ξ̈2 + k4ξ̇ 1 = 0, (10) ξ̈3 + k5ξ 3 = 0, (11) where k1 = 2Γ ϕ̇, k2 = 2Γ ϕ̇, k3 = Γ 22,1ϕ̇ k4 = 2Γ ϕ̇, k5 = Γ 22,3ϕ̇ 2. (12) A curve γ parametrized by the proper time, s, is time- like when ẋµẋµ = 1. For the curve γ(s) we have that this last condition gives us, ϕ̇2 = (r0 − 3m) . (13) The solution of (9)-(11) is ξ0 = −k1(c3 sin(ωs+ c4)/ω + λs) + c1 s+ c5, ξ1 = c3 cos(ωs+ c4) + λ, ξ2 = −k4(c3 sin(ωs+ c4)/ω + λs) + c2 s+ c6, ξ3 = c7 cos( k5s+ c8), where ci, i = 1, . . . , 8 are integration constants, k3 − k1k2 = [β2(r0 − 6m)ϕ̇2/r0]1/2, (15) and λ = −k2c1/ω2. Thus when r0 > 6m, the constant ω is real and the solution (14) shows the typical behav- ior for stability, i.e., vibrational modes untangled with translational ones that can be eliminated by a suitable choice of the initial conditions. When the black hole is removed, we are left with the spacetime of the spinning string whose line element is, ds2 = (dt− αdϕ)2 − dr2 − r2(dθ2 + β2 sin2 θdϕ2). (16) The closed curve, γ, is timelike when α2− r2 β2 > 0. The ar-component of the four-acceleration is given by ar = −β2r0ϕ̇2. Thus for r < |α/β| we have closed timelike curves, which are not geodesics. For the closed curve (2) the system (7) is written now as in (9)-(11) replacing equation (10) by ξ̈1 + k2ξ̇ 2 + k3ξ 1 = ∂r(Γ ϕ̇2)ξ1, (17) where now k2 = 2Γ ϕ̇ and ϕ̇2 = (α2 − r2 β2)−1. In this particular case the solution of (7) has the same form that (14) with ω2 = 2β2ϕ̇2(2+β2r2 ϕ̇2). Therefore, the CTCs are stable. In summary, there exist linearly stable CTCs in the spacetime related to a spinning string and these curves are restricted to a small region of the spacetime. Closed timelike geodesics do not exist in this spacetime. For the nonlinear superposition of a spinning string with a Schwarzschild black hole the new spacetime has linearly stable CTGs. The region of stability is the same of the usual circular geodesics in the Schwarzschild black hole alone. The presence of the spinning string does not affect the stability of the orbits. It seems that torsion lines defects superposed to matter (not strings, β = 1) is a main ingredient to have stable CTGs. Loosely speak- ing, we have that a torsion line defect alone makes possi- ble the existence of CTCs. When the black hole is present the spinning string spacetime is deformed in such a way to allow the existence of a CTG. This fact is also con- firmed in the case of the two Perjeons solutions studied 0 5 10 15 20 25 30 35 m= 0.3; β= 0.9; α= 25; r = 6.1401 0 5 10 15 20 25 30 35 m= 1; β= 0.9; α= 25; r = 9.172 0 5 10 15 20 25 30 35 m= 4; β= 0.9; α= 25; r = 14.5597 FIG. 1: The function s(r0) for a spinning string (solid line) and for a black hole pierced by the string (dashed line). We see how the presence of the mass shift the maximum of s(r0) for the string that is located at r0 = 0 to a position outside the black hole horizon. The maximum, rm represent the radius of the CTGs, the first two are stable and the second is not. in [10] wherein the torsion line defect is a main ingredient to have CTCs and CTGs. It is instructive to look the previous results in a more direct and graphic way. The length of CTC in (2) only depends on the value of r = r0. We find, s(r0) = 2π gϕϕ(r0), = 2π[(1− 2m/r0)α2 − r20β2]1/2. (18) This function has a local maximum for rm = (mα 2/β2)1/3. (19) Note that this equation is equivalent to (6), the condition to have a geodesic. The role of the black hole mass, in the appearance of CTGs, is to produce a local maximum in the length function, s(r0). This maximum gives us the position of the CTG that in our case is located outside of the source of the spacetime, beyond the black hole horizon. In Fig. 1 we present, as a solid line the function s(r0) for a spinning string, and as a dashed line the same func- tion for the superposition of the black hole with the pre- viously mentioned string for the same values of the pa- 0 10 20 30 40 50 60 70 m= 1; β= 0.4; α= 25; r = 15.749 0 10 20 30 40 50 60 70 m= 1; β= 0.7; α= 25; r = 10.845 0 10 20 30 40 50 60 70 m= 1; β= 0.9; α= 25; r = 9.172 FIG. 2: The function s(r0) for a spinning string (solid line) and for a black hole pierced by the string (dashed line). We see how the size of the deficit angle parameter β changes the region for CTCs and the value of rm. rameters α = 25 (spin parameter) and deficit angle pa- rameter β = 0.9 and different values of the black hole mass (m = 0.3, 1, 4). We see how the presence of the mass shift the maximum for the string located at r0 = 0 to a position r0 > 3m. Also the points under the curves represent the pairs [r0, s(r0)] for CTCs in each case. We note that the region for CTCs for the black hole pierced by the string diminishes when the mass increases. The maximum of the dashed line represents the CTG. We see, that in the first two cases the CTGs are stable (rm > 6m) and in the last case the CTG is not stable (rm < 6m). In Fig. 2 we keep the value of the black hole mass constant, m = 1, as well as, the spin parameter, α = 25, and change the deficit angle parameter β = 0.4, 0.7, 0.9. We see that the larger the string density, λ = (1− β)/4, the larger the region for CTCs. In Fig. 3 we keep the value of the black hole mass constant, m = 1, as well as, the deficit angle parameter β = 0.9 and change the spin parameter α = 15, 20, 25. We see that the regions where the CTCs appear are larger for bigger spin parameter. This parameter is essential to have CTCs and CTGs in this case. As we said before the existence of a CTGs does not put restrictions on the energy to travel along this curve. Fur- thermore, the force needed to move near a stable geodesic 0 5 10 15 20 25 30 35 m= 1; β= 0.9; α= 15; r = 6.5248 0 5 10 15 20 25 30 35 m= 1; β= 0.9; α= 20; r = 7.9042 0 5 10 15 20 25 30 35 m= 1; β= 0.9; α= 25; r = 9.172 FIG. 3: The function s(r0) for a spinning string (solid line) and for a black hole pierced by the string (dashed line). We see how the size of the spin parameter α changes the region for CTCs and the value of rm. The spin parameter, in this case, is the essential ingredient to have CTCs and CTGs. is small. Therefore, the energy required will be also small. In principle this small force can be provided by and en- gine, say a rocket. Hence there will be not a severe en- ergy restriction to travel near to a geodesic. Furthermore, when moving along a stable CTG the control problem is a trivial one. Small trajectory corrections require small energy, also we do not have the danger to enter into a run away situation. A result from Galloway [5] states that in a compact Lorentzian manifold, each stable free t-homotopy class contains a longest closed timelike curve, and this curve is necessarily a closed timelike geodesic. The assumption thatM be compact can be weakened, it is sufficient to as- sume that there exists an open set U in M with compact closure such that each curve γ ∈ C (the free t-homotopy class) is contained in U . In our case th Gödel universe and other apacetimes e region containing the CTCs in C is not compact. Therefore Galloway’s conditions do not apply in this case, they too strong. We want to point out that the stability of the circu- lar orbits does not depend on the fact of the orbit be a CTG. We found the same region of stability of the usual circular geodesics. This result is not surprising since our pierced black hole is locally identical to a usual black hole. Moreover one can consider black holes surrounded by different axially symmetric distributions of matter [27] pierced by a spinning string. In this case, depending on the different parameters of the solution, we can also have CTGs and their stability will be the same as the usual circular orbits considered in [27]. Furthermore, we analyze if the CTGs studied in the present work satisfy the sufficient conditions of Gal- loway’s theorem for the existence of CTGs. We found that ours CTGs do not satisfy these conditions. The pos- sibility of an example that satisfy exactly the conditions of this theorem is under study. We want to mention that the solution of Einstein equations considered in this work is much simpler that the ones listed in the introduction. Finally, we notice that the spacetime associated to the black hole pierced by a spinning string is not a counter example to the Chronology Protection Conjecture [28] that essentially says that the laws of the physics do not allow the appearance of closed timelike curves. A valid dynamic to built this spacetime is not known. V.M.R. thanks Departamento de Matemática-UFV for giving the conditions to finish this work which was par- tially supported by PICDT-UFV/CAPES. P.S.L. thanks the partial financial support of FAPESP and CNPq. [1] J. Pfarr, Gen. Rel. Grav., 13, 1073 (1981). [2] J. Hadamard, J. Math. Pures Appl. 4, 27 (1896). [3] E. Cartan, Leçons sur la géométrie de Riemann (Gauthier-Villars, Paris, 1928). [4] F.J. Tipler, Proc. Am. Math. Soc., 76, 145 (1979). [5] G. J. Galloway, Trans. Am. Math. Soc., 285, 379 (1984). [6] M. Guerdiri, Trans. Am. Math. Soc., 359, 2663 (2007). [7] I.D. Soares, J. Math. Phys., 21, 591 (1980). [8] V.M. Rosa and P.S. Letelier, Stability of closed timelike geodesics in different spacetimes; gr-qc/0706.3212. [9] B.R. Steadman, Gen. Rel. Grav., 35, 1721 (2003). [10] W.B. Bonnor and B.R. Steadman, Gen. Rel. Grav., 37, 1833 (2005). [11] V.M. Rosa and P.S. Letelier, Phys. Lett. A, 370, 99 (2007). [12] M. Gürses, A. Karasu, O. Sarioglu, Class. Quantum. Grav., 22, 1527 (2005). [13] R.J. Gleiser, M. Gürses, A. Karasu and Ö. Sarroğlu, Class. Quantum. Grav., 23, 2653 (2006). [14] Ø. Grøn and S. Johannesen, Closed timelike geodesics in a gas of cosmic strings; gr-qc/0703139. [15] S. Deser, R. Jackiw and G. ’t Hooft, Phys. Rev. Letts., 68, 267 (1992). [16] Y.M. Cho and D.H. Park, Phys. Rev. D, 46, R1219 (1992). [17] J. R. Gott, Phys. Rev. Letts., 66, 1126 (1991). [18] B. Jensen and H.H. Soleng, Phys.Rev.D, 45, 3528 (1992). [19] H.H. Soleng, Phys. Rev. D, 49, 1124 (1994). http://arxiv.org/abs/gr-qc/0703139 [20] P.S. Letelier, Class. Quantum. Grav., 12, 471 (1995). [21] P.S. Letelier, Class. Quantum. Grav., 12, 2221 (1995). [22] P. de Sousa, Nucl. Phys. B, 346, 440 (1990). [23] R.J. Petti, Gen. Rel. Grav., 18, 1124 (1986). [24] P.S. Letelier and S.R. Oliveira, Phys. Lett. A, 238, 101 (1998). [25] M. Aryal, L.H. Ford and A. Vilenkin, Phys. Rev. D 34, 2263 (1986). [26] V.M. Rosa and P.S. Letelier, Gen. Rel. Grav. 39, 1419 (2007); gr-qc/0703100. [27] P.S. Letelier, Phys. Rev. D 68 (2003) 104002 . [28] S. Hawking, Phys. Rev. D, 46, 603 (1991) http://arxiv.org/abs/gr-qc/0703100

0704.1110

On spherically symmetrical accretion in fractal media

Mon. Not. R. Astron. Soc. 000, 1–5 (yyyy) Printed 8 November 2018 (MN LATEX style file v2.2) On spherically symmetrical accretion in fractal media Nirupam Roy ⋆ NCRA-TIFR, Post Bag 3, Ganeshkhind, Pune 411 007, India Accepted yyyy mmm dd. Received yyyy mmm dd; in original form yyyy mmm dd ABSTRACT We use fractional integrals to generalize the description of hydrodynamic accretion in fractal media. The fractional continuous medium model allows the generalization of the equa- tions of balance of mass density and momentum density. These make it possible to consider the general case of spherical hydrodynamic accretion onto a gravitating mass embedded in a fractal medium. The general nature of the solution is similar to the “Bondi solution”, but the accretion rate may vary substantially and the dependence on central mass may change significantly depending on dimensionality of the fractal medium. The theory shows consis- tency with the observational data and numerical simulation results for the particular case of accretion onto pre-main-sequence stars. Key words: Accretion, accretion discs – hydrodynamics – turbulence – stars: pre-main- sequence – ISM: clouds – ISM: structure. 1 INTRODUCTION The interstellar medium (ISM) is believed to have a self- similar hierarchical structure over several orders of magni- tude in scale (Larson 1981; Falgarone, Puget & Perault 1992; Heithausen et al. 1998). Direct H I absorption observations and interstellar scintillation measurements suggest that the structure extends down to a scale of 10 AU (Crovisier, Dickey & Kazès 1985; Langer et al. 1995; Faison et al. 1998) and possibly even to a scale of sub-AU (Hill et al. 2005). However, the latter is limited by the spatial resolution of the observations. Hence the issue is far from being definite even after observational detec- tion of lower limit of self-similarity scale in some ISM com- ponents (Goodman, Barranco, Wilner & Heyer 1998). Numerous theories have attempted to explain the origin, evolution and mass distribution of these clouds (began with the hierarchi- cal fragmentation picture, Hoyle 1953) and it has been estab- lished, from both observations (Elmegreen & Falgarone 1996) and numerical simulations (Burkert, Bate & Bodenheimer 1997; Klessen, Burkert & Bate 1998; Semelin & Combes 2000), that the interstellar medium has a clumpy hierarchical self-similar structure with a fractal dimension 2.5 . D . 2.7 (Sánchez, Alfaro & Pérez 2005) in 3 dimensional space. The main reason for this is still not properly understood but can result from the underlying fractal ge- ometry that may arise due to turbulent processes in the medium. Here we have investigated physical processes, in particular hy- drodynamics, in such a medium with fractal dimension. We have considered the simplest situation of hydrodynamic spherical steady accretion of the medium onto a large gravitating mass embedded in this medium. We have assumed the dimensionality of the medium ⋆ E-mail: [email protected] to be isotropic and homogeneous. It implies that the mass den- sity (or, equivalently, dimensionality) is same for any surface inde- pendent of the orientation. The famous solution of the continuous medium case is the “Bondi Solution” for spherical hydrodynamic accretion (Bondi 1952). Here we have extended that analysis for a medium with fractal dimension D (= 3d) 6 3. To describe physical processes in fractal medium, we have used fractional integration and differentiation (Zaslavsky 2002, and references therein). We replace the fractal medium by some contin- uous medium and the integrals on the network of fractal medium is approximated by fractional integrals (Ren et al. 2003). The inter- pretation of fractional integration is connected with fractional mass dimension (Mandelbrot 1983). Fractional integrals can be consid- ered as integrals over fractional dimension space (up to a numerical factor; Tarasov 2004). We have chosen the numerical factor prop- erly to get the right dimension of any physical parameter and to de- rive the standard expression in the limit d → 1. The isotropic and homogeneous nature of dimensionality is also incorporated prop- erly. This model allows us to describe the hydrodynamics in a self- consistent way in a fractal medium. In this paper we first derive, in §2, the steady state mass accre- tion rate for fractal spherical hydrodynamic accretion using sim- ple dimensional analysis. In §3 we derive the steady state hydro- dynamic equations to describe the spherical accretion in a fractal medium and the steady state accretion rate in terms of boundary conditions at infinity is derived in §4. We discuss the actual astro- physical implication of our analysis and compare our central re- sult with observational data and numerical simulation results in §5. Possible limitations of our analysis is discussed in §6. Finally, we summarize and present our conclusions in §7. c© yyyy RAS http://arxiv.org/abs/0704.1110v2 2 N. Roy 2 DIMENSIONAL ANALYSIS OF FRACTAL ACCRETION We assume a medium that, on a range of length scale R, has a frac- tal structure of dimensionality D = 3d < 3 embedded in 3 dimen- sional space. Here D refers to the mass dimension and it implies that in such a medium of constant density ρ, the mass enclosed in a sphere of radius r will be MD = kr 3(d−1) 3d (1) where lc is a characteristic inner length of the medium and can take arbitrary value in the limit d → 1. It is the scale below which the medium will be continuous. We can define the modified “density” for this fractal medium as ρ ≡ ρ/l3(d−1)c so that MD ∼ ρR3d. For steady state hydrodynamic spherical accretion onto a cen- tral mass M from its surroundings fractal medium with a mass di- mension of D, the relevant physical parameters will be (1) sound speed at a large distance away from the accretor (a∞), (2) modified “density” of the fractal medium at a large distance away from the accretor (ρ = ρ∞/l 3(d−1) c ) and (3) mass of the accretor scaled by the gravitational constant (GM ). The dimensions of these three parameters are [a∞] = [M ] 1 (2) ] = [M ] 0 (3) [GM ] = [M ] . (4) It is possible to uniquely construct, from these parameters, a quan- tity Ṁ with dimension [M ]1[L]0[T ]−1. So, from simple dimen- sional analysis we get the mass accretion rate Ṁ ∼ ρ 3(d−1) . (5) Dimension analysis can not be used to fix the dimensionless con- stant C in the above equation and does not give a detailed physical picture. But we have used the fractional integrals to derived, in a more detailed analysis in the following sections, the mass accretion rate and found that to be consistent with the accretion rate derived from the dimensional analysis. 3 FRACTIONAL INTEGRALS AND HYDRODYNAMIC EQUATIONS The integrals on network of fractals can be approximated by frac- tional integrals (Ren et al. 2003) and the interpretation of the frac- tional integration is connected with fractional mass dimension. We consider the fractional integrals as integrals over fractional dimen- sion space (up to a numerical factor) and the fractional infinitesimal length for a medium with isotropic mass dimension will be given ld−1c dr, d = D/3 < 1 (6) where the constant is chosen to derive the standard expression in the limit d → 1. Note that the infinitesimal area and volume elements in this “fractional continuous” medium of mass dimension D = 3d will be dAr = 2(d−1) r2(d−1) sin θdθdφ (7) dAθ = 2(d−1) r2(d−1) rdr sin θdφ (8) dAφ = 2(d−1) r2(d−1) rdrdθ (9) 3(d−1) r3(d−1) dr sin θdθdφ (10) and hence the mass enclosed in a sphere of radius R for constant density ρ will be ρdV = 3(d−1) ∼ RD (11) which will give the standard expression in the limit d → 1. Below we consider the mass density and momentum density balance in such a fractal medium of mass dimension 3d in the case of accretion onto a central gravitating mass M for the case of our interest (i.e. the steady state hydrodynamic spherical accretion). We consider the infinitesimal volume dV in E3 and the bal- ance of mass density or the conservation of mass in that volume in the case of steady state spherical accretion will imply [ρu(r/lc) dAr] = 0 =⇒ ) = 0. (12) This is the generalized mass density balance equation or the gener- alized continuity equation that can be integrated over a surface of constant r to give a constant radial accretion rate 3(1−d) = Constant (independent of r). (13) In the infinitesimal volume dV in E3, the balance of momen- tum density will imply (ρVdV ) = F S (14) where V is the velocity vector, FM is the total gravitational force acting on the mass contained in the infinitesimal volume dV and S is the total surface force due to pressure acting on the surfaces bounding the volume dV and are given by r = − dV (15) r = ber.(r/lc) [(pdArber)r − (pdArber)r+dr + (pdAθbeθ)θ − (pdAθbeθ)θ+dθ + (pdAφbeφ)φ − (pdAφbeφ)φ+dφ] (16) and, for spherical accretion where p = p(r), it will be reduced to r = − dArdr(r/lc) . (17) The total change of radial momentum is given by (ρudV ) = (ρudV ) + (r/lc) dAr]dr. (18) Combining this and equation (12), in case of steady state we get (r/lc) dArdr = − dArdr(r/lc) d−1 − dV (19) and this can be simplified to the generalized momentum density balance equation = 0. (20) This equation, using the equation of state p = Kργ (1 6 γ 6 5 and the boundary condition at infinity, can also be integrated to give c© yyyy RAS, MNRAS 000, 1–5 Accretion in fractal media 3 0.001 0.010 0.100 1.000 10.000 0.01 0.1 1 10 log (r/r0) λ=0.25λc λ=1.00λc λ=4.00λc Figure 1. Velocity profile (u/a–r/r0, r0 = GM/a2∞) for D = 2.55 and γ = 7 at λ = 0.25λc , 1.00λc , 4.00λc γ − 1 γ − 1 ∞ (21) where a is the sound speed given by a ≡ (dp/dρ)1/2 and a∞ is the sound speed at infinity. This is a local conservation law and, as expected, has exactly the same form as that of the continuous medium. 4 MASS ACCRETION RATE We have solved equations (12) and (20) for a smooth, monotonic solution without any singularities in the flow. Following the stan- dard derivation of the “Bondi solution”, we found that there exist unique eigen value solution of the problem for a given d and γ and that the solution must pass through a critical point where c = a c = S(d, γ)a ∞, S(d, γ) = (6d− 1)− 3γ(2d− 1) . (22) The mass accretion rate may not be steady in a small timescale because of the “clumpy” structure of the medium but a time average mass accretion rate can also be calculated following the standard derivation and is given by Ṁ = 4πλc(d, γ) 3(d−1) ”3d−1 = 4πλc(d, γ)ρ∞a∞(GM/a −3(1−d) (23) where F (M,a∞, lc) and λc(d, γ) are dimensionless parameters F (M,a∞, lc) = GM/a ∞lc (24) λc(d, γ) = (3d− 1)(1−3d) [S(d, γ)] 2(γ−1) −(3d−1) . (25) This is consistent with the accretion rate derived from the dimen- sional analysis (see Eqn.(5) in Sec.2). The solution also uniquely corresponds to the physically likely situations properly matching the boundary conditions (small velocity at large distance and high velocity at small r in accretion flow and the opposite in “wind flow”). The velocity profile is shown in figure (1) for a particu- lar case where D = 3d = 2.55 and γ = 7 . The general nature of the solution is similar to the “Bondi solution” (Bondi 1952). Note that the smooth, physically selected solution exists only for the transonic case with λ = λc. It also corresponds to the max- imum accretion rate as there exists no meaningful solution either for λ > λc or for λ < λc. Note that value of λc is of order unity. In particular, for d = 1 and γ = 5 , λc = (Bondi 1952) and for γ = 1, λc is given by λc(d, γ = 1) = (3d− 1)(1−3d) 3d− 3 2 (26) which is the limit of equation (25) as γ → 1, so that λc is continu- ous at γ = 1. The most interesting consequence is the dependence of accre- tion rate on the factor F (M,a∞, lc). In the interstellar medium typical ambient sound speed (which goes as T ) is ∼ 10 km s−1 for a temperature of ∼ 104 K. In this case the value of F is given F (M,a∞, lc) = 8.907 10 km s ”2“1 AU . (27) This is simply the ratio of the length-scale of “sphere of influence” of the accreting object to the scale length of the fractal structure. As we are mainly interested in the medium inside the sphere of influ- ence, our approach to the problem of accretion in fractal medium will be valid only if F & 1. In such situations, this factor can change the mass accretion rate substantially and it will crucially depend on the fractional di- mensionality d. More interestingly, for the cases where this fac- tor is important, the accretion rate Ṁ will not be proportional to M2 but will be significantly different from that. For example, for 2.5 . D . 2.7 (Sánchez, Alfaro & Pérez 2005) or equiv- alently 0.83 . d . 0.90, the accretion rate Ṁ ∼ Mα where 1.5 . α . 1.7. In the actual astrophysical situation, however, the Bondi-Hoyle accretion rate will probably be more rel- evant (Bondi & Hoyle 1944; Bondi 1952; Edgar 2004, and references therein). Bondi (1952) proposed the in- terpolation formula (corrected upto a numerical factor by Shima, Matsuda, Takeda & Sawada 1985) replacing a∞ by (a2∞ + v 1/2 and Bondi radius rB = GM/a ∞ by Bondi-Hoyle radius rBH = GM/(a ∞ + v ∞) where v∞ is the gas velocity relative to the star at a large distance. For a medium with density and velocity structure, it is not possible to define a unique v∞ and the present analysis can not be extended to such a situation. But it may be extended to a more idealized case where the medium has a fractal structure and the accretor is moving in a velocity v∞ relative to the medium at a large distance. Following a similar interpolation method, the accretion rate in this case will be ṀBH = 4πλcG (a2∞ + v −3(1−d) where F ∼ rBH/lc and λc is a factor of the order of unity. Here we have assumed that the gas velocity relative to the star at a large distance will modify the accretion rate in fractal medium in a simi- lar way that of in the continuous medium. Though a more detailed analysis is required to get the Bondi-Hoyle accretion rate and, in particular, F in turbulent medium, it is most likely that it will not change our main result that the accretion rate will be proportional to M3d−1. c© yyyy RAS, MNRAS 000, 1–5 4 N. Roy -1.5 -1.0 -0.5 0.0 0.5 log M M ~ M1.49 M ~ M2.00 Figure 2. Comparison of data and theory for accretion rates of PMS stars and brown dwarfs. The stellar mass is in M⊙ and the accretion rate is in M⊙ yr 5 ASTROPHYSICAL IMPLICATIONS Our results have important implications for a number of astro- physical problems. One of these, for example, is the problem of pre-main-sequence (PMS) accretion. Numerical simulation shows that approximating the PMS accretion process as Bondi-Hoyle ac- cretion leads to agreement between simulation and observation (Padoan et al. 2005). For a solar mass star and for typical sound speed a∞ ∼ 0.2 km s−1 in ambient molecular cloud filaments (Padoan et al. 2005), a fractal scale length of ∼ 10 AU will be smaller than the size of the sphere of influence and hence our ap- proach should be valid in this particular case. Here we compare our central result with observational data and numerical simulation re- sults. For that we have taken the accretion rates of PMS stars and brown dwarfs compiled by Padoan et al. (2005) (from Natta et al. 2004; White & Hillenbrand 2004; Muzerolle et al. 2004, and refer- ences therein) includes all detections but no upper limit. Though the the data do not provide any strong support for one theory as op- posed to another one due to scatter in the accretion rate attributed to (i) an age dependence of the accretion rate, (ii) variation of ρ∞, a∞ and v∞ and (iii) interaction of accretion flow with jets and out- flows on smaller scale (Padoan et al. 2005), we find that our the- ory is consistent with these data. For example, the best fit value of α is 1.49 ± 0.13 for log (Ṁaccr/M⊙ yr−1) > −10. Even when we include all the compiled data of Padoan et al. (2005), the best fit value of α is consistent with our theory within 3σ error for 2.5 . D . 2.7. In figure (2) we have shown the data and the best fit for log (Ṁaccr/M⊙ yr −1) > −10. Data from Muzerolle et al. (2004) and White & Hillenbrand (2004) are shown as filled squares and empty squares respectively and the rest of the data are shown as empty circles. We also found that for higher accretion rate and higher central mass, the exponent is significantly different from that of the Bondi-Hoyle accretion. For a smaller central mass, the self- gravity that we have neglected in our analysis may change the ac- cretion rate significantly. It is possible to use a high resolution simulation (e.g. Krumholz et al. 2006) to find out the fractional dimensionality and the scale length lc by using equation (11) and counting the num- ber of particle n(R) on boxes of different scale length (R). On the other hand, comparing the result with existing numerical simula- tion results is not very straightforward as in most of the cases it is assumed that Ṁaccr = AM 2 and the coefficient A is computed for the assumed M2 dependence on accretion rate. But with the existing published simulation results, it is still possible to check if the variation of this coefficient A within computational uncertainty can consistently accommodate our result and we found that the best fit of the observational data and our theoretical estimate are within 1.5σ scatter around the computed accretion rate of Padoan et al. (2005). On the other hand, to get a normalized mean accretion rate (see Krumholz et al. 2006, for details) of 0.1 for a Mach number of 5 and D = 2.55, we get, from Eqn.(28), rB/lc ∼ 17740. For a solar mass star and for a typical sound speed a∞ ∼ 0.2 km s−1, this translates to lc ∼ 1.3 AU. Fractal structure at this scale is not observationally ruled out. But one should take this as an order of magnitude estimate of the scale length as factors like velocity struc- ture or magnetic field in real complicated astrophysical situations may alter the accretion rate significantly. The other potential implication is the black hole accretion. The model growth of galactic center black holes assume that the black holes accrete at Bondi rate (e.g., Springel, Di Matteo & Hernquist 2005). If the gas around the black hole is fractal in nature, then one should rather use the modified accretion rate for fractal medium. Here we would like to mention the caveat that there is no defini- tive observational evidence against or for the fractal nature of the medium in this case. Even if the medium is fractal, the validity of our approach, as mentioned in §3, crucially depends on the mass of the accretor, the scale length of the fractal and the ambient sound speed. 6 DISCUSSION We have derived the steady state hydrodynamic equations to de- scribe the spherical accretion in a fractal medium by replacing the fractal medium by a “fractional continuous” model. We have de- rived the steady state accretion rate in terms of boundary condi- tions at infinity in this simplified situation without considering the self-gravity of the material for a medium where the dimensionality is independent of position and orientation. Magnetic fields, which we have not included in these models, will certainly play a major role in determining the dynamics. But even without the inclusion of magnetic field we have got the following results We have found that there exists a unique solution with max- imum mass accretion rate and the general nature of the solution is similar to the “Bondi solution” (Bondi 1952) even for a fractal medium with D = 3d < 3. The mass accretion rate, in cases, may differ substantially depending on the fractional dimensionality. In particular, the accretion rate is likely to be significantly different from the “Bondi accretion rate” and will be proportional to MD−1 in case of accretion in fractional medium with scale length lc very different from GM/a2∞. One limitation of our analysis is that we have not considered the self-gravity of the medium. This is justified provided the central mass M is very large. In cases where self-gravitation is not negligi- ble, it can change the accretion rate significantly. A more important limitation is that the present analysis does not account for turbu- lent velocity structure. It will certainly play an important role to determine the accretion rate but is probably unlikely to change the c© yyyy RAS, MNRAS 000, 1–5 Accretion in fractal media 5 mass dependence. In a real astrophysical situation with both density and velocity structure in the medium, the analytical mass accretion rate will hence not be applicable. Neglecting the magnetic fields, as mentioned earlier, is the other major limitation of the analysis. The presence of magnetic field is more likely to suppress the accretion rate and magnetohydrodynamic simulation in fractal medium will be required to make a more definitive statement. But the central re- sult, that the accretion rate will be proportional to MD−1 will not be affected by the addition of magnetic fields. Our assumption of the equation of state of the form p = Kργ may seems to be a considerable limitation but that is not likely to be the case. In ordinary Bondi-Hoyle accretion in continuous medium, changing the equation of state changes changes the ac- cretion rate by a numerical factor of the order unity (Ruffert 1994; Ruffert & Arnett 1994) and hence will not change the result much. A more detail justification of the assumption can be found in Krumholz, McKee & Klein (2006). 7 CONCLUSION We have shown that the accretion rate onto a mass embedded in a fractal medium may differ, in some cases, significantly from the Bondi accretion rate even in the simplest situation. We have used the simple model of accretion onto a mass from a fractal medium of mass dimensionality D 6 3 and derived a self-consistent so- lution that matches the “Bondi solution” as D → 3. The primary result of our investigation is that theoretically accretion rate will be proportional to MD−1. The observational accretion rate data and the numerical simulation for particular astrophysical problem of accretion onto PMS stars is consistent with our result. Our find- ings suggest that the fractal structure of the medium around the accreting mass is playing a major role to determine the accretion rate and its dependence on the central mass. The agreement of the theoretical prediction with existing numerical simulation implies the consistency of the approach. A number of previously published numerical and analytical results have not considered this effect and may need to be reconsidered. We leave a more detailed treatment of the problem, including the effects discussed in this work, the effect of self-gravity and stability of fractal accretion, to a future work. 8 ACKNOWLEDGMENTS We thank Paolo Padoan and Vasily E. Tarasov for useful discus- sions and Paolo Padoan for kindly providing us the compiled data of the accretion rate of PMS stars and brown dwarfs. We are grate- ful to Rajaram Nityananda for reading an earlier version and sug- gesting the dimensional argument as a cross check of the calcula- tions. Thanks to Kandaswamy Subramanian for his comments on an earlier version of the paper. We also thank Jayaram N. Chen- galur, Raghunathan Srianand and Ranjeev Misra for useful discus- sion. We are grateful to the anonymous referee for useful com- ments and for prompting us into substantially improving this paper. This research was supported by the National Centre for Radio As- trophysics (NCRA) of the Tata Institute of Fundamental Research (TIFR). REFERENCES Bondi H., 1952, MNRAS, 112, 195 Bondi H. & Hoyle F., 1944, MNRAS, 104, 273 Burkert A., Bate M. R., Bodenheimer P., 1997, MNRAS, 289, 497 Crovisier J., Dickey J. M., Kazès I., 1985, A&A, 146, 223 Edgar R., 2004, NewA Rev., 48, 843 Elmegreen B. G., Falgarone E. 1996, ApJ, 471, 816 Faison M.D., Goss W.M., Diamond P.J., Taylor G.B., 1998, AJ, 116, 2916 Falgarone E., Puget J-L., Perault M., 1992, A&A, 257, 715 Goodman A. A., Barranco J. A., Wilner D. J., Heyer M. H., 1998, ApJ, 504, 223 Heithausen A., Bensch F., Stutzki J., Falgarone E., Panis J.F., 1998, A&A, 331, L65 Hill A. S., Stinebring D. R., Asplund C. T., Berkwick D. E., Ev- erett W. B., Hinkel, N. R., 2005, ApJ, 619, L171 Hoyle F., 1953, ApJ, 118, 513 Klessen R.S., Burkert A., Bate M.R, 1998, ApJ, 501, L205 Krumholz M. R., McKee C. F., Klein R. I., 2006, ApJ, 638, 369 Langer W. D., Velusamy T., Kuiper T. B. H., Levin S., Olsen E., Migenes V., 1995, ApJ, 453, 293 Larson R.B., 1981, MNRAS, 194, 809 Mandelbrot B., 1983, The Fractal Geometry of Nature, W.H. Free- man, New York Muzerolle J., Luhman K. L., Briceño C., Hartmann L., Calvet N., 2005, ApJ, 625, 906 Natta A., Testi L., Muzerolle J., Randich S., Comerón F., Persi P., 2004, A&A, 424, 603 Padoan P., Kritsuk A., Norman M. L., Nordlund Å., 2005, ApJ, 622, L61 Ren F-Y., Liang J-R., Wang X-T., Qiu W-Y., Chaos, Solitons and Fractals, 2003, 16, 107 Ruffert M., 1994, ApJ, 427, 342 Ruffert M. & Arnett D, 1994, ApJ, 427, 351 Sánchez N., Alfaro E. J., Pérez E., 2005, ApJ, 625, 849 Semelin B., Combes F., 2000, A&A, 360, 1096 Shima E., Matsuda T., Takeda H., Sawada K., 1985, MNRAS, 217, 367 Springel V., Di Matteo T., Hernquist L., 2005, MNRAS, 361, 776 Tarasov V. E., 2004, Chaos, 14, 123 White R. & Hillenbrand L. A., 2004, ApJ, 616, 998 Zaslavsky G. M., 2002, Phys. Rep., 371, 461 This paper has been typeset from a TEX/ LATEX file prepared by the author. c© yyyy RAS, MNRAS 000, 1–5 Introduction Dimensional Analysis of Fractal Accretion Fractional Integrals and Hydrodynamic Equations Mass accretion rate Astrophysical Implications Discussion Conclusion acknowledgments

0704.1111

QCD in One Dimension at Nonzero Chemical Potential

QCD in One Dimension at Nonzero Chemical Potential L. Ravagli Cyclotron Institute and Physics Department TEXAS A&M University, College Station, Texas 77843-3366, USA J.J.M. Verbaarschot Niels Bohr International Academy, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark, Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark and Department of Physics and Astronomy, SUNY, Stony Brook, New York 11794, USA Using an integration formula recently derived by Conrey, Farmer and Zirnbauer, we calculate the expectation value of the phase factor of the fermion determinant for the staggered lattice QCD action in one dimension. We show that the chemical potential can be absorbed into the quark masses; the theory is in the same chiral symmetry class as QCD in three dimensions at zero chemical potential. In the limit of a large number of colors and fixed number of lattice points, chiral symmetry is broken spontaneously, and our results are in agreement with expressions based on a chiral Lagrangian. In this limit, the eigenvalues of the Dirac operator are correlated according to random matrix theory for QCD in three dimensions. The discontinuity of the chiral condensate is due to an alternative to the Banks-Casher formula recently discovered for QCD in four dimensions at nonzero chemical potential. The effect of temperature on the average phase factor is discussed in a schematic random matrix model. I. INTRODUCTION In spite of significant recent progress, QCD at nonzero chemical potential remains a notoriously hard problem [1, 2, 3]. In particular, first principle nonperturbative results at low temperature are absent because the phase of the fermion determinant invalidates probabilistic methods to evaluate the partition function. This problem is known as the sign problem. The sign problem is particularly severe if the phase of the fermion determinant results in a different free energy, i.e. 〈detNfD〉 〈|detNfD|〉 V (Fpq−FNf ) with FNf > Fpq. (1) If the phase quenched free energy, Fpq differs from the free energy of QCD with Nf flavors, FNf , the number of required gauge field configurations grows exponentially with the volume. The ratio defined in equation (1) can also be interpreted as the average phase factor of the fermion determinant [4, 5] 〈eiNfθ〉pq = 〈eiNf θ|detNfD|〉 〈|detNfD|〉 . (2) Alternatively, one can define the average phase factor with respect to the full QCD partition function [4, 5] 〈e2iθ〉Nf = 〈e2iθdetNfD〉 〈detNfD〉 , (3) which has also been used in the literature [4, 5, 6, 7, 8, 9, 10, 11, 12] as a measure for the severity of the sign problem. When the chemical potential is sufficiently small, the average phase factor can be studied by means of chiral perturbation theory [4, 5]. Recently, the average phase factor was analyzed in the microscopic domain of QCD [13, 14], where only the constant fields in the chiral Lagrangian contribute to the mass and chemical potential dependence of the partition function. It was found that the sign problem is not serious for µ < mπ/2. For µ > mπ/2 the chiral condensate of the phase quenched theory rotates into a pion condensate resulting in a free energy that is different from the full theory and a severe sign problem. In this paper we study the sign problem for Euclidean QCD in one dimension. In one dimension, the only effect of the gauge field is in the boundary conditions, and lattice QCD simplifies to a matrix integral. A general formula for exactly this type of integrals was recently derived by Conrey, Farmer and Zirnbauer [15]. Using what we will call the CFZ formula, exact analytical results for the one dimensional QCD partition function and the average phase factor will be obtained. The study of QCD in one dimension at finite chemical potential has had a long history. One reason to study this model is that it is an effective model for the large Nc limit of strong coupling QCD [16, 17, 18, 19, 20, 21]. Among http://arxiv.org/abs/0704.1111v2 others, it has been successfully used to explain [22, 23, 24] puzzling lattice results for quenched QCD at nonzero chemical potential. More recently, one dimensional QCD was used to study the complex zeros of the partition function [1, 25, 26]. The sign problem of QCD at nonzero chemical potential arises because of the nonhermiticity of the Dirac operator. QCD in one dimension is not truly nonhermitean: Instead of being scattered in a two-dimensional domain of the complex plane [27], the eigenvalues of the Dirac operator are localized on an ellipse in the complex plane [6, 22]. Another manifestation of the mild nonhermiticity is that the chemical potential can be absorbed into the quark masses. For each flavor in one dimension with quark mass m, we can associate flavors with mass m+ µ and −m+ µ. In the large Nc limit, this results in spontaneous chiral symmetry breaking according to U(2Nf) → U(Nf )× U(Nf ). Therefore, QCD in one dimension is in the same chiral symmetry class as QCD in three dimensions. The microscopic limit of QCD in one dimension is equivalent to the microscopic limit of QCD in three dimensions [28]. Another peculiarity of QCD in one dimension is that, already in the free theory, the eigenvalues near zero are spaced inversely proportional to the volume. Therefore, in the thermodynamic limit (i.e. at zero temperature) the chiral condensate is discontinuous across the imaginary axis. However, this type of symmetry breaking is not a collective phenomenon, and there are no associated Goldstone bosons. It is reminiscent to the alternative to Goldstone’s theorem proposed by McKane and Stone [29]. Spontaneous symmetry breaking with Goldstone bosons takes place for a fixed number of lattice points in the limit Nc → ∞. A chemical potential excites color singlet excitations with baryon charge qk and mass Mk when qkµ > Mk. This is also the case for one-dimensional QCD. For gauge group U(Nc), the only color singlet excitations are mesons that are uncharged with respect to the chemical potential, whereas for SU(Nc) gauge group we have both neutral mesonic and charged baryonic color singlet excitations. The complex conjugate of the fermion determinant can be interpreted in terms of conjugate quarks [23, 30] which have a baryon charge that is opposite to that of regular quarks. Therefore, both for U(Nc) and SU(Nc), the phase quenched partition function, where the fermion determinant has been replaced by its absolute value, has charged mesonic excitations made out of quarks and conjugate anti-quarks. This results in a phase transition at µ = µc = mπ/2. The SU(Nc) theory with Nf flavors has only charged baryonic excitations and will have a phase transition at µ = µc = mB/Nc. In one dimension it turns out that the pion and the baryon have the same mass per quark number so that the critical chemical potential of the phase quenched U(Nc) theory and the SU(Nc) theory is the same. For the U(Nc) partition function we expect the sign problem to be severe when µ > µc because the phase quenched partition function has a phase transition at µ = µc whereas the normal theory remains in the same phase. For SU(Nc), both the normal partition function and the phase quenched partition function do have a phase transition to a phase of free quarks at µ = µc and have the same free energy not only for µ < µc but also for µ > µc. Therefore, in one dimension there is no severe sign problem for SU(Nc) QCD. Because one dimensional QCD with gauge group U(Nc) does not have a phase transition, the chiral condensate in the thermodynamic limit or large Nc limit is discontinuous at m = 0, independent of the value of the chemical potential. Because the Dirac eigenvalues are located on an ellipse for µ 6= 0, this discontinuity cannot be related to the Dirac spectrum by means of the Banks-Casher formula. In [31] a different mechanism to explain the discontinuity was discovered . It was found that an oscillating contribution to the spectral density with an amplitude that diverges exponentially with the volume, is responsible for the discontinuity of the chiral condensate. We will show that a similar mechanism is at work for U(Nc) gauge theory in one dimension. Lattice QCD in one dimension will be introduced in section II, where we also discuss its continuum limit, mean field results and the Conrey-Farmer-Zirnbauer formula. In section III we will evaluate the U(Nc) partition function and its phase quenched version. The average phase factor for U(Nc) is calculated in section IV. In section V we study one dimensional QCD with SU(Nc) as gauge group and evaluate the regular and phase quenched partition functions and the average phase factor. The effect of temperature will be illustrated with results from a schematic random matrix model in section VI. The connection with the microscopic domain of QCD in three dimensions is discussed in section VII, and the Dirac spectrum and its relation with chiral symmetry breaking is analyzed in section VIII. Concluding remarks are made in section IX. II. QCD IN ONE DIMENSION A. Lattice QCD The staggered lattice QCD Dirac operator in one dimension is given by mI eµU12/2 . . . e −e−µU †12/2 mI · · · 0 0 · · · mI eµUn−1n/2 −eµUn1/2 · · · −e−µU †n−1n/2 mI . (4) We have used anti-periodic boundary conditions and the gauge fields on the links are taken to be in U(Nc) or SU(Nc). The chemical potential µ is an imaginary vector field introduced according to the Hasenfratz-Karsch prescription [32]. In general, the lattice QCD partition function for Nf flavors with mass m is given by ZNf (µc, µ) = dUldet NfD e−SYM/g , (5) where the product is over the links and SYM is the Yang-Mills action. In one dimension, the partition function simplifies substantially. First, there is no Yang-Mills action, and second, by unitary transformations the integral over the links can be reduced to a single integral. In the gauge where all gauge fields except Un 1 ≡ U are equal the unity, the fermion determinant reduces to [24] detD = 2−nNc det[enµc + e−nµc + enµU + e−nµU †], (6) with µc given by µc = sinh −1m. (7) This value will turn out to be the critical value of the chemical potential. From now on, the overall factor 2−nNc will be absorbed into the normalization of the partition function. To avoid sign factors, n is taken to be even throughout this paper. The standard method to evaluate the QCD partition function in one dimension is to use the eigenvalues of U as integration variables [6, 16, 17, 18, 24]. For example, one finds this way that the result for Nf = 1 with U(Nc) as gauge group is given by [24] ZNf=1(µc, µ) = U(Nc) dU detD = sinh((Nc + 1)nµc) sinh(nµc) . (8) For most partition functions that are considered in this paper it is not possible to obtain analytical results by means of this method. Instead we will use powerful integration formulas for unitary integrals that were recently derived in [15]. These integrals are based on the color-flavor transformation [33] which has also been applied to lattice QCD with baryons in the canonical ensemble [34, 35]. B. Continuum Theory The continuum limit of the staggered lattice action is given by Dcont = m ∂0 + iA0 + µ ∂0 + iA0 + µ m , (9) where A0 is a Hermitean Nc × Nc matrix and the off-diagonal blocks connect even and odd lattice sites. What is special in one dimension is that the off-diagonal blocks are identical. The eigenvalue equation (∂0 + iA0)ψk = iukψk is solved by ψ = Pe−i A0dtχ0 with ∂0χ0 = iEχ0, (10) with Pe the path ordered exponent. Nontrivial eigenvalues are obtained by imposing boundary conditions on ψ. At nonzero chemical potential the eigenvalues of the Dirac operator are given by λk = m± (iuk + µ) with uk ∈ R. (11) This is very different from QCD with d ≥ 2 where the eigenvalues of the QCD Dirac operator at µ 6= 0 are scattered in the complex plane. Also the eigenvalues of the one dimensional staggered lattice Dirac at nonzero chemical potential are localized on a curve in the complex plane. Another consequence of the structure of the Dirac operator (9) is that the fermion determinant can be rewritten as detDcont = det(∂0 + iA0 + µ+m) det(∂0 + iA0 + µ−m), (12) which can be interpreted as a two-flavor partition function with masses m + µ and −m + µ. This Dirac operator has the same structure as QCD in three dimensions: In the large Nc limit, chiral symmetry is broken spontaneously according to U(2) → U(1) × U(1) with the squared Goldstone masses proportional to the difference of the positive and negative quark masses, i.e. m2π ∼ m− (−m) = 2m. At low energy, these Goldstone modes interact according to a chiral Lagrangian determined by the pattern of chiral symmetry breaking. In the large Nc limit and mπβ ≪ 1 (with β the length of the one-dimensional box) the QCD partition function in one dimension is therefore equivalent to the low-energy limit of QCD in three dimensions. In section VII this will be worked out explicitly for the microscopic limit of the partition function. This is the limit ρ(0) → ∞ with mπρ(0) = fixed and µπρ(0) = fixed, (13) where ρ(0) is density of the eigenvalues, or their projections onto the imaginary axis, close to zero. Since ρ(0) = nNc/2π (see section VIII), this is the limit nNc → ∞ with nNcm and nNcµ fixed. When we use the term microscopic limit, we always mean the universal microscopic limit. This is the limit (13) associated with the formation of Goldstone bosons, i.e. the limit Nc → ∞ at fixed n. C. Mean Field Limit for Large Nc In the large Nc limit, chiral symmetry is broken spontaneously even in one dimension, so that its low-energy limit is a theory of Goldstone bosons. In this section we give general arguments that determine the chemical potential dependence of the partition function in the microscopic domain where the Compton wavelength of the Goldstone bosons is much larger than β. As was argued in [4, 5], in the microscopic domain, the mean field limit of the partition function is given by Z = J mπ(µ) e−V F , (14) where J is the value of the integration measure at the saddle point, F is the free energy, V is the space time volume, and mπ are the masses of the Goldstone bosons. Let us apply this result to the average phase factor in one dimension. For µ < µc, in the limit T → 0, only the vacuum state contributes to the partition function, so that the free energy is independent of µ. For quark mass m, the equivalent QCD3 mass matrix of the Dirac operator D in (12) is given by diag(−m+µ,m+µ), whereas the hermitean conjugate Dirac operator D†, has diag(−m−µ,m−µ) as equivalent QCD3 mass matrix. For Nf flavors, the average phase factor is the partition function with Nf + 1 fermionic quarks and one conjugate bosonic quark. We thus have 2(Nf +1) 2 Goldstone bosons made out of two fermionic quarks with squared mass m2π = 2mG (with G a constant), 4(Nf +1) fermionic Goldstone modes, half of them with squared mass m2π = 2(m − µ)G, and the other half with squared mass m2π = 2(m + µ)G. Finally, we have 2 Goldstone bosons composed out of two bosonic quarks with squared mass equal to m2π = 2mG. The fermionic partition function ZNf has 2N2f Goldstone bosons all with squared mass m π = 2mG. Using (14) we thus find 〈e2iθ〉Nf = ZNf+1|1∗(µc, µ) ZNf (µc, µ) )Nf+1 for µ < µc. (15) For phase quenched partition functions with Nf flavors and Nf conjugate flavors, the equivalent QCD3 symmetry breaking pattern is U(4Nf )/(U(2Nf)× U(2Nf)), so that the total number of Goldstone bosons is equal to 8N2f . Of these, 4N2f Goldstone bosons have squared massm π = 2mG, half made out of two quarks and the other half out of two conjugate quarks. The other 4N2f Goldstone bosons are composed out of a quark and a conjugate anti-quark quark, half with squared mass m2π = 2(m−µ)G and the other half with squared mass m2π = 2(m+µ)G. All equivalent QCD3 Goldstone bosons of the full QCD partition function with 2Nf flavors have a squared mass equal to m π = 2mG. For µ < µc the free energy of the phase quenched partition function and the full QCD partition function is the same so that, using (14), the phase quenched average phase factor is given by 〈e2iθ〉Nf+N∗f = ZNf+1+(Nf−1)∗(µc, µ) Z2Nf (µc, µ) for µ < µc. (16) Below we will show that the results derived in this section also follow from the zero temperature microscopic limit of the exact evaluation of the partition function. D. The Conrey-Farmer-Zirnbauer formula Exactly the integrals that are required for the evaluation of the average phase factor of QCD at nonzero chemical potential in one dimension were studied in a recent paper by Conrey, Farmer and Zirnbauer [15]. They considered the partition function Z({ψk, φk}) = U(Nc) det(1− eψjU) det(1− eφjU) l=p+1 det(1− e−ψlU †) det(1− e−φlU †) , (17) with dU the Haar measure of U(Nc) and ψk, φk complex parameters with Re(φj) < 0 < Re(φl). Using the color flavor transformation [33] and Howe’s theory of supersymmetric dual pairs, they derived the following formula Z({ψk, φk}) = π∈Sp+q/(Sp×Sq) eNc(π(ψl)−ψl) (1− eφj−π(ψl))(1 − eπ(ψj)−φl) (1− eπ(ψj)−π(ψl))(1 − eφj−φl) . (18) The sum is over permutations in Sp+q/(Sp×Sq) that interchange any of the ψ1, · · · , ψp with any of the ψp+1, · · · , ψp+q. Partition functions with an unequal number of bosonic and fermionic determinants can be obtained from special limits of (18). In the case of only fermionic determinants, an equivalent expression was first obtained in [36, 37]. Expressions for degenerate parameters can be derived by carefully taking limits of the above formula. Unitary matrix integrals can also be calculated by using an eigenvalue representation of the unitary matrices. The integrals we are interested in are of the form U(Nc) F (eiθk), (19) where exp(iθk) are the eigenvalues of U . Using an eigenvalue representation of the unitary matrices, the orthogonal polynomial method can be used to express Z as [17] Z = det(Bk−l)k,l=0,···,Nc−1, (20) where dθeikθF ({eiθ}). (21) This is the method that was used in the literature on one-dimensional QCD prior to this paper [6, 16, 17, 18, 24]. In a few cases the determinant in (20) could be evaluated explicitly resulting in expressions that are similar to those derived directly from the CFZ-formula. We have used (20) to numerically check the results obtained by means of the CFZ-formula. III. EXACT EVALUATION OF THE ONE-DIMENSIONAL U(Nc) PARTITION FUNCTION A. Partition Function for arbitrary Nf In this section we evaluate the one-dimensional U(Nc) QCD partition function for Nf flavors. To apply the CFZ formula (18) we rewrite the determinant (6) as detD = eµcnNc det(1 + en(µ−µc)U) det(1 + en(−µ−µc)U †). (22) For arbitrary Nf we then find the remarkably simple answer ZNf (µc, µ) ≡ U(Nc) dUdetNfD = σ∈S2Nf /SNf ×SNf eNcmσ(+ k) 1− exp(mσ(− l) −mσ(+ k)) , (23) m− k = −nµc k and m+ k = nµc k. (24) The sum is over all permutations that interchange positive and negative masses. Notice that the µ-dependence has canceled from this expression. This also follows from an expansion of the determinant in powers of U and U †. Only terms with an equal number of factors U and factors U † are non-vanishing. Although the result for degenerate positive and negative masses can be obtained by carefully taking limits of (23), it is simpler to start from a different representation [36, 37] of partition function (23) given by ZNf (µc, µ) = 1≤k<l≤2Nf (eMl − eMk) 1 em− 1 · · · e(Nf−1)m− 1 e(Nc+Nf )m− 1 · · · e(Nc+2Nf−1)m− 1 m−Nf · · · e(Nf−1)m−Nf e(Nc+Nf )m−Nf · · · e(Nc+2Nf−1)m−Nf 1 em+1 · · · e(Nf−1)m+1 e(Nc+Nf )m+1 · · · e(Nc+2Nf−1)m+1 m+Nf · · · e(Nf−1)m+Nf e(Nc+Nf )m+Nf · · · e(Nc+2Nf−1)m+Nf . (25) For convenience we have introduced the mass matrix Mk = (m1 1, · · · ,m−Nf ,m+1, · · · ,m+Nf ). To obtain an expres- sion for degenerate masses, we Taylor expand the exponential functions exp(m+ k) and exp(m− k) to order Nf − 1 about exp(−nµc) and exp(nµc), respectively, and write the resulting matrix as the product of two matrices, one containing the Taylor coefficients, and the other containing powers of the expansion parameters. The determinant of the second matrix can be written as a Vandermonde determinant which cancels against part of the prefactor in (25). Our final expression for the partition function with degenerate masses is given by ZNf (µc, µ) = ∏Nf−1 k=0 k!) (enµc − e−nµc)N 1 · · · 0 1 · · · 0 e−nµc · · · e−nµc enµc · · · enµc e−2nµc · · · δNf−1− e−2nµc e2nµc · · · δ e−n(Nf−1)µc · · · δNf−1− e−n(Nf−1)µc en(Nf−1)µc · · · δ n(Nf−1)µc e−n(Nc+Nf )µc · · · δNf−1− e−n(Nc+Nf )µc en(Nc+Nf )µc · · · δ n(Nc+Nf )µc e−n(Nc+2Nf−1)µc · · · δNf−1− e−n(Nc+2Nf−1)µc en(Nc+2Nf−1)µc · · · δ n(Nc+2Nf−1)µc , (26) d(−nµc) , δ+ = d(nµc) . (27) The k’th column (with k ≤ Nf) is given by the δ(k−1)− derivative of the first column, and the Nf + k’th column is given by the δ (k−1) + derivative of the Nf ’th column. From this result one can easily derive explicit expressions for small values of Nf . The partition function for Nf = 2 reads ZNf=2(µc, µ) = (en(Nc+2)µc − e−n(Nc+2)µc)2 (enµc − e−nµc)4 − (Nc + 2) (enµc − e−nµc)2 . (28) The microscopic limit of this partition function is given by ZmicroNf=2(µc, µ) = (eNcµc − e−Ncµc)2 16µ4c . (29) For Nf = 3 the partition function is given by: ZNf=3(µc, µ) = (en(Nc+3)µc − e−n(Nc+3)µc)3 (enµc − e−nµc)9 − (Nc + 3)3 (en(Nc+1)µc − e−n(Nc+1)µc) (enµc − e−nµc)7 +Nc(Nc + 2) n(Nc+3)µc − e−n(Nc+3)µc) (enµc − e−nµc)7 (Nc + 2) 2(Nc + 3) n(Nc+3)µc − e−n(Nc+3)µc) (enµc − e−nµc)5 B. The Phase Quenched Partition Function To calculate the average phase factor according to the definition (2) for Nf = 2, we also need the one-dimensional phase quenched QCD partition function for two flavors which will be evaluated in this subsection. It is defined by Z1+1∗(µc, µ) = U∈U(Nc) dU detD detD† (32) = e2µnNc U∈U(Nc) dU det(1− en(µ−µc)U) det(1 − en(−µ−µc)U †) det(1− en(µ−µc)U †) det(1 − en(−µ−µc)U). This partition function is easily evaluated using the integration formulae of [15]. We find Z1+1∗(µc, µ) = cosh(2n(Nc + 2)µc) 8 sinh(n(µc − µ)) sinh(n(µ+ µc)) sinh2(nµc) cosh(2n(Nc + 2)µ)) 8 sinh(n(µ− µc)) sinh(n(µ+ µc)) sinh2(nµ) 8 sinh2(nµ) sinh2(nµc) . (33) For µ = 0 this result agrees with the Nf = 2 partition function given in (28). Its microscopic limit simplifies to Zmicro1+1∗ (µc, µ) = e2nNcµc + e−2nNcµc 16n4(µ2c − µ2)µ2c 16n4µ2µ2c 2nNcµ + e−2nNcµ 16n4(µ2c − µ2)µ2 . (34) In section VII we will show that this result is equal to the microscopic limit of the QCD3 partition function with masses −(µc + µ), −(µc − µ), µc − µ, µc + µ. The large Nc limit of the phase quenched partition function is given by Z1+1∗(µc, µ) = e2n(Nc+2)µc 16 sinh(n(µc − µ)) sinh(n(µ+ µc))(sinh(nµc))2 for µ < µc, Z1+1∗(µc, µ) = e2n(Nc+2)µ 16 sinh(n(µ− µc)) sinh(n(µ+ µc))(sinh(nµ))2 for µ > µc, Z1+1∗(µc, µ) = 2n(Nc+2)µc 8 sinh2(nµc) sinh(2nµc) 1 +O(N−1c ) for µ = µc, (35) which will be used to calculate the phase quenched average phase factor in this limit. IV. AVERAGE PHASE FACTOR FOR U(Nc) In this section we will evaluate the average phase factor, first from the ratio of the full QCD partition function and the phase quenched partition function and in the next subsection starting from the definition in Eq. (3). A. Phase Quenched Average Phase Factor For Nf = 2, the average phase factor with the absolute value of the fermion determinant as weight is given by 〈e2iθ〉pq = 〈e2iθ| det(D)|2〉 〈| det(D)|2〉 ZNf=2(µc, µ) Z1+1∗(µc, µ) . (36) The two-flavor partition function was evaluated in subsection III A, and the phase quenched partition function was calculated in subsection III B. The large Nc limit of the phase quenched average phase factor is given by 〈e2iθ〉pq = sinh(n(µc−µ)) sinh(n(µc+µ)) sinh2(nµc) for µ < µc, sinh2(nµ) sinh(n(µ−µc)) sinh(n(µ+µc)) sinh4(nµc) e−2n(Nc+2)(µ−µc) for µ > µc, 2 cosh(nµc) Nc sinh(nµc) for µ = µc. The large µcNc, µNc limit of the microscopic phase quenched partition function (34) for µ < µc is given by Zpq(µc, µ) = e2nNcµc 16n4(µ2c − µ2)µ2c , (38) whereas the two-flavor partition function in this limit reads ZNf=2(µc, µ) = e2nNcµc 16n4µ4c , (39) so that the average phase factor simplifies to 〈e2iθ〉pq = 1− for µ < µc. (40) This result is in agreement with the mean field result (16). Indeed, the denominator of (38) is the product of the Goldstone masses µc−µ, µc+µ, µc and µc, whereas the denominator of (40) is the product of four Goldstone masses µc. The free energy of both partition functions is equal to 2nNcµc. For µ > µc, the large Nc limit of the two flavor partition function is still given by (39), but the large Ncµ limit of the microscopic phase quenched partition function is now given by Zpq = e2nNcµ 16n4(µ2 − µ2c)µ2 , (41) resulting in the average phase factor 〈e2iθ〉pq = ZNf=2 − 1)e−2nNc(µ−µc) for µ > µc . (42) This result can also be obtained from the small nµ, nµc expansion of the large Nc limit of the average phase factor (see Eq. (37)) at fixed nNcµc. We conclude that the sign problem becomes exponentially hard for µ > µc. In Fig. 1 we show the average phase factor for different values of n and Nc and a critical chemical potential of µc = 0.1. In the thermodynamic limit at fixed Nc, the average phase factor is one for µ < µc and jumps to zero for µ > µc. In the right figure we observe a rapid convergence to the mean field result discussed in section II C. B. Average phase factor for full QCD The average phase factor with the Nf -flavor quark determinant as weight is defined by 〈e2iθ〉Nf = ZNf (µc, µ) detD† detNfD enNfNcµc ZNf (µc, µ) det(1− Uenµ−nµc) det(1− U †e−nµ−nµc) det(1− Ue−nµ−nµc) det(1− U †enµ−nµc) detNf (1 − Uenµ−nµc)detNf (1− U †e−nµ−nµc). FIG. 1: The average phase factor for one-dimensional QCD with gauge group U(Nc) calculated as the ratio of the two flavor flavor partition function and the phase quenched partition function. In the left figure the number of lattice points varies as indicated for fixed Nc = 3, and in the right figure the number of colors varies as indicated for fixed n = 4. We evaluate this integral exactly for Nf = 0 and Nf = 1 but only give its large Nc limit for other values of Nf . Using the CFZ-formula (18) [15] we obtain in the quenched case 〈e2iθ〉Nf=0 = (1− e−2n(µ+µc))(1− e2n(µ−µc)) (1− e−2nµc)2 + e−2nNcµc (1 − e−2nµ)(1− e2nµ) (1 − e2nµc)(1 − e−2nµc) for µ < µc . (43) The large Nc limit of this result coincides with the large Nc limit of the phase quenched average phase factor. In the microscopic limit where µcnNc and µnNc remain fixed for Nc → ∞ we obtain 〈e2iθ〉Nf=0 = 1− e−2nNcµc . (44) In the large nNcµc limit the second term does not contribute resulting in 〈e2iθ〉Nf=0 = 1− , (45) in agreement with the results obtained in [4] (see subsection II C). In the thermodynamic limit at fixed Nc the average phase factor converges to one for µ < µc. After rewriting the determinants in (43) as det(1− U †e−nµ−nµc) det(1− U †enµ−nµc) → e−2nNcµ det(1− Ue nµ+nµc) det(1− Ue−nµ+nµc) , (46) the denominator of (43) can be expanded in powers of U for µ > µc. For Nf = 0 only the constant term in the integrand yields a non-vanishing result. We thus find 〈e2iθ〉Nf=0 = e−2nNcµ for µ > µc, (47) so that the average phase factor vanishes in the large nNcµ limit. In Fig. 2 we show the quenched average phase factor for various values of n and Nc. Also in this case the average phase factor jumps from 1 to 0 at µ = µc in the thermodynamic limit. The convergence to the mean field result for increasing Nc (right figure) is very rapid. This can be understood from the expansion of the microscopic result given by 1− (µ2/µ2c)(1− 13 (nµ) 2 + · · ·. FIG. 2: The quenched average phase factor for one-dimensional QCD with gauge group U(Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4. Finally we calculate the average phase factor for an arbitrary number of flavors in the large Nc limit. The CFZ formula can again be applied after inserting the factor detNf (1− Ue−α)detNf (1− U †e−β) in the denominator and taking the limit α → ∞ and β → ∞ at the end of the calculation. For µ < µc, the leading order large Nc result is given by the identity permutation in the sum over permutations in the CFZ formula. This results in 〈e2iθ〉Nf = (1− e−2n(µ+µc))(1− e2n(µ−µc)) (1− e−2nµc)2 )Nf+1 for µ < µc. (48) The microscopic limit of this result is given by 〈e2iθ〉Nf = ]Nf+1 for µ < µc, (49) in agreement with the discussion given in section II C. For µ > µc the CFZ formula can again be applied after rewriting the partition function according to (46) and the insertion (48). Because of degenerate prefactors, all critical chemical potentials that occur in the combination µ+ µc have to be taken different. After carefully taking limits we find ZNf+1|1∗(µc, µ) = e−2µnNc eµcnNcNf (1− e−2µcn)N (1− e−2n(µ+µc))Nf (1− e−2nµ)Nf (1− e−2nµc)Nf . (50) The average phase factor given by 〈e2iθ〉Nf = ZNf+1|1∗ ∼ e−2nNcµ for µ > µc (51) vanishes in the thermodynamic limit. The exact result becomes rather cumbersome even for small values of Nf . As an illustration we give the result for Z2|1∗(µc, µ) in Appendix A . V. AVERAGE PHASE FACTOR FOR SU(Nc) To calculate integrals over SU(Nc) we use the identity SU(Nc) dU · · · = U(Nc) dUdetp(U) · · · . (52) For the partition functions discussed below, with a few exceptions the sum truncates to a small number of terms. A. SU(Nc) QCD Partition Function Let us first calculate the 1d SU(Nc) QCD partition function for Nf = 1. The partition function is defined by SU(Nc) (µc, µ) = SU(Nc) dUdetNfD = U(Nc) dUdetpUenNcµcdetNf (1 + Uenµ−nµc)detNf (1 + U †e−nµ−nµc). (53) For p ≥ Nf the integrand can be rewritten in terms of determinant of matrices U only ZNf ,p(µc, µ) = U(Nc) dUdetpUdetNfD U(Nc) dUenNcNfµcdetp−Nf (U)detNf (1 + Uenµ−nµc)detNf (U + e−nµ−nµc) . (54) For p > Nf it follows immediately that the integral vanishes. For p = Nf only the constant term inside the determinants contributes to integral resulting in ZNf ,p=Nf (µc, µ) = e −nNfNcµ, ZNf ,p>Nf (µc, µ) = 0. (55) For p < 0 we combine detpU = det−pU † with the factor detNf (1− Uenµ−nµc). We then find ZNf ,p=−Nf (µc, µ) = e nNfNcµ, ZNf ,p<−Nf (µc, µ) = 0. (56) Using that the hermitean conjugate of a unitary matrix is also unitary we obtain ZNf ,−p(µc, µ) = ZNf ,p(µc,−µ). (57) For |p| < Nf the integral can be calculated by means of the CFZ formula by choosing all µc different and introducing the limit detU = lim det(U − e−α). (58) The limit of degenerate µc and α → ∞ is taken at the end of the calculation. We will only give exact results for Nf = 1 and Nf = 2. Using (55,56) and the result for U(Nc) we obtain for Nf = 1 SU(Nc) (µc, µ) = e −nNfNfµ + enNfNfµ + sinh(n(Nc + 1)µc) sinh(nµc) , (59) in agreement with earlier work by Bilic and Demeterfi [24]. For Nf = 2, the only integrals that have not yet been calculated are those for p = ±1 which are related by (57). For the p = 1 contribution we find ZNf=2,p=1(µc, µ) = e nNc(µc−µ) e−2n(Nc+1)µc + e2nµc (enµc − e−nµc)2 −2nNcµc(e−3nµc − 3e−nµc)− e3nµc + 3enµc (enµc − e−nµc)3 . (60) For µ < µc the large nNcµ limit of the Nf = 2 partition function is dominated by the p = 0 term, whereas for µ > µc the large nNcµc limit is given by the p = −2 term. For µ = µc the large Nc limit is dominated by the p = −1 with 1/Nc corrections from the p = 0 and p = −2 terms. We thus find as leading large Nc result SU(Nc) (µc, µ) = e2n(Nc+2)µc 16 sinh4(nµc) for µ < µc, SU(Nc) (µc, µ) = e 2nNcµ for µ > µc, SU(Nc) (µc, µ) = 2n(Nc+1)µc 4 sinh2(nµc) 1 +O(N−1c ) for µ = µc. (61) Both for µ < µc and µ > µc the terms that are canceled by the integration over the unitary group are subleading in the thermodynamic limit. In other words there is no serious sign problem. By inspection one can easily show that the dominance of the p = 0 terms for µ < µc and the p = −Nf term for µ > µc is a feature of the large Nc limit that is valid for any number of flavors. Therefore, for µ < µc, the chiral condensate is the same as for the U(Nc) theory. For µ > µc though, the dominant p = −Nf term is mass-independent. This results in a vanishing chiral condensate so that the large Nc limit of the U(Nc) theory and the SU(Nc) theory is different. B. The Phase Quenched Partition Function for SU(Nc) A second ingredient for the average phase factor is the phase quenched SU(Nc) partition function. Using the same arguments as for the full QCD partition function, one easily obtains Z1+1∗,|p|>2(µc, µ) = 0, Z1+1∗,|p|=2(µc, µ) = 1, Z1+1∗,−p(µc, µ) = Z1+1∗,p(µc,−µ). (62) The partition function for p = 0 is the U(Nc) partition function which was already given in (33). What remains to be calculated is the partition function for p = +1. Using the CFZ formula one easily arrives at Z1+1∗,p=1(µc, µ) = 4 sinh(nµ) sinh(nµc) sinh(n(Nc + 2)(µ+ µc)) sinh(n(µ+ µc)) − sinh(n(Nc + 2)(µ− µc)) sinh(n(µ− µc))) . (63) Both for µ < µc and µ > µc, the leading large Nc result of the phase quenched partition function resides in the p = 0 term given in Eq. (35). C. Average Phase Factor for SU(Nc) The average phase factor can be either obtained from the ratio of the full QCD partition function and the phase quenched partition function or can be calculated with the full QCD partition function as weight (see Eqs. (2) and (3)). In the first case we find for large Nc, 〈e2iθ〉SU(Nc)pq = sinh(n(µc−µ)) sinh(n(µc+µ)) sinh2(nµc) for µ < µc, 16 sinh(n(µ− µc)) sinh(n(µc + µ)) sinh2(nµ) for µ > µc, 1− e−4nµc for µ = µc. The microscopic limit of the average phase factor is obtained by expanding this result for small µc, µ. We again find the usual mean field result 〈e2iθ〉SU(Nc)pq = θ(µc − µ). (65) In the thermodynamic limit, the average phase factor converges to one in both for µ < µc and µ > µc. FIG. 3: The phase quenched average phase factor for one-dimensional QCD with gauge group SU(Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4. Exact results for the average phase factor are displayed in Fig. 3. At finite Nc, we observe a rapid convergence to the asymptotic value of 1. At fixed n, the approach to the large Nc limit is slow. Also clearly visible is that for µ ≈ µc the corrections to the microscopic limit are large. As was already discussed before, this is due to the p = −1 contribution. In the quenched case the average phase factor for SU(Nc) defined according to Eq. (3) can be written as 〈e2iθ〉SU(Nc)Nf=0 = U∈U(Nc) dUdetp(U) det(1 + Uenµ−nµc) det(1 + U †e−nµ−nµc) det(1 + Ue−nµ−nµc) det(1 + U †enµ−nµc) . (66) The integrals can again be calculated by means of the CFZ formula. For Nc ≥ 3 one can easily show that the limit introduced in (58) gives vanishing results for |p| > 2. However, for Nc = 1 and Nc = 2 additional terms may contribute to this limit. For Nc = 2, where the average phase factor is equal to one, contributions for |p| > 2 vanish for µ < µc, but terms with p < −2 are nonzero for µ > µc. The formulae for |p| ≤ 2 given below are valid for Nc = 2. For Nc = 1 the integrals in (66) only vanish for p ≥ 2 and µ > µc. Also the general answer for p = −2 is not correct in this case. For p = −1 we obtain 〈e2iθ〉Nf=0 = e−Ncn(µc−µ) (1−e−2n(µc+µ))(1−e−2nµ) 1−e−2nµc for µ < µc, enNc(µc−µ)(1−e−2nµ) 1−e−2nµc [(1− e−2nµc−2nµ)− e−2nNcµc(e−2nµc − e−2nµ)] for µ > µc. For p = 1 we find 〈e2iθ〉Nf=0 = e−Ncn(µc+µ) (1−e−2n(µc−µ))(1−e2nµ) 1−e−2nµc for µ < µc, 0 for µ > µc. Finally, for p = −2 the result is 〈e2iθ〉Nf=0 = 0 for µ < µc, (1− e−2nµ)(1 − e−2n(µ+µc))(1 − e2n(µc−µ))(1 − e−2nµ) for µ > µc. For p = 2 the average phase factor vanishes. Notice that the relation 〈e2iθ〉Nf=0,−p(µc, µ) = 〈e2iθ〉Nf=0,p(µc,−µ) (70) FIG. 4: Average phase factor for one-dimensional QCD with gauge group SU(Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4. is also valid in this case. It can be used to derive results for negative values of µ. In the large Nc limit, the average phase factor is dominated by the p = 0 contribution for µ < µc and by the p = −2 contribution for µ > µc. For µ = µc, the p = 0, p = −1 and p = −2 contributions are of equal order in Nc, but only the p = −1 term is nonvanishing. As large Nc limit of the average phase factor we thus find 〈e2iθ〉SU(Nc)Nf=0 = sinh(n(µc−µ)) sinh(n(µc+µ)) sinh2(nµc) for µ < µc, 16 sinh(n(µ− µc)) sinh(n(µc + µ)) sinh2(nµ) for µ > µc, 1− e−4nµc for µ = µc, which is the same expression as obtained in (64) for the phase quenched average phase factor. The microscopic limit of the average phase factor is again given by the small µ-µc expansion of this result which is equal to the usual mean field result 〈e2iθ〉SU(Nc)Nf=0 = θ(µc − µ). (72) In Fig. 4 we show the exact result for the SU(Nc) average phase factor. We observe a rapid approach to the thermodynamic limit at fixed Nc just like in Fig. 3. In the large Nc limit the result for the quenched average phase calculated according to (66) is also given by (64). As already was argued in the introduction, we conclude that QCD in one dimension with SU(Nc) as gauge group does not have a serious sign problem, not only for µ < µc, but also for µ > µc. Indeed one-dimensional QCD at nonzero chemical potential could be simulated reliably by the Glasgow method [26]. VI. THE SIGN PROBLEM AT NONZERO TEMPERATURE The results of the previous two sections show that the sign problem is severe for QCD in one dimension with gauge group U(Nc) and µ > µc. Although this theory does not have a critical temperature, it is clear from Fig. 1 that the average phase factor for µ > µc increases significantly for higher temperatures (i.e. for lower values of n). However, this is a geometric effect due to the Boltzmann factor. Recent lattice simulations [10, 11, 12, 26, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] suggest that the sign problem becomes much milder for T around Tc. Because of the absence of a critical temperature, this temperature dependence of the average phase factor cannot be investigated in FIG. 5: Scatter plot of the eigenvalues of the random matrix Dirac operator for µ =0.3 and T = 0, 0.7, 0.9, 1.0, 1.1 . The quark mass at m = 0.1 is indicated by the black dot. one dimensional QCD. Instead we study this problem in a schematic random matrix model at nonzero temperature and chemical potential [30, 48, 49]. This model has a severe sign problem as well as a critical temperature and its eigenvalues are scattered in a finite domain of the complex plane, just like in QCD. The quenched average phase factor in this model is given by the matrix integral 〈e2iθ〉(m,µ, T ) = dCe−NTrCC † det(D +m) det(D† +m) , (73) with Dirac operator given by 0 iC + µ+ it iC† + µ+ it 0 , (74) and t is the traceless diagonal matrix t = diag(−T, · · · ,−T, T, · · · , T ). The integral is over the real and imaginary parts of the matrix elements of the complex N ×N matrices C. The phase quenched average phase factor is defined 〈e2iθ〉pq = dCdet2(D +m)e−NTrCC dCdet(D +m) det(D† +m)e−NTrCC If at all possible, it certainly takes a significant effort to evaluate these integrals analytically. Therefore we have studied the quenched (73) and phase quenched (75) average phase factor numerically. In Fig. 5, we give scatter plots of the eigenvalues of the random matrix Dirac operator. For each of the temperatures T = 0, 0.7, 0.9, 1, 1.1, the figures show results of 40 matrices with N = 100 and µ = 0.3. The black dot represents the quark mass for which the average phase factor is calculated. In Fig. 6 we show the µ-dependence of the phase quenched (left) and quenched (right) average phase factor obtained by averaging over 10,000 to 80,000 matrices C with N = 100. The results for the phase quenched average phase factor for T = 0 and T = 0.7 and µ > µc are not displayed because they do not converge because of the severity of the sign problem. The average phase factor depends only weakly on the size of FIG. 6: Phase quenched (left) and quenched (right) average phase factor versus chemical potential. Results have been obtained from the chiral random matrix model (74). the matrices. The conclusion of Figs. 5 and 6 is that the sign problem is not serious when the quark mass is outside the domain of the eigenvalues. This confirms the conjecture that was made in [4, 5, 50]. It also agrees with the observation in [30] that the free energy of the full theory and the phase quenched theory are the same for T = 0 if the quark mass is outside the domain of the eigenvalues (this is the case for µ < µc). The µ-dependence is due to the curvatures at the saddle point. VII. COMPARISON WITH THE QCD3 PARTITION FUNCTION According to the general arguments given in section II B, the microscopic limit of the one dimensional QCD partition function at nonzero chemical potential is equal to the microscopic limit of the QCD3 partition function at zero chemical potential, but with shifted quark masses. In this section we show this explicitly for some of the results derived before. We first discuss the microscopic limit of the QCD3 partition function which can be derived from either a chiral Lagrangian or a random matrix model with the same global symmetries [28]. For a discussion of QCD3 in terms of chiral Lagrangians we refer to [51]. A. Random Matrix Model From the continuum Dirac operator (9) it follows that the random matrix model that describes the fluctuations of the low-lying Dirac eigenvalues is given by dHP (H) det(iH + µ+m) det(iH −m+ µ), (76) where the probability distribution can conveniently be taken to be the Gaussian distribution P (H) = e−2NΣ 2TrH2 . (77) Such random matrix partition functions have been studied elaborately in the literature to analyze the microscopic limit of QCD in three dimensions [28, 52, 53, 54, 55]. The microscopic limit of the partition function (76) with quark masses given by M = diag(−m1, · · · ,−mNf ,m1, · · · ,mNf ) is equal to [53, 54] ZQCD3 = A(m) A(−m) A(−m) A(m) , (78) where the matrix elements of the Nf ×Nf matrix A(m) are given by A(m)kl = m −mk , (79) and ∆ is the Vandermonde determinant ∆(M) = (Mkk −Mll). (80) Although general expressions for QCD3 partition functions with an arbitrary number of bosonic and fermionic determinants are also known [56], for our purposes we only need the partition function ZNf+2|2 = det(D +mk) det(D −mk) det(D + z1) det(D − z2) det(D + z̄1) det(D − z̄2) . (81) This partition function was evaluated in [57] by means of the supersymmetric method and is given by ZNf+2|2 = (z1 − z̄1)(z2 − z̄2) ∆2Nf (m) expNc(z̄1 + z̄2) Nc(z̄1 + z̄2) (mk − z̄1)(mk − z̄2) (mk − z1)(mk − z2) sinhNc(mk+ml) Nc(mk+ml) sinhNc(mk+z1) Nc(mk+z1) sinhNc(z2+ml) Nc(z2+ml) sinhNc(z1+z2) Nc(z1+z2) e−Nc(z1−z̄1)+Nc(z2−z̄2) ∆2Nf (m) sinhNc(mk +ml) Nc(mk +ml) . (82) The last term is a so-called Efetov-Wegner term, and the Vandermonde determinant is over positive masses only ∆(m) = (mk −ml). (83) B. Microscopic Limit We will now show that the microscopic limit of the partition function (25) is equal to the microscopic limit of the QCD3 partition function. To this end we multiply row k of the first Nf rows of the determinant by exp(−Ncm− k/2) and row k of the second Nf rows of the determinant in (25) by exp(−Ncm+ k/2). In the microscopic limit we keep Ncm±,k fixed so that we can expand the masses that do not occur in this combination. By subtracting successive columns starting with the first one we obtain ZNf = 1≤k<l≤2Nf (Ml −Mk) e−Ncm− 1/2 m− 1e −Ncm− 1/2 · · · mNf−1− 1 e−Ncm− 1/2 m +Ncm− 1/2 · · · m2Nf−1− 1 eNcm− 1/2 −Ncm−Nf /2 m−Nf e −Ncm−Nf /2 · · · mNf−1−Nf e −Ncm−Nf /2 m +Ncm−Nf /2 · · · m2Nf−1−Nf e Ncm−Nf /2 e−Ncm+1/2 m+1e −Ncm+1/2 · · · mNf−1+ 1 e−Ncm+1/2 m +Ncm+1/2 · · · m2Nf−1+1 eNcm+1/2 −Ncm+Nf /2 m+Nf e −Ncm+Nf /2 · · · mNf−1+Nf e −Ncm+Nf /2 m +Ncm+Nf /2 · · · m2Nf−1+Nf e Ncm+Nf /2 where the masses MK are defined below Eq. (25). By multiplying the columns l = Nf + 1, · · · , 2Nf by (−1)l−1 and introducing microscopic masses, we obtain exactly the expression for the QCD3 partition function given in (78). For masses −(µc + µ), −(µc − µ), µc − µ, µc + µ corresponding to the microscopic limit of the two-flavor phase quenched partition function given in (34) the determinant in (84) is given by 4nµ2(e2nNcµc + e−2nNcµc)− 4nµ2c(e2nNcµ + e−2nNcµ) + 8n(µ2c − µ2). (85) and the prefactor is equal to 1≤k<l≤2Nf (Ml −Mk) = 64n6(µ2c − µ2)µ2cµ2. (86) FIG. 7: The number variance for n = 4 and Nc = 100. Their ratio coincides with the microscopic phase quenched partition function given in (34). As second example, we consider the quenched average phase factor for U(Nc) given in (43). For Nf = 0 the partition function (82) simplifies to Z2|2 = (z1 − z̄1)(z2 − z̄2) eNc(z̄1+z̄2) Nc(z̄1 + z̄2) sinhNc(z1 + z2) Nc(z1 + z2) + e−Nc(z1−z̄1−z2+z̄2). (87) The masses in this partition function corresponding to (43) are given by z1 = nµc + nµ, z2 = nµc − nµ, z̄2 = nµ− nµc, z̄1 = −nµ− nµc. (88) Substituting them into (87) we obtain Z2|2 = 1− e−4Ncnµ (89) which is exactly the microscopic limit of the quenched average phase factor (43) given in Eq. (44). VIII. DIRAC SPECTRUM We consider the Dirac operator (4) in a gauge where all gauge fields except Un1 ≡ U and U1,n = U † are equal to unity. If the eigenvalues of U are equal to exp(iθk) the eigenvalues of the Dirac operator are given by λk,l = 2πi(k+1/2)+iθl +µ − e− 2πi(k+1/2)−iθl −µ), k = 1, · · · , n, l = 1, · · · , Nc. (90) Contrary to QCD at µ 6= 0 in more than one dimension, where the eigenvalues are scattered in the complex plane [27, 30, 58, 59, 60, 61], the eigenvalues are located on an ellipse in the complex plane with real and imaginary parts related by Re(λk,l) eµ − e−µ Im(λk,l) eµ + e−µ = 1. (91) A. Universal Fluctuations In section II B we have shown that the continuum limit of the staggered Dirac operator in one dimension is in the same universality class as QCD in three dimensions. The continuum limit of the staggered three dimensional Dirac operator, though, is in the chiral symmetry universality class of QCD in four dimensions. For example, in [62] this was found for the distribution of the small Dirac eigenvalues. The reason for the chiral structure is that the staggered Dirac operator only couples even and odd lattice sites. This is also the case in one dimension, but the off-diagonal blocks, containing the gauge fields, are the same in the continuum limit (or occur in the combination Un + U †n), resulting in a two flavor theory with opposite masses (see (12)). The secular equation is given by det[(∂0 + iA0 + µ+ λ)(∂0 + iA0 + µ− λ)] = 0. (92) This corresponds to the superposition of the spectrum of ∂0 +A0 + µ and −∂0 − iA0 − µ. In the domain of the Dirac spectrum where ∂0 can be neglected, the eigenvalues on each of the lines ±µ are given by the Hermitean random matrix ensemble A0. For µ = 0 we have the superposition of the Hermitean random matrix ensembles of A0 and −A0. From the eigenvalues (90) it is clear that the staggered lattice Dirac eigenvalues show a similar superposition. One ensemble is localized on the left half of the ellipse and the other ensemble on its right half. Superpostions of more ensembles arise for correlations on scales larger than Nc level spacings. Eigenvalues of U(Nc) matrices are correlated according to the circular unitary ensemble. Therefore, the eigenvalues λk,l are correlated according the Gaussian Unitary Ensemble (GUE) on a scale where the variations in the average spectral density can be neglected, i.e. on a scale much less than Nc level spacings. By correcting for the variation of the average level spacing, the scale on which universal random matrix correlations are found can be extended, but this scale should always remain well below Nc level spacings. Then we are necessarily within one of the n successive copies of the eigenvalues along the ellipse. At a scale of a finite number of average level spacings, we expect convergence to universal random matrix correlations in the large Nc limit. As measure of the correlations between eigenvalues, we use the number variance Σ2(n̄) defined as the variance on the number of eigenvalues in an interval that contains n̄ eigenvalues on average. We calculate the number variance for intervals starting from zero. Because of the superposition of an enemble and its negative, this is equal to the number variance of a single ensemble for an interval that is symmetric about zero which is given by the GUE result. In Fig. 7 we show the number variance obtained from an average over 105 gauge field configurations for Nc = 100 and µ = 0. The number variance is calculated for an interval starting at zero. The discrepancy between the analytical result and the GUE (solid curve) is only barely visible. The increase of the domain of validity of chiral random matrix theory with increasing Nc was also observed for Dirac operator of QCD in four dimensions [63]. This can be explained as follows: The number of eigenvalues that is described by random matrix theory is equal to ≈ F 2π V (with V the space-time volume), but F 2π scales with Nc in the large Nc limit. B. Chiral Symmetry Breaking at µ 6= 0 We first discuss the case of µ = 0. Then the average spacing of the eigenvalues scales as 1/(nNc). This means that the chiral condensate develops a discontinuity at m = 0 when nNc → ∞. Two cases of interest where a nonzero chiral condensate can be obtained are the limit n → ∞ for fixed Nc, i.e. at zero temperature, or at fixed n in the limit Nc → ∞. In both cases the chiral condensate is discontinuous for nNc → ∞ when the quark mass crosses the imaginary axis where the eigenvalues are located. However, Goldstone bosons and universal behavior is only found in the large Nc limit. At fixed Nc, chiral symmetry breaking is not associated with universal behavior. For µ 6= 0 the Dirac eigenvalues are located on the ellipse (91). When the quark mass is inside the ellipse of eigenvalues the chiral condensate is zero for nNc → ∞ for the SU(Nc) partition function and the phase quenched U(Nc) partition function [24]. For gauge group U(Nc) with Nf ≥ 1 flavors, though, the partition function is µ- independent so that the chiral condensate is also nonzero if the mass is inside the eigenvalue ellipse. The Banks-Casher formula fails in this case which is known as the “Silver Blaze Problem” [64]. The resolution is the same as for QCD in four dimensions [31]: the re-weighted eigenvalue distribution defined as ρfull(z) = U(Nc) dUdetNfD δ2(z − λk) (93) shows oscillations with an amplitude that diverges exponentially with n and a period that is proportional to 1/n. Below we will illustrate this for U(1). For Nc = 1 one easily derives that the spectral density is given by ρfull(z) = 4 enµc + e−nµc − en(iα+µ) − e−n(iα+µ) (enµc + e−nµc)(e2r + e−2r + e2iα + e−2iα) δ(r − µ), (94) where z is parameterized as (er+iα − e−r−iα), r > 0, α ∈ [0, 2π], (95) and we have used that ZNf=1 = 2 cosh(nµc) (see eq. (8)). For µ > µc this spectral density has oscillations with an amplitude that diverges exponentially with n and a period of order 1/n which are the essential properties of the spectral density of the Dirac operator of QCD in four dimensions [31, 65, 66]. It can be decomposed as ρfull(z) = ρq(z) + ρosc(z). (96) The average quenched spectral density is defined by ρq(z) = δ2(z − λk) (97) with the eigenvalues λk given by Eq. (90). After changing variables according to (95) and integrating over θ we find ρq(z) = (e2r + e−2r + e2iα + e−2iα) δ(r − µ). (98) The oscillatory part of the spectral density, ρosc(z) is equal to the difference ρfull(z)− ρq(z). The chiral condensate is given by Σfull(m) = ρfull(z) , (99) and can also be decomposed as Σfull(m) = Σq(m) + Σosc(m). (100) The Jacobian for the transformation of d2z to drdα is given by d2z = (e2r + e−2r + e2iα + e−2iα)drdα, (101) so that the chiral condensate after integration over r can be simplified to Σfull(m) = enµc + e−nµc − en(iα+µ) − e−n(iα+µ) (enµc + e−nµc)(eµ+iα − e−µ−iα − 2m) tanh(nµc) coshµc . (102) This result remains finite for n → ∞. We remind the reader that the mass is parameterized as m = sinhµc. This expression represents the resolvent at the quark mass which was evaluated in [49] and is in agreement with earlier work [6, 24]. Decomposing the spectral density and the chiral condensate according to (96) we obtain Σosc(m) = θ(sinh(µ)−m)Σfull(m), Σq(m) = θ(m− sinh(µ))Σfull(m), (103) so that when the quark mass is inside the ellipse of eigenvalues, the entire chiral condensate is due to the oscillatory part of the spectral density. The discontinuity in the chiral condensate is reminiscent to a Stokes phenomenon. The alternative to the Banks-Casher relation proposed in [31] is also at work for QCD in one dimension. This solves the “Silver Blaze Problem” [64]. IX. CONCLUSIONS We have studied QCD in one dimension at nonzero chemical potential. Both the full theory, its quenched and phase quenched versions and gauge groups U(Nc) and SU(Nc) have been considered. In one dimension, the QCD or QCD-like partition functions can be reduced to a single matrix integral, which because of recent advances by Conrey, Farmer and Zirnbauer, could be evaluated analytically. In this paper we have analyzed the small mass behavior of the partition function, the nature of the sign problem, and the relation between the Dirac spectrum and chiral symmetry breaking. To put our results in perspective, we emphasize that QCD in one dimension, is quite different from QCD in more dimensions. In particular, we wish to mention the following three points. First, instead of being scattered in the complex plane, the eigenvalues of the staggered Dirac operator are located on an ellipse in the complex plane. Second, phase transitions can only take place for zero temperature or for Nc → ∞. At the critical chemical potential, a transition from the vacuum state to a state of free quarks takes place. Third, color singlets are made out of noninteracting quarks so that the critical chemical potential is given by the quark mass. In particular, the critical chemical potential for the meson state and the baryon state is the same. Because the Dirac eigenvalues are located on a curve, the large Nc limit of staggered lattice QCD in one dimension is in the same chiral symmetry class as QCD in three dimensions. We have shown this both by an explicit evaluation of the partition function, and by analyzing the fluctuations of the Dirac eigenvalues for large Nc. We have used this equivalence to explain the behavior of the partition functions and the average phase factor in the microscopic limit. Contrary to QCD in four dimensions, the sign problem for QCD in one dimension is not severe in the case of gauge group SU(Nc). One reason is that the critical chemical potential for full QCD and phase quenched QCD is the same, so that the parameter domain mπ/2 < µ < mN/3, where the sign problem becomes severe in four dimensions, is absent in one dimension. A second reason is that for µ > µc both the full theory and the phase quenched theory become a theory of free quarks with the same free energy in the thermodynamic limit. For gauge group U(Nc), on the other hand, the sign problem is severe when µ > mπ/2 both in one dimension and in four dimensions. The reason is that the U(Nc) theory does not have charged excitations, whereas the phase quenched theory has charged mesons resulting in a phase transition at µ = mπ/2. We have evaluated the average phase factor both by averaging with respect to the phase quenched partition function and the full partition function, and similar conclusions have been reached. The condititon µ > mπ/2 for having a severe sign problem can be rephrased as the quark mass being inside the ellipse of eigenvalues. In more dimensions this condition is that the quark mass is inside the support of the Dirac spectrum. It also applies to nonzero temperature as we have demonstrated explicitly in the framework of a chiral random matrix model. Also for U(Nc) QCD in one dimension, the chiral condensate is discontinuous across the imaginary axis in spite of the fact that there are no Dirac eigenvalues. This can only mean that the phase of the fermion determinant is responsible for the discontinuity. This is what happens in four dimensions where the discontinuity is due to a contribution to the spectral density that oscillates with a period of the inverse volume and amplitude that diverges exponentially with the volume. Exactly the same mechanism, where the discontinuity in the chiral condensate arises due to a Stokes like phenomenon, is at work in one dimension. This suggests that this is a universal mechanism for theories with a sign problem. We end by repeating that QCD in one dimension with SU(Nc) as gauge group has no serious sign problem. Our hope is that part of this conclusion translates to four dimensions ameliorating the sign problem for µ > MN/3. Acknowledgments. We wish to thank K. Splittorff for numerous comments and criticism and P.H. Damgaard, H. Neuberger and R.Pisarski are thanked for valuable discussions. K. Splittorff is also thanked for a careful reading of the manuscript. This work was supported by U.S. DOE Grant No. DE-FG-88ER40388 (JV), the Angelo Della Riccia Foundation (LR), the Villum Kann Rassmussen Foundation (JV) and the Danish National Bank (JV). APPENDIX A: AVERAGE U(Nc) PHASE FACTOR FOR Nf = 1 In this appendix we give explicit results for the average phase factor for Nf = 1 defined by 〈e2iθ〉Nf=1 = Z2|1∗(µc, µ) ZNf=1(µc, µ) . (A1) The partition function ZNf=1(µc, µ) was already given in Eq. (8). The numerator can be obtained from the CFZ formula. After taking the limit of degenerate critical chemical potentials at the end of the calculation, we obtain for µ < µc Z2|1∗(µc, µ) = e n(Nc+1)µce−4nµ (1− e2n(µ+µc))2(1− e2n(µ−µc))2 (enµc − e−nµc)5 + e−3n(Nc+1)µc (enµ − e−nµ)4 (enµc − e−nµc)5 +e−n(Nc+1)µc [f0 +Ncf1 +N c f2]. (A2) 2e2n(µ+µc) + 2e2n(µ−µc) − e4nµ − e−4nµ + 4e2nµ + 4e−2nµ − e4n(µ−µc) − e4n(µ+µc) + 2e6nµc−2nµ + 2e6nµc+2nµ (enµc − e−nµc)5 −10e2nµc + 6e4nµc − 2− 5e6nµc − e−2nµc (enµc − e−nµc)5 (1− e−2nµ)2(3e2nµc + e−2nµc − 2e2nµ − 2e−2nµ) (enµc − e−nµc)3 f2 = − (1− e−2nµ)2(1 − en(µ+µc))(1 − en(µ−µc)) (enµc − e−nµc)3 . (A3) For µ > µc a similar calculation results in Z2|1∗ = Nce nNc(µc−2µ) (1− e−2n(µ+µc))(1− e−2nµ) (1− e−2nµc)2 ) +Nce −nNc(2µ+µc) (1− e2n(µc−µ))(1− e−2nµ) (1− e2nµc)2 g1 = 1 + e2nµ − 1 e2n(µ+µc) − 1 e2nµc − 1 g2 = 1 + e2n(µ−µc) − 1 e2nµ − 1 − e−2nµc − 1 . (A5) We have checked that the results in this appendix agree with the large Nc limit of the average phase factor given in section IV B. [1] M. P. Lombardo, arXiv:hep-lat/0612017. [2] C. Schmidt, PoS LAT2006, 021 (2006) [arXiv:hep-lat/0610116]. [3] K. Splittorff, arXiv:hep-lat/0610072. [4] K. Splittorff and J. J. M. Verbaarschot, [arXiv:hep-lat/0702011]. [5] K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 98, 031601 (2007) [arXiv:hep-lat/0609076]. [6] P. E. Gibbs, Preprint PRINT-86-0389-GLASGOW, 1986; P. E. Gibbs, Phys. Lett. B 182 (1986) 369. [7] D. Toussaint, Nucl. Phys. Proc. Suppl. 17, 248 (1990). [8] P. de Forcrand and V. Laliena, Phys. Rev. D 61, 034502 (2000) [arXiv:hep-lat/9907004]. [9] Y. Sasai, A. Nakamura and T. Takaishi, Nucl. Phys. Proc. Suppl. 129, 539 (2004) [arXiv:hep-lat/0310046]. [10] S. Ejiri, Phys. Rev. D 69, 094506 (2004) [arXiv:hep-lat/0401012]; S. Ejiri, Phys. Rev. D 73, 054502 (2006) [arXiv:hep-lat/0506023]. [11] C. Schmidt, Z. Fodor and S. D. Katz, arXiv:hep-lat/0512032. [12] C. R. Allton et al., Phys. Rev. D 66, 074507 (2002) [arXiv:hep-lat/0204010]; C. R. Allton, S. Ejiri, S. J. Hands, O. Kacz- marek, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D 68, 014507 (2003) [arXiv:hep-lat/0305007]; C. R. Allton et al., Phys. Rev. D 71, 054508 (2005) [arXiv:hep-lat/0501030]. [13] E. V. Shuryak and J. J. M. Verbaarschot, Nucl. Phys. A 560, 306 (1993) [arXiv:hep-th/9212088]. [14] J. J. M. Verbaarschot, Phys. Rev. Lett. 72, 2531 (1994) [arXiv:hep-th/9401059]. [15] J.B. Conrey, D.W. Farmer and M.R. Zirnbauer, [arXiv:math-ph/0511024], 2005. [16] E. M. Ilgenfritz and J. Kripfganz, Z. Phys. C 29, 79 (1985). [17] P. H. Damgaard, N. Kawamoto and K. Shigemoto, Phys. Rev. Lett. 53 (1984) 2211; Nucl. Phys. B 264, 1 (1986). [18] A. Gocksch and M. Ogilvie, Phys. Rev. D 31 (1985) 877. [19] P. H. Damgaard, D. Hochberg and N. Kawamoto, Phys. Lett. B 158 (1985) 239. [20] N. Bilic, K. Demeterfi and B. Petersson, Nucl. Phys. B 377, 651 (1992). [21] P. de Forcrand and S. Kim, arXiv:hep-lat/0608012. http://arxiv.org/abs/hep-lat/0612017 http://arxiv.org/abs/hep-lat/0610116 http://arxiv.org/abs/hep-lat/0610072 http://arxiv.org/abs/hep-lat/0702011 http://arxiv.org/abs/hep-lat/0609076 http://arxiv.org/abs/hep-lat/9907004 http://arxiv.org/abs/hep-lat/0310046 http://arxiv.org/abs/hep-lat/0401012 http://arxiv.org/abs/hep-lat/0506023 http://arxiv.org/abs/hep-lat/0512032 http://arxiv.org/abs/hep-lat/0204010 http://arxiv.org/abs/hep-lat/0305007 http://arxiv.org/abs/hep-lat/0501030 http://arxiv.org/abs/hep-th/9212088 http://arxiv.org/abs/hep-th/9401059 http://arxiv.org/abs/math-ph/0511024 http://arxiv.org/abs/hep-lat/0608012 [22] P. E. Gibbs, Phys. Lett. B 172, 53 (1986). [23] A. Gocksch, Phys. Rev. D 37, 1014 (1988). [24] N. Bilic and K. Demeterfi, Phys. Lett. B 212, 83 (1988). [25] S. Gupta, Phys. Lett. B 588, 136 (2004) [arXiv:hep-lat/0307007]. [26] M. P. Lombardo, Nucl. Phys. Proc. Suppl. 83, 375 (2000) [arXiv:hep-lat/9908006]. [27] I. Barbour, N. E. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone and H. W. Wyld, Nucl. Phys. B 275, 296 (1986). [28] J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 73, 2288 (1994) [arXiv:hep-th/9405005]. [29] A. J. McKane and M. Stone, Annals Phys. 131, 36 (1981). [30] M. A. Stephanov, Phys. Rev. Lett. 76, 4472 (1996) [arXiv:hep-lat/9604003]. [31] J. C. Osborn, K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 94, 202001 (2005) [arXiv:hep-th/0501210]. [32] P. Hasenfratz and F. Karsch, Phys. Lett. B 125, 308 (1983). [33] M.R. Zirnbauer, [arXiv:chao-dyn/9609007]. [34] B. Schlittgen and T. Wettig, Nucl. Phys. B 632, 155 (2002) [arXiv:hep-lat/0111039]. [35] J. Budczies, S. Nonnenmacher, Y. Shnir and M. R. Zirnbauer, Nucl. Phys. B 635, 309 (2002) [arXiv:hep-lat/0112018]. [36] J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein and N.C. Snaith, [arXiv:math-ph/0208007]. [37] E.L. Basor and P.J. Forrester, Mathematische Nachrichten, 170, 5 (1194). [38] Z. Fodor and S. D. Katz, JHEP 0203, 014 (2002) [arXiv:hep-lat/0106002]. [39] Z. Fodor and S. D. Katz, JHEP 0404, 050 (2004) [arXiv:hep-lat/0402006]. [40] P. de Forcrand and O. Philipsen, Nucl. Phys. B 642, 290 (2002) [arXiv:hep-lat/0205016]. [41] P. de Forcrand and O. Philipsen, Nucl. Phys. B 673, 170 (2003) [arXiv:hep-lat/0307020]. [42] M. D’Elia and M. P. Lombardo, Phys. Rev. D 67, 014505 (2003) [arXiv:hep-lat/0209146]. [43] C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D 68, 014507 (2003) [arXiv:hep-lat/0305007]. [44] C. R. Allton et al., Phys. Rev. D 71, 054508 (2005) [arXiv:hep-lat/0501030]. [45] R. V. Gavai and S. Gupta, Phys. Rev. D 68, 034506 (2003) [arXiv:hep-lat/0303013]. [46] R. V. Gavai and S. Gupta, Phys. Rev. D 71, 114014 (2005) [arXiv:hep-lat/0412035]. [47] V. Azcoiti, G. di Carlo and A. F. Grillo, Phys. Rev. Lett. 65, 2239 (1990); J. Ambjorn, K. N. Anagnostopoulos, J. Nishimura and J. J. M. Verbaarschot, JHEP 0210, 062 (2002) [arXiv:hep-lat/0208025]. [48] A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 53, 7223 (1996) [arXiv:hep-ph/9509324]. [49] M. A. Halasz, A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 56 (1997) 5140 [arXiv:hep-lat/9703006]. [50] K. Splittorff, arXiv:hep-lat/0505001. [51] G. V. Dunne and S. M. Nishigaki, Nucl. Phys. B 670, 307 (2003) [arXiv:hep-ph/0306220]. [52] G. Akemann and P. H. Damgaard, Nucl. Phys. B 576, 597 (2000) [arXiv:hep-th/9910190]. [53] P. H. Damgaard and S. M. Nishigaki, Phys. Rev. D 57, 5299 (1998) [arXiv:hep-th/9711096]. [54] G. Akemann, D. Dalmazi, P. H. Damgaard and J. J. M. Verbaarschot, Nucl. Phys. B 601, 77 (2001) [arXiv:hep-th/0011072]. [55] T. Andersson, P. H. Damgaard and K. Splittorff, Nucl. Phys. B 707, 509 (2005) [arXiv:hep-th/0410163]. [56] G. Akemann and Y. V. Fyodorov, Nucl. Phys. B 664, 457 (2003) [arXiv:hep-th/0304095]. [57] R. J. Szabo, Nucl. Phys. B 598, 309 (2001) [arXiv:hep-th/0009237]. [58] D. Toublan and J. J. M. Verbaarschot, Int. J. Mod. Phys. B 15, 1404 (2001) [arXiv:hep-th/0001110]. [59] K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 683, 467 (2004) [arXiv:hep-th/0310271]. [60] G. Akemann and T. Wettig, Phys. Rev. Lett. 92, 102002 (2004) [Ibid. 96, 029902 (2006)] [arXiv:hep-lat/0308003]. [61] J. Bloch and T. Wettig, Phys. Rev. Lett. 97, 012003 (2006) [arXiv:hep-lat/0604020]. [62] P. H. Damgaard, U. M. Heller, A. Krasnitz and T. Madsen, Phys. Lett. B 440, 129 (1998) [arXiv:hep-lat/9803012]. [63] R. Narayanan and H. Neuberger, Nucl. Phys. B 696, 107 (2004) [arXiv:hep-lat/0405025]. [64] T. D. Cohen, Phys. Rev. Lett. 91, 222001 (2003) [arXiv:hep-ph/0307089]. [65] J. C. Osborn, Phys. Rev. Lett. 93, 222001 (2004) [arXiv:hep-th/0403131]. [66] G. Akemann, J. C. Osborn, K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 712, 287 (2005) [arXiv:hep-th/0411030]. http://arxiv.org/abs/hep-lat/0307007 http://arxiv.org/abs/hep-lat/9908006 http://arxiv.org/abs/hep-th/9405005 http://arxiv.org/abs/hep-lat/9604003 http://arxiv.org/abs/hep-th/0501210 http://arxiv.org/abs/chao-dyn/9609007 http://arxiv.org/abs/hep-lat/0111039 http://arxiv.org/abs/hep-lat/0112018 http://arxiv.org/abs/math-ph/0208007 http://arxiv.org/abs/hep-lat/0106002 http://arxiv.org/abs/hep-lat/0402006 http://arxiv.org/abs/hep-lat/0205016 http://arxiv.org/abs/hep-lat/0307020 http://arxiv.org/abs/hep-lat/0209146 http://arxiv.org/abs/hep-lat/0305007 http://arxiv.org/abs/hep-lat/0501030 http://arxiv.org/abs/hep-lat/0303013 http://arxiv.org/abs/hep-lat/0412035 http://arxiv.org/abs/hep-lat/0208025 http://arxiv.org/abs/hep-ph/9509324 http://arxiv.org/abs/hep-lat/9703006 http://arxiv.org/abs/hep-lat/0505001 http://arxiv.org/abs/hep-ph/0306220 http://arxiv.org/abs/hep-th/9910190 http://arxiv.org/abs/hep-th/9711096 http://arxiv.org/abs/hep-th/0011072 http://arxiv.org/abs/hep-th/0410163 http://arxiv.org/abs/hep-th/0304095 http://arxiv.org/abs/hep-th/0009237 http://arxiv.org/abs/hep-th/0001110 http://arxiv.org/abs/hep-th/0310271 http://arxiv.org/abs/hep-lat/0308003 http://arxiv.org/abs/hep-lat/0604020 http://arxiv.org/abs/hep-lat/9803012 http://arxiv.org/abs/hep-lat/0405025 http://arxiv.org/abs/hep-ph/0307089 http://arxiv.org/abs/hep-th/0403131 http://arxiv.org/abs/hep-th/0411030 Introduction QCD in One Dimension Lattice QCD Continuum Theory Mean Field Limit for Large Nc The Conrey-Farmer-Zirnbauer formula Exact Evaluation of the One-Dimensional U(Nc) Partition Function Partition Function for arbitrary Nf The Phase Quenched Partition Function Average Phase Factor for U(Nc) Phase Quenched Average Phase Factor Average phase factor for full QCD Average Phase Factor for SU(Nc) SU(Nc) QCD Partition Function The Phase Quenched Partition Function for SU(Nc) Average Phase Factor for SU(Nc) The Sign Problem at Nonzero Temperature Comparison with the QCD3 Partition Function Random Matrix Model Microscopic Limit Dirac Spectrum Universal Fluctuations Chiral Symmetry Breaking at =0 Conclusions Average U(Nc) Phase Factor for Nf = 1 References

0704.1112

Can GLAST detect gamma-rays from the extended radio features of radio galaxies?

Can GLAST detect gamma-rays from the extended radio features of radio galaxies? R. M. Sambruna∗, M. Georganopoulos†, D. Davis∗∗,‡ and A. N. Cillis∗ ∗NASA/GSFC, Code 661, Greenbelt, MD 20771 †Department of Physics, Joint Center for Astrophysics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 ∗∗CRESST and Astroparticle Physics Laboratory NASA/GSFC, Greenbelt, MD 20771 ‡Department of Physics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 Abstract. A few FRI radio galaxies were detected at GeV gamma-rays with CGRO EGRET, with peroperties suggesting that the gamma-ray flux originates from the core. Here we discuss the possibility that the extended radio features of radio galaxies could be detected with the LAT, focusing on the particularly promising case of the nearby giant radio galaxy Fornax A. Keywords: Gamma-rays; Radio Galaxies; Lobes PACS: 95.85.Pw; 98.54.Gr GAMMA-RAYS FROM RADIO GALAXIES Previous observations with CGRO EGRET established that the dominant extragalactic sources of GeV gamma-rays are blazars, providing model-independent evidence for beaming in these sources [8]. However, gamma-rays were also detected around the location of the FRI radio galaxies Centaurus A [11] and NGC 6251 [9]. If the GeV flux comes, indeed, from these sources, a most obvious interpretation is that it originates from the compact core of the radio sources. In fact, analysis of the Spectral Energy Distributions (SEDs) from radio to gamma-rays shows a double- humped structure, similar to blazars, but significantly de-beamed [6]. Gamma-ray flux was also detected from the core of more powerful FRIIs, specifically, the Broad-Line Radio Galaxy 3C 111 [10]. Again, the core SED is double-dumped resembling a misaligned blazar. There are theroetical reasons to expect the cores of radio galaxies to be GeV emitters [10, 5]. Indeed, rescaling for the radio-to-GeV flux ratio of Centaurus A leads to the prediction of significant GeV emission from the cores of roughly two dozen radio sources [6], which GLAST should be able to detect in the first 6–18 months of operations. EXTENDED RADIO LOBES: THE CASE OF FORNAX A Can the extended features of radio galaxies be a source of GeV gamma-rays? If so, can they be detected with the LAT? Our analysis of the nearby giant source Fornax A provides a resounding "Yes" to both questions. The optimal candidates for a GLAST LAT detection of the radio lobes should satisfy the following observational criteria: • Large angular size (> 30′) to be resolved with the LAT PSF; • A weak core, to avoid contamination of the total GeV flux; • Being located at high Galactic latitudes, to avoid contamination from the unresolved galactic sources. Finally, good characterization of the radio spectrum is desirable, to constrain the particle energy distribution. Previous detection at X-rays would also be a plus. One source that fits all these criteria is the nearby (D=18 Mpc) radio galaxy Fornax A, hosted by a massive elliptical. The core of this source is a LINER and its angular size from the E to the W lobe is 33′. Figure 1a shows the VLA image of Fornax A. The lobes were detected with WMAP [1] showing a cutoff of the radio flux above 10 MHz. FIGURE 1. (a, Left): The VLA 5 GHz image of Fornax A [4], showing the bright E and W lobes and a weak core. (b, Right): Simulation of the LAT image of Fornax A during the first year of the survey (see text). North is up and East to the right. Interestingly, X-ray emission from both the lobes was previously detected with ROSAT [3], ASCA [12], and more recently with XMM-Newton [7]. The brighter E lobe has a powerlaw X-ray spectrum with slope αX ∼ 0.6, consistent with the radio slope below the WMAP cutoff. This supports the idea that the X-rays are produced by Inverse Compton scattering of the CMB photons (IC/CMB) off low-energy (γ ∼ 103) electrons in the lobes [3]. The inferred magndetic field is 12µGauss [7]. The W lobe is also detected with 1/3 of the X-ray flux. Can the lobes of Fornax A be detected by the LAT? Assuming the above magnetic field and electron energies, if the electron energy distribution is unbroken (contrary to the WMAP results), the predicted flux for the E lobe is 4×10−8 ph/cm2/s above 100 MeV. Interestingly, this is the flux measured by EGRET at 2.2σ from a reanalysis of the 3rd EGRET catalog [2]. However, if we take into account the WMAP break at 10 GHz, the predicted break energy at gamma-rays is 5 MeV (1.2×1021 Hz). If we assume that the spectrum breaks by ∆α=0.5 (from αr=0.62 to αr=1.12), the predicted gamma-ray flux is 1.5×10−9 ph/cm2/s above 100 MeV. Using the latter predicted flux and a similar break in the gamma-ray spectrum above 5 MeV, we simulated LAT observations of Fornax A. We assumed that both lobes contribute to the EGRET emission, with the E lobe containing 2/3 of the flux. In 12 months we can detect and resolve gamma-ray emission from the E and W lobe (Fig. 1b), with a total of 28 and 7 counts, respectively. CONCLUSIONS It seems clear, from EGRET observations, that the cores of radio galaxies are gamma-ray sources, easily detectable with the LAT. Using the nearby giant radio galaxy Fornax A we demonstrated that another contender for the origin of gamma-rays from radio sources are the diffuse radio lobes. A simulation shows that both the W and E radio lobes are detected with the LAT in 12 months. Detection of the GeV flux will constrain the electron energy distribution and the magnetic field. How common is gamma-ray emission from the radio lobes? This issue can be addressed by systematic studies of lobe-dominated radio galaxies with GLAST, such as e.g., the 3CRR sample. Feasibility studies are underway. ACKNOWLEDGMENTS We thank the organizers for putting together an interesting and informative meeting. RMS, DD, and AC are supported by the GLAST project at GSFC. REFERENCES 1. Cheung, C.C. 2006, astroph//0612372 2. Cillis, A. N., Hartman, R. C., & Bertsch, D. L. 2004, ApJ, 601, 142 3. Feigelson, E. et al. 1995, ApJ, 445, L149 4. Fomalont, E.B. et al. 1989, ApJ, 346, L17 5. Georganopoulos, M. & Kazanas, D. 2003, ApJ,589, L5 6. Ghisellini, G., Tavecchio, F., & Chiaberge, M. 2004, A&A, 432, 401 7. Isobe, N. et al. 2006, ApJ, 645, 256 8. Maraschi, L. 2007, these proceedings 9. Mukherjee, R., Halpern, J., Mirabal, N., & Gotthelf, E. V. 2002, ApJ, 574, 693 10. Sguera, V. et al. 2005, A&A, 430, 107 11. Steinle, H. et al. 1998, A&A, 330, 97 12. Tashiro, M. et al. 2001, ApJ, 546, L19

0704.1113

The VLT-FLAMES survey of massive stars: Wind properties and evolution of hot massive stars in the LMC

_lines.ps Astronomy & Astrophysics manuscript no. 6489 c© ESO 2018 October 28, 2018 The VLT-FLAMES survey of massive stars: Wind properties and evolution of hot massive stars in the LMC M. R. Mokiem1, A. de Koter1, C. J. Evans2, J. Puls3, S. J. Smartt4, P. A. Crowther5, A. Herrero6,7, N. Langer8, D. J. Lennon9,6, F. Najarro10, M. R. Villamariz11,6, and J. S. Vink12 1 Astronomical Institute Anton Pannekoek, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands 2 UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK 3 Universitäts-Sternwarte München, Scheinerstr. 1, D-81679 München, Germany 4 The Department of Pure and Applied Physics, The Queen’s University of Belfast, Belfast BT7 1NN, Northern Ireland, UK 5 Department of Physics and Astronomy, University of Sheffield, Hicks Building, Hounsfield Rd, Shefffield, S3 7RH, UK 6 Instituto de Astrofı́sica de Canarias, E-38200, La Laguna, Tenerife, Spain 7 Departamento de Astrofı́sica, Universidad de La Laguna, Avda. Astrofı́sico Francisco Sánchez, s/n, E-38071 La Laguna, Spain 8 Astronomical Institute, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands 9 The Isaac Newton Group of Telescopes, Apartado de Correos 321, E-38700, Santa Cruz de La Palma, Canary Islands, Spain 10 Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientı́ficas, CSIC, Serrano 121, E-28006 Madrid, Spain 11 Grantecan S.A., E-38200, La Laguna, Tenerife, Spain 12 Astrophysics Group, Lennard-Jones Laboratories, Keele University, Staffordshire, ST55BG, UK Accepted: 24 January 2007 ABSTRACT We have studied the optical spectra of a sample of 28 O- and early B-type stars in the Large Magellanic Cloud, 22 of which are associated with the young star forming region N11. Our observations sample the central associations of LH9 and LH10, and the surrounding regions. Stellar parameters are determined using an automated fitting method (Mokiem et al. 2005), which combines the stellar atmosphere code fastwind (Puls et al. 2005) with the genetic algorithm based optimisation routine pikaia (Charbonneau 1995). We derive an age of 7.0 ± 1.0 and 3.0 ± 1.0 Myr for LH9 and LH10, respectively. The age difference and relative distance of the associations are consistent with a sequential star formation scenario in which stellar activity in LH9 triggered the formation of LH10. Our sample contains four stars of spectral type O2. From helium and hydrogen line fitting we find the hottest three of these stars to be ∼49−54 kK (compared to ∼45−46 kK for O3 stars). Detailed determination of the helium mass fraction reveals that the masses of helium enriched dwarfs and giants derived in our spectroscopic analysis are systematically lower than those implied by non-rotating evolutionary tracks. We interpret this as evidence for efficient rotationally enhanced mixing leading to the surfacing of primary helium and to an increase of the stellar luminosity. This result is consistent with findings for SMC stars by Mokiem et al. (2006). For bright giants and supergiants no such mass discrepancy is found; these stars therefore appear to follow tracks of modestly or non- rotating objects. The set of programme stars was sufficiently large to establish the mass loss rates of OB stars in this Z ∼ 1/2 Z⊙ environment sufficiently accurate to allow for a quantitative comparison with similar objects in the Galaxy and the SMC. The mass loss properties are found to be intermediate to massive stars in the Galaxy and SMC. Comparing the derived modified wind momenta Dmom as a function of luminosity with predictions for LMC metallicities by Vink et al. (2001) yields good agreement in the entire luminosity range that was investigated, i.e. 5.0 < log L/L⊙ < 6.1. Key words. Magellanic Clouds – stars:atmospheres – stars: early-type – stars: fundamental parameters – stars: mass loss 1. Introduction Massive stars play an intricate role in the evolution of galax- ies. Because of the large energies associated with their stel- lar winds, ionising radiation, and life-ending supernova ex- plosions, they dictate galactic structuring processes such as star formation and the creation and evolution of supperbub- bles (e.g. Oey 1999). Mounting evidence also points to a direct link between massive stars and exotic phenomena such as γ-ray bursts (e.g. Hjorth et al. 2003) and the reionisation of the early universe (Bromm et al. 2001). Accordingly, understanding the properties of these stars, both in terms of their fundamental pa- rameters as well as their evolution, is fundamental. The initial metal composition (Z) of the gas out of which massive stars form has a strong impact on their global prop- erties and characteristics. Many studies have shown that pa- rameters such as the effective temperature and ionising fluxes are strong functions of Z (e.g. Kudritzki 2002; Mokiem et al. 2004; Massey et al. 2005; Mokiem et al. 2006), adding an ex- tra dimension to the conversion of morphological properties http://arxiv.org/abs/0704.1113v1 2 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC such as spectral type to physical quantities. Theoretical and ob- servational arguments (e.g. Kudritzki & Puls 2000; Vink et al. 2001) also point to a relation between the strengths of the stel- lar winds of these objects and metallicity. As wind mass loss leads to partial evaporation of the star (e.g. Chiosi & Maeder 1986) – with possible consequences for the nature of the com- pact object that is left behind after the final supernova explosion – and to loss of angular momentum (e.g. Meynet & Maeder 2000), it also dictates to a large extent its evolutionary path and fate (e.g. Yoon & Langer 2005; Woosley & Heger 2006). Quantifying the mass loss versus metallicity dependence Ṁ(Z), therefore, is an important quest in astrophysics. Due to their proximity and low metal content the Magellanic Clouds provide us with unparallelled laboratories to test and enlarge our knowledge of massive stars. These galaxies, therefore, have been in the focal point of many stud- ies analysing their massive star content. Early studies (e.g. Conti et al. 1986; Garmany et al. 1987; Massey et al. 1989; Parker et al. 1992; Walborn et al. 1999) predominantly relied on photometric data and spectral type calibrations. Only rel- atively recent the advent of large telescopes and the develop- ment of sophisticated stellar atmosphere models has allowed for more detailed analyses of individual stars (e.g. Puls et al. 1996; Hillier & Miller 1999; Crowther et al. 2002; Bouret et al. 2003; Martins et al. 2004). Though all these studies have con- tributed enormously to our understanding of massive stars, the samples analysed so far have been rather limited in size (a few objects at a time) and have been focused predominantly on ob- jects in the Small Magellanic Cloud (SMC) – because its metal deficiency is more extreme than that of the Large Magellanic Cloud (LMC) (but see Massey et al. 2004, 2005). The limited sizes of the samples that have so far been stud- ied are at the root cause of the perhaps somewhat disheartening conclusion that – in spite of all the progress that has been made – we still cannot provide robust and sound answers to the ques- tion: what is the role of metal content, stellar winds and rota- tion in the evolution of massive stars? To help attack this prob- lem, our research group has conducted a VLT-FLAMES Survey of Massive Stars (see Evans et al. 2005). In this ESO Large Program the Fibre Large Array Multi-Element Spectrograph at the Very Large Telescope was used to obtain optical spec- tra of more than 50 O- and early B-type stars in the Magellanic Clouds. Here we present the homogeneous analysis, employing au- tomated spectral fitting methods, of a sample of 28 O-type and early B-type stars in the Large Magellanic Cloud; 22 targets from the FLAMES survey and 6 from other sources. This is so far the largest sample of massive stars studied in the LMC and it almost doubles the amount of massive objects in this galaxy for which parameters have been derived from quantita- tive spectroscopy. Specifically, we will try to establish the mass loss rates of OB stars in this Z ∼ 1/2 Z⊙ environment to a level of precision that allows for a quantitative comparison with sim- ilar objects in the SMC and our Galaxy. This will provide a new Z-point in testing the fundamental prediction provided by radi- ation driven wind theory for the mass-loss – metallicity depen- dence: Ṁ(Z) ∝ Z0.5−0.7 (e.g. Kudritzki et al. 1987; Puls et al. 2000; Vink et al. 2001). The majority of our LMC sample is associated with the spectacular star forming region N11 (Henize 1956). It has a Hα luminosity only surpassed by that of 30 Doradus (Kennicutt & Hodge 1986), ranking it as the second largest H ii region in the Magellanic Clouds. N11 is host to several OB associations of apparently different ages, the formation of which is believed to have been triggered by stellar activity in the central OB cluster (Parker et al. 1992; Walborn & Parker 1992; Walborn et al. 1999). Our observations sample both the central cluster LH9 as well as the younger cluster LH10, allow- ing for an investigation of a possible sequential star formation scenario. We will use our N11 stars to test predictions of mas- sive star evolution, including the role of rotation, and the star formation history. This paper is organised as follows: in Sect. 2 we describe the LMC data set that was analysed using our automated ge- netic algorithm based fitting method. A short description of this method is given in Sect. 3 and the results obtained are presented in Sect. 4, with fits and comments on individual objects given in the appendix. In Sect. 5 we investigate the discrepancy be- tween spectroscopically determined masses and those derived from evolutionary tracks. The evolutionary status of N11 is dis- cussed in Sect. 6. Finally, Sect. 7 summarises and lists our most important findings. 2. Data description Our OB-type star sample is mainly drawn from the targets ob- served in the LMC within the context of the VLT-FLAMES survey of massive stars (see Evans et al. 2005). Two fields in the LMC centred on the clusters N11 and NGC 2004 were observed in the survey. Here we analyse a subset of the ob- jects observed in the N11 field. This set consists of all O-type spectra obtained, excluding those that correspond to confirmed binaries, and five early B-type spectra of luminous giant and supergiant stars. To improve the sampling in luminosity and temperature, we supplemented the FLAMES targets with six relatively bright O- type field stars. These objects are part of the Sanduleak (1970, hereafter Sk) and Brunet et al. (1975, hereafter BI) catalogues, and were observed as part of the programs 67.D-0238, 70.D- 0164 and 074.D-0109 (P.I. Crowther) using the Ultraviolet and Visual Echelle Spectrograph (UVES) at the VLT. The observations of the FLAMES targets are described ex- tensively by Evans et al. (2006), to which we refer for full de- tails. Here we only summarise the most important observa- tional parameters. Basic observational properties of the pro- gramme stars together with common aliases are given in Tab. 1. Note that N11-031, BI 237, BI 253 and Sk−67 166 were stud- ied recently using line blanketed stellar atmosphere models. In the appendix a comparison with these analyses is provided. The FLAMES targets were observed with the Giraffe spectrograph mounted at UT2. For six wavelength settings a spectrum was acquired six times for each object with an effective resolving power of R ≃ 20 000. These multiple exposures, often at differ- ent epochs, allowed for the detection of variable radial veloc- ities. As a result, a considerable number of binaries could be M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 3 Table 1. Basic parameters. Primary identification numbers for N11 are from Evans et al. (2006). Identifications starting with “Sk”, “PGMW” and “BI” are from, respectively, Sanduleak (1970), Parker et al. (1992) and Brunet et al. (1975). Photometric data for these objects are from Evans et al. (2006) and Parker et al. (1992), the latter are flagged with an asterisk. For non-N11 objects these are from Ardeberg et al. (1972), Issersted (1979) and Massey (2002). Wind velocities given without brackets are from Crowther et al. (2002), Massa et al. (2003) and Massey et al. (2005). For Sk −66 18 the wind velocity was measured from O iv (1031-1037 Å) in its FUSE spectrum. Values between brackets are calculated from the escape velocity at the stellar surface. For N11-031 the value of v∞ is from Walborn et al. (2004), though this is actually based on the value obtained for Sk-68 137 by Prinja & Crowther (1998). Primary ID Cross-IDs Spectral V AV MV v∞ Type [km s−1] N11-004 Sk −66 16 OC9.7 Ib 12.56 0.74 −6.68 [2387] N11-008 Sk −66 15 B0.7 Ia 12.77 0.84 −6.57 [1619] N11-026 ... O2 III(f*) 13.51 0.47 −5.46 [3116] N11-029 ... O9.7 Ib 13.63 0.56 −5.43 [1576] N11-031 PGMW 3061 ON2 III(f*) 13.68∗ 0.96 −5.78 3200 N11-032 PGMW 3168 O7 II(f) 13.68∗ 0.65 −5.47 [1917] N11-033 PGMW 1005 B0 IIIn 13.68 0.43 −5.25 [1536] N11-036 ... B0.5 Ib 13.72 0.40 −5.18 [1714] N11-038 PGMW 3100 O5 II(f+) 13.81∗ 0.99 −5.68 [2601] N11-042 PGMW 1017 B0 III 13.93 0.22 −4.79 [2307] N11-045 ... O9 III 13.97 0.50 −5.03 [1548] N11-048 PGMW 3204 O6.5 V((f)) 14.02∗ 0.47 −4.95 [3790] N11-051 ... O5 Vn((f)) 14.03 0.19 −4.66 [2108] N11-058 ... O5.5 V((f)) 14.16 0.28 −4.62 [2472] N11-060 PGMW 3058 O3 V((f*)) 14.24∗ 0.81 −5.07 [2738] N11-061 ... O9 V 14.24 0.78 −5.04 [1898] N11-065 PGMW 1027 O6.5 V((f)) 14.40 0.25 −4.35 [2319] N11-066 ... O7 V((f)) 14.40 0.25 −4.35 [2315] N11-068 ... O7 V((f)) 14.55 0.28 −4.23 [3030] N11-072 ... B0.2 III 14.61 0.09 −3.98 [2098] N11-087 PGMW 3042 O9.5 Vn 14.76∗ 0.62 −4.36 [3025] N11-123 ... O9.5 V 15.29 0.16 −3.37 [2890] BI 237 ... O2 V((f*)) 13.89 0.62 −5.23 3400 BI 253 ... O2 V((f*)) 13.76 0.71 −5.45 3180 Sk −66 18 ... O6 V((f)) 13.50 0.37 −5.37 2200 Sk −66 100 ... O6 II(f) 13.26 0.34 −5.58 2075 Sk −67 166 HD 269698 O4 Iaf+ 12.27 0.31 −6.54 1750 Sk −70 69 ... O5 V 13.95 0.28 −4.83 2750 detected (Evans et al. 2006), which we subsequently excluded from our analysis. To allow for a sky subtraction a master sky-spectrum was created from combining the sky fibres in the Giraffe spectro- graph (typically 15), individually scaled by their relative fibre throughput. Even though in general the sky background is low and this approach successfully removes the background con- tribution, in crowded regions such as N11 accurate subtraction of nebular features remains very difficult. As a result of this, the line profiles of many of our programme stars still suffer from nebular contamination. This in principle does not hamper our analysis. For most stars the core nebular emission is well- resolved and we simply disregard this contaminated part of the profile in the automated line fits. Mokiem et al. (2006) showed by performing tests using synthetic data that with this reduced amount of information, the automated method can still accu- rately determine the correct fit parameters. Also tests assessing the impact of possible residual nebular contamination in the 4 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC line wings or over-subtraction of sky components showed that its effect is negligible. For each wavelength range the individual sky-subtracted spectra were co-added and then normalised using a cubic- spline fit to the continuum. A final spectrum covering 3850- 4750 and 6300-6700 Å was obtained by merging the nor- malised data. Depending on the magnitude of the target these combined spectra have typical signal-to-noise ratios of 100- The first four field stars listed in Tab. 1 were observed with the VLT-UVES spectrograph in service mode on 29 and 30 November 2004 under program 74.D-0109. UVES is a two armed cross-dispersed echelle spectrograph allowing for simul- taneous observations in the blue and red part of the spectrum. In the blue arm standard settings with central wavelengths of 390 and 437 nm were used to observe the spectral ranges 3300- 4500 and 3730-5000 Å. For the red arm standard settings with central wavelengths of 564 and 860 nm provided coverage be- tween 4620-5600 and 6600-10400 Å. A 1.2′′ wide slit was used, providing a spectral resolution of 0.1 Å at Hγ, corre- sponding to an effective resolving power of R ≃ 40 000, a value which applies to all UVES setups. Individual exposure times ranged from 1500 to 2200 seconds. Sk −67 166 was observed on 27 and 29 September 2001 under program 67.D-0238 with UVES using a 1′′ slit. A stan- dard blue setting with central wavelength 437 nm provided con- tinuous coverage between 3730-5000 Å. A non-standard red setup with central wavelength 830 nm was used to observe the range between 6370-10250 Å. Three exposures, each of 1000 s were obtained for each setup. Note that the data for this target was previously presented by Crowther et al. (2002). Finally, Sk −70 69 was observed on 1 and 2 December 2002 using UVES under program 70.D-0164. Standard blue and red settings with central wavelengths 390 and 564 nm were used in simultaneous 2400 s exposures using dichroic. A second non-standard red setup with central wavelength 520 nm (4170- 6210 Å) without dichroic was used in a 1500 s exposure. The typical two pixel S/N ratios obtained for all spectra are 80 at Hγ and 60 at Hα. Spectral types for FLAMES stars were determined by vi- sual inspection of the spectra, using published standards. In particular the atlas of Walborn & Fitzpatrick (1990) was used while considering the lower metallicity environment of the LMC. The classifications given in Tab. 1 are from Evans et al. (2006). A comparison with previously published spectral types is also given in the latter paper. For the field stars we adopted the classifications given by Walborn (1977), Walborn et al. (1995, 2002a) and Massey et al. (1995, 2005). Photometric data for the FLAMES targets was obtained predominantly from B and V images of the N11 field taken with the Wide Field Imager at the 2.2-m Max Planck Gesellschaft/ESO telescope on 2003 April 2004 (Evans et al. 2006). The photometry for stars flagged with an asterisk was adopted from Parker et al. (1992). The caption of Tab. 1 lists the references to the various sources of the photometric data of the field stars. To calculate the interstellar extinction (AV ) given in Tab. 1 we adopted intrinsic colours from Johnson (1966, and refer- ences therein) and a ratio of total to selective extinction of RV = 3.1. With these AV values, extinction corrected visual magnitudes (V0) were calculated from the observed V-band magnitudes. Finally, we calculated the absolute visual magni- tude MV , while adopting a distance modulus of 18.5 for the LMC (Panagia et al. 1991; Mitchell et al. 2002). 3. Analysis method All optical spectra are analysed using the automated fitting method developed by Mokiem et al. (2005, hereafter referred to as Paper I). Here we will suffice with a short description of the method and refer to the before mentioned paper and to Mokiem et al. (2006, hereafter Paper II) for the details. In short the automated fitting method uses the genetic algo- rithm based optimisation routine pikaia (Charbonneau 1995) to determine the set of input parameters for the stellar atmosphere code fastwind (Puls et al. 2005) which fit an observed spec- trum the best. This best fitting model is constructed by evolving a population of fastwind models over a course of generations. At the end of every generation the parameters of the models which relatively fit the observed spectrum the best are used to construct a new population of models. By repeating this proce- dure a natural optimisation is obtained and after a number of generations (see below) the best fitting model, i.e. global opti- mum in parameter space, is found. Using the concept of a unified model atmosphere the fast performance code fastwind incorporates non-LTE and an ap- proximate approach to line blanketing to synthesise hydrogen and helium line profiles. Consequently, given the spectral range of the observed data set in this study we will focus on the mod- elling of the optical hydrogen and helium lines. To account for the accuracy with which each individual line can be reproduced by fastwind we adopt the line weighting scheme as described in Paper I. In the fitting of the spectra we allow for seven free parame- ters. These are the effective temperature Teff, the surface gravity g, the helium number density defined as YHe ≡ N(He)/N(H), the microturbulent velocity vturb, the projected rotational veloc- ity vr sin i, the mass loss rate Ṁ and the exponent of the beta- type velocity law describing the supersonic regime of the stellar wind. For the terminal velocity of the wind v∞, which cannot be accurately determined from the optical spectrum, we adopt values determined from the analysis of ultraviolet (UV) wind lines. If no UV determination of v∞ is available a scaling re- lation of v∞ with the escape velocity (vesc) defined at the stel- lar surface is used throughout the fitting process. For our pro- gramme stars we adopt the ratio as determined by Lamers et al. (1995) of v∞/vesc = 2.6. Similar as in Paper II we adopt fixed values for the atmospheric abundances of the background met- als. These were scaled with respect to mass ratios based on the Solar abundances of Grevesse & Sauval (1998, and ref- erences therein). The metallicity scaling factor was set equal to the mean metal deficiency of 0.5 as found for the LMC (Russell & Bessell 1989; Rolleston et al. 2002). For our current data set the spectral range and quality, with exception of the signal-to-noise ratio, is similar to the set anal- ysed in Paper II. Consequently, we adopt the same minimum M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 5 number of generations that have to be calculated to determine the best fit, i.e. to assure that the global optimum in parameter space is found. In Paper II, using formal tests that accounted for nebular contamination of the line profiles and a signal-to-noise ratio of 50, this number was determined to be 150. The uncertainties of the fit parameters are determined us- ing so-called optimum width based error estimates. In Paper I we have argued that an error estimate for a parameter can be defined as the maximum variation of this parameter within the global optimum in parameter space. By measuring the width of this optimum in terms of fit quality one can determine which models, based on their fit quality, are associated with the global optimum. The maximum variation of the individual fit param- eters within this group of models then gives the error estimates (see Sect. 4 in Paper I). In Tab. 3 the optimum width based error estimates of the fit parameters for our programme stars are listed. The uncertain- ties in R⋆, L⋆, Ms and Mev, that are derived from the fit param- eters, were calculated using the same approach as in Paper II. The adopted uncertainty in the visual magnitude is ∆MV = 0.13 (Panagia et al. 1991) for all our programme stars. 3.1. The assumption of spherical symmetry fastwind assumes a spherically symmetric star and wind. For very high rotational speeds, vrot, this may potentially be a prob- lematic assumption. As it is well known, a high rotation rate leads to a distortion of the stellar surface and, via the von Zeipel theorem, to a decrease in flux and effective temperature from the pole to the equator (von Zeipel 1924, (slightly) modified by Maeder (1999) for the relevant case of shellular, i.e., radi- ally dependent, differential rotation, ω = ω(r)). This so-called “gravity” darkening does not only affect the stellar parameters, but also the wind (e.g. Cranmer & Owocki 1995; Owocki et al. 1996; Petrenz & Puls 2000). Thus, it might be questioned in how far the derived properties (which then depend on incli- nation angle and have a somewhat local character) are repre- sentative for the global quantities (e.g., mass, luminosity and mass-loss rate) which refer to integral quantities. As shown by various simulations (Cranmer & Owocki 1995; Petrenz & Puls 1996, 2000), the difference between local and global quantities remains small unless the star rotates faster than >∼ 60. . .70 % of its critical speed. Unfortunately, however, we cannot directly access the actual rotational speed, vrot, but only its projected value, vr sin i. Accounting for the average value of < sin i >= π/4, we have calculated the ratio of average to critical rotational speed, Ω ≈ vr sin i/(< sin i > vcrit), and found that only two objects lie above this value, namely N11-033 and N11-051, both with Ω ≈ 0.7. All other objects lie well below Ω = 0.3 (except for N11-087 withΩ ≈ 0.4). Of course, we cannot exclude that also the latter objects lie above the decisive threshold (if observed pole-on), but the rather large sample size implies that such a possibility should be actually present only for a minority of ob- jects. In conclusion, we suggest that the majority of our objects is, if at all, only weakly affected by a distortion of surface and wind. Note, e.g., that Ω = 0.3 leads to a difference in Teff and R⋆ between pole and equator of less than 5%, i.e., of the same order or less than our error estimates (cf. Tab. 3). Regarding the derived gravities and masses, finally, we have applied a consis- tent centrifugal correction anyhow (cf. Sect. 4.3). For the two objects with large rotational speeds, on the other hand, it is rather possible that we observe them al- most equator-on, i.e., the observed profiles contain an intrin- sic averaging over the complete stellar disk and thus corre- spond, at least in part, to the global, polar angle averaged val- ues. Howarth & Smith (2001) analyzed two galactic fast rota- tors, HD 149757 (ζ Oph) and HD 191423, accounting for ef- fects of non-sphericity and von Zeipel’s theorem. Their study showed that for these objects, rotating at Ω = 0.9, differ- ences in effective temperature and radius between pole and equator can amount to 20–30 %. Interestingly, analysis of the same objects by Herrero et al. (2002), Villamariz et al. (2002), and Villamariz & Herrero (2005) using both hydrostatic and spherically symmetric expanding atmospheres yielded (aver- age) parameters that agreed to within the standard deviation (5–10%) with the results obtained by Howarth & Smith. For stellar parameters it thus appears justified to conclude that non- sphericity has a very small impact even on stars with extreme vr sin i. But note also that with respect to global mass-loss rates the situation is much more unsecure and more-D simulations would be required to constrain their actual values. As shown, e.g., by Petrenz & Puls (2000), the modified wind-momentum rate from a rapidly rotating (Ω = 0.85) B-supergiant might be underestimated up to one magnitude if seen equator on. As the wind properties of N11-031 and N11-051 are rather normal it seems however unlikely that in these cases rotational effects have this type of dramatic impact. 4. Fundamental parameters 4.1. Effective temperatures In Fig. 1 the distribution of the effective temperatures of our programme stars as a function of spectral type is shown. The different luminosity classes are denoted using circles, trian- gles and squares, respectively, for class V, III and I-II objects. Similar as in Paper II we see that for a given spectral type the dwarfs are systematically hotter than the giants and su- pergiants. This separation can be interpreted as the result of the reduced surface gravities of the more evolved objects. A lower surface gravity results in an increased helium ionisation (e.g. Kudritzki et al. 1983), reducing the Teff needed for a given spectral type (e.g. Mokiem et al. 2004). A second reason is that the more evolved objects have stronger winds. These denser winds induce an increased line blanketing effect, further reduc- ing the required temperature (e.g. Schaerer & de Koter 1997). Massey et al. (2005), who also analysed a sizeable sample of LMC stars, do not find evidence for the dwarfs being hotter than the giants. Though, we note that their analysis only con- tained two giant Teff determinations for spectral type later than For comparison we also show in Fig. 1 as a dashed line the observed Teff vs. spectral calibration for Galactic O-type dwarfs from Martins et al. (2005a). With a dotted line the av- 6 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC Table 2. Fundamental parameters of the LMC sample determined using GA optimised spectral fits, with Teff in kK, log g and log gc in cm s −2, R⋆ in R⊙, L⋆ in L⊙, vturb and vr sin i in km s −1, Ṁ in M⊙yr −1and Ms and Mev in M⊙. Results were obtained using a population of 72 fastwind models evolved over a minimum of 150 generations. Gravities corrected for centrifugal acceleration (log gc) were used to calculate the spectroscopic masses (Ms). Evolutionary masses (Mev) were derived from the tracks of Schaerer et al. (1993). ID ST Teff log g log gc R⋆ log L⋆ YHe vturb vr sin i Ṁ β Ms Mev N11-004 OC9.7 Ib 31.6 3.36 3.37 26.5 5.80 0.10 5.7 81 1.78·10−6 1.18 59.9 47.5 N11-008 B0.7 Ia 26.0 2.98 2.99 29.6 5.55 0.10 18.0 83 4.96·10−7 1.87 31.4 31.7 N11-026 O2 III(f*) 53.3 4.00 4.00 10.7 5.92 0.11 19.0 109 1.81·10−6 1.08 42.4 81.9 N11-029 O9.7 Ib 29.4 3.23 3.24 15.7 5.21 0.07 19.1 77 1.73·10−7 1.63 15.5 24.9 N11-031 ON2 III(f*) 45.0 3.85 3.86 13.7 5.84 0.10 20.0 116 3.88·10−6 0.89 49.3 60.8 N11-032 O7 II(f) 35.2 3.45 3.46 14.0 5.43 0.09 11.3 96 8.06·10−7 1.03 20.6 33.8 N11-033 B0 IIIn 27.2 3.21 3.35 15.6 5.07 0.08 17.9 256 2.44·10−7 1.03 19.8 21.0 N11-036 B0.5 Ib 26.3 3.31 3.32 15.6 5.02 0.08 13.6 54 1.06·10−7 0.80a) 18.4 20.0 N11-038 O5 II(f+) 41.0 3.72 3.74 14.0 5.69 0.10 9.2 145 1.52·10−6 0.98 38.8 48.3 N11-042 B0 III 30.2 3.69 3.70 11.8 5.01 0.10 4.0 42 1.89·10−7 1.19 25.1 21.4 N11-045 O9 III 32.3 3.32 3.35 12.0 5.15 0.07 16.8 105 5.48·10−7 0.80a) 11.8 24.6 N11-048 O6.5 V((f) 40.7 4.19 4.20 9.9 5.38 0.06 1.0 130 1.67·10−7 0.80a) 56.6 36.6 N11-051 O5 Vn((f)) 42.4 3.75 3.88 8.4 5.31 0.08 19.7 333 1.01·10−6 0.60 19.5 36.4 N11-058 O5.5 V((f) 41.3 3.89 3.90 8.4 5.27 0.10 14.5 85 1.52·10−7 1.42 20.3 34.4 N11-060 O3 V((f*)) 45.7 3.92 3.93 9.7 5.57 0.12 19.3 106 5.22·10−7 1.26 29.2 49.4 N11-061 O9 V 33.6 3.51 3.52 11.7 5.20 0.09 19.8 87 2.14·10−7 1.80 16.7 26.6 N11-065 O6.5 V((f) 41.7 3.89 3.90 7.4 5.17 0.17 9.4 83 3.63·10−7 0.80a) 15.8 32.9 N11-066 O7 V((f)) 39.3 3.87 3.88 7.7 5.10 0.11 4.8 71 4.08·10−7 0.80a) 16.2 29.5 N11-068 O7 V((f)) 39.9 4.13 4.13 7.1 5.06 0.10 15.6 54 3.43·10−7 1.12 25.2 29.2 N11-072 B0.2 III 30.8 3.78 3.78 7.9 4.70 0.12 7.6 14 2.35·10−7 0.84 13.8 17.7 N11-087 O9.5 Vn 32.7 4.04 4.09 8.9 4.91 0.10 15.5 276 1.38·10−7 0.80a) 35.6 20.9 N11-123 O9.5 V 34.8 4.22 4.23 5.4 4.58 0.09 9.0 110 7.62·10−8 0.80a) 17.8 18.8 BI 237 O2 V((f*)) 53.2 4.11 4.11 9.7 5.83 0.10 12.8 126 7.81·10−7 1.26 44.6 75.0 BI 253 O2 V((f*)) 53.8 4.18 4.19 10.7 5.93 0.09 18.6 191 1.92·10−6 1.21 64.6 84.1 Sk −66 18 O6 V((f)) 40.2 3.76 3.76 12.2 5.55 0.14 10.8 82 1.07·10−6 0.94 31.5 40.7 Sk −66 100 O6 II(f) 39.0 3.70 3.71 13.6 5.58 0.19 8.7 84 8.81·10−7 1.27 34.7 41.4 Sk −67 166 O4 Iaf+ 40.3 3.65 3.66 21.3 6.03 0.28 20.0 97 9.28·10−6 0.94 75.0 70.4 Sk −70 69 O5 V 43.2 3.87 3.88 9.0 5.41 0.17 16.1 131 1.03·10−6 0.78 22.7 39.7 a) assumed fixed value erage temperature of the SMC dwarfs studied in Paper II is shown. The LMC dwarfs are found to occupy the region in between these two average temperature scales, with an average behaviour intermediate to that of the Galactic and SMC dwarfs. We interpret this as the result of the metallicity of the LMC that is lower than the Galactic value and higher than in the SMC. As the amount of line blanketing in a stellar atmosphere depends on metallicity the LMC objects have temperatures in between that of objects in the other two galaxies. 4.2. The Teff scale of O2 stars Our sample contains four O2 type stars. This spectral type was introduced by Walborn et al. (2002b) and is assigned based pri- marily on the ratios of selective emission lines of N iv and N iii. By modelling these lines Walborn et al. (2004) have shown that indeed, as had been hypothesised, the O2 spectra correspond to higher effective temperatures. However, the correct treatment of nitrogen lines in stellar atmosphere models is notoriously difficult. The relevant ionisation stages of this atom represent much more complex ion models compared to the relatively simple hydrogen and helium ions. Moreover, for the higher ions of nitrogen the ionisation depends on the extreme-UV radiation M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 7 Table 3. Optimum width based error estimates for the seven fit parameters. The ND entries correspond to error in vturb that reach up to the maximum allowed value of vturb and, therefore, are formally not defined. See text for details on the calculation of the uncertainties in the derived parameters. Units: Teff in kK, log g and log gc in cm s −2, R⋆ in R⊙, L⋆ in L⊙, vturb and vr sin i in km s−1, Ṁ in M⊙yr −1and Ms and Mev in M⊙. ID ∆Teff ∆log gc ∆R⋆ ∆ log L⋆ ∆YHe ∆vturb ∆vr sin i log∆Ṁ ∆β ∆Ms ∆Mev N11-004 −0.5 −0.06 +0.06 ±1.7 ±0.06 −0.01 +0.02 −0.14 +0.05 −0.00 +0.19 N11-008 −0.6 −0.06 +0.05 ±1.9 ±0.07 −0.01 +0.04 −0.20 +0.16 −0.31 +1.08 N11-026 −3.9 −0.05 +0.05 ±0.8 ±0.14 −0.01 +0.02 −0.05 +0.10 −0.12 +0.03 N11-029 −0.6 −0.05 +0.06 ±1.0 ±0.07 −0.01 +0.02 −0.13 +0.16 −0.42 +0.31 N11-031 −1.6 −0.06 +0.07 ±0.9 ±0.10 −0.02 +0.02 −0.05 +0.06 −0.04 +0.06 N11-032 −0.7 −0.07 +0.06 ±0.9 ±0.06 −0.02 +0.02 −0.16 +0.09 −0.11 +0.28 N11-033 −0.9 −0.06 +0.09 ±1.0 ±0.08 −0.01 +0.03 −0.26 +0.10 −0.20 +0.74 N11-036 −0.3 −0.09 +0.06 ±1.0 ±0.09 −0.01 +0.02 −1.99 +0.30 − N11-038 −1.0 −0.06 +0.07 ±0.9 ±0.07 −0.01 +0.03 −0.16 +0.10 −0.13 +0.22 N11-042 −0.8 −0.07 +0.09 ±0.7 ±0.07 −0.01 +0.03 −0.47 +0.40 −0.47 +0.77 N11-045 −1.2 −0.09 +0.21 ±0.8 ±0.08 −0.02 +0.03 −0.20 +0.14 − N11-048 −2.0 −0.21 +0.06 ±0.7 ±0.10 −0.01 +0.02 +13.2 −1.48 +0.74 − N11-051 −1.9 −0.06 +0.06 ±0.6 ±0.09 −0.02 +0.03 −0.28 +0.11 −0.09 +0.52 N11-058 −0.5 −0.06 +0.05 ±0.5 ±0.06 −0.01 +0.01 −0.36 +0.08 −0.20 +0.60 N11-060 −1.0 −0.05 +0.09 ±0.7 ±0.10 −0.02 +0.03 −13.1 −0.22 +0.06 −0.21 +0.20 N11-061 −0.6 −0.09 +0.07 ±0.7 ±0.06 −0.01 +0.03 −0.21 +0.14 −0.49 +0.38 N11-065 −0.9 −0.07 +0.15 ±0.5 ±0.06 −0.03 +0.04 −1.06 +0.14 − N11-066 −1.3 −0.15 +0.19 ±0.5 ±0.09 −0.03 +0.05 −0.49 +0.22 − N11-068 −1.0 −0.19 +0.07 ±0.4 ±0.07 −0.02 +0.03 −14.8 −0.79 +0.16 −0.24 +0.36 N11-072 −0.6 −0.07 +0.09 ±0.5 ±0.06 −0.02 +0.02 −1.29 +0.24 −0.32 +0.31 N11-087 −0.6 −0.09 +0.11 ±0.6 ±0.07 −0.03 +0.02 −2.82 +0.27 − N11-123 −0.7 −0.11 +0.00 ±0.3 ±0.06 −0.01 +0.02 −2.03 +0.35 − BI 237 −3.8 −0.08 +0.15 ±0.8 ±0.18 −0.01 +0.03 −12.6 −0.16 +0.12 −0.17 +0.16 BI 253 −5.5 −0.15 +0.14 ±0.9 ±0.19 −0.02 +0.03 −18.0 −0.05 +0.10 −0.21 +0.07 Sk −66 18 −1.1 −0.13 +0.09 ±0.8 ±0.07 −0.03 +0.04 −10.4 −0.05 +0.06 −0.10 +0.11 Sk −66 100 −1.0 −0.09 +0.08 ±0.9 ±0.07 −0.03 +0.05 −0.12 +0.12 −0.19 +0.18 Sk −67 166 −0.8 −0.08 +0.00 ±1.3 ±0.07 −0.04 +0.06 −0.05 +0.02 −0.04 +0.09 Sk −70 69 −1.4 −0.14 +0.13 ±0.6 ±0.08 −0.03 +0.05 −13.1 −0.11 +0.14 −0.25 +0.15 field, which may be affected by non-thermal processes, such as shocks. Adding to the complexity is the fact that the nitro- gen abundance has to be treated as a free parameter, as many early type stars show evidence of atmospheric abundance en- hancements (e.g. Crowther et al. 2002; Bouret et al. 2003). In our analysis method we solely model the hydrogen and helium lines and self consistently allow for abundance enhancements by treating the helium abundance as a free parameter. In prin- ciple our temperature determination should, therefore, not be affected by such problems. As a result of this, our analysis can provide an independent confirmation for the hot nature of O2 stars. In Fig. 1 we see that the objects with an O2 spectral type indeed correspond to the hottest stars in our sample. With ex- ception of the giant N11-031, which based on its helium lines we find to be cooler (see below), they have temperatures in ex- cess of 50 kK and, therefore, are significantly hotter than the O3 star at Teff = 46 kK. The error bars, though, are consider- able. Compared to the average error of ∼3 percent the O2 ef- fective temperatures have an uncertainty of 5 up to 11 percent. This large uncertainty can be explained by the weak or even absent neutral helium lines in the spectra of these objects. As a result of this, our fitting method has to predominately rely on the He ii line profiles to determine the correct helium ionisation equilibrium. Based on a single ionisation stage the determina- tion of this equilibrium is more uncertain, which explains the larger error bars. One should also be wary for a possible degen- eracy effect between Teff and YHe that can occur as a result of 8 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC O2 O3 O4 O5 O6 O7 O8 O9 B0 B1 spectral type Fig. 1. Effective temperatures as a function of spectral type for the investigated LMC sample. Different luminosity classes are denoted using circles, triangles and squares for class V, III and I-II objects, respectively. Shown as a dashed line is the Teff cal- ibration of Martins et al. (2005a) for Galactic O-type dwarfs. The dotted line corresponds to the average effective tempera- ture of the SMC dwarfs studied in Paper II. The LMC dwarfs are found to lie in between these two average scales, with a typical ∼2 kK offset from both the average Galactic and SMC relations. the missing neutral helium lines (see e.g. Paper I). This, how- ever, is not an issue as the helium abundances derived for all O2 stars correspond to normal values close to 0.10. It is also possible to question whether the fact that the He i lines are so weak could result in a systematically higher Teff determination by the automated method compared to “by eye” fits. In other words, would a “by eye” fit prefer a lower tem- perature solution? For the O2 giant N11-026 this is a relevant question, as the weak He i λ4471 line, shown in Fig. A.1 in the appendix, is not fitted perfectly. Note, however, that due to the relative plot scale the discrepancy is exaggerated and that the fit quality is good and is comparible to that of the He ii lines. Still, to assess whether the fit of this particular line could be improved, i.e. to search for a solution predominantly using the He i diagnostic, we ran test fits with increasing relative weight of He i λ4471. We found that a solution with an improved fit could be obtained for an increase of the relative line weight with a factor of five. This solution has a Teff lower by 3.7 kK and all other fit parameters approximately equal to the solution with the low He i λ4471 weight. The overal fit quality of the other lines has somewhat decreased. As these lines are rela- tively strong, this is difficult to discern by eye. Also note that within the lower error estimate for Teff of 3.9 kK the two solu- tions agree. Similar results were also obtained for the O2 stars BI 237 and BI 253, where a reduction of Teff by 3.8 kK and 5.4 kK was found, respectively, for an increase in the relative weight of He i λ4471 by a factor of six and two, respectively. Consequently, based on our analysis we can only give a range for the Teff scale of the O2 stars of 49 − 54 kK, the bounds being set by the temperatures based on the He i λ4471 and hy- drogen/He ii diagnostics. The upper end of our hydrogen/helium based O2 effec- tive temperature range is in good agreement compared to the average effective temperature of 54 kK as determined by Walborn et al. (2004). As we already mentioned the giant N11- 031 with a Teff lower by approximately 8 kK compared to the other O2 stars forms an exception to this. Its effective tempera- ture compares better to the temperature of the O3 star N11-060. A close inspection of the spectra of the two stars reveals why this is so. Both stars have exactly the same equivalent width ratio of the He i λ4471 and He ii λ4541 lines, which results in similar values for Teff . To also test whether the relatively strong He i λ4471 line could be dominating the fit, forcing a relatively low Teff, we refitted the spectrum of N11-031 ignoring the neu- tral helium lines. This again resulted in an effective temperature of 45 kK. Interestingly, the nitrogen line analysis of Walborn et al. for N11-031 did result in a higher effective temperature of 55 kK. We are not sure whether this discrepancy is the re- sult of a systematic offset between the N v λ4603-20/N iv and He i λ4471/He ii temperature scales. A more recent compari- son by one of us (P.A.C.) of the spectral fit of Walborn et al. to new VLT data, however, showed relatively large discrepancies in the helium line fits compared to the nitrogen line fits, indi- cating that this could be the case. The discrepancy could also be due to differences in fitting assumptions and approaches. Walborn et al. adopted a surface gravity of log g = 4.0 and estimated a mass loss rate of Ṁ = 1.0 × 10−6 M⊙yr −1 from the He ii λ4686 line. Our fit indicates that the former param- eter should be lower by 0.15 dex, consequently lowering Teff. With respect to the mass loss rate we find a value higher by a factor four. An increase in Ṁ of this magnitude can have a significant effect on the strength of different nitrogen lines (e.g. Crowther et al. 2002) and, therefore, on the derived effective temperature. Massey et al. (2005) analysed a total of 11 O2 stars. They could determine the effective temperature for three dwarfs and one giant, with 47.0 kK . Teff . 54.5 kK. For the supergiants only lower limits of Teff & 42 kK and higher could be derived. These results are in agreement with our findings. However, these authors note that the correlation with Teff for the O2–3.5 spectral types is not tight. In particular no good agreement was found between the ratios of the N iii and N iv emission lines and the He i and He ii lines. Consequently, a more thorough investigation of the O2 stars based on both the nitrogen and helium spectrum is necessary to resolve this degenerate class adequately. 4.3. Gravities The distribution of our programme stars in the log Teff – log gc plane is presented in Fig. 2. To calculate the surface gravity corrected for centrifugal acceleration (log gc) the method dis- cussed by Herrero et al. (1992) and Repolust et al. (2004) was adopted. Different luminosity classes are denoted using circles, triangles and squares for, respectively, class V, III and I-II ob- jects. In this figure we see that the dwarfs, with exception of two objects, form a group clearly separated from the latter two M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 9 4.40 4.50 4.60 4.70 4.80 log Teff Fig. 2. The log Teff – log gc plane for the analysed LMC objects. Different luminosity classes are denoted using circles, triangles and squares for dwarfs, giants, and bright giants and super- giants, respectively. With exception of two objects the dwarfs are clearly separated from the luminosity class I-II objects. The giants seem to overlap with both the luminosity class V and I-II objects. Shown as a dotted line is the average log Teff – log gc relation of the SMC O-type stars of luminosity class I-II-III from Paper II. groups. In contrast to our findings in Paper II we do not find a clear separation between the dwarfs and giants. Instead the latter group of objects shows an overlap with both luminosity class V and I-II objects. Shown in Fig. 2 as a dotted line is the average log Teff – log gc relation of the SMC O-type stars of luminosity class I- II-III from Paper II. The majority of the evolved LMC objects seem to agree with this trend, that illustrates the evolutionary correlation between effective temperature and surface gravity. However, note that some evolved objects are at considerable distance from the average relation. Thus, a calibration of the two parameters for a given luminosity class should be taken with care (see also Repolust et al. 2004). The comparison of spectroscopically determined masses (Ms) with masses obtained from evolutionary calculations is presented in Fig. 3. Using the same symbols as in Fig. 2 dwarfs, giant and supergiants are distinguished. Evolutionary masses (Mev) were derived from the evolutionary tracks calculated for a metallicity of Z = 0.4 Z⊙ from Schaerer et al. (1993). The er- rors on these masses correspond to the maximum mass interval in the error box spanned by the uncertainties in luminosity and effective temperature. The tracks from Schaerer et al. do not include the effects of rotation. Consequently, this additional source of error is not included. Calculations including vrot show that in some cases one can no longer assign an unambiguous M(L, Teff). This is a result of rotationally enhanced mixing or the unknown incli- nation angle, if the star has a non-spherically symmetric dis- tribution of R⋆ and Teff , causing complicated tracks including loops during the secular redward evolution (Meynet & Maeder 2005). Assessing the impact of vrot on the Mev determination we showed in Paper II from a comparison of masses derived from 0 20 40 60 80 100 120 Spectroscopic Mass [Msun] Fig. 3. Comparison of spectroscopic masses with masses de- rived from the evolutionary tracks of Schaerer et al. (1993). The one-to-one correlation between the two mass scales is given by the dashed line. Symbols have the same meaning as in Fig. 2. The objects with the highest evolutionary masses ex- hibiting a mass discrepancy correspond to the four O2 stars in our sample. non rotating tracks to those obtained from tracks calculated for vrot = 300 km s −1 that the error in Mev will not increase by more than approximately ten percent. In Fig. 3 we see that the majority of the objects are lo- cated left of the one-to-one correlation, given by the dashed line. The error bars of twelve objects do not even touch this correlation. Consequently, we find a significant mass dis- crepancy. Similar mass discrepancy problems have been dis- cussed by e.g. Herrero et al. (1992), de Koter & Vink (2003), and Repolust et al. (2004). Most of these classical problems were attributed to limitations in the stellar atmosphere mod- els (Herrero 1993) and to potential biases in the fitting process (see Paper I). Here we cannot explain the found discrepancy in such a manner. We will provide a more thorough investigation and discussion in Sect. 5. 4.4. Helium abundances For the SMC sample that we analysed in Paper II we found a correlation between the helium surface abundance and surface gravity. This could partly be explained by evolutionary effects. As the surface gravity decreases when a star evolves away from the ZAMS, objects with lower gravities would correspond to more evolved objects and are more likely to have atmospheres enriched with helium. To investigate whether the scenario discussed above also applies to the current LMC sample, we plot the helium abun- dance as determined with the automated method as a function of log gc in Fig. 4. Also shown as a dashed line is a measure of the initial helium abundance. This value of YHe = 0.09 was calculated by averaging the surface helium abundances of the dwarf type objects with a helium abundance smaller than the total sample average. Compared to this measure for the ini- tial helium abundance a correlation between the surface gravity 10 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 2.80 3.20 3.60 4.00 4.40 log gc Fig. 4. Helium abundances as a function of surface gravity. Symbols have the same meaning as in Fig. 2. The dashed line at YHe = 0.09 is a measure for the initial helium abundance and corresponds to the mean of the helium abundances of the dwarfs with YHe smaller than the sample average. Starting at the highest gravities an increase of the helium fraction is seen down to log gc ≈ 3.6. and helium enrichment can be observed. Starting at the high- est gravities, we find an increase in the average helium abun- dance towards lower log gc. Note that the two objects exhibit- ing the largest helium fractions are supergiants. The increase can again be partially explained as an evolutionary effect. In Paper II a similar correlation between average helium abun- dance and gravity was found to exist down to the lowest gravi- ties investigated. Interestingly, in Fig. 4 we see that for our LMC sample no helium enrichment is found at log g . 3.6. Not even the super- giants show evidence of enrichment below this gravity. Why is this so? In Fig. 5, where we present the HR-diagram for our sample, the answer to this question is given. Shown as a grey area in this figure is the region in which the rotating evolu- tionary models of Meynet & Maeder (2005) predict a helium surface enrichment of at least ten percent. The majority of the evolved objects are located outside this region. Consequently, based on their specific evolutionary phase these objects are not expected to show any enrichment. Also important is the fact that we have selected the hottest objects from the N11 cluster. As a result of this our sample is biased and does not contain the low gravity objects with a high luminosity, i.e. those more likely to be enriched, as these have evolved into cool B-type stars. Apart from the supergiants also four dwarfs are found to be enriched. Figure 5, in which we have highlighted stars with a helium abundance of at least 0.12 using open symbols, shows that one these dwarfs, Sk −66 18, is located relatively close to the region in which enrichment is predicted. Consequently, ro- tationally enhanced mixing is a possible explanation. The three remaining dwarfs, N11-60, N11-065 and Sk −70 69, in contrast lie relatively close to the ZAMS. Therefore, “normal” mixing cannot explain their enrichment. Instead, a possible explana- tion is given by chemically homogeneous evolution. As this 4.24.34.44.54.64.74.8 log Teff Fig. 5. Hertzsprung-Russell diagram for the LMC sample. Symbols have the same meaning as in Fig. 2. Over plotted as grey lines are the evolutionary tracks of Schaerer et al. (1993) for Z = 0.4 Z⊙, with a black line representing the ZAMS. Open symbols indicate objects with YHe ≥ 0.12. The grey area corre- spond to the region in which the rotating evolutionary models of Meynet & Maeder (2005) predict a relative helium enhance- ment of at least 0.01. may also be linked to the mass discrepancy, we will return to it in Sect. 5. 4.5. Microturbulence The microturbulent velocities determined with the automated fitting method are also given in Tab. 2. Though the error es- timates are relatively large (see Tab. 3), we find that for the current data set the vturb determinations were sufficiently accu- rate to reveal a weak correlation between this parameter and the surface gravity. This is shown in Fig. 6, where it can be seen that for log g . 3.6 the average microturbulence recovered from the line profiles increases systematically. The situation for log g & 3.6 is less clear, as the error bars are on average larger and the values for vturb are more or less randomly distributed between 0 and 20 km s−1. The reason for this is that for larger values of the surface gravity the line profiles become intrinsi- cally broader due to the increased Stark broadening, making it more difficult to accurately recover vturb from the line profiles only. Based on samples predominantly consisting of unevolved early B-type Galactic stars other authors have also found a relation between microturbulence and surface gravity, e.g. Kilian et al. (1991), Gies & Lambert (1992) and Daflon et al. (2004). More recently Hunter et al. (2006) analysed a sample of early B-type stars in the Magellanic clouds and also found a trend of increasing vturb for decreasing log g. To derive the values for vturb all these authors relied on curve-of-growth tech- niques, which were applied to metal lines such as those of Si iii and O iii calculated using plane parallel models. Consequently, our line profile based analysis is an independent confirmation of the existence (or requirement by lack of a physical explana- M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 11 2.8 3.2 3.6 4.0 4.4 log g [cm/s2] Fig. 6. Microturbulent velocities determined using line profile fits as a function of surface gravity. Symbols have the same meaning as in Fig. 2. For log g . 3.6 a trend of increasing vturb with decreasing log g is visible. For larger gravities the uncertainties in the vturb determinations are too large to discern any possible relation. tion or failures in the line broadening mechanisms) of micro- turbulence in the atmospheres of these type of stars. The physical mechanism explaining the observed microtur- bulence or its relation to the surface gravity is still poorly un- derstood. Kudritzki (1992) and Lamers & Achmad (1994) have argued that the observed microturbulence might be the result of a stellar outflow, implying the existence of a velocity gradient in the photospheric layers, which can mimic microturbulence- like desaturation effects. Smith & Howarth (1998), however, showed by applying a simple core-halo model to the Galactic O9.7 supergiant HD 152003 that this effect would not be suffi- cient to explain the observed vturb. Indeed, our analysis employ- ing a unified photosphere and wind model confirms that the mi- croturbulence cannot be explained as an artifact of a transonic velocity field. A possible explanation for the gravity dependence could be related to instabilities in the wind. These instabilities are reflected in the large turbulent velocities (∼100–200 km s−1) necessary to fit the wind lines in the intermediate and outer wind (e.g. Groenewegen & Lamers 1989; Haser et al. 1998; Evans et al. 2004b) and could be related to shocks due to the in- trinsic line-driven instability (Lucy 1983; Owocki et al. 1988; Owocki & Puls 1999). For a low surface gravity the wind starts at larger Rosseland optical depth compared to the high grav- ity case. Therefore, the line forming region of low gravity ob- jects contains a relatively large contribution originating from (the base of) the wind. Consequently, this region could be af- fected by the onset of the line-driven instability introducing wind-turbulence into the line profiles. Tentative support for the above described scenario may be implied by Fig. 7, where we show vturb as function of the line forming region of He i λ4471 in units of the stellar radius. The location of the line forming region is defined as the position at which the radial optical depth in the line core reaches a value of 2/3. This figure shows that when the radial distance 1.00 1.02 1.04 1.06 1.08 1.10 R(τline=2/3) [Rstar] Fig. 7. Microturbulent velocity as a function of the location of the line forming region of He i λ4471, which is defined as the location where the radial optical depth in the line core reaches a value of τ = 2/3. A weak trend is visible that suggests that for increasing extension of the atmosphere larger values of vturb are necessary to fit the line profiles. to this position increases also the average microturbulent ve- locity increases. Though the trend is weak, it does seem to indicate that when the atmosphere becomes more extended, higher values of vturb are necessary to reproduce the line pro- files. It may appear that this implies that the line forming re- gion enters in to the regime where wind turbulence develops. However, for this statement no compelling evidence is avail- able, as we do not find any correlation between vturb and the distance between the line forming region and (for instance) the sonic point. Moreover, for most objects at R(τline = 2/3) > 1.04 values of vturb close to the maximum allowed value are found. Consequently, these values should be interpreted as lower lim- its. Therefore, we can only conclude that our analysis points towards a gradient in the turbulent velocity, possibly connected to a link between microturbulence and wind instabilities, and suggest further investigation in this direction. 4.6. Wind parameters The wind parameters and their uncertainties determined using the automated method are listed in Tabs. 2 and 3. Compared to our SMC analysis we find that we were able to accurately determine these parameters for a significantly larger number of objects. This is mainly the result of an on average higher signal-to-noise ratio of the spectra as well as of denser winds for the LMC objects compared to their SMC counterparts. In total we determined 22 mass loss rates and 6 upper limits. The upper limits are defined (and can be identified in Tab. 3) by an error bar − log Ṁ > 1.0 dex. To provide a meaningful comparison of the mass loss rates we place the LMC objects in the modified wind momentum luminosity diagram. This diagram shows as function of stellar luminosity the distribution of the so-called modified wind mo- mentum, which is defined as Dmom ≡ Ṁv∞R ⋆ . Not only does this allow for an assessment of the behaviour of Ṁ within our 12 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 4.5 5.0 5.5 6.0 6.5 log(L/Lsun) Fig. 8. Modified wind momentum (Dmom) in units of g cm s−2 R⊙ 1/2 vs. luminosity. Symbols have the same mean- ing as in Fig. 2. Upper limits are shown as inverted triangles. Dashed lines correspond to the predicted wind-momentum lu- minosity relations (WLR) from Vink et al. (2000, 2001). The upper, middle and lower relation, respectively, correspond to predictions for Galactic, LMC and SMC metallicity. The ob- served modified wind momenta show a strong correlation with luminosity with an average relation that lies in between the pre- dicted Galactic and SMC WLR. This is quantified by the em- pirical WLR that was constructed for the LMC objects and that is shown by the dotted line. The open square corresponds to the wind momentum corrected for clumping of the supergiant Sk −67 166. Shown as a dashed-dotted is the empirical WLR obtained using this corrected Dmom. sample, it also provides a convenient method to compare the observed wind strengths to the predictions of line driven wind theory. According to this theory Dmom is predicted to behave as log Dmom = x log L⋆/L⊙ + log D◦ , (1) where L⋆ is the stellar luminosity (Kudritzki et al. 1995; Puls et al. 1996). In this equation x is the inverse of the slope of the line-strength distribution function corrected for ionisation effects (Puls et al. 2000). The vertical offset D◦ is a measure for the effective number of lines contributing to the accelera- tion of the outflow. In Fig. 8 the distribution of the modified wind-momenta for our programme stars are presented. Indicated using circles, triangles and squares are objects of, respectively, luminosity class V, III and I-II. Upper limits are shown as grey inverted triangles. A clear correlation between L⋆ and Dmom can be ob- served in this figure. Over an order of magnitude in L⋆ the av- erage modified wind-momentum decreases by approximately 1.5 dex. A comparison of the behaviour of Dmom with pre- dictions is facilitated by the theoretical WLRs calculated by Vink et al. (2000, 2001) that are shown as a set of dashed lines. The upper, middle and lower of these predicted power laws were calculated for, respectively, Solar, LMC and SMC metal- licity. Compared to these predictions we find that the LMC wind-momenta are approximately confined between the theo- retical WLR for Galactic and SMC metallicity. In other words, compared to the Galactic and SMC case, stars in the LMC have intermediate wind strengths. To further quantify the behaviour of LMC winds relative to that of Galactic and SMC outflows we have fitted a power law to the observed modified wind-momentum distribution, consis- tently accounting for both the symmetric errors in L⋆ and the asymmetric errors in Dmom, This yielded the following empiri- cal WLR log Dmom = (1.81 ± 0.18) log L⋆/L⊙ + (18.67 ± 1.01) . (2) The theoretical LMC relation from Vink et al. is given by x = 1.83 and Dmom = 18.43. The error bars of the theoretical and empirical relations are in agreement. More importantly, in Fig. 8 the empirical relation, shown as a dotted line, is found to lie between the predicted Galactic and SMC relations. These predictions have been found to be in good agreement with the observed Galactic WLR (Repolust et al. 2004, Paper I) and observed SMC WLR (Paper II). Consequently, our empirical LMC WLR is quantitative evidence for the fact that massive stars in this system have mass loss rates intermediate between those of massive stars in the Galaxy and SMC. The differences between the empirical and theoretical LMC WLR at the start and end of the observed luminosity range are, respectively, 0.17 and 0.16 dex. In Fig. 8 this seems to imply a systematic offset between the two relations. However, we note that these differences are still smaller than the typi- cal uncertainty in Dmom of 0.2 dex. More importantly, no cor- rection was applied for the possibility that the winds of our sample stars are, in contrast to our assumption, not smooth but structured. In recent years evidence has been mounting for this so-called clumping in the winds of O- and early B-type stars. In particular spectroscopic modelling of UV (resonance) lines (Crowther et al. 2002; Hillier et al. 2003; Bouret et al. 2003; Massa et al. 2003; Martins et al. 2004, 2005b; Bouret et al. 2005; Fullerton et al. 2006) seems to suggest the existence of clumping factors in the range of 10-100, implying correspond- ing reductions of Ṁ by factors 3 up to 10. More recently, Puls et al. (2006) also found clumping factors of the order 5 to 10 (normalized to the unknown clumping properties in the outermost, radio-emitting wind) from the analysis of Hα, in- frared, millimetre and radio fluxes. In the present study we try to account for possible wind clumping effects by correcting the mass loss rates of stars with Hα in emission. Markova et al. (2004) and Repolust et al. (2004) argue that the mass loss rates of these stars could be overestimated as a result of the fact that Hα emission lines are formed over a relatively large volume where clumping might have set in. In contrast, for stars with Hα in absorption the line is formed relatively close to the stel- lar surface, where clumping effects are negligible. Based on the comparison of dwarfs and supergiants in their Galactic sample Repolust et al. derived a numerical correction factor of 0.44 for the mass loss rates of supergiants with Hα in emission. We have applied the clumping correction to the super giant Sk −67 166, which has a Hα emission profile. In Fig. 8 its new M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 13 wind momentum is indicated using an open symbol. Using this value the following empirical WLR is obtained1 log Dmom = (1.43 ± 0.17) log L⋆/L⊙ + (20.77± 0.97) . (3) In Fig. 8 we see that for log L⋆/L⊙ & 5.3 the new WLR com- pares better to the Vink et al. relation. For lower luminosities the situation is less clear. Due to the large uncertainties, this range has a relatively low weight in the fit. Consequently, a discrepancy for low L⋆ is less significant than the good agree- ment obtained for the higher luminosities. For this reason it is difficult to investigate the existence of a “weak wind prob- lem” for stars at log L⋆/L⊙ . 5.3, first reported by Bouret et al. (2003). This study, as well as later studies (Hillier et al. 2003; Evans et al. 2004a; Martins et al. 2004, 2005b) report a steep- ening in the WLR relation relative to predictions starting at about the above mentioned L⋆, leading to an over prediction of the wind strength by up to a factor 100 at log L⋆/L⊙ ∼ 4.5. Our LMC results do not appear to confirm this break be- tween observations and seem to follow the predictions down to log L⋆/L⊙ ≈ 5.0. In a forthcoming paper we will present a comprehensive overview of the observed WLR relations in our Galaxy and the Magellanic Clouds as well as a thorough dis- cussion of the successes and failures of the theory of radiation driven winds in predicting the WLR, including possible causes for the weak wind problem (Mokiem et al. in preparation). 5. The mass discrepancy In Fig. 9 we investigate the mass discrepancy of our LMC tar- gets as a function of the helium surface abundance. On the ver- tical axis a measure for this discrepancy is shown, which is scaled to the mean of the evolutionary and spectroscopic mass. This ensures that positive and negative discrepancies follow the same linear scale. For non-enriched stars, i.e. YHe . 0.10, approximately three times as many objects lie above the one- to-one correlation than below it. Therefore, in contrast to our finding for the equivalent SMC case (Paper II) a significant mass discrepancy is found for our sample of non-enriched LMC stars. The reason for this is unclear. As we stated before our analysis employs state-of-the-art atmosphere models, and is in principle not hampered, as were previous studies, by poten- tially unoptimised fits. As stated, in our SMC dataset, analysed in an identical manner, no evidence was found for a mass dis- crepancy for YHe < 0.10. Massey et al. (2005) also study the Ms vs. Mev problem in a set of LMC stars. For objects hotter than 45 kK they find a mass discrepancy that is even stronger than what we find. This behaviour could be the result of an underestimate of the photospheric line pressure in this high temperature regime. In Fig. 9 we have highlighted the objects with Teff ≥ 45 kK using open symbols. Although their mass discrepancy is considerable 1 The small errors in the relevant parameters of Sk −67 166, which dominates the WLR at very high luminosity, cause the significant dif- ference between Eqs. 2 and 3. Note that our least square fitting does not account for the correlation of both quantities (due to R⋆). In view of the well known distance these effects are likely small compared to the Galactic case (see Markova et al. 2004; Repolust et al. 2004). -1.00 -0.50 0.05 0.10 0.15 0.20 0.25 0.30 Fig. 9. Mass discrepancy as a function of helium abundance for the LMC sample. Evolutionary masses used to calculate the discrepancy were derived from the non-rotating tracks of Schaerer et al. (1993). Symbols have the same meaning as in Fig. 2. The open symbols correspond to stars with Teff ≥ 45 kK. they do not stand out as a separate group.To further illustrate this, note that the hottest supergiant in our sample Sk −67 166 (Teff = 40 kK) at YHe = 0.28 has a Ms of 75 M⊙ which is in very good agreement with its Mev of 70 M⊙. Consequently, though we can not explain the mass discrepancy we do not an- ticipate that it is connected to a flawed treatment of the photo- spheric radiation pressure that manifests itself at metallicities as high as that of the LMC environment (but not yet at values typical for the SMC). If we accept the analysis of stellar and photospheric pa- rameters, the presence of a mass discrepancy may point to an oversimplified picture of the evolution of massive stars used to determine Mev and/or, possibly, to a breakdown of the as- sumption of a spherically symmetric atmosphere. One obvious simplification may be that we have used tracks for non-rotating stars. One of the effects introduced by rotation is a wide bi- furcation in the evolutionary tracks (Maeder 1987). Stars rotat- ing faster than roughly half the surface break-up velocity will essentially follow tracks representative of homogeneous evo- lution. Langer (1992) showed that as a result of this the M/L- ratio will be a monotonically decreasing function for increasing helium enrichment. Consequently, stars evolving along homo- geneous tracks are expected to be increasingly under-massive for increasing helium abundance. One would therefore derive a positive mass discrepancy if the evolution of a rapidly rotating star was incorrectly described using non-rotating or modestly rotating model tracks. So, can rapid rotation be used to explain the observed mass discrepancy? Let us first focus on the supergiants. For these stars the mass discrepancy problem appears absent, though we note two exceptions at YHe = 0.07 and YHe = 0.09. This ab- sence of signs of a distinct mass discrepancy was also found in our study of SMC bright giants and supergiants (Paper II). The conclusion seems to be that the supergiants follow non- rotating or modestly rotating evolutionary tracks. This implies that, apparently, all supergiant targets that we have selected in 14 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC both the SMC (Paper II) and LMC happen to have started out their evolution with low or modest initial rotational velocities. As the initial rotational velocity distribution derived in Paper II for the NGC 346 cluster implies that only some 5 − 15 percent of stars are expected to evolve along homogeneous tracks this should not be alarming. One supergiant, however, might not fit this picture: Sk -67 166. This source does not feature a mass discrepancy, but does show an increased helium abundance. Given its position in the HRD this is quite difficult to explain in term of evolution without rotation. A scenario accounting for a more delicate interplay between mass loss and rotation may be required to explain its properties (see Herrero & Lennon 2004). Now let us turn to the dwarfs and giants. For our SMC sample we found a correlation between the helium surface en- richment and the mass discrepancy for class V and III stars, with Ms being systematically smaller than Mev for YHe & 0.11. Interestingly, this dependence was only found for dwarfs and giants (see above). It was suggested that this behaviour was the result of rotationally enhanced mixing, enriching the at- mospheres with primary helium. The dwarfs and giants at YHe > 0.11 displayed in Fig. 9 also systematically suffer from a positive mass discrepancy. Consequently, this scenario also seems to be a good explanation for the situation in the LMC sample. However, we realise that these seven objects are a rel- atively small sample and that the typical uncertainties in YHe are 0.03. This statement should therefore be seen as a working hypothesis. For helium enriched dwarfs and giants the above scenario seems a logical one, as helium enrichment early on in the evo- lution must imply efficient mixing. For the dwarfs and giants that do not show excess helium in their surface layers such a clue or indication is not present (unless they evolve left of the main sequence as do chemically homogeneous stars, see Sect. 5.1). In principle these non YHe-enriched stars could ro- tate sufficiently rapid to cause a mass discrepancy, but not to the extent of chemically homogeneous evolution. However, if so, a very significant fraction of the stars should have started their evolution at super-critical rotation. At least for the stars studied in the SMC cluster NGC 346 (Paper II) this is not the case. 5.1. Chemically homogeneously evolving O2 stars Three out of the four O2 stars are found to lie to the left of the ZAMS in Fig. 5 and exhibit relatively large mass discrepan- cies (see open symbols in Fig. 9). These are N11-026, BI 237 and BI 253. For a discussion of the effective temperature of the fourth O2 object, N11-031 (Teff ∼ 45 kK), we refer to Sect. 4.1. Even though the hottest three (Teff ∼ 53 − 54 kK) are not sig- nificantly enriched in helium it is tempting to speculate on a possible (near) homogeneous evolution of these objects. This would not only provide an explanation for their mass discrep- ancy, it would also explain their peculiar location in the HRD. The possibility that these are true ZAMS stars is very exciting. However, due to the short evolutionary time scales in this part of the HR-diagram, it would also be unlikely. Based on the non- rotating model tracks of Schaerer et al. (1993) we estimate that a 60 M⊙ star will within 2 Myr evolve away from the ZAMS to a location in the HRD at Teff ≈ 45 kK. This is well beyond the error bars on the parameters of these objects. Given the fact that N11-026 and BI 253 are thought to be associated with the LH 10 and 30 Doradus clusters, respectively, which have ages of approximately 3 (see Sect. 6.2) and 2 Myr (de Koter et al. 1998), a normal evolutionary scenario appears unlikely. Based on its large radial velocity the field star BI 237 probably is a runaway star (Massey et al. 2005), therefore it is also likely to be relatively old. The problems with a reconciliation of the hottest three O2 stars with fully homogeneous evolution and an age of at least 2 Myr, are i) that their surface helium abundances are not sig- nificantly enriched, and ii) that their rotational velocities are not extreme. The latter issue need not be “a smoking gun” con- sidering the possibility that we may see them relatively pole- on and the fact that their vr sin i values, ranging between 110- 190 km s−1, are above the sample average. Concerning the first point, it appears that we must concede to the possibility that stars may evolve along tracks similar to those for homogeneous evolution while in fact they are not fully homogeneous – i.e. the near surface layers are not yet strongly affected by the mix- ing that must occur deeper in. Having said this, we do point to the fact that the error bars on YHe do allow for relatively large ages even within the hypothesis of fully homogeneous evolu- tion. Within the error bars the maximum YHe is ∼ 0.13 for all three stars. It takes fully homogeneously evolving stars of 60 and 40 M⊙ about ∼ 1.5 and ∼2.0 Myr, respectively, to build up this amount of helium enrichment (S. -C. Yoon, private com- munication). This is close to the derived cluster ages of LH 10 and 30 Doradus. For reference, if the star would evolve along near-homogeneous tracks they would be older. We tentatively conclude that the HRD position, helium abundance, and rotational velocity of the three hottest O2 stars in our sample could be consistent with (near-)homogeneous evolution. Evidence for efficient rotation-induced mixing dur- ing the main sequence phase of O stars has also been presented by Lamers et al. (2001), based on the chemical abundance pat- tern of ejected circumstellar nebulae. 5.2. Large sample trends To firmly establish our findings with respect to the mass dis- crepancy problems, we compare in Fig. 10 spectroscopic and evolutionary masses for all stars that have been analysed using our automated fitting method so far. Presented in this figure are the combined Galactic (Paper I), SMC (Paper II) and the cur- rent LMC samples, corresponding to a total of 71 O- and early B-type stars. The figure confirms our two main findings. First, the good agreement between Ms and Mev for the supergiants is clearly visible. This is an encouraging finding and we believe that this means that the improvements in the stellar atmosphere mod- els, evolutionary calculations and spectral analysis techniques have finally resolved the long standing mass discrepancy as found by Herrero et al. (1992). The correspondence between the spectroscopic and evolutionary mass scale, in particular for M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 15 Fig. 11. The FLAMES field for the star forming region N11 with the associations LH9 (south of centre) and LH10 (north of centre). The FLAMES targets are identified using circles and their star identification is given. From Evans et al. (2006). [Included as separate JPG file in astro-ph submission.] -1.00 -0.50 0.05 0.10 0.15 0.20 0.25 0.30 Fig. 10. Mass discrepancy as a function of helium abundance for the combined samples from Paper I (Galactic), Paper II (SMC) and the current LMC sample. Symbols have the same meaning as in Fig. 2. Note the good correspondence between Ms and Mev for the bright giants and supergiants (square sym- bols). increasing helium enrichment is striking. The reason for this is probably related to the fact that the class I-II objects are found in the region of the HR-diagram in which stars are not expected to undergo extreme evolutionary phases, such as homogeneous evolution. Consequently, the enriched bright giants and super- giants have evolved along relatively simple evolutionary tracks, which (in this respect) appear well understood. Our second finding, i.e. that of a correlation of mass dis- crepancy with the helium abundance, is also corroborated by the large sample in Fig. 10. For YHe > 0.10 all dwarfs and gi- ants, except for one dwarf, are found above the Ms = Mev line. In all fairness, given the typical uncertainty of 0.03 in YHe, from a statistical point of view the region 0.09 < YHe < 0.12 should be regarded with care. For larger helium abundance, however, the correlation can be regarded to be statistically significant. A total of 11 objects show a positive mass discrepancy, with only a single counter example. Moreover, the magnitude of the dis- crepancy seems to be related to the amount of enrichment. This strongly points to efficient mixing in the main sequence phase, leading to (near-)chemically homogeneous evolution. 6. The evolutionary status of N11 In this section we explore the evolutionary status of the N11 field. We will first briefly outline the current understanding of N11 with respect to the OB associations LH9 and LH10 and in particular the sequential evolutionary link between the two. Based on the analysis of the 22 stars in our sample associated with these clusters we will then estimate their ages and discuss whether they are compatible with a sequential star formation scenario. 6.1. LH9 and LH10 in N11 N11 is an intricate giant H ii region containing several massive star forming complexes. The largest of these are the OB associ- ations LH9 and LH10 (Lucke & Hodge 1970). An image of the region, including the adopted star identifications, is presented in Fig. 11 (Evans et al. 2006). Several studies have suggested that the star forming activity in these associations are linked. In particular the study by Parker et al. (1992) has provided a key to understanding of the structure and formation of LH9 and LH10. Their analysis of the stellar content of N11 revealed the presence of several O3-O5 stars and possible ZAMS stars in LH10. In contrast the earliest spectral type associated with LH9 was found to be O6. They also found the slope of the ini- tial mass function of LH10 to be significantly flatter than that of LH9, indicating that the former contains a higher ratio of high mass to low mass stars. Combined with the fact that the reddening of LH10 is larger than that of LH9, it was concluded that LH10 is the younger of the two clusters. Based on these findings the authors propose an evolutionary link between the two, where star formation in LH10 could possibly be triggered by the stellar winds and supernovae of massive stars in LH9. Further evidence for a sequential star formation sce- nario was given by Walborn & Parker (1992), who found a dual structural morphology of N11 analogous to that of the 30 Doradus H ii region. In the latter substantial evidence sug- gests the presence of current star formation in regions sur- rounding the central star cluster (e.g. Walborn & Blades 1987; Hyland et al. 1992, also see Walborn & Blades 1997). This sec- ondary burst seems to be set off approximately 2 Myr after the initial star formation took place, possibly initiated by the en- ergetic activity of the evolving cluster core. Walborn & Parker argue that this is very similar to N11 where the stellar content of LH9 and LH10 also suggests an age difference of ∼2 Myr (also see Walborn et al. 1999). The process in N11, however, would be advanced by∼2 Myr, classifying it as an evolved 30 Doradus analogue, though less massive. More recently, several bright IR sources showing charac- teristics of young stellar objects were discovered in the N11B nebula surrounding LH10 by Barbá et al. (2003). These objects are probably intermediate mass (pre-)main-sequence Herbig Ae/Be stars belonging to the same generation as do the LH10 objects (the pre-main-sequence evolutionary timescales of in- termediate mass stars being longer than that of massive OB stars). Barbá et al. also found that the massive stars in LH10 have blown away their ambient molecular material and are cur- rently disrupting the surface of the parental molecular cloud material surrounding LH10 (i.e. the material in N11B). 16 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 4.44.54.64.7 log Teff 65 66 Fig. 12. Comparison of the programme stars located in N11 with isochrones derived from the evolutionary tracks of Schaerer et al. (1993). The symbols used to denote dwarf, giants and bright- and supergiants are, respectively, circles, triangles and squares. Filled and open symbols are used to distinguish stars associated with LH9 and LH10, respectively. Black symbols refer to objects located within a radius of two arc minutes of the LH9 and LH10 core, whereas the grey symbols correspond to stars found outside this radius. Labels correspond to the N11 identification numbers given in Tab. 1. Isochrones (dashed lines) are shown from one up to ten million years with 1 Myr intervals. For reference evolutionary tracks from Schaerer et al. (1993) are drawn as grey lines, where for clarity purpose only the blueward part of the evolution for the most massive tracks is shown. The ZAMS corresponding to these tracks is given by the black solid line. 6.2. The ages of LH9 and LH10 To estimate the ages of our programme stars we compare their location in the HR-diagram to theoretical isochrones in Fig. 12. The luminosity classes V, III and I-II are represented using, re- spectively, circles, triangles and squares. To differentiate be- tween stars associated with LH9 and LH10, respectively, filled and open symbols are used. Membership is defined on the ba- sis of minimum distance to either the LH9 or LH10 cluster core. This, therefore, should be taken with some care: the cores are about 4 arcmin or 60 parsec apart, which can be traversed in 2 Myr if the proper motion of the star is some 30 km s−1. Runaway O- and B-type stars have typical velocities of 50– 100 km s−1, therefore, it is entirely possibly that (a few) indi- vidual objects are assigned to the wrong association. For posi- tion reference see Fig. 11. Isochrones in Fig. 12 are shown as dashed lines for one up to ten million years with 1 Myr intervals and were derived from the evolutionary tracks of Schaerer et al. (1993). Note that these tracks do not account for the effects of rotation. The distribution of the stars in Fig. 12 is such that they can be separated in a group of objects younger than three million years and a group of objects older than this age. We also see that the oldest objects are predominantly found in LH9, whereas LH10 contains the largest fraction of young objects, suggest- ing that the clusters can indeed be separated in terms of age. At first sight it seems that they can be characterised by an age of ∼7 Myr and ∼2 Myr, respectively. However, a significant num- ber of objects in both clusters are found to be several million years younger and older than these preliminary ages. A possi- ble explanation for this large age scatter could be “contamina- tion” by field stars. To assess this possibility we have assigned grey symbols to all objects outside a radius of two arc minutes from the cluster cores. Disregarding these objects reduces the age scatter significantly; still a number of stars appear to con- tradict with the notion of two coeval populations. To investigate the age distributions in more detail the indi- vidual age estimates are shown in Fig. 13. LH9 and LH10 ob- jects, which are shown using the identical symbols as in Fig. 12, are placed in, respectively, the left and right part of the diagram and are separated using a dashed line. First concentrating on the M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 17 8 29 33 36 42 45 58 65 66 123 4 26 31 32 38 48 51 60 61 68 72 87 N11-# Fig. 13. Individual age estimates for stars in N11 based on the non-rotating tracks of Schaerer et al. (1993). Symbols have the same meaning as in Fig. 12. The left and right part of the diagram, respectively, contain stars associated with LH9 and LH10 and are separated using a dashed line. The upward point- ing arrow denotes a lower age limit for N11-065 determined from its surface helium abundance using chemically homoge- neous evolutionary models (Yoon et al. 2006; Yoon & Langer 2005). LH9 objects we see that they are near-coeval with exception of four dwarfs. Of these the two non-core members are located at a distance of approximately six arc minutes from the central concentration. Therefore, it is probable that they are either spa- tially not related to N11 or that possibly their formation was triggered more recently by the stellar activity in LH9. The third dwarf N11-065 occupies a location very close to the ZAMS and in principle would be the youngest member of LH9. However, we also find a considerable helium enrichment for this object (YHe= 0.17). Combined with the fact that N11-065 has a large mass discrepancy, this suggests that it might be evolving chem- ically homogeneously. Consequently, a more appropriate age estimate should be derived from tracks appropriate for this kind of evolution. Adopting such tracks from Yoon et al. (2006, also see Yoon & Langer 2005) we derive a lower limit of 6 Myr for the age of N11-065 based on the surface helium abundance. In Fig. 13 this estimate is indicated using an upward pointing arrow, and is in good agreement with the bulk of the LH9 stars. The dwarf N11-123 exhibits no chemical peculiarities and its location on the sky places it in the central concentration of LH9, suggesting that it should have been formed in the burst of star formation that formed the cluster. However, given the fact that N11-123 is in principle the only star deviating from the coeval nature of LH9 we suspect that it is like N11-058 and N11-066 part of the periphery of the cluster and that its relatively close position to the core is due to a projection ef- fect. Consequently, we conclude that the central concentration of LH9 is coeval with an age of ∼7.0 ±1.0 Myr. The error bar is of the same order of magnitude as the 0.5–1.0 Myr intro- duced by uncertainties in the evolutionary tracks due to effects of (relatively modest) rotation (see Paper II). The right part of Fig. 13 shows that the stars in the cen- tral concentration of LH10 have ages ranging from one up to approximately six million years. Despite this large scatter it is clear that the majority of the stars are younger than 4.5 Myr and that only one object (N11-087) in the cluster core is older than this age. An explanation for the large age of the latter ob- ject might be that it formed in LH9 and, over time, migrated towards LH10. As explained at the start of this section, this is a possibility given that the distance from the centre of LH9 to the current position of N11-087 is only three arc minutes. Considering the error bars on the age determinations and the additional uncertainty of 0.5–1.0 Myr introduced by rotation (see Paper II), we finally estimate an age best describing LH10 of ∼3.0 ±1.0 Myr. 6.3. Sequential star formation? If the starbursts in N11 are sequential the formation of LH10 should be induced by supernova explosions and/or stellar winds in LH9. Given the age difference and distance between the two clusters we find that both means of triggering are possi- ble. In the first case the age difference of approximately four million years is compatible with the time of ∼3 Myr it takes before the most massive stars end their life in a supernova (e.g. Schaller et al. 1992). The time needed for the supernova shock to cross the distance of∼3 arc minutes between LH9 and LH10, corresponding to approximately 40 parsec at the distance of the LMC, is only ∼105 years (see e.g. Falle 1981); hence it is a possible scenario. If we consider triggering through stellar winds, the time scales are also compatible. Garcia-Segura et al. (1996), for instance, using hydrodynamical simulations have shown that a 60 M⊙ star can create a wind-driven bubble in the interstellar medium of ∼50 pc during its main sequence lifetime of 3 Myr. Consequently, this scenario seems to be appropriate as well and in agreement with the determined time scales. In view of the above we conclude that a sequential scenario for LH9 and LH10 seems very likely. A combination of super- novae and stellar winds from stars in LH9 may have initiated star formation in LH10. 7. Summary and conclusions We have analysed a sample of 28 massive OB-type stars lo- cated in the LMC. For a homogeneous and consistent treatment of the data we employed the automated fitting method devel- oped by Mokiem et al. (2006), which combines the genetic al- gorithm based optimisation routine pikaia (Charbonneau 1995) with the fast non-LTE unified stellar atmosphere code fastwind (Puls et al. 2005). The sample is mostly drawn from the targets observed within the context of the VLT-FLAMES survey of massive stars (Evans et al. 2005). In total 22 of these stars are located in the LH9 and LH10 clusters within the giant H ii re- gion N11. This region is believed to have been the scene of sequential star formation, with the stellar activity in LH9 ignit- ing secondary starbursts in different associations in N11. Our main findings are summarised below. 18 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC i) The effective temperature per spectral sub-type of the LMC stars is found to be intermediate between that of Galactic and SMC O- and early B-type stars, with the LMC objects being, respectively, cooler and hotter by typically ∼2 kK. ii) Based on the helium and hydrogen lines it was possible to determine the effective temperatures, though with relatively large error bars, of the four O2 stars in our sample. Three of these are found to be hotter by more than 3 to 7 kK (see Sect. 4.2) compared to the O3 star in our sample, suggesting that O2 stars indeed represent a hotter subgroup within the O-type class. However, we note that for the one ON2 star a relatively low Teff of 45 kK was obtained, indicating that the N iv and N iii classification lines (Walborn et al. 2002b) are not fully compatible with the helium lines traditionally used for classification. iii) The spectroscopically determined masses of the dwarf and giant stars in our set of programme stars are found to be systematically smaller than those derived from non-rotating evolutionary tracks. For helium enriched dwarfs and giants, i.e. those having YHe > 0.11, we find that all show this mass discrepancy. The same was found in an analysis of SMC stars using the same methods (Mokiem et al. 2006). We in- terpret this as evidence for efficient rotationally enhanced mixing leading to the surfacing of primary helium and to an increase of the stellar luminosity. iv) The bright giants and supergiants do not show any mass discrepancy, regardless of the surface helium abundance. This also is consistent with the finding for Galactic and SMC class I-II objects studied with the same methodology (Mokiem et al. 2005, 2006). This suggests that shortly after birth all these stars must have rotated at less than about 30 to 40 percent of the surface break-up velocity. v) A weak correlation is found between microturbulent veloc- ity and surface gravity. More extended atmospheres (i.e. lower gravity stars) require a relatively large vturb to fit the lines. The reason for this relation is unclear, however, it does not seem to be connected to the lines being formed closer to the sonic point of the wind flow in low gravity stars. vi) From a comparison of modified wind momenta Dmom we find that the wind strengths of LMC stars are weaker com- pared to Galactic stars, and stronger compared to SMC stars. Comparing the derived Dmom as a function of lumi- nosity with predictions for LMC metallicities by Vink et al. (2001) yields good agreement in the entire luminosity range that was investigated (5.0 < L/L⊙ < 6.1). vii) We have determined an age of, respectively, ∼7.0±1.0 Myr and ∼3.0±1.0 Myr for the clusters LH9 and LH10. The age difference and relative distances are in good agreement with a sequential star formation scenario, in which stellar activ- ity in LH9 triggered the formation of LH10. Acknowledgements. We would like to thank Sung-Chul Yoon and Alexander van der Horst for constructive discussions, and George Meynet for providing us with the evolutionary models of the Geneva group. M.R.M. acknowledges financial support from the NWO Council for Physical Sciences. S.J.S. acknowledges the European Heads of Research Councils and European Science Foundation EURYI (European Young Investigator) Awards scheme, supported by funds from the Participating Organisations of EURYI and the EC Sixth Framework Programme. A.H. and F.N. acknowledge support from the Spanish MEC through project AYA2004-08271-C02. JSV acknowl- edges RCUK for his Fellowship. Spectral fits were calculated us- ing the LISA computer cluster at SARA Computing & Networking Services. References Ardeberg, A., Brunet, J. P., Maurice, E., & Prevot, L. 1972, A&AS, 6, Barbá, R. H., Rubio, M., Roth, M. R., & Garcı́a, J. 2003, AJ, 125, Bouret, J.-C., Lanz, T., & Hillier, D. J. 2005, A&A, 438, 301 Bouret, J.-C., Lanz, T., Hillier, D. J., et al. 2003, ApJ, 595, 1182 Bromm, V., Kudritzki, R. P., & Loeb, A. 2001, ApJ, 552, 464 Brunet, J. P., Imbert, N., Martin, N., et al. 1975, A&AS, 21, 109 Charbonneau, P. 1995, ApJS, 101, 309 Chiosi, C. & Maeder, A. 1986, ARA&A, 24, 329 Conti, P. S., Garmany, C. D., & Massey, P. 1986, AJ, 92, 48 Cranmer, S. R. & Owocki, S. P. 1995, ApJ, 440, 308 Crowther, P. A., Hillier, D. J., Evans, C. J., et al. 2002, ApJ, 579, 774 Daflon, S., Cunha, K., & Butler, K. 2004, ApJ, 604, 362 de Koter, A., Heap, S. R., & Hubeny, I. 1998, ApJ, 509, 879 de Koter, A. & Vink, J. S. 2003, in IAU Symposium, ed. K. van der Hucht, A. Herrero, & C. Esteban, 164 Evans, C. J., Crowther, P. A., Fullerton, A. W., & Hillier, D. J. 2004a, ApJ, 610, 1021 Evans, C. J., Lennon, D. J., Smartt, S. J., & Trundle, C. 2006, A&A, 456, 623 Evans, C. J., Lennon, D. J., Trundle, C., Heap, S. R., & Lindler, D. J. 2004b, ApJ, 607, 451 Evans, C. J., Smartt, S. J., Lee, J.-K., et al. 2005, A&A, 437, 467 Falle, S. A. E. G. 1981, MNRAS, 195, 1011 Fullerton, A. W., Massa, D. L., & Prinja, R. K. 2006, ApJ, 637, 1025 Garcia-Segura, G., Mac Low, M.-M., & Langer, N. 1996, A&A, 305, Garmany, C. D., Conti, P. S., & Massey, P. 1987, AJ, 93, 1070 Gies, D. R. & Lambert, D. L. 1992, ApJ, 387, 673 Grevesse, N. & Sauval, A. J. 1998, Space Science Reviews, 85, 161 Groenewegen, M. A. T. & Lamers, H. J. G. L. M. 1989, A&AS, 79, Haser, S. M., Pauldrach, A. W. A., Lennon, D. J., et al. 1998, A&A, 330, 285 Henize, K. G. 1956, ApJS, 2, 315 Herrero, A. 1993, Space Science Reviews, 66, 137 Herrero, A., Kudritzki, R. P., Vilchez, J. M., et al. 1992, A&A, 261, Herrero, A. & Lennon, D. J. 2004, in IAU Symposium, ed. A. Maeder & P. Eenens, 209–+ Herrero, A., Puls, J., & Najarro, F. 2002, A&A, 396, 949 Hillier, D. J., Lanz, T., Heap, S. R., et al. 2003, ApJ, 588, 1039 Hillier, D. J. & Miller, D. L. 1998, ApJ, 496, 407 Hillier, D. J. & Miller, D. L. 1999, ApJ, 519, 354 Hjorth, J., Sollerman, J., Møller, P., et al. 2003, Nature, 423, 847 Howarth, I. D. & Smith, K. C. 2001, MNRAS, 327, 353 Hunter, I., Dufton, P. L., Smartt, S. J., et al. 2006, A&A, in press Hyland, A. R., Straw, S., Jones, T. J., & Gatley, I. 1992, MNRAS, 257, Issersted, J. 1979, A&AS, 38, 239 Johnson, H. L. 1966, ARA&A, 4, 193 Kennicutt, R. C. & Hodge, P. W. 1986, ApJ, 306, 130 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 19 Kilian, J., Becker, S. R., Gehren, T., & Nissen, P. E. 1991, A&A, 244, Kudritzki, R. & Puls, J. 2000, ARA&A, 38, 613 Kudritzki, R. P. 1992, A&A, 266, 395 Kudritzki, R. P. 2002, ApJ, 577, 389 Kudritzki, R.-P., Lennon, D. J., & Puls, J. 1995, in Science with the VLT, 246–255 Kudritzki, R. P., Pauldrach, A., & Puls, J. 1987, A&A, 173, 293 Kudritzki, R. P., Simon, K. P., & Hamann, W.-R. 1983, A&A, 118, Lamers, H. J. G. L. M. & Achmad, L. 1994, A&A, 291, 856 Lamers, H. J. G. L. M., Nota, A., Panagia, N., Smith, L. J., & Langer, N. 2001, ApJ, 551, 764 Lamers, H. J. G. L. M., Snow, T. P., & Lindholm, D. M. 1995, ApJ, 455, 269 Langer, N. 1992, A&A, 265, L17 Lucke, P. B. & Hodge, P. W. 1970, AJ, 75, 171 Lucy, L. B. 1983, ApJ, 274, 372 Maeder, A. 1987, A&A, 178, 159 Markova, N., Puls, J., Repolust, T., & Markov, H. 2004, A&A, 413, Martins, F., Schaerer, D., & Hillier, D. J. 2005a, A&A, 436, 1049 Martins, F., Schaerer, D., Hillier, D. J., & Heydari-Malayeri, M. 2004, A&A, 420, 1087 Martins, F., Schaerer, D., Hillier, D. J., et al. 2005b, A&A, 441, 735 Massa, D., Fullerton, A. W., Sonneborn, G., & Hutchings, J. B. 2003, ApJ, 586, 996 Massey, P. 2002, ApJS, 141, 81 Massey, P., Bresolin, F., Kudritzki, R. P., Puls, J., & Pauldrach, A. W. A. 2004, ApJ, 608, 1001 Massey, P., Lang, C. C., Degioia-Eastwood, K., & Garmany, C. D. 1995, ApJ, 438, 188 Massey, P., Parker, J. W., & Garmany, C. D. 1989, AJ, 98, 1305 Massey, P., Puls, J., Pauldrach, A. W. A., et al. 2005, ApJ, 627, 477 Meynet, G. & Maeder, A. 2000, A&A, 361, 101 Meynet, G. & Maeder, A. 2005, A&A, 429, 581 Mitchell, R. C., Baron, E., Branch, D., et al. 2002, ApJ, 574, 293 Mokiem, M. R., de Koter, A., Evans, C. J., et al. 2006, A&A, 456, Mokiem, M. R., de Koter, A., Puls, J., et al. 2005, A&A, 441, 711 Mokiem, M. R., Martı́n-Hernández, N. L., Lenorzer, A., de Koter, A., & Tielens, A. G. G. M. 2004, A&A, 419, 319 Oey, M. S. 1999, in IAU Symp. 190: New Views of the Magellanic Clouds, 78 Owocki, S. P., Castor, J. I., & Rybicki, G. B. 1988, ApJ, 335, 914 Owocki, S. P., Cranmer, S. R., & Gayley, K. G. 1996, ApJ, 472, L115+ Owocki, S. P. & Puls, J. 1999, ApJ, 510, 355 Panagia, N., Gilmozzi, R., Macchetto, F., Adorf, H.-M., & Kirshner, R. P. 1991, ApJ, 380, L23 Parker, J. W., Garmany, C. D., Massey, P., & Walborn, N. R. 1992, AJ, 103, 1205 Petrenz, P. & Puls, J. 1996, A&A, 312, 195 Petrenz, P. & Puls, J. 2000, A&A, 358, 956 Prinja, R. K. & Crowther, P. A. 1998, MNRAS, 300, 828 Puls, J., Kudritzki, R.-P., Herrero, A., et al. 1996, A&A, 305, 171 Puls, J., Markova, N., Scuderi, S., et al. 2006, A&A, 454, 625 Puls, J., Springmann, U., & Lennon, M. 2000, A&AS, 141, 23 Puls, J., Urbaneja, M. A., Venero, R., et al. 2005, A&A, 435, 669 Repolust, T., Puls, J., & Herrero, A. 2004, A&A, 415, 349 Rolleston, W. R. J., Trundle, C., & Dufton, P. L. 2002, A&A, 396, 53 Russell, S. C. & Bessell, M. S. 1989, ApJS, 70, 865 Sanduleak, N. 1970, Contributions from the Cerro Tololo Inter- American Observatory, 89 Schaerer, D. & de Koter, A. 1997, A&A, 322, 598 Schaerer, D., Meynet, G., Maeder, A., & Schaller, G. 1993, A&AS, 98, 523 Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, A&AS, 96, 269 Smith, K. C. & Howarth, I. D. 1998, MNRAS, 299, 1146 Villamariz, M. R. & Herrero, A. 2005, A&A, 442, 263 Villamariz, M. R., Herrero, A., Becker, S. R., & Butler, K. 2002, A&A, 388, 940 Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2000, A&A, 362, Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, A&A, 369, Voels, S. A., Bohannan, B., Abbott, D. C., & Hummer, D. G. 1989, ApJ, 340, 1073 von Zeipel, H. 1924, MNRAS, 84, 665 Walborn, N. R. 1977, ApJ, 215, 53 Walborn, N. R. & Blades, J. C. 1987, ApJ, 323, L65 Walborn, N. R. & Blades, J. C. 1997, ApJS, 112, 457 Walborn, N. R., Drissen, L., Parker, J. W., et al. 1999, AJ, 118, 1684 Walborn, N. R. & Fitzpatrick, E. L. 1990, PASP, 102, 379 Walborn, N. R., Fullerton, A. W., Crowther, P. A., et al. 2002a, ApJS, 141, 443 Walborn, N. R., Howarth, I. D., Lennon, D. J., et al. 2002b, AJ, 123, Walborn, N. R., Lennon, D. J., Haser, S. M., Kudritzki, R.-P., & Voels, S. A. 1995, PASP, 107, 104 Walborn, N. R., Morrell, N. I., Howarth, I. D., et al. 2004, ApJ, 608, Walborn, N. R. & Parker, J. W. 1992, ApJ, 399, L87 Woosley, S. E. & Heger, A. 2006, ApJ, 637, 914 Yoon, S.-C. & Langer, N. 2005, A&A, 443, 643 Yoon, S.-C., Langer, N., & Norman, C. 2006, A&A, 460, 199 20 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC Appendix A: Fits and comments on individual objects The observed spectra shown in this section were corrected for radial velocities. If not noted differently the lines that were fit- ted are the hydrogen Balmer lines Hα, Hγ and Hδ; the He i singlet line at 4387 Å; the He i triplet lines at 4026, 4471 and 4713 Å; and the He ii lines at 4200, 4541 and 4686 Å. Over plotted are the best fit spectra, unless noted differently. For a discussion of the line weighting scheme adopted in our fitting procedure we refer to Paper I (especially to their Table 3.). N11-004 (Fig. A.1) With exception of He i λ4471 good fits for all lines of this OC9.7 supergiant were obtained. The under prediction of the He i λ4471 line is possibly connected to the so-called generalised dilution effect (Voels et al. 1989, see also Repolust et al. 2004). N11-008 (Fig. A.1) All lines are reproduced well. Note that given the weakness of He ii λ4686 and lower weight assigned to this line, the reproduction of its profile can be considered good. N11-026 (Fig. A.1) A high effective temperature of 53 kK was needed to fit the spectrum of this O2 giant. Note that even though on its relative plot scale the very weak He i λ4471 line is not fitted perfectly, the quantitative fit quality is good and is comparible to that of the He ii lines. Still, trying to assess whether the fit could be improved we ran test fits with increas- ing relative weight of the He i λ4471 line. It turned out that an increase of a factor of five was necessary to obtain a solution with an improved He i λ4471 line fit, which is shown in Fig. A.1 as a set of dotted lines. This best fit has a Teff lower by 3.7 kK and all other fit parameters approximately equal compared to the regular fit. The overal fit quality of the other diagnostic lines is somewhat less, though this is difficult to discern by eye be- cause these lines are relatively strong (one has to zoom in to the 0.5 to 1 percent level, as we do for He i λ4471). Note, however, that the error estimates for Teff are relative large and within the lower error estimate of 3.9 kK (cf. Tab. 3) the solutions agree. Also note that even for the lower Teff solution this O2 star is still considerably hotter than an O3 star (see Sect. 4.1). N11-029 (Fig. A.1) A good fit was obtained. The slight un- derestimation of the core strength of the He ii λ4200 and He ii λ4541 lines is, given the relative weakness of these lines, not significant. N11-031 (Fig. A.2) To fit the spectrum of this O2 star a relatively low effective temperature of 45 kK was necessary. Walborn et al. (2004) derived a Teff of 55 kK for this object. However, their determination of this parameter was based on the analysis of the N iv-v ionisation adopting a fixed gravity of log g = 4.0. Based on the analysis of the helium ionisation we exclude a Teff higher than ∼47 kK. This is illustrated by the dot- ted lines, which correspond to a model calculated for an effec- tive temperature higher by the upper error estimate we derived for Teff. For even higher temperatures the He i λ4471 would, in contradiction to the observations, disappear completely (also see Sect. 4.1). N11-032 (Fig. A.2) The He i and He ii blend at 4026 Å was not observed for this object. In the final fit, therefore, also the Hδ Balmer line is shown. N11-033 (Fig. A.2) This B0 giant is fast rotator. To repro- duce the observed line profiles a projected rotational velocity of 256 km s−1 was necessary. In the presented fit the strength of He ii λ4541 seems to be under predicted. However, given the relative weakness of this line this small discrepancy is negligi- ble. Note that the sharp change in the central part of the Hα profile is the result of an over subtraction of the core nebular feature and was not taken into account in the fitting procedure. N11-036 (Fig. A.2) The line profiles of N11-036 could be re- produced quite accurately because of its relatively slow rota- tion. N11-038 (Fig. A.3) The relatively poor reproduction of He i λ4387 in this O5 bright giant is the result of the low weight assigned to this line based on the spectral type of N11-038. A good fit to the He i λ4471 could not be obtained due to its pecu- liar line profile shape. Possibly this triangular profile is related to macroturbulence. N11-042 (Fig. A.3) A relatively low projected rotational ve- locity was found for this B0 giant allowing for a nearly perfect N11-045 (Fig. A.3) A good fit was obtained for this O9 giant. Despite the good fit quality a problem is apparent when Ms and Mev are compared in Tab. 2. The spectroscopic mass is found to be less than half its evolutionary equivalent. N11-048 (Fig. A.3) Based on the fact that the He ii λ4686 is the strongest line in its spectrum Parker et al. (1992) classified this star as a Vz star. The fit parameters, however, place this object at a considerable distance from the theoretical ZAMS in Fig. 5. We also find that the spectrum can not be reproduced ac- curately. The width of the neutral helium lines is systematically unpredicted, whereas the width of the He ii is over predicted. This and the fact that the helium abundance that is recovered from the spectrum is rather low (YHe = 0.06) are indicative of a possible binary nature. N11-051 (Fig. A.4) The spectrum of the O5 dwarf N11-051 shows very broad lines, indicating fast stellar rotation. To ob- tain the final fit a projected rotational velocity of 333 km s−1 was required. M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 21 N11-004 OC9.7 Ib 4330 4340 4350 6552 6562 6572 4022 4026 4030 HeI + HeII 4026 4386 4388 4390 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4192 4200 4208 HeII 4200 4537 4541 4545 HeII 4541 4683 4686 4689 HeII 4686 N11-008 B0.7 Ia 4335 4340 4345 6552 6562 6572 4022 4026 4030 HeI + HeII 4026 4386 4388 4390 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4192 4200 4208 HeII 4200 4537 4541 4545 HeII 4541 4684 4686 4688 HeII 4686 N11-026 O2 III(f*) 4335 4340 4345 6552 6562 6572 4021 4026 4031 HeI + HeII 4026 4386 4388 4390 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4195 4200 4205 HeII 4200 4536 4541 4546 HeII 4541 4681 4686 4691 HeII 4686 N11-029 O9.7 Ib 4335 4340 4345 6552 6562 6572 4023 4026 4029 HeI + HeII 4026 4386 4388 4390 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4194 4200 4206 HeII 4200 4537 4541 4545 HeII 4541 4684 4686 4688 HeII 4686 Fig. A.1. Comparison of the observed line profiles of N11-004, -008, -026 and -028 with best fitting synthetic line profiles obtained using the automated fitting method (grey lines). Wavelengths are given on the horizontal axis in Å. The vertical axis gives the normalised flux. Note that this axis is scaled differently for each line. The dotted line profiles for N11-026 correspond to the best fit obtained adopting a five times larger relative weight for the He i λ4471 line. This best fit has a Teff lower by 3.7 kK and all other fit parameters approximately equal compared to the fit for the nominal He i λ4471 weight. N11-058 (Fig. A.4) A good fit was obtained for all lines of this O5.5 dwarf. The small under prediction of the He i4387 line strength is the result of the relatively low weight assigned to this line for the spectral type of this star. N11-060 (Fig. A.4) Note that the He i λ4471 line of this O3 dwarf has a strength that is comparable to the strength of this line in the spectrum of the O2 giant N11-031. As the effective temperature we find for these two objects are also compara- ble, the helium spectrum suggest a spectral type of O3 for both objects. N11-061 (Fig. A.4) All lines of this O9 dwarf are reproduced correctly. The mass loss could be reliably determined using the automated method at a rate of 2.1 × 10−7 M⊙. Note that the 22 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC N11-031 ON2 III(f*) 4333 4340 4347 6552 6562 6572 4021 4026 4031 HeI + HeII 4026 4385 4388 4391 HeI 4387 4465 4471 4477 HeI 4471 4710 4713 4716 HeI 4713 4195 4200 4205 HeII 4200 4536 4541 4546 HeII 4541 4680 4686 4692 HeII 4686 N11-032 O7 II(f) 4096 4102 4108 4335 4340 4345 6547 6562 6577 4386 4388 4390 HeI 4387 4469 4471 4473 HeI 4471 4711 4713 4715 HeI 4713 4195 4200 4205 HeII 4200 4537 4541 4545 HeII 4541 4682 4686 4690 HeII 4686 N11-033 B0 IIIn 4330 4340 4350 6552 6562 6572 4022 4026 4030 HeI + HeII 4026 4384 4388 4392 HeI 4387 4467 4471 4475 HeI 4471 4709 4713 4717 HeI 4713 4192 4200 4208 HeII 4200 4537 4541 4545 HeII 4541 4681 4686 4691 HeII 4686 N11-036 B0.5 Ib 4330 4340 4350 6552 6562 6572 4022 4026 4030 HeI + HeII 4026 4386 4388 4390 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4192 4200 4208 HeII 4200 4537 4541 4545 HeII 4541 4684 4686 4688 HeII 4686 Fig. A.2. Same as Fig. A.1, however, for N11-031, -032, -033 and -036. Shown as dotted profiles for N11-031 is the effect of and a 2.2 kK increase in Teff . value of 1.8 obtained for the wind acceleration parameter β is rather high for a dwarf type object. N11-065 (Fig. A.5) A relatively high helium abundance of 0.17 was recovered from the spectrum. The good fit quality obtained for all lines, however, does not indicate an overesti- mation of this parameter. A strong mass discrepancy is found with Ms being smaller than Mev by approximately a factor of N11-066 (Fig. A.5) Apart from a slight under prediction of the width of He ii λ4686 all lines of this O7 dwarf could be fitted with good accuracy. Like the previously discussed object, again a large mass discrepancy is found with a ratio of Mev to Ms of N11-068 (Fig. A.5) The final fit reproduces the observed line profiles very accurately. No further comments are required. N11-072 (Fig. A.5) The star has a very sharp lined spectrum. The best fit required a vr sin i of 14 km s −1. Note that given the M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 23 N11-038 O5 II(f+) 4330 4340 4350 6547 6562 6577 4021 4026 4031 HeI + HeII 4026 4383 4388 4393 HeI 4387 4466 4471 4476 HeI 4471 4709 4713 4717 HeI 4713 4195 4200 4205 HeII 4200 4537 4541 4545 HeII 4541 4682 4686 4690 HeII 4686 N11-042 B0 III 4330 4340 4350 6547 6562 6577 4023 4026 4029 HeI + HeII 4026 4386 4388 4390 HeI 4387 4468 4471 4474 HeI 4471 4712 4713 4714 HeI 4713 4192 4200 4208 HeII 4200 4536 4541 4546 HeII 4541 4684 4686 4688 HeII 4686 N11-045 O9 III 4335 4340 4345 6552 6562 6572 4022 4026 4030 HeI + HeII 4026 4386 4388 4390 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4195 4200 4205 HeII 4200 4537 4541 4545 HeII 4541 4683 4686 4689 HeII 4686 N11-048 O6.5 V((f)) 4330 4340 4350 6547 6562 6577 4021 4026 4031 HeI + HeII 4026 4383 4388 4393 HeI 4387 4465 4471 4477 HeI 4471 4710 4713 4716 HeI 4713 4194 4200 4206 HeII 4200 4535 4541 4547 HeII 4541 4682 4686 4690 HeII 4686 Fig. A.3. Same as Fig. A.1, however, for N11-038, -042, -045 and -048. lower error estimate, given in Tab. 3, and the spectral resolution of the data this value can also be interpreted as an upper limit. N11-087 (Fig. A.6) To fit the spectrum of N11-087 a vr sin i of 276 km s−1 was needed. The final fit shows a good repro- duction of the line profiles with a slight under prediction of the He ii λ4541 line. Given the signal-to-noise ratio of the spectrum and the relative strength of this line, we do not believe this to be significant. N11-123 (Fig. A.6) Like the previously discussed object N11- 123 is also classified as an O9.5 dwarf. The effective tempera- ture needed to fit the spectrum is, however, higher by more than 2 kK. This is the result of a log g that is higher by ∼0.2 dex. BI 237 (Fig. A.7) A good fit was obtained for this O2 dwarf star. The He i λ4471 line is hardly visible. Fortunately, the hy- drogen and He ii lines provide the automated fitting method with enough information to determine an effective temperature. The error estimates on this parameter, however, are consider- able (see Tab. 3). The effect of lowering Teff to the lower error estimate of 49 kK is illustrated by the dotted profiles. BI 237 was recently also analysed by Massey et al. (2005). Compared to this study we find large differences for Teff 24 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC N11-051 O5 Vn((f)) 4330 4340 4350 6547 6562 6577 4020 4026 4032 HeI + HeII 4026 4382 4388 4394 HeI 4387 4466 4471 4476 HeI 4471 4706 4713 4720 HeI 4713 4192 4200 4208 HeII 4200 4533 4541 4549 HeII 4541 4678 4686 4694 HeII 4686 N11-058 O5.5 V((f)) 4330 4340 4350 6547 6562 6577 4021 4026 4031 HeI + HeII 4026 4383 4388 4393 HeI 4387 4468 4471 4474 HeI 4471 4709 4713 4717 HeI 4713 4195 4200 4205 HeII 4200 4537 4541 4545 HeII 4541 4682 4686 4690 HeII 4686 N11-060 O3 V((f*)) 4333 4340 4347 6552 6562 6572 4021 4026 4031 HeI + HeII 4026 4385 4388 4391 HeI 4387 4466 4471 4476 HeI 4471 4710 4713 4716 HeI 4713 4195 4200 4205 HeII 4200 4536 4541 4546 HeII 4541 4680 4686 4692 HeII 4686 N11-061 O9 V 4335 4340 4345 6547 6562 6577 4023 4026 4029 HeI + HeII 4026 4386 4388 4390 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4194 4200 4206 HeII 4200 4536 4541 4546 HeII 4541 4683 4686 4689 HeII 4686 Fig. A.4. Same as Fig. A.1, however, for N11-051, -058, -060 and -061. (+5 kK), log g (+0.2 dex) and Ṁ (−0.4 dex). Our solution for higher effective temperature and surface gravity can be ex- plained by the fact that the ionisation structure of the atmo- sphere is set by both these parameters (also see Paper I). The reduced mass loss rate we find is the result of the increased value for β (+0.4) obtained by the automated method. BI 253 (Fig. A.7) The star has as an identical spectral type as the previously discussed object. The photospheric parame- ters determined from the spectrum are also very similar to the parameters of BI 237. Note that in contrast to this agreement BI 253 is found to have a much denser wind. Its mass loss rate is larger by a factor 2.5 and reflects the fact that compared to BI 237 the wind lines Hα and He ii λ4686 are more filled in. In their analysis Massey et al. (2005) could only determine a lower estimate for Teff of 48 kK, which is consistent with our lower error estimate. The other parameters determined by these authors are in fair agreement with our analysis. An exception to this is the surface gravity for which Massey et al. estimate a value lower by 0.3 dex, which is the result of the lower Teff used for their fit. M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 25 N11-065 O6.5 V((f)) 4330 4340 4350 6547 6562 6577 4021 4026 4031 HeI + HeII 4026 4383 4388 4393 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4195 4200 4205 HeII 4200 4537 4541 4545 HeII 4541 4682 4686 4690 HeII 4686 N11-066 O7 V((f)) 4330 4340 4350 6547 6562 6577 4021 4026 4031 HeI + HeII 4026 4385 4388 4391 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4195 4200 4205 HeII 4200 4537 4541 4545 HeII 4541 4682 4686 4690 HeII 4686 N11-068 O7 V((f)) 4330 4340 4350 6547 6562 6577 4021 4026 4031 HeI + HeII 4026 4385 4388 4391 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4195 4200 4205 HeII 4200 4537 4541 4545 HeII 4541 4682 4686 4690 HeII 4686 N11-072 B0.2 III 4335 4340 4345 6547 6562 6577 4024 4026 4028 HeI + HeII 4026 4386 4388 4390 HeI 4387 4468 4471 4474 HeI 4471 4712 4713 4714 HeI 4713 4192 4200 4208 HeII 4200 4536 4541 4546 HeII 4541 4684 4686 4688 HeII 4686 Fig. A.5. Same as Fig. A.1, however, for N11-065, -066, -068 and -072. Sk −66 18 (Fig. A.7) With exception of the He ii lines at 4541 Å and 4686 Å we obtained a near perfect fit. The helium abundance YHe = 0.14 indicates an enriched atmosphere. Sk −66 100 (Fig. A.7) was previously analysed by Puls et al. (1996). Compared to this study we find similar parameters with exception of a lower Teff which is the result of the inclusion of blanketing and an increased helium abundance of YHe = 0.19. The final fit shows that with this large value all lines could be reproduced quite accurately. The slight mismatch of the He ii λ4686 line can be improved by increasing Ṁ to its up- per error estimate. The effect of this increase of 0.12 dex on the line profiles is shown using dotted lines. Sk −67 166 (Fig. A.8) This supergiant has both Hα and He ii λ4686 in emission indicating the presence of a dense stel- lar outflow. To correctly reproduce the line profiles a very high mass loss rate of 9.3×10−6 M⊙yr −1 was required. This value is in good agreement with the findings of Crowther et al. (2002) who studied this objects using the model atmosphere code cm- fgen (Hillier & Miller 1998) in the optical, UV and far-UV. Compared to these authors we also find that the other fit pa- rameters are in good agreement. An exception to this is the he- 26 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC N11-087 O9.5 Vn 4325 4340 4355 6547 6562 6577 4021 4026 4031 HeI + HeII 4026 4383 4388 4393 HeI 4387 4465 4471 4477 HeI 4471 4707 4713 4719 HeI 4713 4192 4200 4208 HeII 4200 4534 4541 4548 HeII 4541 4680 4686 4692 HeII 4686 N11-123 O9.5 V 4330 4340 4350 6542 6562 6582 4022 4026 4030 HeI + HeII 4026 4385 4388 4391 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4192 4200 4208 HeII 4200 4536 4541 4546 HeII 4541 4683 4686 4689 HeII 4686 Fig. A.6. Same as Fig. A.1, however, for N11-087 and -123. lium abundance. Crowther et al. (2002) adopted a fixed value of YHe = 0.2 whereas our automated fitting method was able to self consistently determine a value of YHe = 0.28. Sk −70 69 (Fig. A.8) To fit the spectrum of this O5 dwarf Sk −70 69 a helium abundance of YHe = 0.17 was required. Note that, similar to N11-065, which is also strongly enriched with helium, the spectroscopic mass is found to be much smaller than the evolutionary mass. M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC 27 BI237 O2 V((f*)) 4333 4340 4347 6552 6562 6572 4021 4026 4031 HeI + HeII 4026 4385 4388 4391 HeI 4387 4465 4471 4477 HeI 4471 4710 4713 4716 HeI 4713 4195 4200 4205 HeII 4200 4536 4541 4546 HeII 4541 4680 4686 4692 HeII 4686 BI253 O2 V((f*)) 4333 4340 4347 6552 6562 6572 4021 4026 4031 HeI + HeII 4026 4385 4388 4391 HeI 4387 4465 4471 4477 HeI 4471 4710 4713 4716 HeI 4713 4195 4200 4205 HeII 4200 4536 4541 4546 HeII 4541 4680 4686 4692 HeII 4686 Sk -66 18 O6 V((f)) 4335 4340 4345 6552 6562 6572 4021 4026 4031 HeI + HeII 4026 4385 4388 4391 HeI 4387 4468 4471 4474 HeI 4471 4711 4713 4715 HeI 4713 4195 4200 4205 HeII 4200 4537 4541 4545 HeII 4541 4682 4686 4690 HeII 4686 Sk -66 100 O6 II(f) 4335 4340 4345 6552 6562 6572 4021 4025 4029 HeI + HeII 4026 4385 4388 4391 HeI 4387 4469 4471 4473 HeI 4471 4711 4713 4715 HeI 4713 4196 4200 4204 HeII 4200 4537 4541 4545 HeII 4541 4676 4686 4696 HeII 4686 Fig. A.7. Same as Fig. A.1, however, for BI 237, BI 253, Sk −66 18 and Sk −66 100. The dotted profiles for BI 237 correspond to a model calculated with a Teff reduced by 3.8 kK compared to the best fit value. For Sk −66 100 the dotted profiles show the effect of an increase in Ṁ by 0.12. 28 M. R. Mokiem et al.: Wind properties and evolution of hot massive stars in the LMC Sk -67 166 O4 If+ 4335 4340 4345 6547 6562 6577 4021 4025 4029 HeI + HeII 4026 4385 4388 4391 HeI 4387 4469 4471 4473 HeI 4471 4711 4713 4715 HeI 4713 4196 4200 4204 HeII 4200 4537 4541 4545 HeII 4541 4677 4686 4695 HeII 4686 Sk -70 69 O5 V 4335 4340 4345 6547 6562 6577 4021 4026 4031 HeI + HeII 4026 4383 4388 4393 HeI 4387 4468 4471 4474 HeI 4471 4709 4713 4717 HeI 4713 4195 4200 4205 HeII 4200 4537 4541 4545 HeII 4541 4682 4686 4690 HeII 4686 Fig. A.8. Same as Fig. A.1, however, for Sk −67 166 and Sk −70 69. This figure "6489fig11.jpg" is available in "jpg" format from: http://arxiv.org/ps/0704.1113v1 http://arxiv.org/ps/0704.1113v1 Introduction Data description Analysis method The assumption of spherical symmetry Fundamental parameters Effective temperatures The Teff scale of O2 stars Gravities Helium abundances Microturbulence Wind parameters The mass discrepancy Chemically homogeneously evolving O2 stars Large sample trends The evolutionary status of N11 LH9 and LH10 in N11 The ages of LH9 and LH10 Sequential star formation? Summary and conclusions Fits and comments on individual objects

0704.1114

Identification of Absorption Features in an Extrasolar Planet Atmosphere

Draft version October 29, 2018 Preprint typeset using LATEX style emulateapj v. 7/15/03 IDENTIFICATION OF ABSORPTION FEATURES IN AN EXTRASOLAR PLANET ATMOSPHERE T. Barman Lowell Observatory, 1400 W. Mars Hill Rd., Flagstaff, AZ 86001, [email protected] Draft version October 29, 2018 ABSTRACT Water absorption is identified in the atmosphere of HD209458b by comparing models for the planet’s transmitted spectrum to recent, multi-wavelength, eclipse-depth measurements (from 0.3 to 1 µm) published by Knutson et al. (2007). A cloud-free model which includes solar abundances, rainout of condensates, and photoionization of sodium and potassium is in good agreement with the entire set of eclipse-depth measurements from the ultraviolet to near-infrared. Constraints are placed on condensate removal by gravitational settling, the bulk metallicity, and the redistribution of absorbed stellar flux. Comparisons are also made to the Charbonneau et al. (2002) sodium measurements. Subject headings: planetary atmospheres - extrasolar planets 1. INTRODUCTION The discovery of transiting extrasolar planets has opened the door to direct detections and characteriza- tion of their atmospheres. Observations using the STIS instrument on HST provided the first glimpse of what the photospheric composition is like for a nearby EGP. Charbonneau et al. (2002) measured the relative change in eclipse depth for HD209458b across a sodium doublet (5893Å) resulting in the first detection of atomic absorp- tion in an EGP atmosphere. Following the Na detec- tion, Vidal-Madjar et al. (2003) discovered an extended hydrogen-rich atmosphere surrounding HD209458b using a similar technique as Charbonneau et al., but in the UV. At Lyman-α wavelengths, HD209458b is ∼ 3 times larger than in the optical. These detections were made using a technique called transit spectroscopy as the planet passes in front of its star. Transit spectroscopy uses the fact that the wavelength-dependent opacities in the planet’s atmo- sphere obscure stellar light at different planet radii lead- ing to a wavelength-dependent depth of the light-curve during primary eclipse. Consequently, searching for rela- tive changes in eclipse depth as a function of wavelength directly probes the absorption properties of the planet’s atmosphere with the potential to reveal the presence (or absence) of specific chemical species. In this paper, recent measurements of HD209458b’s radius are combined in a multi-wavelength comparison to model atmosphere predictions. Atmospheric molecular and atomic absorption are identified and constraints are placed on the basic atmospheric properties. 2. THE LIMB MODEL Transit spectroscopy probes the limb of a planet which is the transition region between the day and night sides. One would, therefore, expect that, in the presence of a horizontal temperature gradient between the heated and non-heated hemispheres, the temperatures across the limb would be cooler than the average dayside tem- peratures (Barman et al. 2005; Iro et al. 2005). Re- cent Spitzer observations of both transiting and non- transiting hot-Jupiters showing large flux variations with phase have provided strong evidence supporting such a day-to-night temperature gradient (Charbonneau et al. 2005; Deming et al. 2005b, 2006; Harrington et al. 2006). Describing the limb ultimately requires a multi- dimensional model atmosphere solution; however, as is common practice, a simpler one-dimensional model is used here to represent the average properties of the limb, in both a longitudinal and latitudinal sense. To explore a variety of limb temperature structures, the incident stellar flux has been scaled by a parameter α. A model with α = 0.25 represents an average description of the entire planet in the presence of very efficient horizontal energy transport (Barman et al. 2005). Increasing α to 0.5 increases the heating of the model atmosphere mak- ing it more appropriate for just the dayside which, in this work, will represent an upper limit to the plausible mean temperatures at the limb. The basic chemistry across the limb is modeled assum- ing chemical equilibrium, determined by minimizing the free energy while including grain formation. To under- stand the effects of gravitational settling of grains on the chemistry and the transmission spectrum, the removal of grains from the atmosphere is included via two simple approximations that represent opposite extremes. The first is the “cond” approximation used by Barman et al. (2001) and Allard et al. (2001) which simply ignores the grain opacity without altering the chemistry or abun- dances. The second is the “rainout” approximation which iteratively reduces, at each layer, the elemental abundances (by the appropriate stoichiometric ratios) involved in grain formation and recomputes the chem- ical equilibrium with each new set of stratified elemen- tal abundances. This approach is similar to other rain- out models (Fegley & Lodders 1996; Burrows & Sharp 1999), except that the depletion of elements is contin- ued until grains are no longer present. The transmission of stellar fluxes through the limb of HD209458b is determined by solving the spherically sym- metric radiative transfer equation while fully accounting for scattering and absorption of both intrinsic and ex- trinsic radiation (Hauschildt 1992; Barman et al. 2001, 2002). Spherical, instead of the more traditional plane- parallel geometry, naturally accounts for the curvature of the atmospheric layers and changes in chemistry along the slant paths through the upper atmosphere. The planet radius at a given wavelength (Rλ) is obtained by determining the radial depth at which the transmitted http://arxiv.org/abs/0704.1114v1 Fig. 1.— Monochromatic transit radii over the STIS spectral range for the baseline rainout model with (red) and without (blue) photoionization of Na and K. The solid and dotted red lines are the same, except H2O line opacity is excluded in the latter. Horizontal bars correspond to mean radii across bins with λ-ranges indicated by the width of each bar. STIS measurements by Knutson et al. (2007) are shown in green with 1 σ error bars. Vertical dashed lines mark the narrow λ-range used by Charbonneau et al. (2002). flux is equal to e−1 times the incident starlight along that same path. 3. RESULTS A cloud-free atmosphere with rainout, α = 0.25, and solar abundances is adopted as the baseline model. This model, along with others, is compared to the relative Rλ measurements of Knutson et al. (2007) which have a reported precision high enough to constrain many of the basic atmospheric properties. Since a comparison is being made to relative Rλ values, the model results were uniformly scaled to match the observations in the 4580 to 5120Å wavelength bin; this scaling was always less than 0.005RJup. Overall, the baseline model (red solid line in Fig. 1) reproduces the observed rise in Rλ toward shorter wavelengths, the increase across the Na doublet, and the increase at the far red wavelengths. The baseline model comparison to the data has a χ2 that is 3 times smaller than a constant Rλ. 3.1. Water Absorption Water is predicted to be one of the most abundant species in an EGP atmosphere and, given its broad ab- sorption features in the infrared, plays a crucial role in regulating the temperature-pressure (T-P) profile. The first major H2O absorption band appears between 0.8 and 1 µm, a region covered by the last two wavelength bins of Knutson et al. (2007). As illustrated in Fig. 1, there is excellent agreement between the baseline model and the observations in this part of the spectrum espe- cially across the longest wavelength bin that sits on top of the H2O band. Qualitatively similar water features are seen in the models of Brown (2001) and Hubbard et al. (2001); however these models fall many σ below the ob- servations. The baseline model also predicts mean Rλ- peaks equal to 1.330, 1.343, and 1.341 RJup for the next three water bands (at λ ∼ 1.15, 1.4, and 1.9 µm). A model that excludes H2O line opacity is also shown in Fig. 1 and is greater than 10 σ below both the observa- tions and the baseline model prediction. Removing H2O opacity also produces a significant drop in Rλ near 0.9 µm that further increases the discrepancy between the model and overall red/near-IR observations. No other opacity source could be responsible for the observed rise in Rλ across this part of the spectrum. 3.2. Photoionization After the reported sodium detection in the atmosphere of HD209458b (Charbonneau et al. 2002), there were several attempts to explain why the strength of this fea- ture was much lower than expected based on earlier mod- els (e.g., by Seager & Sasselov, 2000). Barman et al. (2002) explored departures from local thermodynamic equilibrium (LTE) which are capable of producing an in- version in the cores of the Na doublet line profiles. How- ever, the earlier models of Barman et al. (2001, 2002) were constructed under the cond approximation and, consequently, contained a larger number of free met- als along with TiO and VO molecular absorption com- pared to a rainout model. These additional sources of optical/UV opacity lead to a hotter upper atmosphere and a very shallow photoionization depth for Na (only a 5% reduction of Na across the region probed by tran- sit spectroscopy). Fortney et al. (2003) also explored Na photoionization and found that their atmosphere model (which included rainout) could be brought into reason- able agreement with the Na observations. The models presented here account for the angular dependence of ionziation on the limb’s dayside, but not on the night side. However, Fortney et al. (2003) have shown that Na ionization is still present at ∼ 5◦ past the terminator for pressures relevant to the transit spectrum modeled here. Figure 1 compares cloud-free solar abundance rainout models with and without photoionization of Na and K. While these models include photoionization, the LTE ap- proximation on the atomic level populations and line source function is maintained. Ionization (radially) at the limb reaches 50% at P ∼ 0.1 and 1.2 mbar for Na and K, respectively, and stops at P ∼ 2 and 7 mbar, re- spectively, resulting in a reduction of the Rλ-peaks across Fig. 2.— Relative flux differences (with time from center of eclipse) between a wavelength band centered on the Na doublet and the mean of two wavelength bands on either side of the Na doublet. The total wavelength range is indicated by vertical lines in Fig. 1. Each model is cloud-free with solar abundances and α = 0.25. See Figure legend and text for the distinguishing characteristics of each model. Symbols are the observations of Charbonneau et al. (2002) with 1 σ error bars. the Na and K lines. For K, this reduction extends out to the line-wings resulting in a significantly smaller mean Rλ, bringing the model into ∼ 3σ agreement with the Knutson et al. measurement. The impact of Na pho- toionization is mostly confined to the core of the doublet leading to only a small reduction of the mean Rλ, but suf- ficient to bring the model into ∼ 1σ agreement with the observations at these wavelengths. Though not included here, ionization past the terminator onto the night side should further improve the agreement across the Na and K doublets. The two narrow features on the red wing of the K doublet are due to Rb, which should also be affected by photoionization due to its very low first ion- ization potential. Photoionization of Rb resulted in a near complete removal of these features, but reduced the mean Rλ for this bin by less than 0.002 RJup. The broad wavelength bins allowed Knutson et al. to obtain very precise relative Rλ measurements; however, as illustrated by the Na doublet, such broad bins lim- its the constraints that can be placed on atomic absorp- tion features. In contrast, the narrow range analyzed by Charbonneau et al. (2002) resulted in much larger Rλ error-bars, but still precise enough to easily distin- guish between the various models shown in Fig. 1. The flux differences across the Na doublet as a function of time (measured from the transit center) were computed for each model using the same narrow wavelength bins as Charbonneau et al. (2002), indicated in Fig. 1 by vertical dashed lines. Fig. 2 compares model transit curves to the 2002 Na measurements and shows a large discrepancy between the observations and solar abun- dance models with pure equilibrium chemistry (cond or rainout). Including photoionization brings the baseline rainout model into rough agreement with the observa- tions. Note the cond model shown in Fig. 2 does not include photoionization (similar to the LTE model from Barman et al. (2002)) and illustrates how far off the pre- dicted Rλ can be across the Na doublet under simplified assumptions. 3.3. Rainout, Metallicity, and Temperature The rainout model used here reduces the individual metal abundances with depth iteratively until clouds do not form, thus mimicking efficient gravitational settling. However, the removal of metals is not 100% efficient, leaving behind a variety of atoms (in addition to Na and K) to contribute to the line opacity. The impact of rain- out on the planet’s atmosphere is made apparent by com- paring rainout and cond models (Fig. 3). In the cond model grain opacities are simply ignored in the transfer equation while the chemistry remains that of a pure equi- librium model without any actual removal of refractory elements. Thus, the cond model represents the minimum impact of grain formation on the stratified abundances and leaves behind a considerable amount of free metals along with molecules like TiO and VO. This leads to much stronger atomic lines, including Na and K along with TiO and VO bands. Apart from a very minor contribution from molecules, the Rλ features at λ < 0.8µm are all due to atomic line opacity 1. These narrow features are due to blended lines from metals like Ca, Al, Fe, Ni, Mn, and Cr. Note that, while Rayleigh scattering does contribute to the opacity in the blue/UV, a removal of atomic line opacity would drop Rλ by∼ 0.02RJup for λ < 0.45µm. The top panel of Fig. 3 compares the solar abundance rainout model with photoionization (shown in Fig. 1) to a rainout model with 10× solar abundances. Since the metal abundances were uniformly scaled, the grain formation and corre- sponding removal of refractory elements were also uni- formly enhanced. The additional opacity altered the T-P profile as well as the total atmospheric extension. These factors contribute to a similar Rλ pattern in the opti- cal but enhanced molecular absorption features in the red/near-IR. The increase in metals increases the H2O absorption feature along with the wings of the Na and K doublets (which form deeper in the atmosphere where photoionization is less important). Also, FeH and CrH absorption is larger in the metal-rich atmosphere leading to an increase in Rλ on both sides of the K doublet. The metal-enhanced Rλ spectrum is in good agreement with the observations near 0.9µm, but is noticeably too high across both the K doublet and the H2O band. Transit spectroscopy can also help constrain the tem- peratures across the limb. The lower panel of Fig. 3 compares an α = 0.5 model (i.e. a model with a hotter dayside-like T-P profile) to the cooler α = 0.25 base- line model. In the hotter model, grain formation is less pronounced resulting in a greater concentration of most free metals and, thus, stronger UV/blue absorption lines than in the α = 0.25 model. Higher temperatures at depth also result in equilibrium concentrations of Na and K that are several times lower than found at the cooler temperatures of the α = 0.25 model. The net re- sults are two distinctively different Rλ spectra with the α = 0.25 model being more consistent with the Knutson et al. measurements. 4. SUMMARY 1 Ballester et al. (2007) claim the Rλ rise in the blue/UV is due to the Blamer edge of hot hydrogen in an extended atmosphere. While the present model does not support this explanation, if ver- ified, it could indicate that rainout is more efficient than predicted here. Fig. 3.— Monochromatic transit radii over the STIS spectral range. Top panel: solar abundance models, cond (black) and rainout(red), are compared to a model with 10× solar metal abundances including rainout (blue). Lower panel: solar abundances baseline rainout model with α = 0.25 (red) compared to a similar model with α = 0.5 (blue). Horizontal bars have the same meaning as in Fig. 1. Photoionziation plays an important role for both Na and K, and potentially many other species. No evidence is found to support a large metal enhancement (e.g., 10× solar), though smaller Jupiter-like enhancements are not ruled out. Furthermore, the agreement between model and observations demonstrated here alleviates the need for substantial cloud coverage along the limb between 0.05 and 0.001 bars, which would otherwise truncate and flatten the Rλ spectrum (Brown 2001). In the baseline model, the predicted location of clouds (e.g., MgSiO3) lies just below the minimum Rλ(∼ 1.315RJup) across the STIS wavelength range, consequently deep clouds (and also very high clouds) remain a possibility. The models presented here also predict Rλ variations (∼ 0.02 RJup) across the 2-0 R-branch of CO in the K- band. This is inconsistent (at 2.5σ) with a non-detection of CO based on Keck-NIRSPEC transit spectra taken in 2002 (Deming et al. 2005a). It is likely that using a single T-P profile to represent the horizontal and pole- to-pole variations across the limb averages out too much of the upper atmospheric structure; this simplification might explain the K-band discrepancy. High clouds may also be involved. In addition, the Keck and HST obser- vations were taken between 1 and 2 years apart and time- dependent atmospheric variations along the limb cannot be ruled out (Menou et al. 2003). While Knutson et al. (2007) did not attribute their measured Rλ variations to any absorption features (this was not the focus of their paper), the models presented above clearly show that these measurements are consis- tent with strong water absorption near 1 µm. A detection of water in the limb of HD209458b is nominally at odds with a recent Spitzer IRS spectrum that shows no H2O features for this planet(Richardson et al. 2007). These data were taken during secondary eclipse and directly probe the planet’s dayside with negligible contribution from the limb. It is possible that the dayside atmosphere is nearly isothermal (Fortney et al. 2006) which would re- sult in a spectrum with no detectable water absorption features, despite a copious water supply. The transmis- sion spectrum, however, would contain absorption fea- tures independent of an isothermal dayside or limb. The author thanks B. Hansen and the referees for their comments. This research was supported by NASA Ori- gins and Spitzer Theory Grants and made use of NASA’s Project Columbia computer system. REFERENCES Allard, F., Hauschildt, P. H., Alexander, D. R., Tamanai, A., & Schweitzer, A. 2001, ApJ, 556, 357 Ballester, G. E., Sing, D. K., & Herbert, F. 2007, Nature, 445, 511 Barman, T. S., Hauschildt, P. H., & Allard, F. 2001, ApJ, 556, 885 —. 2005, ApJ, 632, 1132 Barman, T. S., Hauschildt, P. H., Schweitzer, A., Stancil, P. C., Baron, E., & Allard, F. 2002, ApJ, 569, L51 Brown, T. M. 2001, ApJ, 553, 1006 Burrows, A. & Sharp, C. M. 1999, ApJ, 512, 843 Charbonneau, D., Allen, L. E., Megeath, S. T., Torres, G., Alonso, R., Brown, T. M., Gilliland, R. L., Latham, D. W., Mandushev, G., O’Donovan, F. T., & Sozzetti, A. 2005, ApJ, 626, 523 Charbonneau, D., Brown, T. M., Noyes, R. W., & Gilliland, R. L. 2002, ApJ, 568, 377 Deming, D., Brown, T. M., Charbonneau, D., Harrington, J., & Richardson, L. J. 2005a, ApJ, 622, 1149 Deming, D., Harrington, J., Seager, S., & Richardson, L. J. 2006, ApJ, 644, 560 Deming, D., Seager, S., Richardson, L. J., & Harrington, J. 2005b, Nature, 434, 740 Fegley, B. J. & Lodders, K. 1996, ApJ, 472, L37+ Fortney, J. J., Cooper, C. S., Showman, A. P., Marley, M. S., & Freedman, R. S. 2006, ApJ, 652, 746 Fortney, J. J., Sudarsky, D., Hubeny, I., Cooper, C. S., Hubbard, W. B., Burrows, A., & Lunine, J. I. 2003, ApJ, 589, 615 Harrington, J., Hansen, B. M., Luszcz, S. H., Seager, S., Deming, D., Menou, K., Cho, J. Y.-K., & Richardson, L. J. 2006, Science, 314, 623 Hauschildt, P. H. 1992, JQSRT, 47, 433 Hubbard, W. B., Fortney, J. J., Lunine, J. I., Burrows, A., Sudarsky, D., & Pinto, P. 2001, ApJ, 560, 413 Iro, N., Bézard, B., & Guillot, T. 2005, A&A, 436, 719 Knutson, H. A., Charbonneau, D., Noyes, R. W., Brown, T. M., & Gilliland, R. L. 2007, ApJ, 655, 564 Menou, K., Cho, J. Y.-K., Seager, S., & Hansen, B. M. S. 2003, ApJ, 587, L113 Richardson, L. J., Deming, D., Horning, K., Seager, S., & Harrington, J. 2007, ArXiv Astrophysics e-prints Seager, S. & Sasselov, D. D. 2000, ApJ, 537, 916 Vidal-Madjar, A., Lecavelier des Etangs, A., Désert, J.-M., Ballester, G. E., Ferlet, R., Hébrard, G., & Mayor, M. 2003, Nature, 422, 143

0704.1117

Condensation of Vortex-Strings: Effective Potential Contribution Through Dual Actions

Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 1 Condensation of Vortex-Strings: Effective Potential Contribution Through Dual Actions Rudnei O. Ramos,∗ Daniel G. Barci,† and Cesar A. Linhares‡ Departamento de Fı́sica Teórica, Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil J. F. Medeiros Neto§ Instituto de Fı́sica, Universidade Federal do Pará, 66075-110 Belem, Pará, Brazil Topological excitations are believed to play an important role in different areas of physics. For example, one case of topical interest is the use of dual models of quantum cromodynamics to understand properties of its vacuum and confinement through the condensation of magnetic monopoles and vortices. Other applications are related to the role of these topological excitations, nonhomogeneous solutions of the field equations, in phase transitions associated to spontaneous symmetry breaking in gauge theories, whose study is of importance in phase transitions in the early universe, for instance. Here we show a derivation of a model dual to the scalar Abelian Higgs model where its topological excitations, namely vortex-strings, become manifest and can be treated in a quantum field theory way. The derivation of the nontrivial contribution of these vacuum excitations to phase transitions and its analogy with superconductivity is then made possible and they are studied here. PACS numbers: 11.10.Wx, 98.80.Cq Keyword: dual models, vortices, phase transitions I. INTRODUCTION Topological excitations, or defects, are nonhomogeneous solutions of the field equations of motion in many types of field theory models [1, 2, 3]. They are finite energy and stable configurations that emerge as a consequence of a spontaneous symmetry breaking process. Mathematically, defects are pre- dicted to appear whenever some larger group of symmetry G breaks into a smaller one H such that there are a nontrivial homotopy group πk(G/H) of the vacuum manifold different from the identity. Well known examples are kinks, or do- main walls (k= 0) [2], that originate from a discrete symmetry breaking, strings, or vortices (k = 1), for example originating from a continuous gauge symmetry breaking U(1) → 1 [4] and magnetic monopoles (k = 2), e.g. from a SO(3)→ U(1) symmetry breaking [5, 6]. Since many phase transitions in na- ture are associated to symmetry breakings, topological excita- tions are a common feature in these processes and are in fact observed in many systems in the laboratory, like in ferromag- netism, helium superfluidity, superconductivity and in many other condensed matter system and they are also expected to have appeared in phase transitions in the early universe as well (for a general review, please see [7]). In the study of phase transitions in quantum field theory one basic quantity usually computed is the effective potential, which is an important tool in the study of phase transitions in scalar and gauge field theories [8]. It is equivalent to the homogeneous coarse-grained free-energy density functional of statistical physics, with its minima giving the stable and, when applicable, metastable states of the system. For inter- acting field theories the effective potential is evaluated per- ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] §Electronic address: [email protected] turbatively, with an expansion in loops being equivalent to an expansion in powers of ~ [9]. The one-loop approximation is then equivalent to incorporating the first quantum corrections to the classical potential. Recall that the effective potential, taking a scalar field theory as an example, is obtained from the effective action Γ[φc], where it is defined in terms of the connected generating functional W [J] as Γ[φc] =W [J]− d4xJ(x)φc(x) , (1) with the classical field φc(~x, t) defined by φc(~x, t)≡ δW [J]/δJ(x), and W [J] =−i~ln Dφ exp S[φ,J] . (2) In order to evaluate Γ[φc] perturbatively, one writes the field as φ(~x, t) → φ0(~x, t) + η(~x, t), where φ0(~x, t) is a field configuration which extremizes the classical action S[φ,J], δS[φ,J] δφ |φ=φ0 = 0, and η(~x, t) is a small perturbation about that extremum configuration. The action S[φ,J] can then be ex- panded about φ0(~x, t) and, up to quadratic order in η(~x, t), we can use a saddle-point approximation to the path integral to obtain for the connected generating functional, W [J] = S[φ0]+~ d4xφ0(x)J(x)+ i ∂µ∂µ+V ′′(φ0) In order to obtain the one-loop expression for Γ[φc], we first note that writing φ0 = φc−η we get to first order in ~, S[φ0] = S[φc]−~ d4xη(x)J(x)+O(~2). Using this result and Eq. (3) into Eq. (1) we find, as J → 0, Γ[φc] = S[φc]+ i ∂µ∂µ+V ′′(φc) . (4) http://arxiv.org/abs/0704.1117v1 mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] 2 Rudnei O. Ramos et al. The effective action can also be computed as a derivative ex- pansion about φc(~x, t), Γ[φc] = −Veff(φc(x))+ (∂µφc) 2 Z(φc(x))+ . . . The function Veff(φc) is the effective potential. For a constant field configuration φc(~x, t) = φc we obtain Γ[φc] =−ΩVeff(φc) , (6) where Ω is the total volume of space-time. Comparing Eqs. (4) and (6) we obtain for the one-loop effective potential, Veff(φc) =V (φc)− i Ω−1Trln ∂µ∂µ+V ′′(φc) . (7) When working at non-vanishing temperature, the same functional techniques can be used. In this case one is in- terested in evaluating the generating functional (the partition function) Zβ[J] which is given by the path integral [10] Zβ[J] = N Dφexp d3x(LE − Jφ) , (8) where the integration is restricted to paths periodic in τ with φ(0,~x) = φ(β,~x), LE is the Euclidean Lagrangian, and N is a normalization constant. Again one expands about an ex- tremum of the Euclidean action and calculates the partition function by a saddle-point evaluation of the path integral. The result for the one-loop approximation to the effective potential Veff(φc,T ) =Veff(φc) 2π2β4 dx x2ln 1− exp x2 +β2V ′′(φc) As the effective potential is equivalent to the free energy func- tional (for a constant field configuration), all thermodynamics functions follow from it. In particular the different phases, critical temperature of phase transition and temperature de- pendence of the field vacuum expectation value can be ob- tained from (9). From the above discussion it is clear that the one-loop ap- proximation to the effective action, Eq. (4), works best when the classical field does not differ much from the configuration that extremizes the classical action, φc = φ0 +η ∼ φ0, since in this case the saddle-point evaluation to the path integral is ad- equate. Also, φc(~x, t) must be nearly constant so that the effec- tive potential can be obtained from Eq. (6). As J → 0, φc(~x, t) is identified with 〈φ〉, the vacuum expectation value. One ma- jor problem we see in this whole approach of studying the phase structure of field theory models from the effective po- tential is when the action functional, determined from Eq. (8) is dominated not by homogeneous, constant field configura- tions but by nonhomogeneous ones. In those situations when other stable, finite energy solutions to the field equations of motion exist, we expect these configurations to dominate the partition function over the homogenous solutions for instance close to the critical temperature [11]. Under these circum- stances the effective potential, which includes only contri- butions to the partition function from constant background field configurations becomes inappropriate to study the phase transition and we must rely in other approaches, for example studying the phase transition directly from the effective action or free energy for the topological configurations [11, 12], or taking directly a field theoretic description for the topological excitations [13]. In either case we are faced with the problem of accounting for nonlocal contributions in the perturbative expansion, which is only amenable of analysis up to the low- est order leading order. To circumvent these difficulties we make use of the techniques of dualization in field theory, from which the degrees of freedom of the topological excitations are explicitly realized in the functional action. This method is described in the following section, Sec. II, where we spe- cialize to the case of the scalar Abelian Higgs model, whose topological solutions are vortex-strings. In Sec. III we evalu- ated the dual action for the model, making explicit the vortex- strings degrees of freedom and how they couple to the matter fields. In Sec. IV we show how an effective potential calcula- tion for an averaged vortex-string field can be computed and the results and interpretation of the phase transition obtained from this quantity. Finally, we give our conclusions in Sec. V. II. THE STRING SOLUTIONS IN THE SCALAR ABELIAN HIGGS MODEL In this work we will use the Abelian Higgs model, with La- grangian density for a complex scalar field φ and gauge field Aµ given by L =−1 µν + |Dµφ|2 −V(φ) , (10) where, Fµν = ∂µAν−∂νAµ, Dµ = ∂µ− ieAµ and V (φ) is a sym- metry breaking potential, for example given by V (φ) =−m2φ |φ| , (11) with positive parameters m2φ and λ. The symmetry breaking U(1) → 1 with homotopy group π1 6= 1 indicates the exis- tence of string-like topological excitations in the system, or Nielsen-Olesen strings [4] (for an extended introduction and review see e.g. Ref. [3]). For example, for a unit winding string solution along the z axis, the classical field equations of motion obtained from the Lagrangian density (10) admit a stable finite energy configuration describing the string given by (using the cylindrical coordinates r,θ,z) φstring = ρ(r)√ eiθ , (12) Aµ,string = A(r) ∂µθ , (13) Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 3 where the functions ρ(r) and A(r) vanish at the origin and have the asymptotic behavior φ(r → ∞) → ρv ≡ A(r → ∞) → 1 . (14) The functions ρ(r) and A(r) can be obtained numerically by solving the classical field equations for φ and Aµ. If we write the field φ as φ = ρexp(iχ)/ 2, then from (12) and (13) for the string, at spatial infinity ρ goes to the vacuum ρv and Aµ becomes a pure gauge. This also gives, in order to get a finite energy for the string configuration, that ∂µχ = eAµ at r → ∞, so Dµφ = 0. This leads then that, by taking some contour C surrounding the symmetry axis, and using Stokes’ theorem, to the nonvanishing magnetic flux ∂µχdxµ = 2π/e . (15) Since φ must be single-valued, the Eq. (15) implies that on the string χ must be singular. Therefore, the phase χ can be separated into two parts: in a regular part and in a singular one, due to the string configuration, χ(x) = χreg(x)+χsing(x) , (16) where the singular (multivalued) part χsing(x) can be related to a closed world-sheet of an vortex-string [14], εµνλρ∂λ∂ρχsing(x) = n dσµν(x)δ4[x− y(ξ)] = ωµν , (17) where n is a topological quantum number, the winding num- ber, which we here restrict to the lowest values, n = ±1, cor- responding to the energetically dominant configurations. The element of area on the world sheet swept by the string is given dσµν(x) = d2ξ (18) and yµ(ξ) represents a point on the world sheet S of the vortex- string, with internal coordinates ξ0 and ξ1. As usual, we con- sider that ξ1 is a periodic variable, since we work with closed strings, whereas ξ0 will be proportional to the time variable (at zero temperature), in such a way that ξ1 parameterizes a closed string at a given instant ξ0. Eq. (17) is known as the vorticity. Eqs. (17) and (18) entails the vortex-string degrees of freedom and then can be used to identify the topological vortex string contributions to the partition function. Let us briefly recall two main previous methods to take into account the effect of topological strings in phase transitions. The first attempt to do so made use of semiclassical methods [2]. In the semiclassical method we use directly the nonho- mogeneous string solutions, Eqs. (12) and (13), when evalu- ating the effective action. In this case the effective action is evaluated after taking fluctuations around the string vacuum solutions, φ→ φstring+φ′ and Aµ →Aµ,string+A′µ and the func- tional integration performed over the fluctuation fields φ′ and A′µ. In the one-loop approximation, this gives the analogous to Eq. (4), with the constant background field now replaced by the scalar string background configuration plus those analo- gous loop corrections for the gauge field string configuration. But from Eqs. (4) and (8), we see that the effective action at finite temperature is just associated with the free energy of the system, where, here is the free energy in the presence of the string field configurations. This is the procedure used for in- stance in the papers in Ref. [11]. The free energy relevant for the study is written as [11] Fstring =− Zstring , (19) where Zstring is the partition function evaluated in the presence (imposing the appropriate boundary conditions for) of strings, Zstring is the partition function for the trivial (constant) vac- uum sector of the model (and then Eq. (19) is actuals the free energy difference between the string and trivial vacuum sec- tors). β, as always, is the inverse of the temperature (we use throughout this work, unless explicitly noted, with the natural units ~,c,kB = 1) and L is the size of the system. The difficult with the approach given by (19), which be- comes evident from Eq. (4) when we are dealing with noncon- stant background fields, is the nonlocal terms that appears in higher order perturbation terms when expanding the effective action (in this case, the effective action for the string back- ground configurations). The only terms amenable of analysis are the one-loop leading order terms. Analogous approach based on the semiclassical method, is the direct evaluation of the classical partition function taking into account the string degrees of freedom, as performed by the authors in Ref. [12] and where the contribution and interpretation of the phase transition based on the picture of string condensation is an- alyzed using known statistical physics results. Another approach that has been used is to define a field creation operator for vortex-string excitations and then work directly with the correlation functions in terms of these opera- tors. This is the approach for instance taken in Refs. [13, 15]. However, also in this approach the evaluation of correlation functions already at tree-level order is involved and results are lacking beyond that order (though in the first reference of [13] results for the asymptotic behavior of the two-point correla- tion function for vortex operators were obtained at one-loop order, but only for the 2+ 1 dimensions case). III. THE DUAL ACTION FOR VORTEX-STRINGS Let us start by writing the partition function for the Abelian Higgs model (10), which, in Euclidean space-time is given by 4 Rudnei O. Ramos et al. DADφDφ∗ exp d3xLE − SGF , (20) where in the above expression LE denotes the Lagrangian den- sity (10) in Euclidian space-time and SGF is some appropriate gauge-fixing and ghost term that must be added to the action to perform the functional integral over the relevant degrees of freedom. A dual action to the original one is obtained from (20) by appropriately performing Hubbard-Stratonovich transformations on the original field in such a way to become explicitly the strings degrees of freedom, like in the form of Eq. (16) and (17). For that, we first write the complex Higgs field φ in the polar parameterization form φ = ρeiχ/ 2. Then, the scalar phase field χ is split in its regular and singular terms, like in Eq. (16). Lets for now, for convenience, omit the gauge fixing term SGF in Eq. (20) and re-introduce it again in the final transformed action. Following e.g. the procedure of Refs. [16, 17, 18, 19, 20], the functional integral over χ in Eq. (20) can then be rewritten as Dχ exp ρ2 (∂µχ+ eAµ) Dχsing DχregDCµ C2µ − iCµ ∂µχreg − iCµ ∂µχsing + eAµ Dχsing DWµν exp V 2µ + eκAµVµ+ iπκWµνωµν , (21) where we have performed the functional integral over χreg in the second line of Eq. (21). This gives a constraint on the functional integral measure, δ(∂µCµ), which can be repre- sented in a unique way by expressing the Cµ in terms of an antisymmetric field, Cµ = −i κ2 εµνλρ∂νWλρ ≡ κVµ, which then leads to the last expression in Eq. (21). κ is some arbitrary pa- rameter with mass dimension and ωµν is the vorticity, defined by Eq. (17) for the singular phase part of χ. Next, in order to linearize the dependence on the gauge field in the action we introduce a new antisymmetric tensor field Gµν through the identity d4xF2µν DGµν exp G2µν − G̃µνFµν ,(22) with G̃µν ≡ 12 εµνλρGλρ. Substituting Eqs. (21) and (22) back into the partition function, we can immediately perform the functional in- tegral over the Aµ field. This leads to the constraint εµναβ∂µ Gαβ −Wαβ = 0 can be solved by setting Gµν = Wµν − 1µW (∂µBν − ∂νBµ), where we defined, for convenience, eκ = µW and Bµ is an arbitrary gauge field. Using these ex- pressions back in the partition function (and re-introducing the gauge fixing term) we then finally obtain the result DWµνDχsing DBµDρ × exp −Sdual Wµν,Bµ,ρ,χsing − SGF , (23) where the dual action is given by Sdual = 2e2ρ2 V 2µ + (µWWµν − ∂µBν + ∂νBµ)2 + (∂µρ) ρ4 + iπ Wµνωµν . (24) The model described by Sdual is completely equivalent to the original Abelian Higgs model, in the polar representation ob- tained from Eq. (20) and so, any calculations done using (23) must lead to the same results as those done with the original action. The advantage of this dual formulation (24) is that it explicitly exhibits the dependence on the singular configura- tion of the Higgs field, making it appropriate to study phase transitions driven by topological defects. At the same time it also show, from the last term in Eq. (24), that the vortex- string’s degrees of freedom coupled to the matter field through the antisymmetric (or Kalb-Ramond) field. Now, if we come to the part concerning the gauge fixing term SGF in (23), we Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 5 see from Eq. (24) that the dual action exhibits invariance un- der the double gauge transformation: the hypergauge transfor- mation δWµν(x) = ∂µξν(x)− ∂νξµ(x) , δBµ = µW ξµ(x) , (25) and the usual gauge transformation δBµ = ∂µθ(x) , (26) where ξµ(x) and θ(x) are arbitrary vector and scalar functions, respectively. Choosing ξµ = Bµ in the first transformation is equivalent to fix the gauge through the condition Bµ = 0 [17] and this is equivalent to choose the unitary gauge in Eq. (23). The complete form for the gauge fixing action accounting for the gauge invariances (25) and (26) was obtained in Ref. [21], which, besides an overall normalization factor independent of the action fields (and the background Higgs field) give for the quantum partition function the complete result [21] Z = N DWµν DρDBµ DηDη exp 2e2ρ2 V 2µ + (µWWµν − ∂µBν + ∂νBµ)2 (∂µρ) ρ4 −ηρ−3η− 1 (∂µWµν) µWWµν (∂µBν − ∂νBµ)+ (∂µBµ) . (27) where η, η are the ghost fields used to exponentiate the Jaco- bian ρ−3 in the functional integration measure in Eq. (23) and θ,u and ξ are gauge parameters. IV. THE EFFECTIVE POTENTIAL FOR LOCAL VORTEX-STRINGS AVERAGED FIELDS Let us turn now to the study of the problem of vortex-strings condensation during a phase transition. Thus, to proceed fur- ther with the evaluation of the string contribution to the par- tition function we introduce a (nonlocal) field associated to the string. Quantizing the vortex–strings as nonlocal objects and associating to them a wave function Ψ[C], a functional field, where C is the closed vortex–string curve in Euclidean space-time, and noting that the interaction term of the vortex- string with the antisymmetric field in Eq. (24) is in the form of a current coupled to the antisymmetric field, following Refs. [22, 23] we can define the string action term in the form Sstring(Ψ[C],Wµν) = |DσµνΨ[C]|2 −M40|Ψ[C]|2 where Dσµν is a covariant derivative term defined by [24] Dσµν(x) = δσµν(x) − i2πµW Wµν(x) , (29) where δσµν(x) is to be considered as an infinitesimal rectan- gular deformation on the string’s worldsheet. It can be eas- ily checked that Eq. (28) is invariant under the combined gauge invariances (25) and (30) if the hypergauge transfor- mation (25) is now supplemented by the vortex–string field transformation Ψ[C]→ exp −i2πµW dxµξµ(x) Ψ[C] . (30) M40 in Eq. (28) is a dynamical mass for the strings, M40 ≡ with τs is the string tension (the total energy per unit length of the vortex-string) [22, 23], which, in terms of the pa- rameters of the Abelian Higgs model, it is given by [25] τs = πρ2c ε(λ/e 2), where ε(λ/e2) is a function that increases monotonically with the ratio of coupling constants. a in Eq. (31) can approximately be given by the string typical radius can be expressed as [12]) 1/a ∼ mφ , (32) where Tc is the mean-field critical temperature, Tc = 12m2φ/(3e 2 + 2λ/3) [26]. ρc, the Higgs vacuum expecta- tion value, can likewise be expressed as . (33) By defining a local string field as [22] ψ̂C ≡ 4 |Ψ[C]|2 , (34) 6 Rudnei O. Ramos et al. where l is the length of a curve C, and Cx,t represents a curve passing through a point x in a fixed direction t. The vacuum expectation value of ψ̂C is denoted by ψC and represents the sum of existence probabilities of vortices in Cx,t . In terms of ψ̂C, it can be shown that the contribution of the vortices to the quantum partition function, indicated by the last term in Eq. (24) and the integration over χsing, can be written as [22] DΨ[C]DΨ∗[C]e M40 ψ̂C+ µνψ̂C . (35) Eq. (35) implies, together with Eq. (24), that an im- mediate consequence of ψC 6= 0 is the increase of the Wµν mass. This is directly associated with a shift in the mass of the original gauge field in the broken phase, MA = eρc, as M2A → M2A(1+ψC). Since the field ψC, defined by Eq. (34), works just like a local field for the vortex-strings, we are al- lowed to define an effective potential for its vacuum expec- tation value ψC in just the same way as we do for a constant Higgs field. Since this vortex-string field only couples directly to Wµν, at the one-loop level the effective potential for ψC will only involve internal propagators of the antisymmetric tensor field. This effective potential, at one-loop order and at T = 0, was actually computed in Ref. [22] in the Landau gauge for the antisymmetric tensor field propagator and it is given by (in Euclidean momentum space and at finite temperatures) 1-loop eff (ψC) = M40ψC (2π)3 ω2n +k 2 +M2A(1+ψC) , (36) where ωn = 2πn/β are the Matsubara frequencies for bosons. When ψC = 0, in the absence of string vacuum contribu- tions to the partition function, we re-obtain the standard result for the one-loop contribution to the Higgs effective potential coming from the gauge field loops. The sum over the Mat- subara frequencies in (36) is easily performed [26]. We can also work with the resulting expression by expanding it in the high-temperature limit MA 1+ψC/T ≪ 1 and for e2/λ≪ 1, which corresponds to deep in the second order regime of phase transition for the scalar abelian Higgs model. This is analogous to the phenomenology of the Landau-Ginzburg theory for superconductors, where the parameter (e2/λ)−1 (also called the Ginzburg parameter), measuring the ratio of the penetration depth and the coherent length, controls the regimes called Type II and Type I superconductors. In our case, the coherent length is governed by a ∼ 1/MH , where MH is here the temperature dependent Higgs mass, while the penetration depth is proportional to 1/MA, where MA is the (temperature dependent) gauge field mass. This way we find a manageable expression for the finite temperature effective potential given by [21] eff,string(ψC)≃ M40 + 3e2ρ2c 16π2a2 e2ρ2c (1+ψC) 3/2 T − 3e 4ρ4c ln [2/aT ] ψ2C , (37) where M0, a and ρc are given by Eqs. (31), (32) and (33). We can then see that the quantum and thermal corrections in the effective potential for strings, Eq. (37), are naturally ordered in powers of α = e2/λ. Therefore, in the regime α ≪ 1 the leading order correction to the tree-level potential in Eq. (37) is linear in ψC, while the second and the third correction terms are O(α3/2) and O(α2), respectively. Thus, the linear term in ψC controls the transition in the deep sec- ond order regime since the other terms are all subleading in α. Thus, near criticality, determined by some temperature Ts where the linear term in Eq. (37) vanishes, V eff,string(ψC) ∼ 0 in the α ≪ 1 regime. Ts is interpreted as the temperature of transition from the normal vacuum to the state of condensed strings, is then determined by the temperature where the lin- ear term in ψC in Eq. (37) vanishes and it is found to be re- lated to the mean field critical temperature, for which the ef- fective mass term of the Higgs field, obtained from V eff (ρc), vanishes. Using again Eqs. (32) and (33), with the result τsa2 ∼ O(1/λ) and in the perturbative regime e2 ≪ λ ≪ 1, after some straightforward algebra, we find the relation Tc −Ts e−1/λ [1+O(α)] , (38) with next order corrections to the critical temperatures dif- ference being of order O(α). This result for Ts allows us to identify it with the Ginzburg temperature TG for which the contribution of the gauge field fluctuations become important. These results are also found to be in agreement with the calcu- lations done by the authors in Ref. [12], who analyzed an anal- ogous problem using the partition function for strings config- urations, in the same regime of deep second order transition. For the case where the gauge fluctuations are stronger, i.e., for α = e2/λ & 1, the second term in Eq. (37) of order α3/2 induces a cubic term ρ3c to the effective potential, favoring the appearance of a first order phase transition instead of a second order one. Here we see that the non-trivial vacuum ψc 6= 0 above the critical temperature Ts enhance the first order phase transition by an amount (1+ψc)3/2. Hence, since Ts ∼ Tc, we see that the driven mechanism of the first order transition can be interpreted as a melting of topological defects. This mechanism is very well known in condensed matter physics [20] and always leads to a first order phase transition (except in two dimensions). V. CONCLUSIONS We have interpreted here the phase transition in the scalar Abelian Higgs model as a process of condensation of vortex- Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 7 strings condensation. Our analysis was based on a dual real- ization of the original model in such a way to make explicitly the vortex-strings degrees of freedom of the nontrivial vacuum of the model. This way, by constructing a field theory model for string fields, the finite temperature effective potential for a local expectation value for the string field was obtained. The transition temperature obtained from this effective potential, the temperature of transition from the normal vacuum to the state of condensed strings, was then obtained and identified with the Ginzburg temperature for which gauge field fluctua- tions become important. Possible extensions of this work could, for example, to in- clude magnetic monopoles, like in the context of the compact Abelian Higgs model [27], in which case monopoles could be added as external fields in the dual transformations. The study of finite temperature effects and possible consequences for the confinement picture in the dual superconductor model, should be possible in the context of the study performed in this work. Acknowledgments The authors would like to thank Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico (CNPq-Brazil), Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) for the financial support. R.O.R. would like to thank the organizers of the conference Infrared QCD in Rio for the invitation to talk about this work at the conference. [1] S. Coleman, Aspects of Symmetry (Cambridge University Press, Cambridge, 1985). [2] R. Rajaraman, Solitons and Instantons (North Holland, Ams- terdam, 1989). [3] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, 2000). [4] H. B. Nielsen and P. Olesen, Nuc. Phys. B 61, 45 (1973). [5] G. ’t Hooft, Nucl. Phys. B 79, 276 (1974). [6] A. M. Polyakov, JETP Lett. 20, 194 (1974); Soviet Phys. JETP 41, 988 (1976). [7] T. W. B. Kibble, Symmetry breaking and defects, in T.W.B. Kib- ble (Imperial Coll., London) . IMPERIAL-02-3-5, Nov 2002. 35pp. Lectures given at NATO Advanced Study Institute and COSLAB School: Patterns of Symmetry Breaking, Crakow, Poland, 15-29 Sep 2002. ArXive: cond-mat/0211110. [8] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974); S. Wein- berg, Phys. Rev. D 9, 3357 (1974). [9] S. Coleman and E. Weinberg, Phys. Rev. D 3, 1888 (1973); P. Ramond, Field Theory: A Modern Primer, (Addison-Wesley, 1990). [10] J. Kapusta, Finite Temperature Field Theory, (Cambridge Uni- versity Press, 1989). [11] C. A. de Carvalho, D. Bazeia, O. J. P. Eboli and G. C. Marques, Phys. Rev. D 32, 3256 (1985); J. M. Guerra Jr., G. C. Marques and S. J. Rodrigues, Phys. Rev. D 42, 2022 (1990). [12] E. Copeland, D. Haws and R. Rivers, Nuc. Phys. B 319, 687 (1989). [13] E. C. Marino, G. C. Marques, R. O. Ramos and J. S. Ruiz, Phys. Rev. D 45, 3690 (1992); E. C. Marino and R. O. Ramos, Phys. Rev. D 49, 1093 (1994). [14] P. A. M. Dirac, Phys. Rev. D 74, 817 (1948). [15] E. C. Marino, Phys. Rev. D 53, 1001 (1996). [16] K. Lee, Phys. Rev. D 48, 2493 (1993). [17] P. Orland, Nuc. Phys. B 428, 221 (1994). [18] E. T. Akhmedov, M. N. Chernodub, M. I. Polikarpov and M. A. Zubkov, Phys. Rev. D 53, 2087 (1996). [19] D. Antonov and D. Ebert, Eur. Phys. J. C 12, 349 (2000); D. Antonov, Int. J. Mod. Phys. A 14, 4347 (1999). [20] H. Kleinert, Lett. Nuovo Cimento 35, 405 (1982); Phys. Lett. B 293, 168 (1992); see also, Chapter 13 in the second volume of H. Kleinert, Gauge fields in Condensed Matter, Vol I: Su- perflow and Vortex lines (World Scientific, Singapore, 1989); Gauge fields in Condensed Matter, Vol II: Stresses and Defects, Differential geometry, Crystal defects (World Scientific, Singa- pore, 1989). [21] R. O. Ramos, J. F. Medeiros Neto, D. G. Barci, C. A. Linhares, Phys. Rev. D 72, 103524 (2005). [22] K. Seo and A. Sugamoto, Phys. Rev. D 24, 1630 (1981). [23] H. Kawai, Progr. Theor. Phys. 65, 351 (1981). [24] Y. Nambu, in Broken Symmetry (World Scientific, Hong Kong, 1995). [25] M. B. Hindmarsh and T. W. B. Kibble, Rept. Progr. Phys. 48, 477 (1995). [26] L. Dolan and R. Jackiw, Phys. Rev. D 9, 2904 (1974). [27] M. N. Chernodub, Phys. Rev. D 69 094504 (2004). http://arxiv.org/abs/cond-mat/0211110

0704.1118

Observation of Anti-correlation between Scintillation and Ionization for MeV Gamma-Rays in Liquid Xenon

Observation of Anti-correlation between Scintillation and Ionization for MeV Gamma-Rays in Liquid Xenon E. Aprile, K.L. Giboni, P. Majewski∗, K. Ni†, and M. Yamashita Physics Department and Columbia Astrophysics Laboratory, Columbia University, New York, New York 10027 (Dated: November 4, 2018) Abstract A strong anti-correlation between ionization and scintillation signals produced by MeV γ-rays in liquid xenon has been measured and used to improve the energy resolution by combining the two signals. The improvement is explained by reduced electron-ion recombination fluctuations of the combined signal compared to fluctuations of the individual signals. Simultaneous measurements of ionization and scintillation signals were carried out with 137Cs, 22Na and 60Co γ rays, as a function of electric field in the liquid. A resolution of 1.7%(σ) at 662 keV was measured at 1 kV/cm, significantly better than the resolution from either scintillation or ionization alone. A detailed analysis indicates that further improvement to less than 1%(σ) is possible with higher light collection efficiency and lower electronic noise. PACS numbers: 29.30.Kv, 34.80.Gs, 95.55.Ka Keywords: gamma-ray astronomy, dark matter, liquid xenon ∗ Current address: Department of Physics and Astronomy, University of Sheffield, UK. † Current address: Physics Department, Yale University, New Haven, CT 06511. http://arxiv.org/abs/0704.1118v1 I. INTRODUCTION Liquid xenon (LXe) is an excellent medium for radiation detection, with high stopping power, good ionization and scintillation yields. Currently liquid xenon detectors are being developed for several fundamental particle physics experiments, from neutrinoless double beta decay [1] and dark matter weakly interactive massive particles (WIMPs) detection [2, 3, 4, 5], to spectroscopy and imaging of gamma-rays in physics, astrophysics and nuclear medicine [6, 7, 8, 9]. A more precise energy measurement than currently demonstrated with liquid xenon ionization and scintillation detectors would largely benefit all these exper- iments. The best experimental energy resolution is not only orders of magnitude worse than that expected from the Fano factor [10] but even worse than that predicted by the Poisson statistics, based on the measured W-value of 15.6 eV [11], as average energy to produce an electron-in pair. The reason for the discrepancy is yet to be fully understood but fluc- tuations in electron-ion pair recombination rate are known to play a dominant role. Both electron-ion pairs and excitons are produced by the passage of an ionizing particle in liquid xenon. In the presence of an electric field, some of the electron-ion pairs are separated before recombination, providing the charge signal as electrons drift freely in the field (positive ions do not contribute as their drift velocity is several orders of magnitude slower). Recombina- tion of the remaining electron-ion pairs lead to excited xenon molecules, Xe∗2. Excitons that are directly produced by the incident particle also become Xe∗2 molecules. De-excitation of these molecules to the ground state, Xe∗2 → 2Xe + hν, produce scintillation photons with a wavelength of 178 nm [12]. The ionization and scintillation signals in liquid xenon are thus complementary and anti-correlated as the suppression of recombination by the external field results in more free electrons and less scintillation photons. This anti-correlation was first observed by Kubota et al. [13]. Large fluctuations in the number of collected electrons due to their reduction by recombination lead to poor energy resolution of the ionization signal. A way to increase the ionization signal and thus the energy resolution via the photo-ionization effect in LXe doped with triemethylamine (TEA) yielded good results, but only at low elec- tric fields [14]. Another way to improve the energy resolution is to reduce recombination fluctuations by combining ionization and scintillation signals. Since recombination also pro- duce scintillation photons, fluctuations of the combined signal should be reduced. This was originally suggested by Ypsilantis et al. many years ago [15], but the simultaneous detection ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ ������� ������� ������� ������� ������� ������� ������� ������� ������ ������ ������ ������ ������ ������ ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ ������ ������ ������ ������ ������� ������� ������� ������� ������� ������� ������� ������� ����� ����� ����� ����� ����� ����� ����� ����� ������� ������� ������� ������� ������� ������� ������� ������� �������������������������� ��� 3. Signal line Vacuum Pump Cathode Radiation Source Grid Anode Screening mesh at ground 1.9LED PMT PMT Gas Out Anode signal redout Opening for Pumping and Gas Filling Cathode and grid HV feedthroughs Vacuum Insulation HV connection LXe level FIG. 1: The detector schematics (see text for details). of scintillation and ionization in LXe has been hard to realize because of the difficulty to efficiently detect VUV light under the constraints of efficient charge collection. In the Liquid Xenon Gamma-Ray Imaging Telescope (LXeGRIT) [7], which we developed for Compton imaging of cosmic γ-rays, both ionization and scintillation were detected, but the fast scin- tillation signal merely provided the event trigger, while the ionization signal provided the energy measurement, with a resolution of 4.2%(σ) at 1 MeV. The fair resolution has been a major limitation of the LXe time projection chamber (TPC) technology for astrophysics. In recent years, the development of VUV-sensitive photomultiplier tubes (PMTs), capable to operate directly in the cryogenic liquid and to withstand overpressure up to 5 atm, has enabled to significantly improve the Xe scintillation light collection efficiency with good uniformity across the liquid volume. A light readout based on these novel PMTs, coupled with a lower-noise charge readout, was proposed to measure both signals event-by-event and thus to improve the energy resolution and the Compton imaging of the LXeGRIT telescope [16]. The work presented here was carried out with two of the first PMTs developed for operation in LXe and is our first attempt in this direction. Further optimization of these VUV PMTs has continued, driven largely by our XENON dark matter detector develop- ment [2, 17]. These improved PMTs, along with other VUV sensor technologies such as Large Area Avalanche Photodiodes (LAAPDs) and Si photomultipliers (SiPMs), which we are also testing for LXe scintillation detection [18, 19], promise further energy resolution improvement. II. EXPERIMENTAL SET-UP AND SIGNALS The detector used for this study is a gridded ionization chamber with two VUV sensitive PMTs (two-inch diameter Hamamatsu R9288) viewing the sensitive liquid xenon volume from the anode and cathode side. The two PMTs, and the transparent meshes serving as anode, cathode and shielding grid, are mounted in a structure made of Teflon for its VUV reflectivity [20] (Fig. 1). The drift gap, between cathode and grid, is 1.9 cm while the distance between grid and anode is 3 mm. Separate high voltage is supplied to the cathode and the grid, keeping a ratio between the field in the drift gap and the field in the collection gap such as to maximize electron transmission through the grid. The electrons collected on the anode are detected by a charge sensitive amplifier (ClearPulse Model 580). The charge signal is subsequently digitized with 10 bit resolution and a sampling time of 200 ns (LeCroy Model 2262). The scintillation signal from each of the two PMTs is recorded with a digital oscilloscope (LeCroy Model LT374) with 1 ns sampling time. The time difference between the scintillation and ionization signals is the electron drift time which gives the event depth-of-interaction information. The coincidence of the two PMT signals is used as event trigger. Fig. 2 shows the scintillation and ionization waveforms recorded at 1 kV/cm for a 662 keV γ-ray event from 137Cs. The number of photoelectrons, Npe, detected by the PMTs is calculated based on the gain calibration with a light emitting diode (LED). The charge waveform is well described by the Fermi-Dirac threshold function in equation 1, as shown in [21]The pulse height A, drift time td, rise time tr and fall time tf are determined from fitting equation 1 to the charge waveform. A known test pulse is used to calibrate the charge readout system, and the number of collected electrons, Ne, is calculated from the pulse height of the charge waveform. Q(t) = A · e−(t−td)/tr 1 + e−(t−td)/tf The Teflon structure holding the PMTs and the meshes is mounted in a stainless steel vessel filled with liquid xenon at about –95oC during the experiment. A vacuum cryostat surrounds the vessel for thermal insulation. The xenon gas filling and purification system, as well as the “cold finger” system used for this set-up is described in a previous publication[21]. The set-up was modified for these measurements by adding a gas recirculation system [17] in order to purify the xenon continuously until sufficient charge collection is reached. 0 100 200 300 400 500 Time [ns] 0 20 40 60 80 100 120 Time [µs] FIG. 2: Waveforms of scintillation signal (left, sum of two PMTs) and ionization signal (right) of a 662 keV γ-ray event from 137Cs at 1 kV/cm drift field. Field [kV/cm] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Field [kV/cm] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Light Charge Combine FIG. 3: Light and charge yield as a function of drift field for 662 keV γ-rays from 137Cs. The uncertainty for charge measurement is due to the uncertainty in pre-amplifier calibration. The light yield is relative to that at zero field, for which the systematic uncertainty is negligible. III. RESULTS AND DISCUSSION A. Field dependence of scintillation and ionization Fig. 3 shows our measurement of the field dependence up to 4 kV/cm of the light and charge yield for 662 keV γ rays from 137Cs. With increasing drift field, the charge yield increases, while the light yield decreases. This behavior has been known for a long time, and was originally reported in [13]. A parametrization of the field dependence of the light yield, S(E)/S0, was proposed by Doke et al. [22] introducing the model of escaping electrons to explain the scintillation light reduction at low LET(linear energy transfer). In this parametrization, expressed by equation 2, the light yield S(E) at drift field E, normalized by the light yield at zero field, S0, depends on the charge yield Q(E) normalized to the charge at infinite field, Q0, and on the ratio Nex/Ni of the number of excitons and ion pairs produced by a γ-ray. χ is the fraction of escaping electrons, i.e. the fraction of Ni electrons which do not recombine with positive ions for an extended time (>ms) even at zero field, when the probability of recombination is highest. 1 +Nex/Ni −Q(E)/Q0 1 +Nex/Ni − χ Nex/Ni and χ can be determined from a fit of equation 2 to the charge and light yields data, knowing Q0 which is given by Eγ/W , where Eγ is the γ-ray deposited energy and W = 15.6 eV [11] is the average energy required to produce an electron-ion pair in liquid xenon. Fig. 4 shows the result of such a fit to our 662 keV data which gives Nex/Ni = 0.20±0.13 and χ = 0.22 ± 0.02. The errors are from the uncertainty on charge collection only. A ratio of 0.06 for Nex/Ni was originally estimated from the optical approximation, using the absorption spectrum of solid rare gases [23]. In [22], Nex/Ni = 0.20 and χ = 0.43 as estimated from 1 MeV conversion electrons data in LXe. This Nex/Ni value is consistent with that obtained from our data. The difference in χ might be due to the limited range of electric fields used in our study. The charge and light signals can be combined by the following equation, C(E) = a with a = 1/(1 + Nex/Ni) and b = 1 − aχ, which gives a constant C(E) = 1, regardless of applied field. The proportion of charge and light is different at different fields but their sum is constant, as verified by our data in in Fig. 3. Note that at very low fields, equation 2 and 3 are not valid as escaping electrons are not fully collected. B. Combined Energy from Scintillation and Ionization The observed field dependent anti-correlation between charge and light signals and its linear relationship offers a way to improve the energy resolution by combining the two signals with proper coefficients. This was first shown in a measurement of energy loss of relativistic Charge Yield Q(E)/Q 0 0.2 0.4 0.6 0.8 1 1.2 Charge Yield Q(E)/Q 0 0.2 0.4 0.6 0.8 1 1.2 =0.20i/NexN =0.22χ FIG. 4: Correlation between light yield and charge yield for 662 keV γ rays. La ions in liquid argon [25]. More recently, Conti et al. [26] applied the same method to improve the energy measurement of relativistic electrons in a liquid xenon detector using a single UV PMT to detect the scintillation signal. For 570 keV γ-rays from 207Bi, an energy resolution of 3%(σ) was measured at 4 kV/cm, by combining the charge and light signals. In our study, the improved light collection efficiency with two PMTs immersed in the liquid gives even better results. Fig. 5 shows the strong anti-correlation of charge and light signals measured with our detector for 662 keV γ-rays from 137Cs at 1 kV/cm. The energy resolution inferred from the light and the charge signals separately is 10.3% (σ) and 4.8%(σ), respectively. The resolution from the charge signal is consistent with previously measured values [7] and [24]. The charge-light correlation angle, θ, also shown in Fig. 5 is defined as the angle between the major axis of the charge-light ellipse and the X-axis for light. θ can be roughly calculated as tan−1(Rq/Rs), where Rs and Rq are the energy resolutions of the 662 keV peak from scintillation and ionization spectra separately. θ can also be found by a 2D gaussian fit to the charge-light ellipse of the 662 keV peak. A better energy resolution can be achieved by combining the charge and light signals as, sin θ · εs + cos θ · εq sin θ + cos θ where εc is the combined signal, in unit of keV. εs and εq are scintillation light and charge based energy in units of keV. The charge-light combined energy resolution of 662 keV line Energy from Light [keV] 0 100 200 300 400 500 600 700 800 900 10000 662 keV Energy from Charge [keV] 0 100 200 300 400 500 600 700 800 900 10000 662 keV Charge-Light Combined Energy [keV] 0 100 200 300 400 500 600 700 800 900 10000 662 keV Energy from Light [keV] 0 200 400 600 800 1000 1200 1400 1600 662 keV FIG. 5: Energy spectra of 137Cs 662 keV γ-rays at 1 kV/cm drift field in liquid xenon. The top two plots are from scintillation and ionization separately. The strong charge-light anti-correlation is shown in the bottom-right plot. The straight line indicates the charge-light correlation angle. A charge-light combined spectrum (bottom-left) shows a much improved energy resolution of 1.7%(σ). is significantly improved to 1.7%(σ). The energy resolution from the charge-light combined spectrum, Rc, can be derived from equation 4 as [27], R2c = sin2 θ · R2s + cos 2 θ · R2q + 2 sin θ cos θ · Rsq (sin θ + cos θ)2 where Rs and Rq are the energy resolutions from scintillation and ionization spectra sep- arately. The covariance Rsq is the contribution from the correlation of the two signals. The magnitude of Rsq indicates the strength of anti-correlation (or correlation) between the scintillation and ionization signals. It is usually expressed in terms of correlation coefficient ρsq = Rsq/(RsRq) (6) A value of ρsq close to -1 (or 1) indicates a very strong anti-correlation (or correlation) of scintillation and ionization signals, while a zero ρsq means no correlation. In equation 5, Rs and Rq can be expressed as, 2 = R2si +R sg +R ss ≈ R si +R ss (7) ≈ R2qi +R qe (8) where Rsi and Rqi are the energy resolution of scintillation and ionization, respectively contributed by liquid xenon itself. They include the liquid xenon intrinsic resolution and the contribution from fluctuations of electron-ion recombination. Rsg is from the geometrical fluctuation of light collection. It is negligible in our result since only events in the center of the detector were selected for the analysis. Rss is from the statistical fluctuation of the number of photoelectrons Npe in the PMTs. Rss can be calculated roughly as Rs = [1 + ( )2]/Npe, which includes the statistical fluctuations of the number of photoelectrons and the PMTs gain variation (σg/g ∼ 0.67, based on the single-photoelectron spectrum). Rqe is from the noise equivalent charge (ENC) of the charge readout. ENC was measured to be between 600 and 800 electrons, depending on the drift field, from a test pulse distribution. Rqe = ENC/Ne, where Ne is the number of collected charges from the 662 keV peak. We note that we have neglected other contributions to the resolution of the charge measurement, such as from shielding grid inefficiency or pulse rise time variation, as they are sub-dominant compared to the electronic noise contribution. Table I lists the energy resolution of the 662 keV γ-ray peak inferred from ionization, scintillation, and charge-light combined spectra at different drift fields. The quoted errors are statistical only. The correlation angle and the correlation coefficient at each field are also presented. The energy resolution improves with increasing field for both scintillation and ionization, while the charge-light combined energy resolution is about the same at different fields. The best value achieved in this study is 1.7%(σ) at 1 kV/cm drift field. We should mention that during this work, we observed improvement of the energy resolution from both light and combined energy spectra with improved light collection efficiency by using Teflon, while the energy resolution from the charge spectrum did not change. The different value of the charge-light correlation coefficient at different fields indicates a more fundamental correlation coefficient between ionization and scintillation in liquid xenon. In fact, the energy resolution Rc from charge-light combined signals comes from two factors. One is the liquid xenon intrinsic energy resolution Rci. Another factor, Rce, is contributed by external sources, such as the fluctuation of light collection efficiency on the light signal and fluctuation of electronic noise on the charge signal. The charge-light combined energy Field [kV/cm] Rs(%) Rq(%) Rc(%) θ ρsq 0 7.9±0.3 1 10.3±0.4 4.8±0.1 1.7±0.1 24.8o -0.87 2 10.5±0.3 4.0±0.1 1.8±0.1 20.8o -0.80 3 10.0±0.3 3.6±0.1 1.9±0.1 19.7o -0.74 4 9.8±0.3 3.4±0.1 1.8±0.1 19.1o -0.74 TABLE I: Measured energy resolutions (σ) and correlation coefficients for 662 keV gamma rays at different electric field values. resolution can be written as below, R2c = R ci +R ce (9) R2ci = sin2 θ · R2si + cos 2 θ · R2qi + 2 sin θ cos θ ·Rsqi (sin θ + cos θ)2 R2ce ≈ sin2 θ · R2ss + cos 2 θ · R2qe (sin θ + cos θ)2 In these equations Rsi, Rqi are the liquid xenon energy resolution from scintillation and ion- ization separately, as previously discussed. Rsqi indicates the correlation between ionization and scintillation signals in liquid xenon. We can define the intrinsic correlation coefficient, ρsqi, of liquid xenon scintillation and ionization, similar to equation 6 for the measured charge-light correlation coefficient, but without instrumental noise contributions. ρsqi = Rsqi/(RsiRqi) (12) The energy resolution for scintillation, Rsi, and ionization, Rqi, can be calculated based on equation 7 and 8, from the measured values of correlation angle θ, statistical fluctuation of light detection Rss and electronic noise contribution Rqe. The calculated values are listed in Table II. Table II also shows the intrinsic and external contributions, Rci and Rce, to the charge-light combined energy resolution. The values of Rci and Rce are calculated from equation 9-11. The intrinsic correlation coefficients from equation 12 are also shown. The intrinsic energy resolution in liquid xenon from the combined scintillation and ion- ization signals is estimated to be less than 1%. Only upper limits are given here since the uncertainties become large at such small values. The intrinsic correlation coefficients are Field [kV/cm] Rsi(%) Rqi(%) Rce(%) Rci(%) ρsqi 0 6.0±0.3 1 9.9±0.4 4.3±0.1 1.6±0.1 < 1.0 −1.00 2 10.1±0.3 3.5±0.1 1.7±0.2 < 1.2 −0.98 3 9.5±0.3 3.0±0.1 1.8±0.2 < 1.2 −0.98 4 9.3±0.3 2.8±0.1 1.8±0.2 < 1.0 −1.00 TABLE II: Predicted achievable energy resolutions in liquid xenon for light (Rsi), charge (Rqi) and combined (Rci) measurements, and charge-light correlation coefficient by removing instrumental noise contributions. closer to -1 than those measured from the experimental data including instrumental noise- contributions. This indicates near-perfect anti-correlation between ionization and scintilla- tion in liquid xenon. We therefore expect that further improvement in the combined signal energy resolution can be achieved by increasing light collection efficiency and by minimizing electronic noise. C. Energy dependence of resolution The improvement of the energy resolution by combining scintillation and ionization sig- nals was studied at 3 kV/cm drift field as a function of γ-ray energy, using radioactive sources such as 22Na (511 keV and 1.28 MeV), 137Cs (662 keV) and 60Co (1.17 MeV and 1.33 MeV) (Fig. 6). The energy resolution from charge, light and charge-light combined spectra is shown in Fig. 7. The data was fitted with an empirical function, σ/E = α/ (E/MeV ), yielding for the parameter α (8.6±0.4)%, (3.0±0.4)% and (1.9±0.4)% for light, charge and combined spectra separately. IV. CONCLUSION We have shown that the energy resolution of MeV γ-rays in liquid xenon can be signifi- cantly improved by combining simultaneously measured scintillation and ionization signals. The best resolution achieved is 1.7% (σ) for 662 keV γ-rays at 1 kV/cm. This value is 0 500 1000 1500 2000 Energy [keV] 511 keV 1.27 MeV Na−22, 3 kV/cm 0 500 1000 1500 2000 Energy [keV] 1.17 MeV 1.33 MeV Co−60, 3 kV/cm FIG. 6: Energy spectra of 22Na and 60Co γ ray sources at 3 kV/cm, by combining charge and light signals. Energy [keV] 400 600 800 1000 1200 1400 Energy [keV] 400 600 800 1000 1200 1400 Light Charge Combined FIG. 7: Energy dependence of resolution measured from 22Na, 137Cs, and 60Co at 3 kV/cm drift field. much better than that measured from scintillation [10.3% (σ)] or from ionization [4.8%(σ)], separately. By summing the two signals, which are strongly anti-correlated, recombination fluctuations are reduced, resulting in improved energy resolution. At present, the energy resolution of the combined signal is still limited by external factors, such as light collection efficiency, PMT quantum efficiency and charge readout electronic noise. By reducing the contribution from these factors, we estimate that the intrinsic energy resolution of MeV gamma-rays in liquid xenon from charge-light combined signal should be less than 1%. The simultaneous detection of ionization and scintillation signals in liquid xenon therefore pro- vides a practical way to improve the energy measurement with a resolution better than the Poisson limit, and possibly closer to the Fano limit [10]. On the other hand, the limit to the energy resolution of LXe might well be not determined by Fano statistics but rather connected to the liquid phase, for instance to microscopic density non-uniformities of the liquid itself. It has been shown by Bolotnikov and Ramsey [28] that the energy resolution deteriorates as the density of Xe gas increases. The behavior was attributed to the formation of molecular clusters in high pressure and liquid xenon, but a more quantitative explana- tion is needed. Despite the limitation of energy resolution, the liquid phase offers far too many advantages for high energy radiation detection. The development of a LXeTPC which combines millimeter spatial resolution with 1% or better energy resolution, within a large homogeneous volume, is very promising for particle physics and astrophysics experiments. V. ACKNOWLEDGMENTS This work was supported by a grant from the National Science Foundation (Grant No. PHY-02-01740) to the Columbia Astrophysics Laboratory. The authors would like to express their thanks to Tadayoshi Doke and Akira Hitachi for valuable discussions. [1] M. Danilov et al., Phys. Lett. B 480, 12 (2000). [2] E. Aprile et al. (XENON Collaboration), in the proceedings of the International Workshop on Techniques and Applications of Xenon Detectors (Xenon01), ICRR, Univ. of Tokyo, Kashiwa, Japan, held December 2001, arXiv:astro-ph/0207670. [3] G. J. Alner et al., “First limits on WIMP nuclear recoil signals in ZEPLIN-II: A two phase xenon detector for dark matter detection,” arXiv:astro-ph/0701858. [4] D. Y. Akimov et al., Astropart. Phys. 27, 46 (2007) [arXiv:astro-ph/0605500]. [5] M. Yamashita et al., in the proceedings of the International Workshop on Techniques and Applications of Xenon Detectors (Xenon01), ICRR, Univ. of Tokyo, Kashiwa, Japan, held December 2001, World Scientific (2002) p.136. [6] MEG Collaboration, “Search for µ+ → e+γ down to 10−14 branching ratio”, Research proposal to Paul Scherrer Institut, May 1999. [7] E. Aprile et al., Nucl. Instr. and Meth. A 461, 256 (2001). [8] V. Chepel et al., Nucl. Instr. and Meth. A 392, 427 (1997). http://arxiv.org/abs/astro-ph/0207670 http://arxiv.org/abs/astro-ph/0701858 http://arxiv.org/abs/astro-ph/0605500 [9] T. Doke et al., Nucl. Instr. and Meth. A 569, 863 (2006). [10] U. Fano, Phys. Rev. 70, 44 (1946); U. Fano, Phys. Rev. 72, 26 (1947). [11] T. Takahashi et al., Phys. Rev. A 12, 1771 (1975). [12] J. Jortner et al., J. Chem. Phys. 42, 4250 (1965). [13] S. Kubota et al., Phys. Rev. B 17, 2762 (1978). [14] H. Ichinose et al., Nucl. Instr. and Meth. A 322, 216 (1992). [15] J. Séguinot et al., Nucl. Instr. and Meth. A 323, 583 (1992). [16] E. Aprile et al., IEEE Trans. Nucl. Sci. 50, 1303 (2003). [17] E. Aprile et al. (XENON Collaboration), New Astro. Rev., 49, 289 (2005), astro-ph/0407575. [18] K. Ni et al., Nucl. Instr. and Meth. A 551, 356 (2005). [19] E. Aprile et al., Nucl. Instr. and Meth. A 556, 215 (2006). [20] M. Yamashita et al., Nucl. Instr. and Meth. A 535, 692 (2004). [21] E. Aprile et al., Nucl. Instr. and Meth. A 480, 636 (2002). [22] T. Doke et al., Jpn. J. Appl. Phys. 41, 1538 (2002). [23] M. Miyajima et al., Phys. Rev. A 9, 1438 (1974). [24] A. Curioni et al., “A Study of the LXeGRIT Detection Efficiency for MeV Gamma-Rays during the 2000 Balloon Flight Campaign”, arXiv:physics/0702078. [25] H.J. Crawford et al., Nucl. Instr. and Meth. A 256, 47 (1987). [26] E. Conti et al. (EXO Collaboration), Phys. Rev. B 68, 054201 (2003). [27] P.R. Bevington and D.K. Robinson, Data Reduction and Error Analysis for the Physical Science, 2nd ed., pp.45. [28] A. Bolotnikov and B. Ramsey, Nucl. Instr. and Meth. A 428, 391 (1999). http://arxiv.org/abs/astro-ph/0407575 http://arxiv.org/abs/physics/0702078 Introduction Experimental Set-up and Signals Results and Discussion Field dependence of scintillation and ionization Combined Energy from Scintillation and Ionization Energy dependence of resolution Conclusion Acknowledgments References

0704.1119

Massive Stars: From the VLT to the ELT

_wlr_comp.ps To appear in “Massive Stars: Fundamental Parameters and Circumstellar Interactions (2007)”RevMexAA(SC) MASSIVE STARS: FROM THE VLT TO THE ELT Chris Evans 1 RESUMEN Nuestro conocimiento de las estrellas masivas ha aumentado significativamente en los últimos 30 años gracias a las nuevas instalaciones y tecnoloǵıas. En esta contribución presento un gran survey de estrellas masivas que se ha realizado recientemente mediante el uso de VLT-FLAMES, mostrando los campos observados y remarcando la fracción de estrellas binarias que se ha encontrado. Estos datos se han utilizado para la primera comprobación empirica de la dependencia de la intensidad de los vientos estelares con la metalicidad, encontrandose un buen acuerdo con la teora – un resultado de gran importancia para los modelos de evolución estelar, que se utilizan para la interpretaión de cúmulos lejanos, starburst y galaxias con formación estelar. Dando un paso más, comentaré como en la actualidad se están dedicando grandes esfuerzos al avance de los planes de actuacin y desarrollo de los Telescopios de Gran Tamaño, que serán una realidad en un futuro próximo; este hecho nos ofrecerá una posibilidad más que interesante para obtener observaciones con resolución espacial de estrellas masivas más allá del Grupo Local. ABSTRACT New facilities and technologies have advanced our understanding of massive stars significantly over the past 30 years. Here I introduce a new large survey of massive stars using VLT-FLAMES, noting the target fields and observed binary fractions. These data have been used for the first empirical test of the metallicity dependence of the intensity of stellar winds, finding good agreement with theory – an important result for the evolutionary models that are used to interpret distant clusters, starbursts, and star-forming galaxies. Looking ahead, plans for future Extremely Large Telescopes (ELTs) are now undergoing significant development, and offer the exciting prospect of observing spatially-resolved massive stars well beyond the Local Group. Key Words: GALAXIES: MAGELLANIC CLOUDS — INSTRUMENTATION — STARS: EARLY TYPE 1. INTRODUCTION An excellent example of the progress enabled by new observing capabilities is afforded by our knowl- edge of the stellar content of NGC346 – the largest H II region in the Small Magellanic Cloud (SMC). The first photometric study of NGC 346 was pub- lished by Niemela et al. (1986) using data from the 1-m Yale telescope at the Cerro Tololo Inter- American Observatory (CTIO). This paper also pre- sented spectroscopy of some of the brighter mem- bers, building on the pioneering work of Walborn (1978). In the following years, 4-m class telescopes were used to investigate the cluster initial mass function (Massey et al., 1989), and to obtain high- resolution echelle spectra for detailed atmospheric analysis (Kudritzki et al. 1989; Walborn et al. 2000; Bouret et al. 2003). More recently, the images of the cluster from the Hubble Space Telescope Advanced Camera for Surveys (e.g. Sabbi et al. 2007) offer a dramatic illustration of the current ‘state-of-the-art’. Understanding the role of environment and 1UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK ([email protected]). metallicity on the evolution of massive stars has been a key topic for the past two decades. However, un- til now, we have lacked the facilities to obtain large, homogenous sets of observations (in a sensible allo- cation of telescope time) to provide robust empirical constraints to evolutionary models. The delivery of the Fibre Large Array Multi-Element Spectrograph (FLAMES) to the Very Large Telescope (VLT) was the catalyst for such a survey – to address questions such as the metallicity dependence of stellar rota- tional velocities and wind mass-loss rates. 2. THE VLT-FLAMES SURVEY The FLAMES survey of massive stars has ob- served over 800 targets in 7 fields, centered on stel- lar clusters in the Galaxy and SMC/LMC, as listed in Table 1; the inclusion of NGC 346 was an obvi- ous choice. The full content of the survey has been presented by Evans et al. (2005, 2006). Our observations provide lower limits to the bi- nary fraction of the O- and early B-type targets, finding ∼25% in NGC346 and ∼35% in N11. Al- though the programme was not originally concerned with binary detection, the FLAMES spectra are of http://arxiv.org/abs/0704.1119v1 2 CHRIS EVANS TABLE 1 SUMMARY OF VLT-FLAMES FIELDS ‘Young’ clusters ‘Old’ clusters (<5 Myrs) (10-20 Myr) Milky Way NGC 6611 NGC 3293 & 4755 LMC N11 NGC 2004 SMC NGC 346 NGC 330 sufficient quality that the nature and properties of many of the newly discovered systems can be deter- mined. Specific systems will be the subject of future papers, and we are seeking further monitoring of our fields to better constrain the binary fraction. The first science papers from the survey have now been published, presenting analyses of the O- and early B-type spectra (Mokiem et al. 2006, 2007; Hunter et al. 2007). One of our primary motivations for the FLAMES survey was to test the theoretical prediction that wind mass-loss rates are dependent on metallicity (Kudritzki et al. 1987; Vink et al. 2001). Figure 1 shows the wind momentum – lumi- nosity relations (WLR) obtained from our LMC and SMC targets, compared with those from contempo- rary Galactic results. This is the first comprehensive empirical test of the metallicity dependence. For lu- minosities greater than ∼105.2 L⊙, the relative off- sets between different metallicity regimes are in good agreement with theory. 3. ELT SCIENCE CASE With instruments such as the FOcal-Reducer low-dispersion Spectrograph (FORS) on the VLT, detailed analysis of stars in galaxies at 1-2 Mpc is currently possible (e.g. Urbaneja et al., 2003; Evans et al., 2007). But ideally we want to reach out to more distant systems (thereby sampling different en- vironments – elliptical galaxies, lower metallicities etc.), or we need spectral resolutions that are greater than those available from FORS. Moving into 2007, the European Southern Ob- servatory (ESO) is starting a full phase A design of its Extremely Large Telescope (ELT). The current baseline reference design is a 42-m primary, with a novel 5-mirror solution that provides correction for ground-layer atmospheric turbulence as its minimum operating mode. An ELT will allow us to observe the resolved massive-star populations in a wide range of systems beyond the Local Group, exploring chemical abundances, stellar kinematics and so on. An ELT will also be able to probe distant clusters and star- 4.5 5.0 5.5 6.0 log(L/Lsun) Fig. 1. The observed wind momentum – luminosity rela- tions (solid lines) compared with theoretical predictions (Vink et al., 2001; dotted lines). The top, middle and bottom lines correspond, respectively, to Galactic, LMC and SMC results (Mokiem et al., submitted). bursts at new levels of detail to study their initial mass functions and kinematic structure. There will be many exciting new opportunities in the ELT era, but it is worth noting for our commu- nity that the most significant observational advances will almost certainly arise in the near-infrared (be- cause of the demands of adaptive optics). To fully exploit an ELT, a large effort will be required to de- velop further diagnostic tools in this wavelength re- gion – both in terms of atomic data (e.g. Przybilla, 2005), and comparison studies in the Milky Way and Magellanic Clouds. REFERENCES Bouret, J.-C., et al. 2003, ApJ, 595, 1182 Evans, C. J., et al. 2005, A&A, 437, 467 Evans, C. J., et al. 2006, A&A, 456, 623 Evans, C. J., et al. 2007, ApJ, astro-ph/0612114 Hunter, I., et al. 2007, A&A, astro-ph/0609710 Kudritzki, R.-P., Pauldrach, A. W. A. & Puls, J. 1987, A&A, 173, 293 Kudritzki, R.-P., et al. 1989, A&A, 226, 235 Massey, P., Parker, J. W. & Garmany, C. D., 1989, AJ, 98, 1305 Mokiem, R. M., et al. 2006, A&A, 456, 1131 Mokiem, R. M., et al. 2007, A&A, accepted Niemela, V. S., Marraco, H. G., & Cabanne, M. L. 1986, PASP, 98, 1133 Przybilla, N. 2005, A&A, 443, 293 Sabbi, E., et al. 2007, AJ, 133, 44 Urbaneja, M. A. et al. 2003, ApJ, 584, 73 Vink, J. S., de Koter, A. & Lamers, H. J. G. L. M. 2001, A&A, 369, 574 Walborn, N. R. 1978, ApJ, 224, L133 Walborn, N. R. et al. 2000, PASP, 112, 1243 http://arxiv.org/abs/astro-ph/0612114 http://arxiv.org/abs/astro-ph/0609710

0704.1120

"Illusion of control" in Minority and Parrondo Games

The efficient punishment theory is a theory of mind, which attempts to give a rational answer to the paradoxal evidence that h “Illusion of Control” in Minority and Parrondo Games J.B. Satinover1,a and D. Sornette2,b 1Laboratoire de Physique de la Matière Condensée, CNRS UMR6622 and Université des Sciences, Parc Valrose, 06108 Nice Cedex 2, France 2Department of Management, Technology and Economics, ETH Zurich, CH-8032 Zurich, Switzerland Abstract: Human beings like to believe they are in control of their destiny. This ubiquitous trait seems to increase motivation and persistence, and is probably evolutionarily adaptive [1,2]. But how good really is our ability to control? How successful is our track record in these areas? There is little understanding of when and under what circumstances we may over-estimate [3] or even lose our ability to control and optimize outcomes, especially when they are the result of aggregations of individual optimization processes. Here, we demonstrate analytically using the theory of Markov Chains and by numerical simulations in two classes of games, the Minority game [4,5] and the Parrondo Games [6,7], that agents who optimize their strategy based on past information actually perform worse than non-optimizing agents. In other words, low-entropy (more informative) strategies under-perform high-entropy (or random) strategies. This provides a precise definition of the “illusion of control” in set-ups a priori defined to emphasize the importance of optimization. PACS. 89.75.-k Complex systems – 89.65Gh Economics; econophysics, financial markets, business and management – 02.50.Le Decision theory and game theory 1 Introduction The success of science and technology, with the development of ever more sophisticated computerized integrated sensors in the biological, environmental and social sciences, illustrate the quest for control as a universal endeavor. The exercise of governmental authority, the managing of the economy, the regulation of financial markets, the management of corporations, and the attempt to master natural resources, control natural forces and influence environmental factors all arise from this quest. Langer’s phrase, “illusion of control” [3] describes the fact that individuals appear hard-wired to over-attribute success to skill, and to underestimate the role of chance, when both are in fact present. Whether control is genuine or merely perceived is a prevalent question in psychological theories. The sequel presents two rigorously controlled mathematical set-ups demonstrating generic circumstances in which optimizing agents perform worse than non-optimizing or random agents. 2 Minority Games We first study Minority games (MGs), which constitute a sub-class of market-entry games. MGs exemplify situations in which the “rational expectations” mechanism of standard economic theory fails. This mechanism in effect asks, “what expectational model would lead to collective actions that would on average validate the model, assuming everyone adopted it,”?[8]. In minority games, a large number of interacting decision-making agents, each aiming for personal gain in an artificial universe with scarce resources, try to anticipate the actions of others on the basis of incomplete information. Those who subsequently find themselves in the minority group gain. Therefore, expectations that are held in common negate themselves, leading to anti-persistent behavior both for the aggregate behavior and for a [email protected] b [email protected] mailto:[email protected] mailto:[email protected] individuals. Minority games have been much studied as repeated games with expectation indeterminacy, multiple equilibria and inductive optimization behavior. Consider the Time-Horizon MG (THMG), where N players have to choose one out of two alternatives at each time step based on information represented as a binary time series A(t). Those who happen to be in the minority win. Each agent is endowed with S strategies. Each strategy gives a prediction for the next outcome A(t) based on the history of the last m realizations A(t-1), …, A(t-m) (m is called the memory size of the agents). Each agent holds the same number S of (in general different) strategies among the total number 22 strategies. The S strategies of each agent are chosen at random but once and for all at the beginning of the game. At each time t, in the absence of better information, in order to decide between the two alternatives for A(t), each agent uses her most successful strategy in terms of payoff accumulated in a rolling window of finite length τ up to the last information available at the present time t. This is the key optimization step. If her best strategy predicts A(t)=+1 (resp. -1), she will take the action ai(t) = - 1 (resp. +1). The aggregate behavior A(t) = Σi=1N ai(t) is then added to the information set available for the next iteration at time t+1. The corresponding instantaneous payoff of agent i is given by – sign[ai(t) A(t)] (and similarly for each strategy for which it is added to the τ-1 previous payoffs). As the name of the game indicates, if a strategy is in the minority (ai(t) A(t) < 0), it is rewarded. In other words, agents in MG try to be anti-imitative. The richness and complexity of minority games stem from the fact that agents strive to be different. Previous investigations have shown the existence of a phase transition between an inefficient regime and an uncorrelated phase as the parameter 2m/N is increased, with the size of the fluctuations of A(t) (as measured by its normalized variance σ2/N) falling below the random coin-toss limit for large m’s when agents always use their highest scoring strategy [5]. The phenomenon discussed here is not directly related to these effects as it is observed in all regimes. Our main result may be stated concisely from the perspective of utility theory: Throughout the space of parameters (N, m, S, τ), the mean payoff of strategies not only surpasses the mean payoff of supposedly-optimizing agents, but the respective cumulative distribution functions (CDF) of payoffs show a first-order stochastic dominance of strategies over agents. Thus, were the option available to them, agents would behave in a risk-averse fashion (concave utility function) by switching randomly between strategies rather than optimizing. This result generalizes when comparing optimizing agents with S>1 strategies with agents having only one strategy (or equivalently m identical strategies), when the single strategies are actually implemented (which takes into account the impact of the strategy itself). The same result is also found when comparing optimizing agents with agents flipping randomly between their S strategies. Agents are supposed to enhance their performance by choosing adaptively between their available strategies. In fact, the opposite is true: The optimization method is found to be strictly a method for worsening performance! This “illusion of control” effect is observed for all N, m, S and τ . Indeed, extensive numerical simulations show that all the phenomena we derive for the THMG with the simplest parameter values are found in the MG with arbitrary parameters. We use the Markov chain formalism for the THMG [8,9] to obtain the following theoretical prediction for the average gains ΔWAgent and ΔWStrategy, respectively of optimizing agents and of their strategies [10]: Agent DNW A μΔ = ⋅ (1) ( )12 ˆStrategy NW μ κ μΔ = ⋅s ⋅ (2) where the brackets denote a time average. μ is a ( )m τ+ -bit “path history” (sequence of 1-bit states); μ is the normalized steady-state probability vector for the history-dependent ( ) (m mτ τ+ × + transition matrix , where a given element T̂ 1,t t Tμ μ − represents the transition probability that 1tμ − will be followed by tμ ; DA is a (2 m )τ+ -element vector listing the particular sum of decided values of A(t) associated with each path-history; ˆμs is the table of points accumulated by each strategy for each path-history; κ is a (2 m )τ+ -element vector listing the total number of times each strategy is represented in the collection of N agents. As shown in the supplementary material, may be derived from T̂ DA , ˆμs and , the number of undecided agents associated with each path history. Thus agents’ mean gain is determined by the non-stochastic contribution to A(t) weighted by the probability of the possible path histories. This is because the stochastic contribution for each path history is binomially distributed about the determined contribution. Strategies’ mean gain is determined by the change in points associated with each strategy over each path-history weighted by the probability of that path. We find excellent agreement between the numerical simulations and the analytical predictions (1) and (2) for the THMG. For instance, for m=2, S=2, τ=1 and N=31, ‚∆W Ú=-0.22 for both analytic and numerical methods (Agent payoff per time step averaged over time and over the optimizing agents) compared with ‚∆W Ú=-0.06 also for both analytic and numerical methods (corresponding payoff for individual strategies). In this numerical example, the average payoff of individual strategies is larger than for optimizing agents by 0.16 units per time step. The numerical values of the predictions (1) and (2) are obtained by implementing each agent individually as a coded object. Strategy Taking into account the impact of strategies modifies these results quantitatively but not qualitatively. The mean per-agent per-step payoff ‚∆WNon-OptÚ accrued by non-optimizing agents (because they have only one fixed strategy, or equivalently their S strategies are identical) is larger than the payoff ‚∆WAgentÚ of optimizing agents. In general, this comparative advantage decreases with their proportion. For example, with m=2, S=2, τ=1 and N=31, and 2500 random initializations and n optimizing agents, ‚∆WNon-OptÚ-‚∆WAgentÚ= (2.380, 2.270, 2.289, 2.275, 2.145, 2.060, 2.039, 1.994, 1.836, 1.964) for n=1, 2, …, 10. More generally, the following ordering holds: payoff of individual strategies > payoff of non- optimizing agents > payoff of optimizing agents. The first inequality is due to the fact that not all individual strategies are implemented and the theoretical payoff of the non-implemented strategies does not take into account what their impact would have been (had they been implemented). Implementation of a strategy tends to decrease its performance (this is similar to the market impact of trading strategies in financial markets associated with slippage and market friction). Non-optimizing agents by definition always implement their strategies. However, the higher payoff of non-optimizing compared with optimizing agents shows that the illusion-of-control effect is not due to the impact mechanism, but is a genuine observable effect. 310−× The amplitude of the illusion-of-control effect highlights the distinction between optimizing agents with S maximally distinct strategies (in the sense of Hamming distance) and non-optimizing agents with S identical strategies. It is helpful to generalize this dichotomy by characterizing the degree of similarity between the S strategies of a given agent using the Hamming distance dH between strategies (the Hamming distance between two strings of equal length is the number of positions for which the corresponding symbols are different). Non-optimizing agents with S identical strategies corresponds to dH=0. In contrast, optimizing agents with S maximally distinct strategies have large dH’s. Since agents with dH=0 out-perform agents with large dH, it is natural to ask whether the ranking of dH could be predictive of the ordering of agents’ payoffs. A non-zero Hamming dH implies that there are at least two strategies among the S strategies of the agent which are different. But, if dH is small, the small difference between the S strategies makes the optimization only faintly relevant and one can expect to observe a payoff similar to that of non-optimizing agents, therefore larger than for optimizing agents with large dH’s. This intuition is indeed confirmed by our calculations: the average payoff per time step is a decreasing function of dH, as originally discussed in [11,12]. The illusion-of-control effect suggests that the initial set-up of MG in terms of S fixed strategies per agent is evolutionarily unstable. It is thus important to ask what happens when agents are allowed to replace strategies over time based on performance. A number of authors have investigated this issue, adding a variety of longer-term learning mechanisms on top of the short-term adaptation that constitutes the basic MG [12-18]. Inter alia, Ref.[13] demonstrates that if agents are allowed to replace strategies over time based on performance, they do so by ridding themselves of those composed of the more widely Hamming-distant tuples. Agents that start out composed of identical strategies do not change at all; those composed of strategies close in Hamming space change little. Similarly, the authors of [12] explicitly fixed agents with tuples of identical strategies and found they performed best. Another important finding in [12] is that the best performance attainable is equivalent to that obtained by agents choosing their strategies at random. Note that learning only confers a relative advantage. In general, agents that learn out-perform agents that don’t. This is certainly true for this privileged subset of agents among standard ones. But the performance of learning agents approaches a maximum most closely attained by agents where the hamming distance between strategies is 0. These agents neither adapt (optimize) nor learn. One might say that when learning is introduced, the system learns to rid itself of the illusory optimization method that has been hampering it. There are exceptions, of course. Carefully designed privileges and certain kinds of learning can yield superior results for a subset of agents, and occasionally for all agents. But the routine outcome is that both optimization and straightforward learning cannot improve on simple chance. The fact that the optimization method employed in the MG yields the opposite of the intended consequence, and that learning eliminates the method, leads to an important question. We pose it carefully so as to avoid introducing either privileged agents or learning: Is the illusion-of-control so powerful in this instance that inverting the optimization rule could yield equally unanticipated and opposite results? The answer is yes: If the fundamental optimization rule of the MG is symmetrically inverted for a limited subset of agents who choose their worst-performing strategy instead of their best, those agents systematically outperform both their strategies and other agents. They also can attain positive gain. Thus, the intuitively self-evident control over outcome proffered by the MG “optimization” strategy is most strikingly shown to be an illusion. Even learning and evolutionary strategies generally at best rid the system of any optimization method altogether. They do not attain the kind of results obtained simply by allowing some agents to reverse the method altogether. We discuss elsewhere the phenomena that arise as the proportion of agents choosing their best performing strategy and of agents choosing their worst performing strategy are varied for different parameters of the MG. We emphasize here only the fact that extensive numerical studies confirm that the phenomenon here indicated persist over a very wide range of parameters for both the MG and the THMG. Hence, having a portfolio of S strategies to choose from is counter-productive: (diversification + optimization) performs on average worse than a single fixed strategy. Intuitively, the illusion-of-control effect in MG results from the fact that a strategy that has performed well in the past becomes crowded out in the future due the minority mechanism: performing well in the recent past, there is a larger probability for a strategy to be chosen by an increasing number of agents, which inevitably leads to its demise. This argument in fact also applies also to all the strategies which belong to the same reduced set; their number is 22 2 m , equal to the ratio of the cardinality of the set of all strategies to the cardinality of the set of reduced strategies. Thus, the crowding mechanism operates from the fact that a significant number of agents have at least one strategy in the same reduced subset among the 2m reduced strategy subsets. Optimizing agents tend on average to adapt to the past but not the present. They choose an action a(t) which is on average out-of-phase with the collective action A(t). In contrast, non-optimizing agents average over all the regimes for which their strategy may be good and bad, and do not face the crowding-out effect. The crowding-out effect also explains simply why anti-optimizing agents over-perform: choosing their worst strategy ensures that it will be the least used by other agents in the next time step, which implies that they will be in the minority. The crowding mechanism also predicts that the smaller the parameter 2m/N, the larger the illusion-of-control effect. Indeed, for large values of 2m/N, it becomes more and more probable that agents have their strategies in different reduced strategy classes, so that a strategy which is best for an agent tells nothing about the strategies used by the other agents, and the crowding out mechanism does not operate. Thus, regions of successful optimization, if they occur at all, are more likely at higher values of 2m/N . (See supplementary material for further details.) This leads to the conclusion that there is a profound clash between optimization on the one hand and minority payoff on the other hand: an agent who optimizes identifies her best strategy, but in so doing by her “introspection”, she somehow knows the fate of the other agents, that it is probable that the other agents are also going to choose similar strategies, … which leads to their underperformance since most of them will then be in the majority. It follows then that an optimizing agent playing a standard minority game should optimize at a second order of recursion in order to win: Her best strategy allows her to identify the class of best strategies of others, which she thus must avoid absolutely to be in the minority and to win (given that other players are just optimizing at the first order as in the standard MG). Generalization to ever more complex optimizing set-ups, in which agents are aware of prior- level effects up to some finite recursive level, can in principle be iterated ad infinitum. Actually, the game theory literature on first-entry games shows that the resulting equilibria depend on how agents learn [19]: with reinforcement learning, pure equilibria involve considerable coordination on asymmetric outcomes where some agents enter and some stay out; learning with stochastic fictitious plays leads to symmetric equilibria in which agents randomize over the entry decisions. There may even exist asymmetric mixed equilibria, where some agents adopt pure strategies while others play mixed strategies. We consider the situation where agents use a boundless recursion scheme to learn and optimize their strategy so that the equilibrium corresponds to the fully symmetric mixed strategies where agents randomize their choice at each time step with unbiased coin tosses. Consider a MG game with N agents total, NR of which employ such a fully random symmetric choice. The remaining NS = N-NR “special” agents (with NR >> NS) will all be one of three possible types: agents with S fixed strategies that choose their best (respectively worst) performing strategy to make the decision at the next step (referred to above as anti-optimizing) and agents with a single fixed strategy. Our simulations confirm that these three types of agents indeed under-perform on average the optimal fully symmetric purely random mixed strategies of the NR agents (see Fig. 4 of the Supplementary materials). Here, pure random strategies are obtained as optimal, given the fully rational fully informed nature of the competing agents. The particular results are sensitive to which strategies are available to the special agents and to their proportion. Their underperformance in general requires averaging over all possible strategies and S-tuples of strategies. (In the supplementary material we show sample numerical results for NS = 1). 3 Parrondo Games We now turn to the illusion-of-control effect in the Parrondo games (PG). The basic Parrondo effect (PE) was first identified as a game-theoretic equivalent to the directional drift of Brownian particles in a time-varying “ratchet”-shaped potential [20,21], wherein two or more individually fair (or losing) games yield a net winning outcome if alternated periodically or randomly. Consider N > 1 s-state Markov games , iG { }1,2, ,i ∈ … N , and their N s s× transition matrices, . For every ( )ˆ iM ( )ˆ iM , denote its vector of s winning probabilities conditional on each of the s states as ( ) ( ) ( ) ( ){ }1 2, ,i i i sip p p=p … and its steady-state equilibrium distribution vector as ( ) ( ) ( ) ( ){ }1 2, , ,i i i sπ π πΠ = i… . For each game, the steady-state probability of winning is therefore ( ) ( ) ( )i iwinP = ⋅Πp . Consider also a sequence of randomly alternating with individual time-averaged proportion of play 0,1 , 1 ∈ =∑ . The transition matrix for the combined sequence of games is the convex linear combination with conditional winning probability vector 1 2( , , , ) ( ) γ γ γ γ ≡ ∑M M… ( , ,..., ) γ γ γ γ = ∑p p and steady-state probability vector ( 1 2, , , n )γ γ γΠ … (which is a complex nonlinear mixture of the Π 's ). The steady-state probability of winning for the combined game is therefore ( ) ( ) (1 2 1 2 1 2 , , , , , , , , ,N N )Nγ γ γ γ γ γ γ γ γ= ⋅ Πp… … … (3) A PE occurs whenever (and in general it is the case that) ( ) ( 1 2 , , , i win win P P )γ γ γγ ≠∑ … i.e. ( ) ( ) ( ) (1 2 1 2, , , , , , γ γ γ γ γ γγ ⋅ Π ≠ ⋅ Π∑ p p … )N… (4) hence the PE, or “paradox”, when the left hand sides of (4) are less than zero and the right- hand sides greater. Many variants have been studied, including capital-dependent multi-player PG (MPPG) [22,23]: At (every) time-step t, a constant-size subset of all participants is randomly re- selected actually to play. All participants keep individual track of their own capital but do not alternate games independently based on it. Instead this data is used to select which game the participants must use at t. The chosen game is the one which, given the individual values of the capital at and the known matrices of the two games and their linear convex combination, has the most positive expected aggregate gain in capital, summed over all participants. This rule may be thought of as a static optimization procedure—static in the sense that the “optimal” choice appears to be known in advance. It appears exactly quantifiable because of access to each player’s individual history. If the game is chosen at random, the change in wealth averaged over all participants is significantly positive. But when the “optimization” rule is employed, the gain becomes a loss significantly greater than that of either game alone. The intended “optimization” scheme actually reverses the positive (collective) PE. The reversal arises in this way: the “optimization” rule causes the system to spend much more time playing one of the games, and individually, any one game is losing. Here, we present a more natural illustration of the illusion-of-control: while the MG is intrinsically collective, PGs are not. Neither the capital- nor the history-dependent variations require a collective setting for the PE to appear as shown from (4). Thus, the effect is most clearly demonstrated in a single-player implementation with two games under the most natural kind of optimization rule: at time t, the player plays whichever game has accumulated the most points (wealth) over a sliding window of τ prior time-steps from to 1t − t τ− . Under this rule, a “current reversal” (reversal of a positive PE) appears. By construction, the individual games ( )1M̂ and ( )2M̂ played individually are both losing; random alternation between them is winning (the PE effect (4)), but unexpectedly, choosing the previously best- performing game yields losses slightly less than either ( )1M̂ and ( )2M̂ individually: the PE is almost entirely eliminated. Furthermore, if instead the previously worst performing game is chosen, the player does better than either game and even much better than the PE from random game choice. Under the choose-best optimization rule, two matrices and ( )1M̂ ( )2M̂ do not form a linear convex sum. Instead, the combined game is represented by an ( ) (s s )τ τ+ × + transition matrix ( )1,2Q̂ with conditional winning probabilities ( ){ }1 1 2 2 1 212 1 1j j j j j jq p p p p p pα β β α β β⎡ ⎤ ⎡ ⎤= + − + − +⎣ ⎦ ⎣ ⎦j with 1,2, , 2j s= … and indices ( ) [ ] [ ] [ ](121,4 1, 1,2 1j Mod j j j Mod jα β= − + = − − )+ . (Under the choose-worst rule ( ){ }1 1 2 2 1 212 1 1j j j j j jq p p p p p pα β β α β β⎡ ⎤ ⎡= − + + + −⎣ ⎦ ⎣ j ⎤⎦ ). Using matrices with the same values as studied in [24,25], the one-player two-game history-dependent PE is as follows: ( )1M̂ and ( )2M̂ have respective winning probabilities and . Alternated at random in equal proportion , . If the previously winning game is selected, , while if the previously losing one is, . The mechanism for this illusion-of-control effect characterized by the reversing of the PE under optimization is not the same as for the MG, as there is no collective effect and thus no- crowding out of strategies or games. As seen from 4), the PE results from a distortion of the steady-state equilibrium distributions 1 0.494winP = 2 0.495winP = ( )1 2 0.5γ γ= = 1 2 0.5, 0.5 0.501winP γ γ= = = (1,2) 0.496bestwinP = (1,2) 0.507bestwinP = Π and Π into a vector Π (for the n=2 version) which is more co-linear to the conditional winning probability vector p than in the case of each individual game (this is just a geometric restatement of the fact that the combined game is winning). One can say that each game alternatively acts at random so as to better align these two vectors on average under the action of the other game. Choosing the previously best performing game amounts to removing this combined effect, while choosing the previously worst performing game tends to intensify this effect. We have identified two classes of mechanisms operating in the Minority games and in the Parrondo games in which optimizing agents obtain suboptimal outcomes compared with non- optimizing agents. These examples suggest a general definition: the “illusion of control” effect occurs when low-entropy strategies (i.e. which use more information) under-perform random strategies (with maximal entropy). The illusion of control effect is related to bounded rationality as well as limited information [26] since, as we have shown, unbounded rational agents learn to converge to the symmetric mixed fully random strategies. It is only in the presence of bound rationality that agents can stick with optimization scheme on a subset of strategies. Our robust message is that, under bounded rationality, the simple (large-entropy) strategies are often to be preferred over more complex elaborated (low-entropy) strategies. This is a message that should appeal to managers and practitioners, who are well-aware in their everyday practice that simple solutions are preferable to complex ones, in the presence hat it can be ly to help formulate better strategies and tools for management. , Gilles Daniel and especially Yannick Malevergne for elpful feedback on the manuscript. the ubiquitous uncertainty. More examples should be easy to find. For instance, control algorithms, which employ optimal parameter estimation based on past observations, have been shown to generate broad power law distributions of fluctuations and of their corresponding corrections in the control process, suggesting that, in certain situations [27], uncertainty and risk may be amplified by optimal control. In the same spirit, more quality control in code development often decreases the overall quality which itself spurs more quality control leading to a vicious circle [28]. In finance, there are many studies suggesting that most fund managers perform worse than random [29] and strong evidence that over-trading leads to anomalously large financial volatility [30]. Let us also mention the interesting experiments in which optimizing humans are found to perform worse than rats [301]. We conjecture that the illusion-of-control effect should be widespread in many strategic and optimization games and perhaps in many real life situations. Our contribution is to put this question at a quantitative level so t studied rigorous nowledgements We are grateful to Riley Crane Appendix A. Analytic Methods and Simulations A1 The Minority Game: Choosing the Best Strategy In the simplest version of the Minority Game (MG) with N agents, every agent has S = 2 strategies and m = 2. In the Time Horizon Minority Game (THMG), the point (or score) table associated with strategies is not maintained from the beginning of the game and is not ever growing. It is a rolling window of finite length τ (in the simplest case 1τ = ). The standard MG reaches an equilibrium state after a finite number of steps stt . At this point, the dynamics and the behavior of individual agents for a given initial quenched disorder in the MG are indistinguishable from an otherwise identical THMG with sttτ ≥ . Extensive numerical simulations show that all the phenomenon we discuss in the THMG with the simplest parameters are found in the MG with arbitrary parameters and in the THMG with τ of generally arbitrary length and parameters. In particular, the message of our communication holds true: agents under-perform strategies. The fundamental result of the MG is generally cast in terms of system volatility: 2 Nσ . All variations of agent and strategy reward functions depend on the negative sign of the majority vote. Therefore both agent and strategy “wealth” (points, whether “real” or hypothetical) are inverse or negative functions of the volatility: The lower the value of 2 Nσ , the greater the mean “wealth” of the “system”, i.e., of agents. However, this mean value is scarcely ever compared to the comparable value for the raw strategies of which agents are composed. Yet agents are supposed to enhance their performance by choosing adaptively between their available strategies. In fact, the opposite is true: The optimization method is strictly a method for worsening performance. To emphasize the relation of the MG to market-games and the illusion of optimization, we transform the fundamental result of the MG from statements on the properties of 2 Nσ to change in wealth, i.e., W tΔ Δ for agents and W tΔ Δ for strategies. We again use the simplest possible formulation—if an agent’s actual (or strategy’s hypothetical ) vote places it in the minority, it scores points, otherwise 1+ 1− . Formally: At every discrete time-step t, each agent independently re-selects one of its S strategies. It “votes” as the selected strategy dictates by taking one of two “actions,” designated by a binary value: ( ) { }1,0 , ,ia t i t∈ ∀ (3) The state of the system as a whole at time t is a mapping of the sum of all the agents’ actions to the integer set { }12N N− , where is the number of 1 votes and . This mapping is defined as : 1N 0 1N N N= − ( ) ( ) 1 0A t a t N N N= − = −∑ (4) If ( ) 2NA t > , then the minority of agents will have chosen 0 at time t ( ); if 0N N< 1 ( ) 2NA t < , then the minority of agents will have chosen 1 at time t ( ). The minority choice is the “winning” decision for t . This is then mapped back to 1N N< 0 { }0,1 : ( ) ( ) ( ) { } { }Sgn 1, 1 0,1sys sysD t A t D t= − ∴ ∈ − + →⎡ ⎤⎣ ⎦ (5) For the MG, binary strings of length m form histories ( )tμ , with . For the THMG, binary strings of length ( )dimm μ= ⎡⎣ t ⎤⎦ m τ+ form paths (or “path histories”) [8,9], with ( )dim tm τ μ+ = , where we define ( )tμ as a history in the standard MG and tμ as a path in the THMG. Then as demonstrated in [8,9], any THMG has a Markov chain formulation. For { } { }, , 2,2,31m S N = , the typical initial quenched disorder in the strategies attributed to each of the N agents is represented by the tensor and its symmetrized equivalent Ω̂ (12ˆ ˆ ˆ= +Ψ Ω ΩT ) . Positions along all S edges of represent an ordered listing of all available strategies. The numerical values in indicate the number of times a specific strategy- tuple has been selected. (E.g., for two strategies per agent, S=2, ijΩ … Ω̂ 2,5Ω =3 means that there are 3 agents with strategy 2 and strategy 5.) Without loss of generality, we may express in upper-triangular form since the order of strategies in a agent has no meaning. The example is a typical such tensor for S = 2, N = 31. Ω̂ (6) 1 2 0 0 1 1 0 0 0 0 0 0 3 3 1 1 0 0 2 0 1 0 0 0 0 0 0 1 1 0 0 1ˆ 0 0 0 0 1 0 2 1 0 0 0 0 0 2 2 1 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 ⎜ ⎟⎜ ⎟ Actions are drawn from a reduced strategy space (RSS) [4,32] of dimension 2m. Each is associated with a strategy k and a path tμ . Together they can be represented in table form as a ( ) ( )dim RSS dim tμ× binary matrix with elements converted for convenience from { } { }0,1 1, 1→ − + , i.e., { }1,tka μ ∈ − + , 1 . For m=2, τ=1 ( )tdimτ μ =3, there are 2m + = possible histories and r=22 reduced strategies (and 2r strategies in total). In this case, the table of dimensio ( ) ( )dim RSS dim tμ× s: coding for all possible reduced strategies and paths read 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 − − − − − − + + − + − + − + + − + − − + + − + − + + − − + + + + ⎟ (7) The change in wealth (point gain or loss) associated with each of the 2r =8 strategies for the 8 paths (= allowed transitions between the 4 histories) at any time t is then: ( ) ( ) ( ) ( ) ( ){ }, 1 ˆ 2 1 ,2t t tS a Mod tμ μ μδ μ− = × −⎡ ⎤⎣ ⎦ 1− (8) [ ],Mod x y is “x modulo y”; and ( )tμ ( )1tμ − label each of the 4 histories { }00,01,10,11 hence take on one of values { }1,2,3,4 . Equation (8) picks out from (7) the correct change in wealth over a single step since the strategies are ordered in symmetrical sequence. The change in points associated with each strategy for each of the allowed transitions between paths tμ of the last τ time steps used to score the strategies is: ( ) ( ) t t i t i μ μ μδ − − − = ∑ (9) For example, for m = 2 and t = 1, the strategy scores are kept for only a single time-step. There is no summation so (9) in matrix form reduces to the score: ( ) ( ), 1t t ts Sμ μ μδ −= (10) or, listing the results for all 8 path histories: ˆˆμ δ=s S (11) ˆδS is an 8μ8 matrix that can be read as a lookup table. It denotes the change in points accumulated over t = 1 time steps for each of the 8 strategies over each of the 8 path- histories. Instead of computing ( )A t , we compute ( )tA μ . Then for each of the 2 possible 8 m τ+ = tμ , ( )tA μ is composed of a subset of wholly determined agent votes and a subset of undetermined agents whose votes must be determined by a coin toss: ( ) ( ) ( )t D t UA A A tμ μ= + μ (12) Some agents are undetermined at time t because their strategies have the same score and the tie has to be broken with a coin toss. ( )U tA μ is a random variable characterized the binomial distribution. Its actual value varies with the number of undetermined agents. This number can be explicated (2): [ ]( ) ( ) [ ]( ) ( ) ( ) 1 1Mod 1,4 1 Mod 1,4 1 Mod 1,2 1 ˆˆ ˆ1 t t m a a s sδ μ δ μμ μ − + − + ⎡ ⎤− +⎣ ⎦ ⎧ ⎫⎛ ⎞⎡ ⎤− ⊗ ⊗⎨ ⎬⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎩ ⎭ (13) “ δ⊗ ” is a generalized outer product, with the product being the Kronecker delta. UN constitutes a vector of such values. The summed value of all undetermined decisions for a given tμ is distributed binomially. Similarly (2): ( ){ } 1 Mod 1,2 1 Sgn s s aμ μ = ⎡ ⎤− +⎣ ⎦ ⎛ ⎡ ⎤⎡ ⎤− •⎜ ⎟⎣ ⎦⎣ ⎦⎝ ⎠ ⎞ (14) An example of how (13) and (14) can be deduced is given later in the context of the original definition of alternate types of agents. Details may also be found in Ref.[9]. We define DA as a vector of the determined contributions to ( )A t for each path tμ . In expression (14) tμ numbers paths from 1 to 8 and is therefore here an index. tsμ is the “ tμ th” vector of net point gains or losses for each strategy when at t the system has traversed the path tμ ( i.e., it is the “ tμ th” element of the matrix ˆˆμ δ=s S in (11)). is a generalized outer product of two vectors with subtraction as the product. The two vectors in this instance are the same, i.e., sμ . “ ” is Hadamard (element-by-element) multiplication and “ ” the standard inner product. The index r refers to strategies in the RSS. Summation over r transforms the base-ten code for tμ into { }1, 2,3, 4,1, 2,3, 4 . Selection of the proper number is indicated by the subscript expression on the entire right-hand side of (13). This expression yields an index number, i.e., selection takes place 1 + Modulo 4 with respect to the value of ( )1tμ − . To obtain the transition matrix for the system as a whole, we require the 2 2m mτ τ+ +× adjacency matrix that filters out disallowed transitions. Its elements are 1,t tμ μ (15) 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0ˆ 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 ⎜ ⎟⎜ ⎟ Equations (13), (14) and (15) yield the history-dependent ( ) ( )m mτ τ+ × + matrix with elements 1,t t Tμ μ − , representing the 16 allowed probabilities of transitions between the two sets of 8 path-histories tμ and 1tμ − : ( ) ( ) ( ) ( )( ) { }( ) 1 1, , Sgn 2 2 Mod , 2 1 t t t t U tU t U t D t U t t A x N μ μ μ μ δ μ μ μ = Γ × ⎧ ⎫⎪ ⎪⎛ ⎞ ⎡ ⎤× + − +⎨ ⎬⎜ ⎟ ⎣ ⎦⎝ ⎠⎪ ⎪⎩ ⎭ (16) The expression ( ) ( )1 U tU t μμ ⎛ ⎞ in (16) represents the binomial distribution of undetermined outcomes under a fair coin-toss with mean = ( )D tA μ . Given a specific , Ω̂ ( ) ( ) t D tA A tμ μ μ= ∀ (17) We now tabulate the number of times each strategy is represented in , regardless of coupling (i.e., of which strategies are associated in forming agent S-tuples): (18) ( ) ( ) ( ) ({ 1 2 2 ˆ ˆ2 , , n n n τκ σ σ ≡ + = =∑ ∑Ω Ω Ψ …T )}σ where kσ is the k th strategy in the RSS, and are the kˆ ˆ,k kΩ Ω th element (vector) in each tensor and ( kn )σ represents the number of times kσ is present across all strategy tuples. Therefore ( )1Agent DNW Abs A μΔ = − ⋅ (19) ( )12 ˆStrategy NW μ κ μΔ = ⋅s ⋅ (20) with μ the normalized steady-state probability vector for . Expression T̂ (19) states that the mean per-step change in wealth for agents equals –1 times the probability-weighted sum of the (absolute value of the) determined vote imbalance associated with a given history. Expression (20) states that the mean per-step change in wealth for individual strategies equals the probability-weighted sum of the representation of each strategy (in a given ) times the sum over the per-step wealth change associated with every history. The –1 in (19) reflects the minority rule. I.e., the awarding of points is the negative of the direction of the vote imbalance. No minus sign is required in (20) as it is already accounted for in (7). Figure A1 shows the cumulative mean change in wealth for strategies versus agents over time, given (15). As first studied in [11,12], and discussed in the body of the manuscript, agent performance is inversely proportional to the Hamming distance between strategies within agents. With the variation expected of a single example, our sample given by Ω̂ (6) reproduces this relation as shown in Figure A2. Thus agent performance is distributed within in orderly if complex fashion. The mean over many corresponds to a “flat” . Ω̂ Ω̂ A2 The Minority Game: Choosing the Worst Strategy First, we re-cast the initial quenched disorder on the set of strategies attributed to the N agents in a given game realization as a two-component tensor: { }ˆ ˆ ˆ,= + -Ω Ω Ω . represents standard (S) agents that adapt as before; represents “counteradaptive” (C) agents that instead select their worst-performing strategies. In our example (6) then, suppose we select at random 3 agents to use the C rule, one each at 1,2 2,6,Ω Ω and 7,8Ω : { }ˆ ˆ ˆ, 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 2 1 1 0 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎧⎛ ⎞ ⎛ ⎪⎜ ⎟ ⎜ ⎪⎜ ⎟ ⎜ ⎪⎜ ⎟ ⎜ ⎪⎜ ⎟ ⎜ ⎪⎜ ⎟ ⎜ ⎨⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎝ ⎠ ⎝⎩ + -Ω Ω Ω (21) For any number of C agents in thus redefined, the analytic expression for need only be modified by decomposing Ω̂ T̂ ( )D tA μ accordingly. The new term in ( )D tA μ makes evident the symmetry of the C rule with respect to the S rule, and the lack of privilege of C agents. Thus: ( ) ( ){ } 1 Mod 1,2 1 ˆ ˆ ˆ1 1 t t t t Sgn s s Sgn s s aμ μ μ μ = ⎡ ⎤− +⎣ ⎦ ⎛ ⎞⎡ ⎤⎡ ⎤ ⎡ ⎤− + + •⎜ ⎟⎣ ⎦ ⎣ ⎦⎣ ⎦⎝ ⎠ ∑ Ψ Ψ  (22) with ( ) ( )1 12 2ˆ ˆ ˆ ˆ ˆ ˆ; + −= + = ++ +Ψ Ω Ω Ψ Ω ΩT − − T (23) The number of undetermined agent votes remains unchanged. In (13), need only be replaced with ( )ˆ ˆ++ -Ω Ω : [ ]( ) ( ) [ ]( ) ( ) ( ) ( )1 1Mod 1,4 1 Mod 1,4 1 Mod 1,2 1 ˆ ˆˆ ˆ1 t t m a a s sδ μ δ μμ μ − + − + ⎡ ⎤− +⎣ ⎦ ⎧ ⎫⎛ ⎞⎡ ⎤− ⊗ ⊗ +⎨ ⎬⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎩ ⎭ + -Ω ΩT T (24) Results for numerical simulation and analytic calculation are in close agreement even for a single short simulation, as illustrated in Table A1. The 3 C agents of 31 now perform so well that they significantly raise the overall performance of the system as detailed in Figure A3. They not only outperform both their own strategies and the other S agents on average, they generate net positive gain. The hypothetical outperformance of unused relative to used strategies in the MG was first observed in (5). But the explicit generation of positive results, by agents simply deploying their unused strategies (without privileging), has not been tested. (In the case of S = 2, “unused” are by definition the “worst-performing”.) We discuss in the manuscript and elsewhere the phenomena that arise as the proportion of S and C agents are varied for different parameters of the MG. We emphasize here only the fact that extensive numerical studies confirm that the phenomenon here illustrated persist over a very wide range of parameters for both the MG and the THMG. A3 The Minority Game: Random Agents We provide in Figure A4 some numerical results for a MG game with N agents total, NR of which employ such a fully random symmetric choice. The remaining NS = N-NR “special” agents (with NR >> NS) will all be one of two possible types: (i) agents with S fixed strategies that choose their worst performing strategy to make the decision at the next step (referred to above as anti-optimizing); (ii) agents with a single fixed strategy. We use the simplest example, that of NS = 1 (with 1τ = ), to illustrate the fact that in the MG, agents allowed/restricted to a fully symmetric random choice outperform agents that attempt to optimize. (Note that the outperformance and absolute positive returns associated with a small proportion of anti-optimizing agents, requires the remaining agents to optimize, as described above. Here the small proportion of or optimizing and anti-optimizing agents compete with fully random agents.) Figures and Tables Figure A1: Mean Strategy versus Agent Cumulative Change in Wealth in the THMG. { } { }, , 2,2,31m S N = ; 100 time steps Figure A2: Agent wealth as a function of Hamming distance between strategy pairs in agents for the example simulation. Figure A3: Average wealth variation per time step for different agents. In red are shown the wealth variations of the three among the 31 agents which use counteradaptive (“C”, choose worst) strategy selection. The usual underperformance of agents compared to individual strategies when using standard selection rule (“S”, choose best) is shown in the blue dots. -0.350 -0.300 -0.250 -0.200 -0.150 -0.100 2 3 4 5 random, S=2 strategic, S=2 random, S=3 strategic, S=3 -0.350 -0.300 -0.250 -0.200 -0.150 -0.100 2 3 4 5 random, S=2 strategic, S=2 random, S=3 strategic, S=3 -0.350 -0.300 -0.250 -0.200 -0.150 -0.100 2 3 4 5 random, S=2 strategic, S=2 random, S=3 strategic, S=3 Figure A4: Performance (mean change in wealth per step) of a single optimizing agent versus all other agents making a symmetric random choice in a MG-like game. From left to right n=11, 21, 31. S=2,3 m= 2,3,4,5 and t=1. Random agents always outperform optimizing agents. Similar results obtain for other values of n, m, S and t.Within statistical fluctuations typical for the number of runs/random selection of strategies comprising the optimizing agent (100 runs), results for anti-optimizing agents are identical. Table A1: Numerical/Analytic Results of THMG with and without 3 C Agents 28 S Agents ‚∆WAgentÚ ‚∆WStrategyÚ With –0.14/–0.14 –0.05/–0.05 Without –0.26/–0.26 –0.05/–0.05 References 1. S.E. Taylor and J.D. Brown, Psychological Bulletin 103 (2), 193-210 (1988). 2. A. Bandura, Self-efficacy: The exercise of control. New York: W.H. Freeman and Company (1997). 3. E.J. Langer, Journal of Personality and Social Psychology 32 (2), 311-328 (1975). 4. D. Challet, D and Y.-C. Zhang, Physica A 246, 407 (1997). 5. D. Challet, M. Marsili and Y.-C. Zhang, Minority Games: Interacting Agents in Financial Markets, Oxford University Press (2005). 6. J.M.R. Parrondo, in EEC HC&M Network on Complexity and Chaos (#ERBCHRX- CT940546), ISI, Torino, Italy (1996). 7. G.P. Harmer, and D. Abbott, Statistical Science 14 (2), 206-213 (1999). 8. M.L. Hart, P. Jefferies, P. M. Hui and N.F. Johnson, The European Physical Journal B 20, 547-550 (2001). 9. M.L. Hart, P. Jefferies, and N. F. Johnson, Physica A 311, 275-290 (2002). 10. The mathematical derivation is given in Appendix A. 11. R. D'Hulst, R. and G. J. Rodgers, Physica A 270, 514-525 (1999). 12. Y. Li, R. Riolo, and R. S. Savit, Physica A, 276, 234-264 (2000) and ibid, 276, 265-283 (2000). 13. M. Andrecut, and M. K. Ali, Physical Review E 64, 67103 (2001). 14. R.M. Araujo and L. C. Lamb, Proceedings of the 16th IEEE International Conference on Tools with Artificial Intelligence (ICTAI'04), 727-731 (2004). 15. W.C. Man and H. F. Chau, Physical Review E 73, 36106 (2006). 16. M. Sysi-Aho, A. Chakraborti, and K. Kaski, European Physical Journal B 34, 373-377 (2003). 17. M. Sysi-Aho, M., A. Chakraborti, and K. Kaski, Physical Review E 69, 36125 (2004). 18. W.S. Yang, B. H. Wang, Y. L. Wu and Y.B. Xie, Physica A 339, 583-590 (2004). 19. J. Duffy and E. Hopkins, Games and Economic Behavior 51, 31-62 (2005). 20. G.P. Harmer, and D. Abbott, Nature 402, 864 (1999). 21. G.P. Harmer, and D. Abbott, Statistical Science 14, 206 (1999). 22. L. Dinis, Europhysics Letters 63, 319 (2003). 23. J.M.R. Parrondo, L. Dinis, J. Buceta, and K. Lindenberg, in Advances in Condensed Matter and Statistical Mechanics, eds. E. Korutcheva and R. Cuerno (Nova Science Publishers, 2003). 24. J.M.R. Parrondo, G.P.Harmer and D. Abbot, Physical Review Letters 85, 5226-5229 (2000) 25. R.J. Kay, and N.F. Johnson, Physical Review E 67, 056128 (2003) 26. J. Huber, M. Kirchler and M. Sutter, working paper (2006). 27. C.W. Eurich, and K. Pawelzik, Int.Conference on Artificial Neural Networks 2, 365-370 (2005). 28. S. Berczuk, and B. Appleton, Software Configuration Management Patterns: Effective Teamwork, Practical Integration, November (2002). 29.B.G. Malkiel, A random walk down Wall Street, 6th ed., W.W. Norton & Co (1996). 30. R.J. Shiller, Market volatility (MIT Press, Cambridge, MA, (1989). 31. T. Grandin and C. Johnson, Animals in translation, Scribner (2004). 32. D. Challet and Y.-C. Zhang, Physica A 256, 514 (1998). http://www.amazon.com/s/002-9956070-3499244?ie=UTF8&index=books&rank=-relevance%252C%252Bavailability%252C-daterank&field-author-exact=Challet%252C%20Damien http://www.amazon.com/s/002-9956070-3499244?ie=UTF8&index=books&rank=-relevance%252C%252Bavailability%252C-daterank&field-author-exact=Marsili%252C%20Matteo http://www.informatik.uni-trier.de/~ley/db/conf/icann/icann2005-2.html#EurichP05 “Illusion of Control” in Minority and Parrondo Games

0704.1121

Classification of Noncommuting Quadrilaterals of Factors

CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS PINHAS GROSSMAN AND MASAKI IZUMI Abstra t. A quadrilateral of fa tors is an irredu ible in lusion of fa tors N ⊂ M with intermediate subfa tors P and Q su h that P and Q generate M and the interse tion of P and Q is N . We investigate the stru ture of a non ommuting quadrilateral of fa tors with all the elementary in lusions P ⊂ M , Q ⊂ M , N ⊂ P , and N ⊂ Q 2-supertransitive. In parti ular we lassify non ommuting quadrilater- als with the indi es of the elementary subfa tors less than or equal to 4. We also ompute the angles between P and Q for quadrilaterals oming from α-indu tion and asymptoti in lusions. 1. Introdu tion If M is a fa tor with a �nite group G a ting by outer automorphisms, then N = MG, the algebra of �xed points of the a tion, is an irredu ible subfa tor of M with index equal to the order of G. It is well known that in this ase the intermediate subalgebras N ⊂ P ⊂ M are pre isely the �xed point algebras of the subgroups of G [48℄. In this spirit one thinks of the lassi� ation of intermediate subfa tors as a �non- ommutative Galois theory�. Motivated by von Neumann's study of proje tion lat- ti es in Hilbert spa e as a � ontinuous geometry�, Watatani proposed studying latti es of intermediate subfa tors, whi h are �nite for irredu ible �nite-index in lusions, as a quantization [56℄. Sano and Watatani introdu ed the notion of angles between subfa tors, a numeri al invariant whi h measures the degree of non ommutativity of pairs of subfa tors [54℄. A major step towards the lassi� ation of intermediate subfa tors was Bis h's har- a terization of intermediate subfa tors as biproje tions in planar algebras- elements of the standard invariant whi h are proje tions in both of the dual algebrai stru tures [4℄. Bis h and Jones then des ribed a generi onstru tion of intermediate subfa - tors by onstru ting a universal planar algebra generated by a single biproje tion, the Fuss-Catalan algebra with parameters orresponding to the indi es [7℄. These intermediate subfa tors are supertransitive in the sense that the standard invariants are minimal. It was hoped that a similar generi planar algebra for multiple intermediate sub- fa tors ould be des ribed in terms of indi es and invariants su h as angles. Indeed, a tensor produ t does yield a generi onstru tion of ommuting pairs of intermediate subfa tors, but onstru ting pairs with nontrivial angles has proven more di� ult. Work supported by JSPS. http://arxiv.org/abs/0704.1121v2 2 PINHAS GROSSMAN AND MASAKI IZUMI In [13℄, Jones and the �rst-named author showed that there are essentially only 2 examples of non ommuting irredu ible quadrilaterals of fa tors su h that all of the elementary in lusions are supertransitive. One is a quadrilateral of subgroups of the symmetri group S3 and the other omes from the GHJ family of subfa tors and all the elementary in lusions have index 2 + It is known that the stru ture of an intermediate subfa tor N ⊂ P ⊂ M is deter- mined by the relationship between the two systems of P -P bimodules arising from N ⊂ P and P ⊂ M (see [6℄,[26℄). After [13℄ appeared, the se ond-named author gave a short proof of the above mentioned result using simple omputation of P −P bimodules (see Remark 4.2). It turns out from the proof that the assumption in [13℄ is too restri tive from the view point of bimodules. Therefore to apture more in- teresting and general stru ture, one should relax the assumption of supertransitivity. Indeed, the �rst named author [12℄ re ently obtained a omplete lassi� ation result of irredu ible non- ommuting quadrilaterals with N ⊂ P and N ⊂ Q isomorphi to the Jones subfa tor of index less than 4 (without posing any assumption on P ⊂ M or Q ⊂ M) and showed that a series of su h quadrilaterals exists. In this paper we investigate quadrilaterals whose elementary in lusions P ⊂ M , Q ⊂ M , N ⊂ P , and N ⊂ Q are ea h 2-supertransitive (an in lusion L ⊂ K is 2-supertransitive if the omplement of L in the L − L bimodule de omposition of K is irredu ible.) In this ase the systems of P − P bimodules for N ⊂ P and for P ⊂ M are ea h generated by a single irredu ible P −P bimodule, and the question is how these two bimodules are related. A key tool in this analysis is the notion of se ond ohomology for subfa tors, introdu ed by the se ond named author and Kosaki in [27℄, whi h ounts the inner onjuga y lasses of subfa tors sharing the same basi extension (as a bimodule lass). For a quadrilateral with N ⊂ P and N ⊂ Q 2-supertransitive, non ommutativity is equivalent to the existen e of an N −N bimodule isomorphism from P to Q [13℄. If N ⊂ P (dual) has trivial se ond ohomology, this N −N bimodule isomorphism an a tually be realized as an algebra isomorhism. In parti ular they showed that any 3-supertransitive subfa tor has trivial se ond ohomology. Ultimately this leads to a lassi� ation of non ommuting quadrilaterals whose ele- mentary subfa tors are 2-supertransitive with trivial se ond ohomology into two ba- si types: those whi h are not o ommuting (� o ommuting� means that the quadri- lateral of ommutants is a ommuting square) and have [M : P ] = [P : N ], like the 2)2 example, and those whi h are o ommuting and have [M : P ] = [P : N ]−1, like the S3 example. The latter type is further subdivided a ording to the Galois group of N ⊂ M , whi h must be a subgroup of S3. In the ase that the elementary subfa tors have index less than or equal to 4 we examine all the possibilities and arrive at the following result. Theorem 1.1. There are exa tly seven non ommuting irredu ible quadrilaterals of hyper�nite II1 fa tors whose elementary in lusions all have indi es less than or equal to 4, up to onjuga y. If (GN⊂P , GP⊂M) are the prin ipal graphs of the elementary subfa tors, the possible on�gurations are: CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 3 (A7, A7), (E 7 , E (A5, A3), (D6, A4), (E 7 , A5), (E 6 , D4) 6 , A3) The fully supertransitive ases are (A7, A7), whi h is the (2 + 2)2 example, and (A5, A3), whi h is the S3 example, and all other ases have some extra stru ture. The new examples in lude a non o ommuting quadrilateral whose elementary subfa tors have index 4 (E 7 , E 7 ), a o ommuting quadrilateral whose indi es are (3 + and (5+ 5)/2 (D6, A4), and several o ommuting quadrilaterals arising from group a tions. Note that any quadrilateral is either non ommuting, non o ommuting (and hen e dual to a non ommuting quadrilateral), or else is both ommuting and o om- muting, in whi h ase there is no obstru tion to the hoi es of N ⊂ P and N ⊂ Q. It is also interesting to study su h quadrilaterals with larger indi es. In fa t the exoti Haagerup subfa tor appears as an elementary subfa tor with maximal super- transitivity in both types of quadrilaterals and the Asaeda-Haagerup subfa tor may appear as well. It is still unknown how many examples there are of the various types of quadrilaterals (although there is an in�nite series of o ommuting quadrilaterals with 3-supertransitive elementary subfa tors, oming from symmetri group a tions.) Se ond ohomolgy also turns out to be losely related to angles. For a non ommut- ing quadrilateral with N ⊂ P and N ⊂ Q 2-supertransitive, there are at most two possibilities for the angle for ea h ohomolgy lass of N ⊂ P (dual). In parti ular, if N ⊂ P is 3-supertransitive, the angle is always cos−1 1/([P : N ] − 1). We also ompute the angles in quadrilaterals arising from α-indu tion (whi h are identi�ed with the GHJ pairs of [13℄) and asymptoti in lusions, the former of whi h in ludes the forked Temperley-Lieb quadrilaterals of [12℄. The paper is organized as follows: Se tion 2 sets forth the basi notation of quadrilaterals, se tors and Q-systems whi h are used throughout. We often deal with in�nite fa tors sin e we are a tually studying properties of the standard invariant. Se tion 3 studies the relationship between se ond ohomology for subfa tors and angles. Se tion 4 proves that quadrilaterals whose elementary subfa tors are 2-supertransitive with trivial se ond ohomology fall into the two basi types dis ussed above, and that for the o ommuting type the Galois group of the total in lusion is a subgroup of S3. Se tion 5 analyzes several lasses of quadrilaterals a ording to the results of the previous se tion and for ea h lass provides an example with maximal supertransi- tivity. Se tion 6 lassi�es all non ommuting irredu ible quadrilaterals whose elementary subfa tors have indi es less than or equal to 4. Se tion 7 omputes the angles for quadrilaterals oming from α-indu tion and asymptoti in lusions. The Appendix onstru ts a Q-system from a se tor in the Haagerup prin ipal graph and shows that it is unique, �lling in a gap for one of the examples in Se tion 5. 4 PINHAS GROSSMAN AND MASAKI IZUMI 2. Preliminaries We �rst �x notation used throughout this paper. We always assume that Hilbert spa es are separable and von Neumann algebras have separable preduals. 2.1. Quadrilaterals. For an in lusion of II1 fa tors N ⊂ M , the Jones index, the Jones proje tion, and the tra e preserving onditional expe tation onto N are de- noted by [M : N ], eN and EN respe tively [29℄. We use the same symbols for an in lusion of properly in�nite fa tors N ⊂ M onsidering a unique minimum ondi- tional expe tation [38℄, [15℄. Although the Jones index [M : N ] does not ne essarily oin ides with the minimum index for a II1 in lusion in general, no onfusion arises as we always onsider extremal in lusions throughout this paper. For a II1 fa tor, we denote by tr the unique normalize tra e. For an in lusion of general fa tors N ⊂ M of �nite index, we use the same symbol tr for the restri tion of EN to the relative ommutant M ∩N ′. Let N ⊂ M be an in lusion of fa tors of �nite index with asso iated tower N = M−1 ⊂ M = M0 ⊂ M1 ⊂ M2 ⊂ · · · , where Mk+1, k ≥ 0 is the von Neumann algebra on the standard spa e of Mk gener- ated by Mk and the Jones proje tion ek+1 = eMk−1. Ea h ek ommutes with N , so {1, e1, .., ek} generates a *-subalgebra, whi h we will all TLk+1, of the kth relative ommutant N ′ ∩Mk. The following de�nition �rst appeared in [32℄. De�nition 2.1. Call a �nite-index subfa tor N ⊂ M k-supertransitive (for k > 1) if N ′ ∩Mk−1 = TLk. We will say N ⊂ M is supertransitive if it is k-supertransitive for all k. Note that N ⊂ M is k-supertransitive if and only if M ⊂ M1 is k-supertransitive. Sano and Watatani [54℄ introdu ed the notion of angles between two subfa tors. De�nition 2.2. For two subfa tors P and Q of a fa tor M , the set of angles Ang(P,Q) between P and Q is the spe trum of the angle operator of the two Jones proje tions eP and eQ, that is, the spe trum of cos −1√eP eQeP − eP ∧ eQ where eP eQeP − eP ∧ eQ is regarded as the operator a ting on its support. Note that Ang(P,Q) = Ang(Q,P ) always holds. De�nition 2.3. A quadrilateral of fa tors Q = P ⊂ M N ⊂ Q is an in lusion of fa tors N ⊂ M of �nite index with two intermediate subfa tors P 6= Q su h that P and Q generate M and P ∩Q = N . The subfa tors N ⊂ P , N ⊂ Q, P ⊂ M , and Q ⊂ M are said to be the elementary subfa tors for Q. When M ∩N ′ = C, the quadrilateral Q is said to be irredu ible. When EP ommutes with EQ, the quadrilateral Q is said to be ommuting. Let M1 = JMN ′JM , P̂ = JMP ′JM , and Q̂ = JMQ ′JM be the basi extensions of M by N , P , and Q respe tively. Then P̂ and Q̂ are intermediate subfa tors between M ⊂ M1, whi h form the dual quadrilateral Q̂ = Q̂ ⊂ M1 N ⊂ P̂ CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 5 The quadrilateral Q is said to be o ommuting if Q̂ is ommuting. The quadrilateral Q is said to be (p, q)-supertransitive if P ⊂ M and Q ⊂ M are p-supertransitive and N ⊂ P and N ⊂ Q are q-supertransitive. Two quadrilaterals Q = P ⊂ M N ⊂ Q P̃ ⊂ M̃ Ñ ⊂ Q̃ are said to be onjugate if there exists an isomorphism π from M onto M̃ su h that π(P ) = P̃ and π(Q) = Q̃. The two quadrilateral Q and Q̃ are said to be �ip onjugate if either they are onjugate or there exists an isomorphism π from M onto M̃ su h that π(P ) = Q̃ and π(Q) = P̃ . We denote by P̂ ⊂ M the in lusion M ⊂ P̂ and by N̂ ⊂ P the in lusion P̌ ⊂ N , where P̌ ⊂ N ⊂ P is the downward basi onstru tion. Note that P̌ is uniquely determined up to inner onjuga y in N . When Q = P ⊂ M N ⊂ Q is a non- ommuting quadrilateral of subfa tors su h that N ⊂ P and N ⊂ Q are 2-supertransitive, the two proje tions eP and eQ are of rank 2 in M1∩N ′ and eP ∧ eQ = eN . Thus Ang(P,Q) onsists of at most one point, whi h will be denoted by Θ(P,Q) if Q is non- ommuting. The following lemma is essentially proved in [13, Proposition 3.2.11, Theorem 3.3.4℄ using Landau's result [41℄: Lemma 2.4. Let Q = P ⊂ M N ⊂ Q be an irredu ible o ommuting quadrilaterals of subfa tors su h that N ⊂ P and N ⊂ Q are 2-supertransitive. If [M : P ] = [P : N ], the quadrilateral Q is ommuting, and if [M : P ] 6= [P : N ], cos2Θ(P,Q) = [P : N ]− [M : P ] [M : P ]([P : N ]− 1) . We show an easy example of a non ommuting quadrilateral oming from a �nite group a tion. For an automorphism group G of a fa tor R, we denote by RG the �xed point subalgebra of R under G. The next lemma follows from Lemma 2.4. Lemma 2.5. Let G be a �nite group and H and K be subgroups of G su h that the natural a tions of G on G/H and G/K are 2-transitive. Assume that [G : H ] = [G : K], [G : H ] 6= [H : H ∩ K], and G a ts on a fa tor R as an outer automorphism group. We set M = RK∩H , P = RH , Q = RK , and N = RG. Then Q = P ⊂ M N ⊂ Q is an irredu ible non ommuting and o ommuting quadrilateral of fa tors su h that N ⊂ P and N ⊂ Q are 2-supertransitive. The angle between P and Q is given by cos2Θ(P,Q) = [G : H ]− [H : H ∩K] [H : H ∩K]([G : H ]− 1) . 6 PINHAS GROSSMAN AND MASAKI IZUMI Example 2.6. For a �nite set F , we denote by SF the set of permutations of F and by AF the set of even permutations of F . Let F0 = {1, 2, · · · , n}, F1 = {1, 2, · · · , n − 1}, and F2 = {2, 3, · · · , n}. Then G = SF0 (G = AF0), H = SF1 (H = AF1), and K = SF2 (K = AF2) with n ≥ 3 (n ≥ 4) satis�es the assumption of Lemma 2.5 and Θ(P,Q) = cos−1(1/(n− 1)). In [13℄ Jones and the �rst-named author obtained the following theorem: Theorem 2.7. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors. If Q is (6,6)-supertransitive, one of the following two o urs: (1) All the elementary subfa tors of Q are the A7 subfa tors and Θ(P,Q) = cos−1( 2 − 1). When M is the hyper�nite II1 fa tor, su h a quadrilateral exists and is unique up to onjuga y. (2) [M : P ] = [M : Q] = 2, [P : N ] = [Q : N ] = 3, Θ(P,Q) = π/3, and the prin ipal graphs of N ⊂ P and N ⊂ Q are A5. In this ase, there exists an outer a tion of the symmetri group S3 of degree 3 on M su h that N is the �xed point algebra of the a tion. When M is the hyper�nite II1 fa tor, su h a quadrilateral is unique up to onjuga y. In Se tion 4, we give a new proof of the above theorem, ex ept for the existen e, relaxing the assumption. The theorem still holds if we assume that Q is (3,4)- supertransitive. 2.2. Se tors. To ompute the angle between P and Q, it is more onvenient to work with properly in�nite fa tors using se tors as we will see below. Thus we re all a few basi fa ts about se tors here. The reader is referred to [20℄ for details. Note that the stru ture of the intermediate subfa tor latti e of an in lusion N ⊂ M , in luding information of the angle, only depends on the standard invariant (or paragroup, plannar algebra) of N ⊂ M . Every standard invariant realized in the type II1 ase is also realized in the properly in�nite ase and vi e versa. For the reason stated above, we always assume in the proofs that the subfa tors involved are properly in�nite though we state results for general quadrilaterals of fa tors. When we show uniqueness results, we need to deal with quadrilaterals of hyper�nite II1 fa tors. Sin e the lassi� ation theory of subfa tors for the hyper�nite II1 fa tor and the hyper�nite II∞ fa tor is the same as far as strongly amenable subfa tors are on erned [52℄, we may assume in the proofs that the fa tors involved are isomorphi to the hyper�nite II∞ fa tors in this ase. Remark 2.8. When one dis usses se tors, it is ustomary to assume that every fa tor involved is of type III. However the whole theory also works for general properly in�nite fa tors. This is based on the following fa ts: (1) for an in lusion of properly in�nite fa tors N ⊂ M , every non-zero proje tion in the relative ommutant N ′ ∩ M is an in�nite proje tion in M , and (2) a properly in�nite fa tor has a unique representation whose ommutant is also properly in�nite. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 7 Let M and N be properly in�nite fa tors. We denote by Mor(N,M) the set of normal unital homomorphisms from N to M . For two morphisms ρ, σ ∈ Mor(N,M), we denote by (ρ, σ) the set of intertwiners between ρ and σ (ρ, σ) = {v ∈ M ; vρ(x) = σ(x)v, ∀x ∈ N}. When ρ is irredu ible, that is M ∩ρ(N)′ = C, the spa e (ρ, σ) is a Hilbert spa e with an inner produ t 〈v1, v2〉 = v∗2v1 for v1, v2 ∈ (ρ, σ). The two morphisms ρ and σ are said to be equivalent if there exists a unitary u ∈ (ρ, σ). We denote by Sect(M,N) the quotient of Mor(N,M) by this equivalen e relation. We denote by [ρ] the equivalen e lass of ρ, whi h we all a se tor. However, we often omit the quare bra ket when no onfusion arises. When M = N , we use the notation Mor(M,M) = End(M) and Sect(M,M) = Sect(M). For M −N bimodule X and ρ ∈ Mor(L,N), we denote by Xρ the M −L bimodule de�ned by x · ξ · y = xξρ(y), ξ ∈ X, x ∈ M, y ∈ L. For σ ∈ Mor(L,M), we denote by σX the L−N bimodule similarly de�ned. Let L2(M) be the standard Hilbert spa e of M , whi h is a M −M bimodule with xξy = xJMy ∗JMξ, where JM is the modular onjugation. Then it is known that for every M − N bimodule X , there exists a unique se tor [ρ] ∈ Sect(M,N) su h that X is equivalent to L2(M)ρ. With this orresponden e, the relative tensor produ t of two bimodules is transformed into the omposition of two morphisms. A dire t sum of bimodules is easily translated into the se tor language too (see [20℄). We denote by Mor0(N,M) the set of ρ ∈ Mor(N,M) whose image has �nite index. For Mor0(N,M), we denote by d(ρ) the square root of [M : ρ(N)], alled the (statisti al) dimension of ρ, whi h is multipli ative under omposition of two se tors and is additive under dire t sum. We use the notation Sect0(M,N) et . in an obvious sense. Corresponding to the omplex onjugate bimodule of L2(M)ρ, the onjugate se tor of ρ is de�ned. For ρ ∈ Mor0(N,M), the (equivalen e lass of) onjugate morphism ρ̄ ∈ Mor0(M,N) is hara terized by existen e of two isometries rρ ∈ (idN , ρ̄ρ) and r̄ρ ∈ (idM , ρρ̄) satisfying the following equation r∗ρρ̄(r̄ρ) = , r̄∗ρρ(rρ) = Throughout this paper, we �x su h rρ and r̄ρ for ea h ρ. When ρ is not self- onjugate, we an and do assume rρ̄ = r̄ρ. This is not the ase for a self- onjugate ρ. In this ase r∗ρρ(rρ) = ±1/d(ρ) holds and so rρ = ±r̄ρ [42℄. A self- onjugate endomorphism ρ is said to be real if rρ̄ = r̄ρ and said to be pseudo-real if rρ̄ = −r̄ρ. We set φρ(x) = r ρρ̄(x)rρ for x ∈ N , whi h is alled the standard left inverse of ρ. We denote by Eρ the minimum onditional expe tation from M onto the image of ρ. Then we have Eρ = ρφρ. For an in lusion of properly in�nite fa tors N ⊂ M of �nite index, we denote by ιM,N the in lusion map ιM,N : N →֒ M . Then the M − N bimodule X = 2(M)N is nothing but L 2(M)ιM,N . Sin e ML 2(M1)M is isomorphi to X ⊗N X∗, we have ML 2(M1)M ∼= L2(M)ιM,N ιM,N . The endomorphism ιM,N ιM,N of M is alled 8 PINHAS GROSSMAN AND MASAKI IZUMI the anoni al endomorphism for N ⊂ M . The endomorphism ιM,N ιM,N of N is alled the dual anoni al endomorphism, whi h is the anoni al endomorphism for the in lusion ιM,N (M) ⊂ N . We often use diagrams for omputation of intertwiners. An intertwiner s ∈ (ρ, στ) is expressed by the diagram For t ∈ (σ, µν) and u ∈ (τ, ξη), the produ ts ts ∈ (ρ, µντ) and σ(u)s ∈ (ρ, σξη) are expressed as µ ν τ , σ(u)s = For spe ial operators 1 ∈ (ρ, ρ), d(ρ)rρ ∈ (id, ρ̄ρ), and d(ρ)r̄∗ρ ∈ (id, ρρ̄), we use the following diagrammati expressions: d(ρ)rρ = d(ρ)r̄∗ρ = Then the equations d(ρ)r̄∗ρρ(rρ) = 1 and d(ρ)ρ̄(r̄ ρ)rρ = 1 are expressed as The following lemma holds (see [28, Se tion 3℄): Theorem 2.9. Let N ⊂ M be an irredu ible in lusion of properly in�nite fa tors with �nite index and let [ιM,N ιM,N ] = ni[ρi] be the irredu ible de omposition, where ni is the multipli ity of ρi in ιM,N ιM,N . Then (1) ni ≤ d(ρi). (2) Let Hi = (ιM,N , ιM,Nρi). For s1, s2 ∈ Hi, EN(s1s 〈s1, s2〉 d(ρi) (3) dimHi = ni. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 9 (4) Let {s(i)j}nij=1 be an orthonormal basis of Hi. Then x ∈ M has the following unique expansion s(i)∗jx(i)j with oe� ients x(i)j ∈ N . The oe� ients x(i)j is given by x(i)j = d(ρi)EN(s(i)jx). The fourth statement above says that { d(ρi)s(i) j}ij is a Pimsner-Popa basis [51℄. The advantage of working on properly in�nite fa tors is that we an hoose ea h element of the basis from an intertwiner spa e, and so (4) is onsidered as a rossed produ t type de omposition. Let Ai be the linear span of {s∗eN t; s, t ∈ Hi}. Then [28, Se tion 3℄ shows that Ai is a simple omponent of M ∩N ′ and M1 ∩N ′ = We an introdu e a right a tion of Ai on Hi by v · (s∗eN t) = d(ρi) 〈v, s〉t, whi h gives a ∗-isomorphism from the opposite algebra of Ai onto B(Hi). Let e(i)jk = d(ρi)s(i) jeNs(i)k. Then {e(i)jk}jk is a system of matrix unit of Ai. Let P be an intermediate subfa tor and let mi = dim(ιP,N , ιP,Nρi). Then we may and do assume that {s(i)j}mij=1 is an orthonormal basis for Hi ∩ P = (ιP,N , ιP,Nρi). Lemma 2.10. Let the notation be as above. Then (1) The restri tion EP |Hi of EP toHi is the proje tion ontoHi∩P = (ιP,N , ιP,Nρi). (2) The Jones proje tion eP is expressed as e(i)jj. (3) Let zi be the unit of Ai. Then s · zieP = EP (s), ∀s ∈ Hi. Proof. (1) Note that EP (Hi) = Hi ∩ P = (ιP,N , ιP,Nρi) holds and EP |Hi is an idem- potent. Thus it su� es to show that EP |Hi is self-adjoint. For s, t ∈ Hi, we have 〈EP (s), t〉 = d(ρi)EN(EP (s)t∗) = d(ρi)EN(EP (EP (s)t∗)) = d(ρi)EN(EP (s)EP (t) ∗)) = 〈EP (s), EP (t)〉, whi h shows the statement. (2) Let ϕ be a faithful normal state of M satisfying ϕ · EN = ϕ. Then (1) and Theorem 2.9,(4) imply the following for x ∈ M and the GNS y li and separating 10 PINHAS GROSSMAN AND MASAKI IZUMI ve tor Ω for ϕ: ePxΩ = EP ( t(i)∗jx(i)j)Ω = EP (t(i)j) ∗x(i)jΩ t(i)∗jx(i)jΩ = d(ρi) t(i)∗jEN(t(i)jx)Ω d(ρi) t(i)∗jeN t(i)jxΩ. This shows (2). (3) follows from (2). � Remark 2.11. The above lemma gives a powerful method to ompute the angle be- tween subfa tors. Let Q be another intermediate subfa tor of N ⊂ M . To ompute the angle between P and Q, it su� es to ompute the angle between two subspa es (ιP,N , ιP,Nρi) and (ιQ,N , ιQ,Nρi) of (ιM,N , ιM,Nρi) for those ρi ontained in ιP,N ιP,N and ιQ,N ιQ,N . Equivalently, it su� es to ompute the spe trum of the restri tion of EPEQ to (ιP,N , ιP,Nρi). Remark 2.12. Even when the in lusion N ⊂ M is redu ible, the angle Ang(P,Q) an be omputed from EPEQ|(ιP,N ,ιP,Nρi) as long as N ⊂ P and N ⊂ Q are irredu ible. Indeed we have M = P ⊕ kerEP and N(ιP,N , ιP,Nρi). Sin e EPEQEP preserves (ιP,N , ιP,Nρi), it su� e to ompute EPEQ|(ιP,N ,ιP,Nρi) to ob- tain the eigenvalues of EPEQEP . 3. Q-systems and angles P ⊂ M N ⊂ Q is an irredu ible non ommuting quadrilateral of subfa tors and N ⊂ P and N ⊂ Q are 2-supertransitive, it is observed in [13, Lemma 4.2.1℄ that L2(P ) and L2(Q) are equivalent as N −N bimodules, or in other words, ιP,N ιP,N is equivalent to ιQ,NιQ,N . Consequently, [P : N ] = [Q : N ] and so [M : P ] = [M : Q]. This follows from the fa t that the two proje tions eP and eQ are equivalent in M1 ∩ N ′, whi h is identi�ed with EndNL2(M)N . Let P̌ and Q̌ be subfa tors of N su h that P̌ ⊂ N ⊂ P and Q̌ ⊂ N ⊂ P are towers. It is known that under a ertain situation, su h as 3-supertransitivity of N ⊂ P and N ⊂ Q, the equivalen e 2(P )N ∼= NL2(Q)N implies inner onjuga y of P̌ and Q̌ in N [27℄, [35℄ though it is not ne essarily the ase in general. To hara terize the anoni al endomorphism, Longo [43℄ introdu ed the notion of Q-systems. De�nition 3.1. Let M be a properly in�nite fa tor and let γ ∈ End0(M). We say that (γ, v, w) is a Q-system if the following three onditions are satis�ed: CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 11 (1) v ∈ (idM, γ) and w ∈ (γ, γ2) are isometries. (2) There exists a positive number d su h that v∗w = w∗γ(v) = 1/d. (3) ww = γ(w)w. Two Q-systems (γ, v, w) and (γ′, v′, w′) are said to be equivalent if there exists a unitary u ∈ (γ, γ′) su h that v′ = uv and w′ = uγ(u)wu∗. It is known that (3) is equivalent to ww∗ = γ(w∗)w under the assumption (1) and (2) [45℄, [27℄. If γ = σσ̄ with σ ∈ Mor0(N ,M) and v = r̄σ ∈ (idM, γ), w = σ(rσ) ∈ σ((idN , σ̄σ)) ⊂ (γ, γ2), then (γ, v, w) is a Q-system with d = d(σ). The equivalen e lass of this Q-system only depends on the inner onjuga y lass of σ(N ) in M. Thus we say that (γ, v, w) arises from σ(N ) ⊂ M. On the other hand, Longo [43℄ showed that any Q-system (γ, v, w) arises from a subfa tor N ⊂ M, determined by EN (x) = w ∗γ(x)w, and equivalen e of two Q-systems exa tly orresponds to inner onjuga y of the orresponding two subfa tors. Let N ⊂ M be an in lusion of properly in�nite fa tors of �nite index and let (γ, v, w) be the Q-system arising from the in lusion. The se ond-named author and Kosaki [27℄ introdu ed the �se ond ohomology" H2(N ⊂ M), whi h is always a �nite set, by the equivalen e lasses of Q-systems (γ1, v1, w2) su h that γ and γ1 are equivalent. When N is the �xed point algebra of M by an outer a tion of a �nite group G, then we an identify H2(N ⊂ M) with the se ond ohomology group H2(G,T). The following is [27, Lemma 6.1℄. Lemma 3.2. Let N ⊂ M be an irredu ible in lusion of fa tors of �nite index. If N ⊂ M is 3-supertransitive, then H2(N ⊂ M) is trivial. When Q = P ⊂ M N ⊂ Q is a non ommuting quadrilateral of subfa tors and N ⊂ P and N ⊂ Q are 2-supertransitive, the two in lusions P̌ ⊂ N and Q̌ ⊂ N give equivalent anoni al endomorphisms as we observed above. De�nition 3.3. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors with 2-supertransitive N ⊂ P and N ⊂ Q. We denote by c(Q) the lass of the Q-system arising from Q̌ ⊂ N in H2(N̂ ⊂ P ). The lass c(Q) is very mu h related to the angle Θ(P,Q) and there are at most two possibilities for Θ(P,Q) for a given lass c(Q) as we will see below. The following lemma is an easy onsequen e of the de�nition above: Lemma 3.4. For an irredu ible non ommuting quadrilateral Q = P ⊂ M N ⊂ Q fa tors with 2-supertransitive N ⊂ P and N ⊂ Q, the following three onditions are equivalent: 12 PINHAS GROSSMAN AND MASAKI IZUMI (1) The lass c(Q) ∈ H2(N̂ ⊂ P ) is trivial. (2) The two subfa tors P̌ and Q̌ are inner onjugate in N . (3) There exists an isomorphism π : P → Q of P onto Q su h that the restri tion π|N of π to N is the identity map. Sin e the Q-systems we en ounter in this paper always have γ de omposed into two irredu ible omponents, we give a detailed des ription of the ondition (3) of De�ni- tion 3.1 in this ase. Let σ ∈ End0(M) be an irredu ible endomorphism satisfying [idM] 6= [σ] and let γ = idM ⊕ σ. Note that for γ to have a Q-system, the endo- morphism σ must be self- onjugate. We �x isometries v ∈ (idM, γ) and v1 ∈ (σ, γ), whi h means γ(x) = vxv∗ + v1σ(x)v 1. We give a des ription of a Q-system of the form (γ, v, w) in terms of an isometry in (σ, σ2). It is easy to show that an isometry w ∈ (γ, γ2) satisfying (2) of De�nition 3.1 is of the following form and vi e versa: d(σ) + 1 v1σ(v)v d(σ) + 1 d(σ) + 1 cv1σ(v1)rσv d(σ)− 1 d(σ) + 1 v1σ(v1)sv where c is a omplex number with |c| = 1 and s ∈ (σ, σ2) is an isometry. Passing from w to uγ(u)wu∗ with u = vv∗ + c̄v1v 1 (or rede�ning rσ) if ne essary, we may always assume c = 1 for a Q-system (γ, v, w). We say that su h a Q-system is normalized (with respe t to (v, v1, rσ)). It is straightforward to show the following: Lemma 3.5. Let σ, γ, v, and v1 be as above. If s ∈ (σ, σ2) is an isometry satisfying (3.1) σ(s)rσ = srσ, (3.2) d(σ)rσ + (d(σ)− 1)s2 = d(σ)σ(rσ) + (d(σ)− 1)σ(s)s, and w is given by (3.3) d(σ) + 1 v1σ(v)v d(σ) + 1 d(σ) + 1 v1σ(v1)rσv d(σ)− 1 d(σ) + 1 v1σ(v1)sv then (γ, v, w) is a Q-system. Conversely, every normalized Q-system is of this form with an isometry s ∈ (σ, σ2) satisfying (3.1) and (3.2). If s and s′ are isometries in (σ, σ2) satisfyingly (3.1) and (3.2), the orresponding Q-systems are equivalent if and only if s = ±s′. Remark 3.6. When the Q-system (γ, v, w) as above arises from a 3-supertransitive subfa tor, we have dim(σ, σ2) = 1. If s, s′ ∈ (σ, σ2) are isometries satisfying (3.1) and (3.2), there exists a omplex number ω of modulus 1 su h that s′ = ωs and (3.2) implies ω2 = 1. This proves Lemma 3.2 again. We keep using the same notation as above and onsider the ase with an in lusion M ⊂ M1 su h that γ = ῑι, v = rι, w = ῑ(r̄ι) where ι = ιM1,M. We set d(ι)√ ι(v∗1)r̄ι, CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 13 whi h is an isometry in (ι, ισ) thanks to Frobenius re ipro ity [20, Proposition 2.2℄. Then Theorem 2.9,(4) shows that for every x ∈ M1, we have x = EM(x) + d(σ)EM(xt In parti ular, t2 = EM(t 2) + d(σ)EM(t 2t∗)t, t∗ = EM(t ∗) + d(σ)EM(t ∗2)t = d(σ)EM(t ∗2)t, where we use the fa t EM(t) ∈ (idM, σ) = {0}. Therefore the ∗-algebra stru ture of M1 is ompletely determined by EM(t2) and EM(t2t∗), whi h an be omputed in terms of the Q-system as follows: d(ι)2 r∗ι ῑ(ι(v 1)r̄ιι(v 1)r̄ι)rι = v∗1 ῑ(ι(v 1)r̄ι)rι σ(v∗1)v 1wv = 2t∗) = d(ι)3 r∗ι ῑ(ι(v 1)r̄ιι(v 1)r̄ιr̄ ι ι(v1))rι = v∗1 ῑ(ι(v 1)r̄ι)v1 σ(v∗1)v 1wv1 = d(σ)− 1 d(σ)3 Lemma 3.7. Let the notation be as above. Then the following hold: d(σ)− 1 d(σ)r∗σt. Corollary 3.8. Let M be a properly in�nite fa tor and let σ ∈ End0(M) be an irredu ible self- onjugate endomorphism. If there exists a Q-system for idM⊕σ, then [σ] is a real se tor, that is, rσ = r̄σ. Proof. Using one of the above formulae, we get t = (t∗)∗ = d(σ)(r∗σt) d(σ)t∗rσ = d(σ)r σtrσ = d(σ)r σσ(rσ)t, whi h shows r∗σσ(rσ) = 1/d(σ). � Now we show how the lass c(Q) determines the angle between P and Q. Let P ⊂ M N ⊂ Q be an irredu ible quadrilateral of fa tors su h that N ⊂ P and N ⊂ Q are 2-supertransitive. We assume that NL2(P )N and NL2(Q)N are equivalent. We apply the above argument to the ase where M1 = P , M = N , ι = ιP,N and γ = ῑι. We set P̌ = ῑ(P ). We hoose two isometries v = rι ∈ (id, γ) and v1 ∈ (σ, γ). Then we may assume that the Q-system (γ, v, wP = ῑ(r̄ι)) is normalized with respe t to (v, v1, rσ). By hoosing an appropriate representative of the onjugate 14 PINHAS GROSSMAN AND MASAKI IZUMI se tor of [ιQ,N ], we may also assume γ = ιQ,N ιQ,N , v = rιQ,N , and that the Q-system (γ, v, wQ = ιQ,N(r̄ιQ,N )) is normalized with respe t to (v, v1, rσ). By de�nition, the lass c(Q) is given by (γ, v, wQ) in H 2(P̌ ⊂ N). Let sP and sQ be isometries in (σ, σ2) orresponding to wP and wQ through (3.3) respe tively. We set d(σ) + 1 ιP,N(v 1)r̄ιP,N ∈ (ιP,N , ιP,Nσ), d(σ) + 1 ιQ,N(v 1)r̄ιQ,N ∈ (ιQ,N , ιQ,Nσ), whi h are isometries. Lemma 3.9. Let Q = P ⊂ M N ⊂ Q be an irredu ible quadrilateral of fa tors su h that N ⊂ P and N ⊂ Q are 2-supertransitive. If L2(P ) and L2(Q) are equivalent as N −N bimodules, then (1) 〈tP , tQ〉 satis�es the following quadrati equation: 〈tP , tQ〉2 − d(σ)− 1 〈sP , sQ〉〈tP , tQ〉 − (2) 〈sP , sQ〉 and 〈tP , tQ〉 are real numbers. Proof. (1) Lemma 3.7 implies 〈tP , tQ〉2 = t2Q t2P = d(σ)− 1 〈sP , sQ〉〈tP , tQ〉. (2) Thanks to Lemma 2.10, we have EP (tQ) = 〈tQ, tP 〉tP and EQ(tP ) = 〈tP , tQ〉tQ. EN(tP tQ) = EN (EP (tP tQ)) = 〈tQ, tP 〉EN(t2P ) = 〈tQ, tP 〉√ EN(tP tQ) = EN (EQ(tP tQ)) = 〈tP , tQ〉EN(t2Q) = 〈tP , tQ〉√ whi h shows that 〈tP , tQ〉 is real. This and (1) show that 〈sP , sQ〉 is also real. � Note that dim(ιP,N , ιP,Nσ) = dim(ιQ,N , ιQ,Nσ) = 1 by Frobenius re ipro ity. Thus Remark 2.11 shows cosΘ(P,Q) = |〈tP , tQ〉|. Theorem 3.10. Let Q = P ⊂ M N ⊂ Q be an irredu ible quadrilateral of fa tors su h that N ⊂ P and N ⊂ Q are 2-supertransitive. If L2(P ) and L2(Q) are equivalent as N −N bimodules, then (1) Q is not ommuting. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 15 (2) The angle Θ(P,Q) is given by cosΘ(P,Q) = (d(σ)− 1)2〈sP , sQ〉2 + 4d(σ) + (d(σ)− 1)|〈sP , sQ〉| 2d(σ) (d(σ)− 1)2〈sP , sQ〉2 + 4d(σ)− (d(σ)− 1)|〈sP , sQ〉| 2d(σ) Proof. (1) If Q were ommuting, we would have 〈tP , tQ〉 = 0, whi h never o urs due to Lemma 3.9,(1). (2) follows from Lemma 3.9 too. � Note that |〈sP , sQ〉| is a numeri al invariant of the lass c(Q). Therefore the above theorem says that there are only two possibilities of the angle Θ(P,Q) given c(Q). Corollary 3.11. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors su h that N ⊂ P and N ⊂ Q are 2-supertransitive. Then 0 < Θ(P,Q) ≤ cos−1 1 [P : N ]− 1 . The equality holds if and only if the lass c(Q) ∈ H2(P̌ ⊂ N) is trivial. In parti ular, if N ⊂ P is 3-supertransitive, Θ(P,Q) = cos−1 [P : N ]− 1 . Proof. Note that |〈tP , tQ〉| = 1 never o urs sin e |〈tP , tQ〉| = 1 would imply P = Q. The statement follows from (d(σ)− 1)2〈sP , sQ〉2 + 4d(σ)− (d(σ)− 1)|〈sP , sQ〉| 2d(σ) where the quality holds if and only if 〈sP , sQ〉 = ±1. � Corollary 3.12. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors su h that N ⊂ P and N ⊂ Q are 2-supertransitive. If Q is o ommuting, [M : P ] ≤ [P : N ]− 1. The equality holds if and only if the lass c(Q) ∈ H2(P̌ ⊂ N) is trivial. In parti ular, if Q is o ommuting and N ⊂ P is 3-supertransitive, [M : P ] = [P : N ]− 1. Proof. This follows from Lemma 2.4 and Lemma 3.11. � Remark 3.13. There is an example of a quadrilateral Q ful�lling the assumption of Corollary 3.12 with non-trivial c(Q). Let q be a prime power and let F be a �nite �eld with q elements. We set G = PSL(n, q) with n ≥ 3, whi h naturally a ts on the proje tive spa e PF n−1. It is known that this a tion is 2-transitive. We denote by 16 PINHAS GROSSMAN AND MASAKI IZUMI F n∗ the dual spa e of F n, on whi h G naturally a ts too. Let {ei}ni=1 be the anoni al basis of F n and let {e∗i }ni=1 be the dual basis of {ei}ni=1. Let H be the stabilizer of Fe1 and let K be the stabilizer of Fe∗1. Then we have [G : H ] = [G : K] = (q n−1)/(q−1) and [H : H ∩K] = [K : H ∩K] = qn−1. Let Q be the quadrilateral onstru ted by Lemma 2.5 with these G, H , and K. Then [M : P ] is stri tly smaller than [P : N ]−1. In this ase we have cosΘ(P,Q) = q−n/2. Lemma 3.14. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors su h that N ⊂ P and N ⊂ Q are 2-supertransitive and H2(N̂ ⊂ P ) is trivial. Let [ιP,N ιP,N ] = [idN ] ⊕ [σ]. If dim(ιM,N , ιM,Nσ) = 2 and there is an intermediate subfa tor N ⊂ R ⊂ M other than P and Q su h that L2(R) is equivalent to L2(P ) as N −N bimodules, then [P : N ] = 3. Proof. Let tP and tQ be as in Lemma 3.9. We may and do assume 〈tP , tQ〉 = 1/d(σ) by repla ing tQ with −tQ if ne essary. Assume that there exists an intermediate subfa tor N ⊂ R ⊂ M other than P and Q su h that L2(R) is equivalent to L2(P ) as N − N bimodules. Then we may hoose an isometry t ∈ (ιR,N , ιR,Nσ) su h that 〈t, tP 〉 = 1/d(σ), 〈t, tQ〉 = ǫ/d(σ) with ǫ ∈ {1,−1}, and EN (t2) = 1/ d(σ)rσ. Sin e dim(ιM,N , ιM,Nσ) = 2, the intertwiner spa e (ιM,N , ιM,Nσ) is spanned by tP and tQ. Therefore, the �rst two equalities imply d(σ)− ǫ d(σ)2 − 1(tP + ǫtQ). Sin e t is an isometry, we get 1 = 〈t, t〉 = (d(σ)− ǫ) (d(σ)2 − 1)2 〈tP + ǫtQ, tP + ǫtQ〉 = (d(σ)− ǫ)2 (d(σ)2 − 1)2 2(d(σ) + ǫ) 2(d(σ)− ǫ) d(σ)(d(σ)2 − 1) If ǫ = 1, we would have d(σ) = 1, whi h ontradi ts dim(ιM,N , ιM,Nσ) = 2 and Theorem 2.9,(1). Therefore we get ǫ = −1 and d(σ) = 2, whi h means [P : N ] = Remark 3.15. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors. Then there exists σ ontained both in ιP,N ιP,N and in ιQ,N ιQ,N so that the multipli ity of σ in ιM,N ιM,N is at least 2. Therefore σ is not an automorphism due to Theorem 2.9,(1). In parti ular, the subfa tor N ⊂ P is not the rossed produ t by a �nite group a tion. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 17 4. (2,2)-supertransitive quadrilaterals In what follows, for a non- ommuting quadrilateral P ⊂ M N ⊂ Q of properly in�nite fa tors with 2-supertransitive elementary subfa tors, we use the following notation: the symbols ι, κ, ιQ, and κQ denote the in lusion maps ι : N →֒ P, κ : P →֒ M, ιQ : N →֒ Q, κQ : Q →֒ M. Sin e ῑι, ιῑ, et are de omposed into two irredu ible se tors, one of whi h is the identity se tor, we hoose representatives ξ, η ∈ End0(P ), ξQ, ηQ ∈ End0(Q), ξ̂, ξ̂Q ∈ End0(N), and η̂, η̂Q ∈ End0(M) satisfying [ιῑ] = [idP ]⊕ [ξ], [κ̄κ] = [idP ]⊕ [η], [ιQῑQ] = [idQ]⊕ [ξQ], [κ̄QκQ] = [idQ]⊕ [ηQ], [ῑι] = [ῑQιQ] = [idN ]⊕ [ξ̂], [κκ̄] = [idM ]⊕ [η̂], [κQκ̄Q] = [idM ]⊕ [η̂Q]. Note that neither ξ nor ξ̂ is an automorphism (see Remark 3.15). When N ⊂ P and N ⊂ Q are 3-supertransitive, the se tors ξι and ξQιQ are de omposed into two irredu ible omponents and we use the following notation: [ξι] = [ι]⊕ [ι′], [ξQιQ] = [ιQ]⊕ [ι′Q]. In this ase, we also have [ιξ̂] = [ι]⊕ [ι′], [ιQξ̂Q] = [ιQ]⊕ [ι′Q]. In the same way, when P ⊂ M and Q ⊂ M are 3-supertransitive, we use the following notation: [κη] = [κ]⊕ [κ′], [κQηQ] = [κQ]⊕ [κ′Q], [η̂κ] = [κ]⊕ [κ′], [η̂QκQ] = [κQ]⊕ [κ′Q]. In the sequel we often use the te hniques developed in [17℄. For example, let ρ, σ be irredu ible endomorphisms of P asso iated with the in lusion N ⊂ P . Then ρξ ontains σ if and only if either [ρ] = [σ] or the distan e between ρ and σ in the dual prin ipal graph of N ⊂ P is two. When [ρ] 6= [σ], the multipli ity of σ in ρξ is the number of the paths from ρ to σ. The multipli ity of ρ in ρξ is i=1 n i − 1 where [ρι] = ⊕mi ni[τi], is the irredu ible de omposition of ρι. In parti ular, the in lusion N ⊂ P is 3-supertransitive if and only if the multipli ity of ξ in ξ2 is one. Lemma 4.1. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors. We assume that Q is (2,2)-supertransitive. Then (1) [ξ] 6= [η]. (2) ξ2 ontains η. Consequently, the depth of N ⊂ P is at least 4. 18 PINHAS GROSSMAN AND MASAKI IZUMI Proof. (1) Sin e N ⊂ M is irredu ible, we have dim(κι, κι) = 1. Thus Frobenius re ipro ity implies 1 = dim(κι, κι) = (κ̄κ, ιῑ) = (idP ⊕ η, idP ⊕ ξ), whi h shows [ξ] 6= [η]. (2) Sin e Q is non ommuting and L2(P ) ∼= L2(Q) ∼= L2(N)⊕ L2(N)ξ̂, as N −N bimodules, the N −N bimodule L2(M) ontains L2(N)ξ̂ with multipli ity at least two. Thus 2 ≤ dim(ῑκ̄κι, ξ̂) = dim(κ̄κ, ιξ̂ῑ) = dim(κ̄κ, ιῑιῑ)− dim(κ̄κ, ιῑ) = dim(idP ⊕ η, ξ ⊕ ξ2) = 1 + dim(η, ξ2), whi h shows the statement. � Remark 4.2. Using the above lemma, we give a short proof of Theorem 2.7 only assuming (4,4)-supertransitivity of Q (see Corollary 4.9 and Corollary 5.18 below for sharper results in this dire tion). Sin e N ⊂ P is 4-supertransitive, Lemma 4.1 shows [ξ2] = [idP ]⊕ [ξ]⊕ [η]. If the prin ipal graph is A5, η is an automorphism and M is the rossed produ t P ⋊η Z/2Z. Therefore the proof of [18, Theorem 3.1℄ implies that there exists an outer a tion of S3 on M su h that N is the �xed point algebra of the a tion. Assume now that the prin ipal graph of N ⊂ M is not A5. Then we have dim(ηι, ηι) ≥ 2. Frobenius re ipro ity implies dim(η2, ξ) = dim(η, ηξ) = dim(η, ηιῑ)− dim(η, η) = dim(ηι, ηι)− 1 Sin e P ⊂ M is 4-supertransitive, we get [η2] = [idP ]⊕[η]⊕[ξ]. Therefore d(ξ) = d(η) and d(ξ)2 = 1+2d(ξ), whi h shows that [M : P ] = [P : N ] = 2+ 2 = 4 cos2(π/8) and all the elementary subfa tors are the A7 subfa tor. Uniqueness of su h a quadrilateral follows from Lemma 3.2 and Lemma 3.14 as η is uniquely determined by N ⊂ P (see Theorem 4.7 below for details). The following is one of the main results in this paper, whi h says that irre- du ible non ommuting (2,2)-supertransitive quadrilaterals with trivial ohomologi al obstru tion are lassi�ed into four lasses and one ex eptional ase. Theorem 4.3. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors. We assume that Q is (2,2)-supertransitive and the lass c(Q) ∈ H2(N̂ ⊂ P ) is trivial. Then Θ(M,N) = cos−1 [P : N ]− 1 . Moreover, CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 19 (1) If Q is not o ommuting and the lass c(Q̂) in H2(P ⊂ M) is trivial, then [M : P ] = [P : N ]. In this ase, there exists an outer automorphism α of P su h that [η] = [αξ] = [ξα−1] and ξ2 ontains η. (In onsequen e, the automorphism α is ontained in ξ3.) (2) If Q is o ommuting, then [M : P ] = [P : N ] − 1 and [ξ] = [κ̄κQπ], where π is an isomorphism from P to Q su h that the restri tion of π to N is the identity map. In this ase, the Galois group Gal(M/N) = {θ ∈ Aut(M); θ(x) = x, ∀x ∈ N} is either trivial or isomorphi to Z/2Z,Z/3Z, or the symmetri group S3 of degree 3. If Gal(M/N) = {idM , θ} ∼= Z/2Z, then θ swit hes P and Q and [ξ] = [κ̄θκ]. If Gal(M/N) = {idM , θ, θ2} ∼= Z/3Z, then either θ(P ) = Q or θ(Q) = P holds and [ξ] = [κ̄θκ] = [κ̄θ2κ]. If Gal(M/N) ∼= S3, then N is the �xed point algebra of an outer a tion of S3 on M and P and Q are the �xed point algebras of two di�erent order two elements in S3. Proof. The angle between P and Q was already omputed in Theorem 3.10. Lemma 3.4 shows that sin e c(Q) is trivial there exists an isomorphism π from P onto Q su h that the restri tion of π to N is trivial. This means ιQ = πι. Thus κι = κQιQ implies κι = κQπι and 1 = dim(κι, κQπι) = dim(κιῑ, κQπ) = dim(κ, κQπ) + dim(κξ, κQπ) = dim(κ, κQπ) + dim(ξ, κ̄κQπ). We laim [κ] 6= [κQπ], and in onsequen e, laim that ξ is ontained in κ̄κQπ. Indeed, if [κ] = [κQπ] were the ase, there would exist a unitary u inM su h that x = uπ(x)u holds for all x ∈ P . In parti ular, the unitary u ommutes with all x ∈ N and u is a s alar. However, this means P = Q, whi h is a ontradi tion. Therefore the laim holds. (1) Assume that Q is not o ommuting and the lass c(Q̂) in H2(P ⊂ M) is trivial. Then Lemma 3.4 applied to the dual quadrilateral Q̂ implies that P and Q are inner onjugate in M and there exists a unitary v ∈ M satisfying vQv∗ = P . We introdu e an automorphism α of P by setting α(x) = vπ(x)v∗ for x ∈ P . This implies [κα] = [κQπ] and so [κ̄κQπ] = [κ̄κα] = [α]⊕ [ηα]. Sin e ξ is ontained in κ̄κQπ, either [ξ] = [α] or [ξ] = [αη] holds. Note that [ξ] = [α] never o urs thanks to Remark 3.15. Thus we have [ξ] = [ηα] and so [η] = [ξα−1], whi h is equal to [αξ] for η is self- onjugate. Sin e ξ2 ontains η and idP , we on lude that ξ3 ontains α. (2) Assume that Q is o ommuting. Then Theorem 3.10,(1) applied to the dual quadrilateral Q̂ implies that [κκ̄] 6= [κQκ̄Q] and so dim(κ̄κQπ, κ̄κQπ) = dim(κκ̄, κQκ̄Q) = 1. 20 PINHAS GROSSMAN AND MASAKI IZUMI This shows that κ̄κQπ is irredu ible and [ξ] = [κ̄κQπ], whi h implies [P : N ]− 1 = d(ξ) = d(κ)d(κQ) = [M : P ]. (This is also shown in Corollary 3.12 with help of Lemma 2.4 though we don't need to use it here.) Note that the Galois group of the in lusion N ⊂ M is isomorphi to the set of 1-dimensional se tors G ontained in κιῑκ̄. (Note that sin e N ⊂ M is irredu ible, the multipli ity of any automorphism in κιῑκ̄ is at most one.) We assume that G is non-trivial now. Using [ξ] = [κ̄κQπ], we get [κιῑκ̄] = [κκ̄]⊕ [κξκ̄] = [idM ]⊕ [η̂]⊕ [κκ̄κQπκ̄] = [idM ]⊕ [η̂]⊕ [κQπκ̄]⊕ [η̂κQπκ̄]. Assume that η̂ is an automorphism. Then [M : P ] = 2 and so [P : N ] = 3. Sin e N ⊂ P is 2-supertransitive, it is the A5 subfa tor. Therefore, the proof of [18, Theorem 3.1℄ shows that there exists a unique outer a tion of S3 on M su h that N is the �xed point algebra of the a tion. Assume that η̂ is not an automorphism. We laim that κQπκ̄ ontains an auto- morphism. By the assumption of non-triviality of Gal(M/N), either κQπκ̄ or η̂κQπκ̄ ontains an automorphism. Assume that the latter ontains an automorphism, say α. Then Frobenius re ipro ity implies 1 = dim(η̂κQπκ̄, α) = dim(κQπκ̄, η̂α), and κQπκ̄ ontains η̂α. Sin e d(κQπκ̄) = d(η̂α) + 1, we on lude that κQπκ̄ ontains an automorphism. Let θ be an automorphism ontained in κQπκ̄. Then Frobenius re ipro ity implies [θκ] = [κQπ]. We may assume that θκ = κQπ holds by hoosing an appropriate representative of [θ]. Then θ(P ) = Q and θκι = κQπι = κQιQ = κι, whi h shows θ ∈ Gal(M,N). Using [θκ] = [κQπ], we get [κιῑκ̄] = [idM ]⊕ [η̂]⊕ [θκκ̄]⊕ [η̂θκκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [η̂θη̂]. If η̂θη̂ does not ontain any automorphism, then Gal(M,N) = {idM , θ} ∼= Z/2Z. Assume that η̂θη̂ does ontain an automorphism, say, β. Then Frobenius re i- pro ity again implies [η̂θ] = [βη̂] (and [θη̂] = [η̂β]), and [κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [βη̂2]. Therefore G = {[idM ], [θ]} ∪ {[βγ]}γ∈Gal(M/η̂(M)). Sin e [β] ∈ G, the group G1 = {[γ]}γ∈Gal(M/η̂(M)) is a subgroup of G su h that #G = 2 + #G1. Sin e #G1 divides #G, either G1 is trivial or #G1 = 2. If G1 is trivial, we get Gal(M/N) ∼= {[idM ], [θ], [β]} ∼= Z/3Z. Suppose that G1 = {[idM ], [γ]}. Then G = {[idM ], [θ], [β], [βγ]} and [θ] = [γ]. This would imply [θη̂] = [γη̂] = [η̂] and [η̂θη̂] = [η̂2] would ontain [idM ]. However, this means that κιῑκ̄ ontain idM with multipli ity two, whi h is a ontradi tion. Therefore G1 is trivial. � CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 21 Corollary 4.4. Let Q = P ⊂ M N ⊂ Q be a (3,3)-supertransitive irredu ible quadrilat- eral of fa tors su h that Q is neither ommuting nor o ommuting. Then Θ(P,Q) = cos−1 1/([P : N ] − 1), [M : P ] = [P : N ] and there exists an outer automorphism α of P su h that [η] = [αξ] = [ξα−1]. Moreover, (1) The automorphism α satis�es [αι′] = [ι′]. (2) dim(κι, κιξ̂) = 2. (3) The Galois group Gal(M/N) is either trivial or isomorphi to Z/2Z. It is isomorphi to Z/2Z if and only if the order of [α] is two. When Gal(M/N) = {id, θ}, the automorphism θ swit hes P and Q. Proof. The �rst part follows from Lemma 3.2 and Theorem 4.3. (1) Sin e αξ is ontained in ι′ῑ, by Frobenius re ipro ity ι′ is ontained in αξι = αι⊕αι′. Sin e d(ι′) = d(ι)(d(ι)2−2), the equality [ι′] = [αι] would imply that N ⊂ P is the A5 subfa tor, whi h does not allow ξ to ontain αξ, and so [αι′] = [ι′]. (2) Thanks to Lemma 4.1, we have [ῑκ̄κι] = [ῑι]⊕ [ῑηι] = [idN ]⊕ [ξ̂]⊕ [ῑαξι] = [idN ]⊕ [ξ̂]⊕ [ῑαι]⊕ [ῑαι′] = [idN ]⊕ [ξ̂]⊕ [ῑαι]⊕ [ῑι′]. Frobenius re ipro ity implies dim(ῑαι, ξ̂) = dim(αι, ιξ̂) = dim(αι, ι) + dim(αι, ι′) = dim(αι, ι), where we use [αι] 6= [ι′] again. Sin e the Galois group of N ⊂ P is trivial, we have dim(αι, ι) = 0 and dim(ῑαι, ξ̂) = 0. Sin e N ⊂ P is 3-supertransitive, the endomorphism ῑι′ ontains ξ̂ with multipli ity one and we get the statement. (3) The Galois group Gal(M/N) is isomorphi to the set of 1 dimensional se tors ontained in κιῑκ̄. By a similar omputation as above, [κιῑκ̄] = [κκ̄]⊕ [κξκ̄] = [idM ]⊕ [η̂]⊕ [κηακ̄] = [idM ]⊕ [η̂]⊕ [κακ̄]⊕ [κ′ακ̄]. Sin e the depth of P ⊂ M is at least 4 due to Lemma 4.1, the endomorphism η̂ is not an automorphism. If κ′ακ̄ ontained an automorphism θ, Frobenius re ipro ity would imply [κ′α] = [θκ] and d(κ) = d(κ′), whi h is a ontradi tion. Assume that Gal(M/N) is not trivial. Then κακ̄ ontains an automorphism, say, θ. Then Frobenius re ipro ity implies [θκ] = [κα] and [idP ]⊕ [αξ] = [κ̄κ] = [α−1κ̄κα] = [idP ]⊕ [α−1αξα] = [idP ]⊕ [ξα]. Sin e [αξ] = [ξα−1], this shows that ξ2 ontains α2. If α2 were outer, we would have [α2ι] = [ι′] as ι′ῑ would ontain α2. However this ontradi ts d(ι) 6= d(ι′). Thus the order of [α] is two. This implies [θ2κ] = [κα2] = [κ]. Sin e the Galois group of P ⊂ M is trivial, we get [θ2] = [idM ]. Let β be another automorphism ontained in κακ̄. Then the above argument shows that [θκ] = [βκ] and [β−1θκ] = [κ], whi h implies [β−1θ] = [idM ]. Therefore Gal(M/N) ∼= Z/2Z. Assume now that the order of [α] is two. Then [(κα)κα] = [α−1(idP ⊕ η)α] = [idP ]⊕ [α−1ηα] = [idP ]⊕ [η] = [κ̄κ]. 22 PINHAS GROSSMAN AND MASAKI IZUMI Sin e H2(P̂ ⊂ M) is trivial, the subfa tors α−1κ̄(M) and κ̄(M) are inner onjugate in P and so there exists an automorphism θ of M su h that [θκ] = [κα]. Sin e the Galois group of κ̄(M) ⊂ P is trivial, we have [κα] 6= [κ] and θ is outer. The same omputation as above implies that [θ] is ontained in [κιῑκ̄] and so Gal(M/N) is not trivial. The above argument shows that when Gal(M/N) is isomorphi to Z/2Z, we have [θκ] = [κα] = [κQπ] and we an hoose a representative θ of [θ] so that θκ = κQπ holds. Then θ(P ) = Q and θκι = κQπι = κQιQ = κι, and so Gal(M/N) = {id, θ}. Corollary 4.5. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilat- eral of fa tors su h that Q is (2,2)-supertransitive and the lass c(Q) is trivial. Assume that Q is o ommuting and the Galois group Gal(M/N) is isomorphi to Z/3Z. Then [M : P ] ≥ (5+ 13)/2 and Q is neither (4,2)-supertransitive nor (2,3)- supertransitive. If Q is (3,2)-supertransitive, the dual prin ipal graph (to be more pre ise, the indu tion-redu tion graph for M − M and M − P bimodules) ontains one of the graphs of the Haagerup subfa tor. Proof. Let Gal(M/N) = {idM , θ, θ2 = θ−1}. The proof of Theorem 4.3 shows that [θη̂] = [η̂θ−1], the endomorphism η̂2 does not ontain any non-trivial automorphism [κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [θ2η̂]⊕ [θ2η̂2]. Note that [η̂], [θη̂], [θ2η̂] are distin t se tors. Sin e η̂2 ontains η̂, the endomorphism κιῑκ̄ ontains θ2η̂ with multipli ity at least 2. Sin e [θκι] = [κι], the multipli ities of η̂ and θη̂ in κιῑκ̄ are also at least 2, whi h implies η̂2 ontains idM ⊕ η̂ ⊕ θη̂ ⊕ θ2η̂. This shows that P ⊂ M is not 4-supertransitive and d(η̂)2 ≥ 1+3d(η̂), whi h implies [M : P ] = 1 + d(η̂) ≥ (5 + 13)/2. Note that N ⊂ P is 3-supertransitive if and only if dim(ξ, ξ2) = 1. Using [ξ] = [κ̄θκ], we get dim(ξ, ξ2) = dim(κ̄θκ, κ̄θκκ̄θκ) = dim(κκ̄θκκ̄, θκκ̄θ) = dim((idM ⊕ η̂)θ(idM ⊕ η̂), θ(idM ⊕ η̂)θ) = dim(θ ⊕ θη̂ ⊕ θ2η̂ ⊕ θ2η̂2, θ2 ⊕ η̂) = dim(θ2η̂2, θ2 ⊕ η̂) ≥ 2, whi h shows that N ⊂ P is not 3-supertransitive. Assume that Q is (3,2)-supertransitive. Then κ′κ̄ ontains η̂ ⊕ θη̂ ⊕ θ2η̂. and the dual prin ipal graph of P ⊂ M ontains the following graph, M − P θη̂ θ whi h is one the prin ipal graphs of the Haagerup subfa tor [1℄. � CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 23 In Se tion 5 and in the Appendix, we will show that there exists a (3,2)-supertransitive quadrilateral of fa tors in the above lass su h that P ⊂ M is the Haagerup subfa tor and that su h a quadrilateral is unique up to �ip onjuga y. Corollary 4.6. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors. Assume that Q is o ommuting and (2,3)-supertransitive. Then Θ(P,Q) = cos−1 1/([P : N ] − 1), [M : P ] = [P : N ] − 1, and the Galois group Gal(M/N) is either trivial or isomorphi to Z/2Z or S3. Moreover, (1) Two equalities [ηι] = [ι′] and [ῑκ̄κι] = [idM ]⊕ [ξ̂]⊕ [ῑι′] hold. In onsequen e, the endomorphism ι′ῑ ontains η with multipli ity one and [ι′ῑ] = [η]⊕ [ηξ]. (2) dim(κι, κιξ̂) = 2. (3) Assume that Gal(M/N) is isomorphi to Z/2Z and Gal(M/N) = {idM , θ}. [κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [η̂θη̂], and dim(κιῑκ̄, κιῑκ̄) ≥ 7. The se tors [η̂], [θη̂], [η̂θ], and [θη̂θ] are distin t. The endomorphism η̂θη̂ ontains θη̂θ with multipli ity one and it does not ontain either any automorphism or any of η̂, θη̂, η̂θ. The endomorphism η̂2 does not ontain any of θη̂, η̂θ, θη̂θ. (4) The endomorphism ξ2 ontains π̄ηQπ and [π̄ηQπι] = [ι ′] holds. Assume that H2(P̂ ⊂ M) is trivial. Then the Galois group Gal(M/N) is trivial if and only if [η] 6= [π̄ηQπ]. Assume further that Gal(M/N) is trivial. Then the dual prin ipal graph of N ⊂ P ontains D6 in su h a way that every endpoint of D6 is also an endpoint of the dual prin ipal graph. (5) Assume that P ⊂ M is 3-supertransitive and Gal(M/N) is trivial. Then Q ⊂ M is 3-supertransitive as well and [κιῑκ̄] = [idM ]⊕ [η̂]⊕ 2[κQπκ̄]⊕ [κ′Qκ̄Q] = [idM ]⊕ [η̂Q]⊕ 2[κQπκ̄]⊕ [κ′κ̄]. The se tor [κQπκ̄] is irredu ible and self- onjugate and [κ ′] = [κ′Qπ] holds. Unless the prin ipal graph of P ⊂ M is A4, the inequality dim(κιῑκ̄, κιῑκ̄) ≥ 8 holds. Proof. The �rst part follows from Lemma 3.4, Theorem 4.3, and Corollary 4.5. (1) Let d = d(ι) = [P : N ]− 1. Then we have d(ξ) = d2 − 1 and d(ι′) = d(d2 − 2). On the other hand, we have d(η) = [M : P ] − 1 = [P : N ] − 2 = d2 − 2. Sin e ηι ontains ι′ and d(ηι) = d(ι′), we get [ηι] = [ι′]. (2) Using (1), we get [ῑκ̄κι] = [ῑι]⊕ [ῑηι] = [idN ]⊕ [ξ̂]⊕ [ῑι′]. Sin eN ⊂ P is 3-supertransitive, ῑι′ ontains ξ̂ with multipli ity one and dim(κι, κιξ̂) = dim(ῑκ̄κι, ξ̂) = 2. (3) The proof of Theorem 4.3 shows that η̂θη̂ ontain no automorphism, [θκ] = [κQπ], [ξ] = [κ̄θκ], and [κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [η̂θη̂]. 24 PINHAS GROSSMAN AND MASAKI IZUMI If [θη̂] = [η̂], we would have [η̂θη̂] = [η̂2], whi h ontains idM . Thus [θη̂] 6= [η̂]. In the same way, we get [η̂θ] 6= [η̂] and [θη̂] 6= [θη̂θ] 6= [η̂θ]. If [η̂] = [θη̂θ], we would have [κκ̄] = [idM ]⊕ [η̂] = [idM ]⊕ [θη̂θ] = [θκκ̄θ] = [κQππ̄κ̄Q] = [κQκ̄Q], whi h ontradi ts the assumption thatQ is o ommuting thanks to Theorem 3.10,(1). Thus [η̂] 6= [θη̂θ] and [θη̂] 6= [η̂θ]. Sin e N ⊂ P is 3-supertransitive, we have dim(ξ, ξ2) = 1, and 1 = dim(ξ, ξ2) = dim(κ̄θκ, κ̄θκκ̄θκ) = dim(κκ̄θκκ̄, θκκ̄θ) = dim((idM ⊕ η̂)θ(idM ⊕ η̂), θ(idM ⊕ η̂)θ) = dim(θ ⊕ θη̂ ⊕ η̂θ ⊕ η̂θη̂, idM ⊕ θη̂θ) = dim(η̂θη̂, θη̂θ), whi h means that η̂θη̂ ontains θη̂θ with multipli ity one. Sin e [θκι] = [κι], the multipli ities of η̂, θη̂, η̂θ and θη̂θ in κιῑκ̄ should be the same, whi h is the same as η̂θη̂ ontains θη̂θ with multipli ity one. Thus η̂θη̂ does not ontain any of η̂, θη̂, η̂θ. Thanks to Frobenius re ipro ity we on lude that η̂2 does not ontain any of θη̂, η̂θ, θη̂θ. (4) Sin e [ξ] = [κ̄κQπ] = [π̄κ̄Qκ], we have [ξ2] = [π̄κ̄Qκκ̄κQπ] = [π̄κ̄QκQπ]⊕ [π̄κ̄Qη̂κQπ], whi h ontains π̄ηQπ. Sin e N ⊂ P is 3-supertransitive and π̄ηQπ (whi h is not equivalent to id) is ontained in ξ2, the morphism ι′ῑ ontains π̄ηQπ and so π̄ηQπι ontains ι′. Sin e d(ι′) = d(π̄ηQπι), we on lude [π̄ηQπι] = [ι The proof of Theorem 4.3 shows that Gal(M/N) is non-trivial if and only if there exists and automorphism θ of M su h that [θκ] = [κQπ]. Assume �rst that su h an automorphism θ exists. Then [κ̄κ] = [π̄κ̄QκQπ] and [η] = [π̄ηQπ] holds. Assume onversely that [η] = [π̄ηQπ] holds. Then [κ̄κ] = [idP ]⊕ [η] = [π̄(idQ ⊕ ηQ)π] = [π̄κ̄QκQπ]. Sin e H2(P̂ ⊂ M) is trivial, the subfa tors κ̄(M) and π̄κ̄Q(M) are inner onjugate in P and there exists an automorphism θ of M su h that [θκ] = [κQπ]. Therefore the Galois group Gal(M/N) is trivial if and only if [η] 6= [π̄ηQπ]. Now we assume that Gal(M/N) is trivial. Then the dual prin ipal graph of N ⊂ P is as follows: η π̄ηQπP − P whi h ontains D6. (5) Sin e [η] 6= [π̄ηQπ], dim(κQπκ̄, κQπκ̄) = dim(π̄κ̄QκQπ, κ̄κ) = dim(idP ⊕ π̄ηQπ, idP ⊕ η) = 1, CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 25 and κQπκ̄ is irredu ible. Sin e [ξ] = [κ̄κQπ] is self- onjugate, 1 = dim(κ̄κQπ, π̄κ̄Qκ) = dim(κQπκ̄, κπ̄κ̄Q), and κQπκ̄ is self- onjugate. This and the proof of Theorem 4.3 imply [κιῑκ̄] = [idM ]⊕ [η̂]⊕ [κQπκ̄]⊕ [η̂κQπκ̄] = [idM ]⊕ [η̂]⊕ [κQπκ̄]⊕ [η̂κπ̄κ̄Q] = [idM ]⊕ [η̂]⊕ 2[κQπκ̄]⊕ [κ′π̄κ̄Q]. On the other hand, repla ing P by Q, we get [κιῑκ̄] = [idM ]⊕ [η̂Q]⊕ [κQπκ̄]⊕ [η̂QκQπκ̄]. Sin e [η̂] 6= [η̂Q], the endomorphism η̂QκQπκ̄ ontains η̂ and 1 ≤ dim(η̂QκQπκ̄, η̂) = dim(η̂QκQπ, η̂κ) = dim(η̂QκQπ, κ⊕ κ′). If η̂QκQπ ontained κ, we would have [η̂Q] = [κπ̄κ̄Q], whi h ontradi ts d(η̂Q) 6= d(κπ̄κ̄Q). Therefore η̂QκQπ ontains κQπ and κ , and so [η̂QκQπ] = [κQπ]⊕ [κ′] as we have d(η̂QκQπ) = d(κQπ) + d(κ ′). This shows that Q ⊂ M is 3-supertransitive and [κ′Qπ] = [κ ′], whi h �nishes the proof. � About uniqueness, we have the following: Theorem 4.7. Let Q = P ⊂ M N ⊂ Q and Q̃ = P̃ ⊂ M̃ Ñ ⊂ Q̃ be irredu ible non om- muting and non o ommuting quadrilaterals of fa tors su h that Q and Q̃ are (3,3)- supertransitive. Assume that there exists an isomorphism Φ from P onto P̃ su h that Φ(N) = Ñ . If there exists no se tor σ ontained in ξ2 su h that [ξ] 6= [σ], [η] 6= [σ], and d(σ) = d(η), then Q and Q̃ are onjugate. Proof. Sin e P ⊂ M and P̃ ⊂ M̃ are 3-supertransitive, H2(P̂ ⊂ M) and H2( ˜̂P ⊂ M̃) are trivial thanks to Lemma 3.2. Note that we have [ιP̃ ,Ñ ιP̃ ,Ñ ] = [ΦιP,N ιP,NΦ Therefore Lemma 4.1 implies [ιM̃,P̃ ιM̃,P̃ ] = [ΦιM,P ιM,PΦ −1], whi h shows that Φ ex- tends to an isomorphism Ψ from M onto M̃ as H2(P̂ ⊂ M) and H2( ˜̂P ⊂ M̃) are trivial. Lemma 3.14 and Corollary 4.4 show Ψ(Q) = Q̃. � Theorem 4.8. Let Q = P ⊂ M N ⊂ Q and Q̃ = P̃ ⊂ M̃ Ñ ⊂ Q̃ be irredu ible non- ommuting and o ommuting quadrilaterals of fa tors su h that H2(P̂ ⊂ M) and H2( ˜̂P ⊂ M̃) are trivial and Q and Q̃ are (2,3)-supertransitive. Assume that there exists an isomorphism Φ from P onto P̃ su h that Φ(N) = Ñ . Then (1) If Gal(M/N) ∼= Z/2Z and η is the only se tor σ ontained in ξ2 su h that [σι] = [ι′], then Q and Q̃ are onjugate. (2) If Gal(M/N) and Gal(M̃/Ñ) are trivial and there exists only two se tors σ ontained in ξ2 su h that [σι] = [ι′], then Q and Q̃ are �ip onjugate. 26 PINHAS GROSSMAN AND MASAKI IZUMI Proof. The proof of (1) is the same as that of Theorem 4.7. In (2), [ιM̃,P̃ ιM̃,P̃ ] = [Φ(idP ⊕ π̄ηQπ)Φ−1] = [Φπ−1ιM,QιM,QπΦ−1] may o ur instead of [ιM̃ ,P̃ ιM̃ ,P̃ ] = [ΦιM,P ιM,PΦ −1]. In this ase, the two quadrilater- als are �ip onjugate. � We an improve the assumption of Theorem 2.7. Corollary 4.9. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors. Then (1) If Q is not o ommuting and (3,4)-supertransitive, then all the elementary subfa tors are isomorphi to A7 subfa tors. There exists a unique su h quadri- lateral up to isomorphism when the fa tors are hyper�nite II1 fa tors. (2) If Q is o ommuting and (2,4)-supertransitive, then there exists an outer a - tion of S3 on M su h that N is the �xed point algebra of the a tion. There exists a unique su h quadrilateral up to isomorphism when the fa tors are hyper�nite II1 fa tors. Proof. Sin e N ⊂ P is 4-supertransitive, we have [ξ2] = [idP ]⊕ [ξ]⊕ [η]. (1) Sin e Q is not o ommuting, there exists an automorphism θ of P su h that [η] = [θξ] and so we get d(ξ)2 = 1 + 2d(ξ). This shows that the prin ipal graphs of N ⊂ P and P ⊂ Q are A7. The uniqueness follows from Theorem 4.7. (2) Sin e Q is o ommuting, we have [ηι] = [ι′] and so the prin ipal graph of N ⊂ P is A5. The rest of the proof has already been stated in Remark 4.2 ex ept for uniqueness, whi h follows from the uniqueness of outer a tions of �nite groups by Jones [29℄ and the Galois orresponden e. � 5. Classifi ation I A ording to our dis ussions in the last se tion, we onsider the following four lasses of quadrilaterals Q of fa tors. We assume that every quadrilateral Q = P ⊂ M N ⊂ Q of fa tors appearing in this se tion is irredu ible and non ommuting. Class I: Q is non o ommuting and (3,3)-supertransitive. Class II: Q is o ommuting and (2,3)-supertransitive. The Galois group Gal(M/N) is trivial. Class III: Q is o ommuting and (2,3)-supertransitive. The Galois group Gal(M/N) is isomorphi to Z/2Z. Class IV: Q is o ommuting and (3,2)-supertransitive. The Class c(Q) is trivial. The Galois group Gal(M/N) is isomorphi to Z/3Z. For ea h lass, we show that there exists an example of a quadrilateral and we seek an example with maximal supertransitivity. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 27 We keep using the notation of se tors su h as ι, κ, et . as in the previous se tion, We often use expli it formulae of fusion rules of subfa tors and the reader is refered to [17℄ for them. 5.1. Class I. The stru ture of quadrilaterals in Class I is relatively easy to des ribe. Theorem 5.1. Let N ⊂ P be an irredu ible 3-supertransitive in lusion of fa tors. Let ι = ιP,N and let [ιP,N ιP,N ] = [idP ] ⊕ [ξ] and [ξι] = [ι] ⊕ [ι′] be the irredu ible de omposition. Assume that there exists an outer automorphism α ∈ Aut(P ) su h that [ξ] 6= [αξ] = [ξα−1] and ξ2 ontains αξ. If idP ⊕ αξ has a Q-system, then there exists a unique irredu ible non ommuting quadrilateral Q = P ⊂ M N ⊂ Q in Class I su h that [η] = [αξ]. Proof. Assume that there exists a Q-system for idP ⊕αξ. We laim that a Q-system with idP⊕αξ is unique up to equivalen e. For this, it su� es to show dim(αξ, (αξ)2) = 1 thanks to Lemma 3.5. As in the proof of Corollary 4.5, we have [αι′] = [ι′], [αι] 6= [ι], and d(ι) 6= d(ι′). Thus dim(αξ, (αξ)2) = dim(αξ, ξ2) = dim(αξ, ξιῑ)− dim(αξ, ξ) = dim(αξι, ξι) = dim(αι⊕ αι′, ι⊕ ι′) = 1, whi h shows the laim. The laim implies that there exists a unique fa tor M on- taining N su h that [κ̄κ] = [idP ]⊕ [αξ] where κ = ιM,P . Sin e [ξ] 6= [αξ], the in lusion N ⊂ M is irredu ible. We next laim that και is irredu ible and [και] = [κι]. Indeed, we have dim(και, και) = dim(κ̄κα, αιῑ) = dim(α⊕ αξα, α⊕ αξ) = 1, dim(και, κι) = dim(κ̄κα, ιῑ) = dim(α⊕ αξα, idP ⊕ ξ) = 1, whi h show the laim. The laim shows that there exists a unitary v ∈ M su h that vα(x)v∗ = x holds for every x ∈ N . We set π(y) = vα(y)v∗ for y ∈ P and set Q = π(P ). By onstru tion Q is an intermediate subfa tor between N and M su h that ιQ,N = πι, [ιQ,N ιQ,N ] = [ῑι], and [M : P ] = [M : Q]. Suppose P = Q. Then π would be an automorphism of P , whi h satis�es [κπ] = [κα]. Frobenius re ipro ity implies that πα−1 is ontained in κ̄κ = idP ⊕ αξ and so [π] = [α]. Thus there exists a unitary u in P su h that vα(y)v∗ = uα(y)u∗ for all y ∈ P and so u∗v ∈ P ′ ∩M = C. This shows that v ∈ P and [αι] = [ι], whi h is a ontradi tion. Therefore P 6= Q. Sin e N ⊂ P is 3-supertransitive, it has no intermediate subfa tor and P ∩Q = N . Sin e dim(αξ, (αξ)2) = 1, the in lusion P ⊂ M is 3-supertransitive too and M is generated by P and Q. Therefore Q = P ⊂ M N ⊂ Q is an irredu ible quadrilateral of fa tors. Theorem 3.10,(1) shows that Q is non ommuting. Sin e P and Q are inner onjugate in M , Theorem 3.10,(1) applied to the dual quadrilateral Q̂ shows that Q is non o ommuting. � 28 PINHAS GROSSMAN AND MASAKI IZUMI The above theorem shows that quadrilaterals in Class I is ompletely determined by the subfa tor N ⊂ P and α. Jones and the �rst-named author [13℄ showed that there exists a unique quadrilat- eral of the hyper�nite II1 fa tors su h that all the elementary subfa tors are the the A7 subfa tor. It is easy to show that this is the only quadrilateral in Class I satis- fying [P : N ] < 4. Other than this example, we know two subfa tors satisfying the assumption of Theorem 5.1, namely, the E 7 subfa tor and the Haagerup subfa tors. Let N ⊂ P be the E(1)7 subfa tor. Then the dual prin ipal graph is as follows: P − P Note that the ategory of P − P bimodules for the E(1)7 subfa tor is isomorphi to the ategory Â4 of the unitary representations of the alternating group A4. It is observed in [24, Corollary 4.2℄ that there exists an automorphism of the ategory Â4 that �ips the two representations orresponding to ξ and αξ. Thus idP ⊕αξ has a Q- system and gives rise to an irredu ible non ommuting quadrilateral in Class I whose elementary subfa tors are the E 7 subfa tor. Note that this quadrilateral satis�es the assumption of Theorem 4.7 and su h a quadrilateral is unique. Sin e it is easy to see that no other subfa tors of index less than or equal to 4 �t into the statement of Theorem 4.3,(1), we get the following: Theorem 5.2. Let Q = P ⊂ M N ⊂ Q be a quadrilateral of fa tors in Class I su h that [M : P ] ≤ 4. Then one of the following two ases o urs: (1) The prin ipal graphs of all the elementary subfa tors are A7. (2) The prin ipal graphs of all the elementary subfa tors are E In ea h ase, su h a quadrilateral of hyper�nite II1 fa tors exists and is unique up to onjuga y. Let N ⊂ P be the Haagerup subfa tor [1℄. Sin e the Haagerup subfa tor is not self-dual, we spe ify the dual prin ipal graph (the indu tion-redu tion graph of the P − P and P −N bimodules) as below. P − P It is known that α3 = idP and αξ = ξα holds. Sin e idP ⊕ ξ has a Q-system, so does α−1(idP ⊕ ξ)α = idP ⊕ αξ. Thus there exists an irredu ible non ommuting CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 29 quadrilateral of fa tors in Class I whose elementary subfa tors are the Haagerup subfa tor. Although the quadrilateral arising from N ⊂ P does not satisfy the assumption of Theorem 4.7, we have [κα] = [κQπ] and so [π̄ηQπ] = [α −1αξα] = [α2ξ]. Thus the same proof of Theorem 4.8,(2) works and we an show uniqueness of su h a quadrilateral up to �ip onjuga y. We don't know if there are in�nitely many mutually non- onjugate quadrilaterals of the hyper�nite II1 fa tors in Class I. 5.2. Class II. The following two are the main theorems of this subse tion: Theorem 5.3. Let Q = P ⊂ M N ⊂ Q be a quadrilateral of fa tors in Class II su h that Q is (5,3)-supertransitive. Then (1) If Q is (6,3)-supertransitive, the prin ipal graphs of P ⊂ M and Q ⊂ M are A4 and those of N ⊂ P and N ⊂ Q are D6. (2) If Q is not (6,3)-supertransitive, the prin ipal graphs of P ⊂ M and Q ⊂ M are E In ea h ase, su h a quadrilateral of the hyper�nite II1 fa tors exists and is unique up to onjuga y. Theorem 5.4. Let Q = P ⊂ M N ⊂ Q be a quadrilateral of fa tors in Class II su h that [M : P ] ≤ 4. Then one of the following holds: (1) The prin ipal graphs of P ⊂ M and Q ⊂ M are A4 and those of N ⊂ P and N ⊂ Q are D6. Su h a quadrilateral of the hyper�nite II1 fa tors exists and is unique up to onjuga y. (2) The prin ipal graphs of P ⊂ M and Q ⊂ M are E6 and the dual prin ipal graphs of N ⊂ P and N ⊂ Q are as below. η π̄ηQπ αξ αP − P Su h a quadrilateral of the hyper�nite II1 fa tors exists. (3) The prin ipal graphs of P ⊂ M and Q ⊂ M are E(1)8 . Su h a quadrilateral of the hyper�nite II1 fa tors exists and is unique up to onjuga y. In the sequel, the dimension of the spa e (κιῑκ̄, κιῑκ̄) often gives important infor- mation about a quadrilateral and we give useful formulae about it �rst. Lemma 5.5. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors su h that Q is o ommuting and (2,3)-supertransitive. Then ι′ῑ ontains ξ 30 PINHAS GROSSMAN AND MASAKI IZUMI and η with multipli ity one, the equality [ι′ῑ] = [η]⊕ [ηξ] holds, and dim(κιῑκ̄, κιῑκ̄) = 5 + dim(ηξ, ηξ). If moreover P ⊂ M is 3-supertransitive, then [ηξ] = [ξ]⊕ [κ̄′κQπ] and dim(κιῑκ̄, κιῑκ̄) = 6 + dim(κ̄′κQπ, κ̄ ′κQπ) = 6 + dim(κ ′κ̄′, κQκ̄Q). Proof. By Corollary 4.6,(1) and Frobenius re ipro ity, we have dim(κιῑκ̄, κιῑκ̄) = dim(ῑκ̄κι, ῑκ̄κι) = dim(idN ⊕ ξ̂ ⊕ ῑι′, idN ⊕ ξ̂ ⊕ ῑι′) = 2 + 2 dim(ιξ̂, ι′) + dim(ῑι′, ῑι′) = 4 + dim(ῑι′, ῑι′). Sin e [ῑιῑι] = 2[ῑι] ⊕ [ῑι′], the se tor [ῑι′] is self- onjugate, and so is [ι′ῑ] for a similar reason. Thus we get dim(ῑι′, ῑι′) = dim(ῑι′, ῑ′ι) = dim(ι′ῑ, ιῑ′) = dim(ι′ῑ, ι′ῑ). The statement follows from Corollary 4.6,(1) now. � In view of Corollary 4.6,(4), a anoni al andidate of a quadrilateral in Class II with the smallest index is that with the D6 subfa tor for N ⊂ P . Proposition 5.6. Let Q = P ⊂ M N ⊂ Q be a quadrilateral in Class II or Class III su h that the prin ipal graph of P ⊂ M is A4. Then Q is in Class II and the prin ipal graph of N ⊂ P is D6. Su h a quadrilateral of the hyper�nite II1 fa tors exists and is unique up to onjuga y. Proof. From Theorem 4.3, we see [P : N ] = [M : P ]+1 = (5+ 5)/2 = 4 cos2(π/10), whi h shows that the prin ipal graph of N ⊂ P is either A9 or D6. Thanks to Theorem 2.7, the former never o urs. If Q were in Class III, Corollary 4.6,(3) would imply that η̂θη̂ does not ontain any automorphism and it ontains θη̂θ. However, this ontradi ts d(η̂θη̂)− d(θη̂θ) = d(η̂)2 − d(η̂) = 1 and Q is in Class II. In the ase of the hyper�nite II1 fa tors, uniqueness up to �ip onjuga y follows from Theorem 4.8,(2) and the uniqueness of the D6 subfa tor. Let [ξ 2] = [idP ] ⊕ [ξ] ⊕ [ξ′] ⊕ [ξ′′] be the irredu ible de omposition of ξ2. To prove uniqueness up to onjuga y, it su� es to show that there exists an automorphism α of P su h that α(N) = N and [αξ′α−1] = [ξ′′] (see the proof of Theorem 4.8,(2)). It is shown in [34℄ that there exists an automorphism β of period two on the A9 subfa tor Ñ ⊂ P̃ su h (N ⊂ P ) ∼= (Ñ ⋊β Z/2Z ⊂ P̃ ⋊β Z/2Z). The dual a tion β̂ of β a ts on the higher relative ommutants of N ⊂ P non-trivially. It is routine work to show that β̂ does the right job. Finally we show existen e. Let N ⊂ M be an in lusion of fa tors whose prin ipal graph is D6. We use the following parameterization of se tors asso iated with N ⊂ CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 31 idM ρ ρ1 ρ2 idN σ σ1 σ2 Re all the fusion rule for the D6 subfa tors [17, Proposition 3.10℄: [ρ2] = [idM]⊕ [ρ]⊕ [ρ1]⊕ [ρ2], [ρ1ρ2] = [ρ2ρ1] = [ρ], [ρ21] = [idM]⊕ [ρ1], [ρ22] = [idM]⊕ [ρ2], [µ] = [ρ1ν] = [ρ2ν] = [νσ1] = [νσ2], where ν = ιM,N . We an take representatives ρ1 and ρ2 satisfying µ = ρ1ν = ρ2ν. We set M = M, P = ρ1(M), Q = ρ2(M), and N = µ(N ) = ρ1(N ) = ρ2(N ). Then d(ρ1) = d(ρ2) = (1 + 5)/2, whi h means [M : P ] = [M : Q] = 4 cos2 π/5, and so the prin ipal graphs of P ⊂ M and Q ⊂ M are A4. By onstru tion, the prin ipal graphs of N ⊂ P and N ⊂ Q are D6. Sin e µ is irredu ible, the in lusion N ⊂ M is irredu ible and Q = P ⊂ M N ⊂ Q is an irredu ible quadrilateral. Sin e [ρ21] = [idM] ⊕ [ρ1] and [ρ22] = [idM] ⊕ [ρ2], we have [ιM,P ιM,P ] = [idM ] ⊕ [ρ1] and [ιM,QιM,Q] = [idM ]⊕ [ρ2], whi h implies that Q is o ommuting. Therefore [M : P ] 6= [P : N ] shows that the quadrilateral Q is not ommuting. � Remark 5.7. The above onstru tion shows that the prin ipal graph and the dual prin ipal graph of N ⊂ M are the same, whi h an be easily omputed from [µµ] = [ρ1νν̄ρ1] = [ρ1(idM ⊕ ρ)ρ1] = [idM ]⊕ [ρ1]⊕ [ρ2]⊕ 2[ρ], [ρ1µ] = [ρ 1ν] = [ν]⊕ [ρ1ν] = [ν]⊕ [µ], [ρ2µ] = [ρ 2ν] = [ν]⊕ [ρ2ν] = [ν]⊕ [µ], [ρµ] = [ρρ1ν] = [ρν]⊕ [ρ2ν] = [ν]⊕ 2[µ], [νµ̄] = [νν̄ρ1] = [ρ1]⊕ [ρρ1] = [ρ1]⊕ [ρ2]⊕ [ρ]. By symmetry of µ and µ̄, there exist two more intermediate subfa tors L and R between N and M su h that the prin ipal graphs of L ⊂ M and R ⊂ M are D6 and those of N ⊂ L and N ⊂ R are A4. Sin e ιM,LιM,L and ιM,RιM,R are ontained in µµ̄, we see that they are equivalent to idM ⊕ ρ. There is no intermediate subfa tor other than P,Q, L,R. Indeed, if S is an intermediate subfa tor, the endomorphism ιM,SιM,S is ontained in µµ̄. Computation of the indi es [M : S] and [S : N ] shows that all the possibilities of ιM,SιM,S are exhausted by P,Q, L,R. Thus Lemma 3.14 applied to the dual quadrilateral implies the laim. 32 PINHAS GROSSMAN AND MASAKI IZUMI Proposition 5.8. Let Q = P ⊂ M N ⊂ Q be a quadrilateral in Class II or Class III su h that the prin ipal graph of P ⊂ M is A5. Then Q is in Class III and the prin ipal graph of N ⊂ P is E(1)7 . When M is the hyper�nite II1 fa tor, su h a quadrilateral exists and is unique up to onjuga y. Proof. Sin e the prin ipal graph of P ⊂ M is A5, there exists an outer automorphism of period two α ∈ Aut(P ) su h that [κη] = [κ]⊕ [κα]. Sin e [ξ] = [κ̄κQπ], we get [ηξ] = [ηκ̄κQπ] = [κ̄κQπ]⊕ [ακ̄κQπ] = [ξ]⊕ [αξ]. Therefore Corollary 4.6,(1) shows that the dual prin ipal graph of N ⊂ P is αηP − P whi h is E 7 . Existen e of su h a quadrilateral follows from Example 2.6 with G = S{1,2,3,4} and uniqueness follows from Theorem 4.8. Corollary 4.6,(4) shows that Q is in Class III. � Proposition 5.9. Let Q = P ⊂ M N ⊂ Q be a quadrilateral in Class II or Class III su h that the prin ipal graph of P ⊂ M is E6. Then Q is in Class II and the dual prin ipal graph of N ⊂ P is as in Theorem 5.4 and [P : N ] = 3+ 3. There exists a quadrilateral of the hyper�nite II1 fa tors satisfying the above property, whi h arises from the GHJ pair [13℄ (see Example 7.10) for E6 with ∗ given by the vertex with the smallest entry of the Perron-Frobenius eigenve tor. Proof. From Theorem 4.3, we see [P : N ] = [M : P ] + 1 = 3 + 3. We re all the fusion rule for the E6 subfa tors [17, p.968℄: [η2] = [idP ]⊕ [α]⊕ 2[η], [αη] = [ηα] = [η], [α2] = [idP ], [κη] = [κ]⊕ [κ′]⊕ [κα], [κ̄κ′] = [η], [η̂2] = [idM ]⊕ [β]⊕ 2[η̂], [η̂κ] = [κ]⊕ [κ′]⊕ [βκ], with [βκ] = [κα], d(κ) = 3, d(κ′) = 2, d(α) = d(β) = 1, d(η) = d(η̂) = 3, d(ξ) = 2 + From [ξ] = [κ̄κQπ], we get [ηξ] = [ηκ̄κQπ] = [ξ]⊕ [κ̄′κQπ]⊕ [αξ]. Sin e dim(κ̄′κQπ, κ̄ ′κQπ) = dim(κ ′κ̄′, κQκ̄Q) = dim(idM ⊕ β, idM ⊕ η̂Q) = 1, the endomorphism κ̄′κQπ is irredu ible, and Lemma 5.5 implies dim(κιῑκ̄, κιῑκ̄) = 8. Suppose that Q is in Class III. Then thanks to Corollary 4.6,(3), we have [κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [η̂θη̂], CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 33 and η̂θη̂ does not ontain any other irredu ible omponents above. Therefore we must have dim(η̂θη̂, η̂θη̂) = 3. However, dim(η̂θη̂, η̂θη̂) = dim(θη̂2θ, η̂2) = dim(idM ⊕ θβθ ⊕ 2θη̂θ, idM ⊕ β ⊕ 2η̂) = 1 + dim(θβθ, β), whi h is a ontradi tion. Therefore Q is in Class II. By Corollary 4.6,(4), we get [π̄ηQπ] = [κ̄ ′κQπ] and the prin ipal graph of N ⊂ P is as stated above. � We ex lude A7, D6, and E8 for the prin ipal graph of P ⊂ M . Lemma 5.10. There exists no quadrilateral Q = P ⊂ M N ⊂ Q in Class II or Class III su h that the prin ipal graph of P ⊂ M is A7. Proof. Suppose there exists su h a quadrilateralQ. Sin e P ⊂ M is the A7 subfa tor, there exist outer automorphisms α of P and β of M and su h that [η̂2] = [idM ]⊕ [η̂]⊕ [βη̂], [κ̄′κ] = [η]⊕ [αη]. Lemma 5.5 shows [ηξ] = [ξ]⊕ [κ̄′κQπ]. Suppose that κ̄′κQπ is irredu ible. Then Lemma 5.5 shows that dim(κικ̄ῑ, κικ̄ῑ) = 7. Sin e κ′κ̄ is redu ible, Corollary 4.6,(3),(5) shows that the quadrilateral Q should be in Class III and η̂θη̂ is de omposed into two mutually inequivalent irredu ibles, whi h implies 2 = dim(η̂θη̂, η̂θη̂) = dim(θη̂2θ, η̂2) = (idM ⊕ θη̂θ ⊕ θβη̂θ, idM ⊕ η̂ ⊕ βη̂). Thus either [η̂] = [θβη̂θ], [βη̂] = [θη̂θ], or [βη̂] = [θβη̂θ] should hold. However, if [η̂] = [θβη̂θ], we would get [η̂2] = [η̂η̂] = [θη̂2θ], whi h is a ontradi tion. The same reasoning shows that the other two do not o ur as well and we on lude that κ̄′κQπ is redu ible. Suppose that κ̄′κQπ is redu ible. Sin e any irredu ible omponent ζ of κ̄ ′κQπ is ontained in ι′ῑ, Frobenius re ipro ity shows that ζι ontains ι′. On the other hand, [κ̄′κQπι] = [κ̄ ′κQιQ] = [κ̄ ′κι] = [ηι]⊕ [αηι] = [ι′]⊕ [αι′]. This implies [ι′] = [αι′] and that κ̄′κQπ is de omposed into two irredu ibles, say, ζ1 and ζ2 su h that [ζ1ι] = [ζ2ι] = [ι ′]. This shows [ι′ῑ] = [ξ]⊕[η]⊕[ζ1]⊕[ζ2] and the set of P −P se tors appearing the dual prin ipal graph of N ⊂ P is {[idP ], [ξ], [η], [ζ1], [ζ2]}. However, sin e [αι′] = [ι′], the automorphism α appears in ι′ῑ′, whi h is a ontradi - tion. Therefore the statement is shown. � Lemma 5.11. There exists no quadrilateral Q = P ⊂ M N ⊂ Q in Class II or Class III su h that the prin ipal graph of P ⊂ M is D6. 34 PINHAS GROSSMAN AND MASAKI IZUMI Proof. Suppose that Q is su h a quadrilateral. We �rst re all the fusion rule for the D6 subfa tor: [κ′κ̄] = [η̂]⊕ [η̂1]⊕ [η̂2], [κ′κ̄′] = [idM ]⊕ 2[η̂]⊕ [η̂1]⊕ [η̂2], [η̂2] = [idM ]⊕ [η̂]⊕ [η̂1]⊕ [η̂2], [η̂21] = [idM ]⊕ [η̂1], [η̂2]2 = [idM ]⊕ [η̂2], [η̂1η̂2] = [η̂2η̂1] = [η̂], [η̂1η̂] = [η̂η̂1] = [η̂]⊕ [η̂2], [η̂2η̂] = [η̂η̂2] = [η̂]⊕ [η̂1], d(η̂) = (3 + 5)/2, d(η̂1) = d(η̂2) = (1 + 5)/2. Lemma 5.5 shows [ηξ] = [ξ]⊕ [κ̄′κQπ]. We laim that κ̄′κQπ is irredu ible. Indeed, dim(κ̄′κQπ, κ̄ ′κQπ) = dim(κ ′κ̄′, κQκ̄Q) = dim(idM ⊕ 2η̂ ⊕ η̂1 ⊕ η̂2, idM ⊕ η̂Q). Sin e Q is o ommuting, we have [η̂] 6= [η̂Q] and we get the laim. Lemma 5.5 and the laim implies that dim(κιῑκ̄, κιῑκ̄) = 7. Corollary 4.6,(3),(5) shows that Q is in Class III and dim(η̂θη̂, η̂θη̂) = 2, and so 2 = dim(θη̂2θ, η̂2) = dim(idM ⊕ θη̂θ ⊕ θη̂1θ ⊕ θη̂2θ, idM ⊕ η̂ ⊕ η̂1 ⊕ η̂2) = 1 + dim(θη̂1θ ⊕ θη̂2θ, η̂1 ⊕ η̂2). If [θη̂1θ] = [η̂2], we would have [θη̂2θ] = [η̂1] too as the period of θ is two, whi h is a ontradi tion. Thus we may assume [θη̂1θ] = [η̂1] and [θη̂2θ] 6= [η̂1], [η̂2]. Sin e η̂θη̂ ontains θη̂θ with multipli ity one, there exists an irredu ible ζ ∈ End0(M) inequivalent to θη̂θ su h that [η̂θη̂] = [θη̂θ]⊕ [ζ ]. We laim that [ζ ] = [η̂θη̂2] holds. Indeed, we have dim(η̂θη̂2, η̂θη̂2) = dim(η̂ 2, θη̂22θ) = dim(idM ⊕ η̂ ⊕ η̂1 ⊕ η̂2, idM ⊕ θη̂2θ) = 1, whi h shows that η̂θη̂2 is irredu ible. Sin e [θη̂θ] 6= [η̂θη̂2] and dim(η̂θη̂, η̂θη̂2) = dim(η̂ 2, θη̂2η̂θ) = dim(idM ⊕ η̂ ⊕ η̂1 ⊕ η̂2, θη̂1θ ⊕ θη̂θ) = 1, we get the laim. Sin e [κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [θη̂θ]⊕ [ζ ], and [θκι] = [κι], we get [θζ ] = [ζ ]. However, dim(η̂θη̂2, θη̂θη̂2) = dim(η̂θη̂, θη̂ 2θ) = dim(θη̂θ ⊕ ζ, idM ⊕ θη̂2θ) = 0, whi h is a ontradi tion. Thus the statement is proven. � Lemma 5.12. There exists no quadrilateral Q = P ⊂ M N ⊂ Q in Class II or Class III su h that the prin ipal graph of P ⊂ M is E8. Proof. Suppose thatQ is su h a quadrilateral. Sin e the E8 fa tor is 4-supertransitive, there exists an irredu ible η̂′ ∈ End0(M) su h that [η̂2] = [idM ]⊕ [η̂]⊕ [η̂′]. Note that we have d(η̂) 6= d(η̂′) and η̂ is the only irredu ible se tor σ ontained in η̂′2 (also in κ′κ̄′) su h that d(η̂) = d(σ) (see [17, Se tion 3.3℄). As before, we have [ξη] = [ξ]⊕ [κ̄′κQπ] and dim(κ̄′κQπ, κ̄ ′κQπ) = dim(κ ′κ̄′, κQκ̄Q) = 1 + dim(κ ′κ̄′, η̂Q) = 1. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 35 Thus Q is in Class III and 2 = dim(η̂θη̂, η̂θη̂) = dim(θη̂2θ, η̂2) = dim(idM ⊕ θη̂θ ⊕ θη̂′θ, idM ⊕ η̂ ⊕ η̂′), whi h shows [θη̂′θ] = [η̂′]. However, this implies [θη̂′2θ] = [η̂′2], whi h ontradi ts [θη̂θ] 6= [η̂] as η̂′2 ontains η̂. � We are ready to prove Theorem 5.3 now. Proof of Theorem 5.3. Let Q be in Class II and (5,3)-supertransitive su h that the prin ipal graph of P ⊂ M is not A4. Corollary 4.6,(5) shows dim(κιῑκ̄, κιῑκ̄) ≥ 8 and Lemma 5.5 shows dim(κ̄′κQπ, κ̄ ′κQπ) ≥ 2. Sin e the prin ipal graph of P ⊂ M is not A5, the depth of P ⊂ M is at least 5. Thus we use the following parameterization of M −M , M − P , and P − P se tors arising from P ⊂ M . κ κ′ κ′′ idM η̂ η̂ idP η η M − P P − P Corollary 4.6,(5) shows [κιῑκ̄] = [idM ]⊕ [η̂]⊕ [η̂Q]⊕ [η̂′]⊕ 2[κQπκ̄]. Sin e Q is o ommuting, [η̂] 6= [η̂Q]. Sin e the prin ipal graph of P ⊂ M is neither A5 nor A7, we have d(η̂Q) 6= d(η̂′), d(κQπ) 6= d(κ′) and d(κQπ) 6= d(κ′′). Sin e dim(κQπκ̄, η̂ ′) = dim(κQπ, η̂ ′κ) = dim(κQπ, κ ′ ⊕ κ′′) = 0, we get [κQπκ̄] 6= [η̂′]. Thus dim(κιῑκ̄, κιῑκ̄) = 8 and dim(κ̄′κQπ, κ̄′κQπ) = 2. Thanks to Corollary 4.6,(4),(5), the endomorphism κ̄′κQπ is de omposed into π̄ηQπ and an irredu ible endomorphism of P , say ξ′. Sin e [κ̄′κQπι] = [κ̄ ′κι] = [ηι]⊕ [η′ι] = [ι′]⊕ [η′ι], and [π̄ηQπι] = [ι ′], we get [ξ′ι] = [η′ι], whi h ontains ι′ as ξ′ is ontained in ι′ῑ. Thus Frobenius re ipro ity implies 1 = dim(ι′, η′ι) = dim(ι′ῑ, η′) = dim(ξ ⊕ η ⊕ π̄ηQπ ⊕ ξ′, η′). 36 PINHAS GROSSMAN AND MASAKI IZUMI If [ξ] = [η′], we would have [ξ′ι] = [ι] ⊕ [ι′] and by Frobenius re ipro ity [ξ′] = [ξ], a ontradi tion. Thus [ξ′] = [η′]. idP η π̄ηQπξ η′ P − P We laim [ηπ̄ηQπ] = [ξ]⊕ [η′]. Thanks to Corollary 4.6(5), we know that κQπκ̄ is self- onjugate. Thus [ξ2] = [κ̄κQπκ̄κQπ] = [κ̄κπ̄κ̄QκQπ] = [(idP ⊕ η)(idP ⊕ π̄ηQπ)] = [idP ]⊕ [η]⊕ [π̄ηQπ]⊕ [ηπ̄ηQπ], whi h shows the laim. Let [η′ι] = [ι′] ⊕ i=1mi[ιi], and [ιiῑ] = mi[η j=1mij [ξj ] be the irredu ible de ompositions. We ompute [ηξ] in two ways: [ηξ][ξ] = [ξ2]⊕ [π̄ηQπξ]⊕ [η′ξ] = [idP ]⊕ [ξ]⊕ [η]⊕ [π̄ηQπ]⊕ [η′]⊕ [ξ]⊕ [η]⊕ [η′] ⊕ [ξ]⊕ [η]⊕ [π̄ηQπ]⊕ m2i [η mimij [ξj] = [idP ]⊕ 3[ξ]⊕ 3[η]⊕ 2[π̄ηQπ]⊕ (2 + m2i )[η mimij [ξj], [η][ξ2] = [η]⊕ [ηξ]⊕ [η2]⊕ [ηπ̄ηQπ]⊕ [ηη′] = [η]⊕ [ξ]⊕ [π̄ηQπ]⊕ [η′]⊕ [idP ]⊕ [η]⊕ [η′]⊕ [ξ]⊕ [η′]⊕ [ηη′] = [idP ]⊕ 2[ξ]⊕ 2[η]⊕ [π̄ηQπ]⊕ 3[η′]⊕ [ηη′]. This shows [ηη′] = [ξ]⊕ [η]⊕ [π̄ηQπ]⊕ ( m2i − 1)[η′]⊕ mimij [ξj]. Sin e P ⊂ M is 5-supertransitive, the multipli ity of η′ in ηη′ is 1, whi h implies k = 2 and m1 = m2 = 1. This shows [κ̄κ′′] = [η′]⊕ [π̄ηQπ]⊕ [ξ]⊕ (m1j +m2j)[ξj] and P ⊂ M annot be 6-supertransitive. Frobenius re ipro ity implies that κπ̄ηQπ and κξ ontain κ′′ with multipli ity one and [κξ] = [κκ̄κQπ] = [κQπ]⊕ [η̂κQπ], CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 37 On the other hand, the endomorphisms κπ̄ηQπ and η̂κQπ are irredu ible as we have dim(κπ̄ηQπ, κπ̄ηQπ) = dim(κ̄κπ̄ηQπ, π̄ηQπ) = dim(π̄ηQπ ⊕ ηπ̄ηQπ, π̄ηQπ) = dim(π̄ηQπ ⊕ ξ ⊕ η′, π̄ηQπ) = 1, dim(η̂κQπ, η̂κQπ) = dim(η̂ 2, κQκ̄Q) = dim(idM ⊕ η̂ ⊕ η̂′, idM ⊕ η̂Q) = 1, and so [κ′′] = [κπ̄η̂Qπ] = [η̂κQπ]. By indu tion, we an get d(η) = d(κ) 2 − 1, d(κ′) = d(κ)(d(κ)2− 2), d(η′) = d(κ)4− 3d(κ2)+ 1, d(κ′′) = d(κ)(d(κ)4− 4d(κ)2+3). Therefore d(κ′′) = d(κπ̄η̂Qπ) implies d(κ) 4−5d(κ)2+4 = 0 and [M : P ] = d(κ2) = 4. This shows that the prin ipal graph of P ⊂ M is E(1)8 . η′ π̄ηQπ ξ κπ̄ηQπ κQπM − P P − P The above omputation shows that ξ is determined solely by the in lusion P ⊂ M . Sin e N ⊂ P is 3-supertransitive, a Q-system for idP ⊕ ξ is unique up to equivalen e and so M is uniquely determined by P ⊂ M up to inner onjuga y in P . Therefore uniqueness of su h a quadrilateral of the hyper�nite II1 fa tors up to onjuga y follows from Theorem 3.14. We show existen e now. Let R be a fa tor and α be an outer a tion of the al- ternating group A5. We regard A4 as a subgroup of A5 and �x it. We hoose two mutually inequivalent 2-dimensional irredu ible proje tive unitary representations σ1 and σ2. Sin e σ1 and σ2 arry a unique non-trivial element in H 2(A5,T), we may assume that they have the same o y le ω. Sin e A4 has a unique 2-dimensional irredu ible proje tive unitary representation up to equivalen e, we may assume that the restri tion of σ1 and that of σ2 to A4 oin ide. Let M2(C) be the 2 by 2 matrix algebra and let βg = αg⊗Adσ1(g) for g ∈ A5, whi h is an a tion of A5 on R⊗M2(C). We set N = (R⊗ C)⋊β A4 ⊂ P = (R⊗ C)⋊β A5 ⊂ M = (R⊗M2(C))⋊β A5. Then the prin ipal graph of P ⊂ M is E(1)8 [11, p.224℄, the in lusion N ⊂ P is 3-supertransitive [39℄, and it is routine work to show that N ⊂ M is irredu ible. Let ug = 1⊗σ2(g)σ1(g)∗, whi h is a β- o y le, and let {λg}g∈A5 be the implementing unitary inM = (R⊗M2(C))⋊βA5. Sin e ug ommutes with R⊗C, we an introdu e a homomorphism π : P → M by setting π(x) = x for x ∈ R ⊗ C and π(λg) = ugλg for g ∈ A5. We set Q = π(P ). The restri tion of π to N is the identity map be ause ug = 1 for g ∈ A4. Sin e σ1 and σ2 are not equivalent, there exists g ∈ A5 \ A4 su h that ug is not a s alar, and so P 6= Q. Sin e A4 is maximal in A5 there is no intermediate subfa tor between N ⊂ P . Sin e the prin ipal graph of P ⊂ M 8 , there is no intermediate subfa tor between P ⊂ M . Therefore we on lude 38 PINHAS GROSSMAN AND MASAKI IZUMI that Q = P ⊂ M N ⊂ Q is a quadrilateral, whi h is non ommuting thanks to Theorem 3.10,(1). Theorem 4.3 shows thatQ is o ommuting as we have [M : P ] = [P : N ]−1. To �nish the proof, we show that the prin ipal graph of Q ⊂ M is also E(1)8 . Sin e (N ⊂ Q) ∼= (R ⋊α A4 ⊂ R ⋊α A5), the ategory of Q − Q bimodules arising from N ⊂ Q is equivalent to the unitary dual of A5. Lemma 4.1 shows that this ategory ontains the ategory of Q − Q bimodules arising from Q ⊂ M . Sin e [M : Q] = 4, we on lude that the prin ipal graph of Q ⊂ M is E(1)8 . � To prove Theorem 5.4, we have to ex lude two more ases. Proposition 5.13. Let Q = P ⊂ M N ⊂ Q be a quadrilateral in Class II or III su h that the prin ipal graph of P ⊂ M is E(1)7 . Then Q is in Class III. There exists a fa tor R and an outer a tion of G = S5 on R with two maximal subgroups H 6= K of G su h that M = RH∩K , P = RH , Q = RK, and N = RG. Su h a quadrilateral of the hyper�nite II1 fa tors exists and is unique up to onjuga y. Proof. We use the following parameterization of M −M , M − P , and P − P se tors arising from P ⊂ M : M − P P − P idM η̂ η̂ ′ βη̂ β κ κ′ βκ = κα η η′ αη α The ategory of M − M bimodules and P − P bimodules arising from P ⊂ M are isomorphi to the unitary dual of S4, whi h gives the fusion rule of the above se tors. We have [η2] = [idP ]⊕ [η]⊕ [η′]⊕ [αη], [η′2] = [idP ]⊕ [α]⊕ [η′], [αη] = [ηα], [αη′] = [η′α] = [η′], [α2] = [idP ], [η̂2] = [idM ]⊕ [η̂]⊕ [η̂′]⊕ [βη̂], [η̂′2] = [idM ]⊕ [β]⊕ [η̂′], [βη̂] = [η̂β], [βη̂′] = [η̂β ′] = [η̂′], [β2] = [idM ], d(η) = d(η̂) = 3, d(η′) = d(η̂′) = 2, d(α) = d(β) = 1. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 39 First we laim that κ̄′κQπ is irredu ible, whi h shows that Q is in Class III thanks to Corollary 4.6,(4) and Lemma 5.5. Indeed, we have dim(κ̄′κQπ, κ̄ ′κQπ) = dim(κ ′κ̄′, κQκ̄Q) = dim(idM ⊕ β ⊕ 2η̂ ⊕ 2βη̂ ⊕ η̂′, idM ⊕ η̂Q) = 1 + 2 dim(βη̂, η̂Q), whi h shows that either [βη̂] 6= [η̂Q] and κ̄′κQπ is irredu ible or [βη̂] = [η̂Q] and κ̄′κQπ is a dire t sum of three irredu ible distin t se tors. Suppose that the latter holds and [κ̄′κQπ] = [ξ1]⊕ [ξ2]⊕ [ξ3]. Then ξiι would ontain ι′ for ea h i = 1, 2, 3. On the other hand, we have [κ̄′κQπι] = [κ̄ ′κι] = [ηι]⊕ [η′ι]⊕ [αηι] = [ι′]⊕ [η′ι]⊕ [αι′]. Sin e d(η′) = 2, d(ι) = 5, and d(ι′) = 3 5, this is impossible and we get the laim. Lemma 5.5 and Corollary 4.6,(3),(5) imply 2 = dim(η̂θη̂, η̂θη̂) = dim(η̂2, θη̂2θ) = dim(idM ⊕ η̂ ⊕ η̂′ ⊕ βη̂, idM ⊕ θη̂θ ⊕ θη̂′θ ⊕ θβη̂θ) = 1 + dim(η̂, θβη̂θ) + dim(βη̂, θη̂θ) + dim(βη̂, θβη̂θ) + dim(η̂′, θη̂′θ). If [η̂] = [θβη̂θ], we would have [η̂2] = [η̂η̂] = [θη̂2θ], whi h is impossible. For the same reason, we get [βη̂] 6= [θη̂θ], [βη̂] 6= [θβη̂θ] and so [η̂′] = [θη̂′θ]. In [21, Corollary 4.8℄, the se ond-named author showed that there exists a fa tor R and an outer a tion µ of S4 on R su h that M = R and P = RS4. Let ν = ιR,M . Then we have [ν̄ν] = [idM ]⊕ 2[σ]⊕ [ϕ] where d(σ) = 2, d(ϕ) = 1 and [σ2] = [idM ]⊕ [ϕ]⊕ [σ], [ϕσ] = [σϕ] = [σ], [ϕ2] = [idM ]. We laim that [η̂′] = [σ] and [β] = [ϕ] hold. Indeed, sin e d(η̂) = 3 and η̂2 does not ontain any non-trivial automorphism, [idM ], [ϕ], [σ], [η̂], and [ϕη̂] are all distin t se tors. Sin e P ⊂ R is of depth 2 and [R : P ] = 24, we get 24 = dim(νκκ̄ν̄, νκκ̄ν̄) = dim(ν̄νκκ̄, κκ̄ν̄ν) = dim((idM ⊕ ϕ⊕ 2σ)(idM ⊕ η̂), (idM ⊕ η̂)(idM ⊕ ϕ⊕ 2σ)) = dim(idM ⊕ ϕ⊕ 2σ ⊕ η̂ ⊕ ϕη̂, idM ⊕ ϕ⊕ 2σ ⊕ η̂ ⊕ η̂ϕ) + 2 dim(idM ⊕ ϕ⊕ 2σ ⊕ η̂ ⊕ ϕη̂, η̂σ) + 2 dim(ση̂, idM ⊕ ϕ⊕ 2σ ⊕ η̂ ⊕ η̂ϕ) + 4 dim(ση̂, η̂σ) = 7 + dim(ϕη̂, η̂ϕ) + 2 dim(σ ⊕ ϕσ ⊕ 2σ2, η̂) + 2 dim(η̂2, σ) + 2 dim(ϕη̂, η̂σ) + 2 dim(η̂, σ ⊕ σϕ⊕ 2σ2) + 2 dim(σ, η̂2) + 2 dim(ση̂, η̂ϕ) + 4 dim(ση̂, η̂σ), whi h shows [ϕη̂] = [η̂ϕ] and 4 = 2 dim(σ2, η̂) + dim(η̂2, σ) + dim(η̂2ϕ, σ) + dim(ση̂, η̂σ) = 2 dim(η̂2, σ) + dim(ση̂, η̂σ) = 2 dim(idM ⊕ η̂ ⊕ η̂′ ⊕ βη̂, σ) + dim(ση̂, η̂σ) = 2 dim(η̂′, σ) + dim(ση̂, η̂σ). If [η̂′] 6= [σ], we would have, dim(ση̂, ση̂) = dim(σ2, η̂2) = dim(id⊕ ϕ⊕ σ, id⊕ η̂ ⊕ η̂′ ⊕ βη̂) = 1, 40 PINHAS GROSSMAN AND MASAKI IZUMI whi h means that ση̂ is irredu ible. A similar argument shows that η̂σ is irredu ible too and dim(ση̂, η̂σ) is at most 1, whi h is a ontradi tion. Thus we get [σ] = [η̂′] and in onsequen e, the equality [ϕ] = [β] holds. Sin e Q is in Class III, the prin ipal graph of Q ⊂ M is E(1)7 too and we an apply the same argument to Q ⊂ M . Then there exists a fa tor R̃ and an outer a tion of S4 on R̃ su h that M = R̃ and Q = R̃S4 . Note that we have [θκ] = [κQπ], whi h implies [η̂Q] = [θη̂θ] and [η̂2Q] = [idM ]⊕ [θη̂θ]⊕ [θη̂′θ]⊕ [θβη̂θ] = [idM ]⊕ [η̂Q]⊕ [η̂′]⊕ [θβη̂θ]. Thus the above argument shows [ιR̃,M ιR̃,M ] = [ιR,M ιR,M ]. The se ond-named author and Kosaki [26, Theorem 3.3℄ showed that the ohomology H2(M ⊂ R) is identi�ed with H2(Ŝ3,T) introdu ed by Wassermann [55℄. Sin e the order of S3 is square free, there is only one ergodi a tion of S3 on a 6-dimensional C -algebra and the latter is trivial. Therefor we may assume R = R̃. We denote by µQ the a tion of S4 on R whose �xed point algebra is Q. We set H = µS4 and K = µ We laim that N ⊂ R is irredu ible. Indeed, we known that η̂θη̂ is a dire t sum of θη̂θ and an irredu ible, say ζ , with d(ζ) = 6. Thus we have dim(νκι, νκι) = dim(ν̄ν, κιῑκ̄) = dim(idM ⊕ β ⊕ 2η̂′, idM ⊕ θ ⊕ η̂ ⊕ θη̂ ⊕ η̂θ ⊕ θη̂θ ⊕ ζ) = 1 + dim(β, θ). Sin e [βη̂] = [η̂β], if [β] = [θ], the endomorphism η̂θη̂ would ontain an automorphism, whi h ontradi ts Corollary 4.6,(3). Therefore the laim is shown. Let G be the group generated by H and K in Aut(R). Then the �xed point algebra for RG is P ∩ Q = N and the a tion of G on R is outer for N ⊂ R is irredu ible. Sin e N ⊂ P is 3-supertransitive, the a tion of G on G/H is 3-transitive. Therefore from [G : H ] = [P : N ] = [M : P ] + 1 = 5, we on lude that G is isomorphi to S5 be ause any 3-transitive transforation group of the �ve point set is either A5 or S5. Uniqueness in the ase of the hyper�nite II1 fa tors follows from Theorem 4.8. � Proposition 5.14. Let Q = P ⊂ M N ⊂ Q be a quadrilateral in Class II or III su h that the prin ipal graph of P ⊂ M is E(1)6 . Then Q is in Class III. There exists a fa tor R and an outer a tion of G = A5 on R with two maximal subgroups H 6= K of G su h that M = RH∩K , P = RH , Q = RK, and N = RG. Su h a quadrilateral of the hyper�nite II1 fa tors exists and is unique up to onjuga y. Proof. We employ the same strategy as in the proof of the previous proposition using A4 instead of S4. Sin e the prin ipal graph of P ⊂ M is E(1)6 , there exists outer automorphisms of period 3 α ∈ Aut(P ) and β ∈ Aut(M) su h that [η2] = [idP ]⊕ [α]⊕ [α2]⊕ 2[η], [αη] = [ηα] = [η], [η̂2] = [idM ]⊕ [β]⊕ [β2]⊕ 2[η̂], [βη̂] = [η̂β] = [η̂]. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 41 idM η̂ β β2 κ βκ = κα β2κ = κα2 η α α2 M − P P − P Sin e [M : Q] = 4, Theorem 5.3 and Proposition 5.13 show that the prin ipal graph of Q ⊂ M is also E(1)6 . Therefore there exists outer automorphism βQ ∈ Aut(M) of period 3 su h that [η̂2Q] = [idM ]⊕ [βQ]⊕ [β2Q]⊕ 2[η̂Q], [βQη̂Q] = [η̂QβQ] = [η̂Q]. Sin e [κη] = [κ] ⊕ [κα] ⊕ [κα2], we have [ηξ] = [ηκ̄κQπ] = [ξ] ⊕ [αξ] ⊕ [α2ξ], and Lemma 5.5 shows dim(κιῑκ̄, κιῑκ̄) = 8. If Q is in Class III, Corollary 4.6,(3) implies 3 = dim(η̂θη̂, η̂θη̂) = dim(η̂2, θη̂2θ) = dim(idM ⊕ β ⊕ β2 ⊕ 2η̂, idM ⊕ θβθ ⊕ θβ2θ ⊕ 2θη̂θ), whi h shows either [β] = [βQ] or [β] = [β Q]. Sin e the se ond ohomology of the y li group {idM , β, β2} ∼= A3 is trivial, a similar proof, using either [16℄ or [21℄, as in the ase of E 7 works . Suppose now that Q is in Class II. (Note that sin e the ohomology of the E subfa tor is not trivial, we annot apply Corollary 4.6,(4) as before.) Corollary 4.6,(4) shows that π̄ηQπ is ontained in ξ , whi h is [ξ2] = [idP ]⊕ [ξ]⊕ [αξ]⊕ [α2ξ]⊕ [η], and so [η] = [π̄ηQπ]. The proof of Theorem 4.3 shows [κιῑκ̄] = [idM ]⊕ [η̂]⊕ [κQπκ̄]⊕ [η̂κQπκ̄]. Sin e dim(κQπκ̄, κQπκ̄) = dim(π̄κ̄QκQπ, κ̄κ) = 2, the endomorphism κπ̄κ̄Q is de om- posed two irredu ible endomorphism, say ρ and σ, neither of whi h is an automor- phism as we assumed that Q is in Class II. By Frobenius re ipro ity, the morphisms ρκQπ and σκQπ ontain κ with multipli ity one. Sin e [κπ̄κ̄QκQπ] = [κ(id⊕ η)] = 2[κ]⊕ [βκ]⊕ [β2κ], we may assume [ρκQπ] = [κ]⊕[βκ] and [σκQπ] = [κ]⊕[β2κ]. By Frobenius re ipro ity again, the morphism βκπ̄κ̄Q ontains ρ and the morphism β 2κπ̄κ̄Q ontains σ, and so κπ̄κ̄Q ontains β 2ρ and βσ. If [βσ] = [σ], we also have [β2ρ] = [ρ] and so [βκπ̄κ̄Q] = [κπ̄κ̄Q]. However, this would imply 2 = dim(βκπ̄κ̄Q, κπ̄κ̄Q) = dim(καπ̄κ̄Q, κπ̄κ̄Q) = dim(κ̄κα, π̄κ̄QκQπ) = dim(α⊕ η, idP ⊕ η), 42 PINHAS GROSSMAN AND MASAKI IZUMI whi h is a ontradi tion. Thus we get [ρ] = [βσ] and d(σ) = 2 and [κιῑκ̄] = [idM ]⊕ [η̂]⊕ [σ]⊕ [βσ]⊕ [η̂σ]⊕ [η̂βσ] = [idM ]⊕ [η̂]⊕ [σ]⊕ [βσ]⊕ 2[η̂σ]. Sin e d(σ) = 2, the endomorphism η̂2 does not ontain σ and by Frobenius re ipro ity η̂σ does not ontain η̂. Therefore the multipli ity of η̂ in κιῑκ is one, and so the multipli ity of η̂Q is also one by symmetry. However, sin e d(η̂Q) = 3 and [η̂] 6= [η̂Q], this is impossible. Therefore Q is in Class III. � Now the proof of Theorem 5.4 follows from the above propositions, lemmas, and Theorem 5.3. We don't know if there are in�nitely many mutually non- onjugate quadrilaterals of the hyper�nite II1 fa tors in Class II. 5.3. Class III. Theorem 5.15. Let Q = P ⊂ M N ⊂ Q be a quadrilateral of fa tors in Class III su h that Q is (5,3)-supertransitive. Then (1) If Q is (6,3)-supertransitive, the prin ipal graphs of P ⊂ M and Q ⊂ M are A5 and those of N ⊂ P and N ⊂ Q are E(1)7 . (2) If Q is not (6,3)-supertransitive, the dual prin ipal graph of P ⊂ M on- tains one of the prin ipal graphs of the Asaeda-Haagerup subfa tor [1℄, and in onsequen e [M : P ] ≥ (5 + 17)/2. Proof. Assume that the prin ipal graph of P ⊂ M is not A5. Sin e it is neither A3 nor A4, the depth of P ⊂ M is at least 5. We use the same symbols η̂′, κ′, κ′′, and η′ for M − M , M − P , and P − P se tors arising from P ⊂ M as in the proof of Theorem 5.3. Corollary 4.6,(3) implies 2 ≤ dim(η̂θη̂, η̂θη̂) = dim(idM ⊕ η̂ ⊕ η̂′, idM ⊕ θη̂θ ⊕ θη̂′θ) = 1 + dim(η̂′, θη̂′θ), whi h shows that [η̂′] = [θη̂′θ] and the endomorphism η̂θη̂ is a dire t sum of θη̂θ and an irredu ible endomorphism, whi h we all ζ̂. Therefore we have [κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [θη̂θ]⊕ [ζ̂]. Sin e θκι = κι, we have [θζ̂] = [ζ̂θ] = [ζ̂], whi h is also self- onjugate. We laim [θη̂′] 6= [η̂′]. Suppose, on the ontrary, that [θη̂′] = [η̂′] holds. Then [κ′]⊕ [κ′′] = [η̂′κ] = [θη̂′κ] = [θκ′]⊕ [θκ′′]. If [θκ′] = [κ′′], then the prin ipal graph of P ⊂ M would be A9 and θ would ommute with η̂, whi h is a ontradi tion. Thus [θκ′] = [κ′] holds. However, this implies [η̂]⊕ [η̂′] = [κ′κ̄] = [θκ′κ̄] = [θη̂]⊕ [θη̂′] = [θη̂]⊕ [η̂′], and [θη̂] = [η̂], whi h is a ontradi tion again. Therefore the laim is proven. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 43 Let [κ′′κ̄] = [η̂′]⊕ i=1mi[η̂i] be the irredu ible de omposition. We ompute η̂θη̂ in two ways: [η̂θη̂][η̂] = [θη̂θη̂]⊕ [ζ̂ η̂] = [η̂θ]⊕ [θζ̂ ]⊕ [ζ̂ η̂] = [η̂θ]⊕ [ζ̂]⊕ [ζ̂ η̂], [η̂θ][η̂2] = [η̂θ]⊕ [η̂θη̂]⊕ [η̂θη̂′] = [η̂θ]⊕ [θη̂θ]⊕ [ζ̂]⊕ [η̂η̂′θ] = 2[η̂θ]⊕ [θη̂θ]⊕ [η̂′θ]⊕ [ζ̂]⊕ mi[η̂iθ], whi h implies [ζ̂ η̂θ] = [η̂]⊕ [θη̂]⊕ [η̂′]⊕ mi[η̂i]. Sin e [θ][ζ̂ η̂θ] = [ζ̂ η̂θ] and [θ][η̂′] 6= [η̂′], we may assume that m1 = 1, [θη̂′] = [η̂1] and mi[θη̂i] = mi[η̂i]. By Frobenius re ipro ity, the endomorphism η̂′θη̂ ontains ζ̂ with multipli ity one. On the other hand, we have [η̂′θη̂] = [θη̂′η̂] = [θη̂]⊕ [θη̂′]⊕ [η̂′]⊕ mi[θη̂i] = [θη̂]⊕ [θη̂′]⊕ [η̂′]⊕ mi[η̂i]. Corollary 4.6,(3) shows [ζ̂] 6= [θη̂], and [θη̂′] 6= [η̂′] implies [ζ̂] 6= [η̂′], [θη̂′] as [θζ̂] = [ζ̂] holds. Thus we may assume that m2 = 1 and [ζ̂] = [η̂2]. This in parti ular shows that Q annot be (6, 3)-supertransitive. Now we have [ζ̂ η̂] = [η̂θ]⊕ [θη̂θ]⊕ [η̂′]⊕ [θη̂′]⊕ [ζ̂]⊕ mi[η̂iθ], and [ζ̂] 6= [η̂iθ] for i ≥ 3. Sin e the multipli ity of ζ̂ in ζ̂ η̂ is one, there are exa tly two verti es in the dual prin ipal graph of P ⊂ M onne ted to ζ̂, one of whi h is κ′′. Therefore there exists an irredu ible morphism κ′′′ su h that [ζ̂κ] = [κ′′]⊕ [κ′′′] with [κ′′′] 6= [κ′′]. We laim [κ′′′] = [θη̂θκ] = [η̂θκ]. Indeed, omputing dim(θη̂θκ, θη̂θκ), dim(η̂θκ, η̂θκ), and dim(η̂θκ, θη̂θκ), we an show that θη̂θκ and η̂θκ are irredu ible and [θη̂θκ] = [η̂θκ]. Sin e θη̂θκκ̄ ontains ζ̂ with multipli ity one, Frobenius re i- pro ity implies that ζ̂κ ontains θη̂θκ. If [θη̂θκ] = [κ′′] we would have d(η̂)d(κ) = d(κ′′), whi h implies [M : P ] = 4. Sin e P ⊂ M is 5-supertransitive, the prin ipal graph of P ⊂ M should be E(1)8 . However, sin e the M − M se tors arising from P ⊂ M is non ommutative, we get a ontradi tion. Therefore the laim is proven. 44 PINHAS GROSSMAN AND MASAKI IZUMI The above omputation shows that the dual prin ipal graph is as follows. idM κ η̂ κ ′ η̂′ κ′′ θη̂′ θκ′ θη̂ θκ θ θη̂θκ = η̂θκ θη̂θ η̂θ whi h ontains one of the prin ipal graphs of the Asaeda-Haagerup subfa tor [1℄. � Remark 5.16. We do not known whether there exists a (5,3)-supertransitive quadri- lateral of fa tors in Class III with [M : P ] = (5 + 17)/2. If su h a quadrilateral of hyper�nite II1 fa tors exists, it must be unique up to onjuga y. Indeed, the above argument shows that the in lusion P ⊂ M must be Asaeda-Haagerup subfa tor and ξ = κ̄θκ is determined by P ⊂ M . Sin e N ⊂ P is 3-supsertransitive, a Q-system for idP ⊕ ξ is unique up to equivalen e and M is uniquely determined by P ⊂ M up to inner onjuga y in P . Sin e Q = θ(P ), where the representative θ of the se tor [θ] is hosen so that the restri tion θ|N is trivial, the quadrilateral is uniquely determined by P ⊂ M . For existen e, the same riterion using a Q-system as in the ase of the Haagerup subfa tor below (Theorem 5.19) should work in prin iple, though it involves heavy omputation in pra ti e. Theorem 5.15 and the propositions and lemmas in the previous subse tion imply the following theorem: Theorem 5.17. Let Q = P ⊂ M N ⊂ Q be a quadrilateral of fa tors in Class III su h that [M : P ] ≤ 4. Then one of the following holds: (1) The prin ipal graphs of P ⊂ M and Q ⊂ M are A5 and those of N ⊂ P and N ⊂ Q are E(1)7 . The quadrilateral is given by the onstru tion in Example 2.6 with S{1,2,3,4}. (2) The prin ipal graphs of P ⊂ M and Q ⊂ M are E(1)6 . The quadrilateral is given by the onstru tion in Example 2.6 with A{1,2,3,4,5}. (3) The prin ipal graphs of P ⊂ M and Q ⊂ M are E(1)7 . The quadrilateral is given by the onstru tion in Example 2.6 with S{1,2,3,4,5}. In ea h ase, su h a quadrilateral of the hyper�nite II1 fa tors exists and is unique up to onjuga y. Corollary 5.18. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting (6,3)-supertransitive quadrilateral of fa tors. Then one of the following holds: (1) The prin ipal graphs of P ⊂ M and Q ⊂ M are A3 and those of N ⊂ P and N ⊂ Q are A5. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 45 (2) The prin ipal graphs of P ⊂ M and Q ⊂ M are A4 and those of N ⊂ P and N ⊂ Q are D6. (3) The prin ipal graphs of P ⊂ M and Q ⊂ M are A5 and those of N ⊂ P and N ⊂ Q are E(1)7 . (4) The prin ipal graphs of P ⊂ M and Q ⊂ M are A7 and those of N ⊂ P and N ⊂ Q are A7. In ea h ase, su h a quadrilateral of the hyper�nite II1 fa tors exists and is unique up to onjuga y. Thanks to Example 2.6, there are ountably many mutually non- onjugate (3,3)- supertransitive quadrilaterals of the hyper�nite II1 fa tors in Class III. 5.4. Class IV. Theorem 5.19. There exists a quadrilateral of fa tors Q = P ⊂ M N ⊂ Q in Class IV with [M : P ] = (3 + 13)/2. Su h a quadrilateral of the hyper�nite II1 fa tors is unique up to �ip onjuga y. Proof. Let P ⊂ M be the Haagerup subfa tor whose prin ipal graph and the dual prin ipal graph are as follows. M − P θη̂ θ idP κ̄θκη η P − P It is known that θ and η̂ satis�es the following relation [1℄: [θ3] = [idM ], [θ −1η̂] = [η̂θ], [η̂2] = [idM ]⊕ [η̂]⊕ [θη̂]⊕ [θ2η̂]. In the Appendix, we will show that there exists a unique Q-system for idP ⊕κ̄θκ up to equivalen e. Therefore there exists a unique subfa tor N of P up to inner onjuga y su h that [ιῑ] = [idP ]⊕ [κ̄θκ], where ι : N →֒ P is the in lusion map. Sin e dim(κι, κι) = dim(κ̄κ, ιῑ) = dim(idP ⊕ η, idP ⊕ κ̄θκ) = 1, the in lusion N ⊂ M is irredu ible. Sin e [κιῑκ̄] = [κκ̄]⊕ [κκ̄θκκ̄] = [idM ]⊕ [η̂]⊕ [(idM ⊕ η̂)θ(idM ⊕ η̂)] = [idM ]⊕ [η̂]⊕ [θ]⊕ [θη̂]⊕ [θ2η̂]⊕ [θ2η̂2] = [idM ]⊕ [θ]⊕ [θ2]⊕ 2[η̂]⊕ 2[θη̂]⊕ 2[θ2η̂], 46 PINHAS GROSSMAN AND MASAKI IZUMI we an take representative θ of the se tor [θ] su h that θκι = κι. Then θ3 is an inner automorphism satisfying θ3κι = κι and so θ3 = idM as N ⊂ M is irredu ible. We set Q = θ(P ). Using a similar argument as in the proof of Theorem 5.1, we an show that P ⊂ M N ⊂ Q is the desired quadrilateral. Uniqueness up to �ip onjuga y follows from uniqueness of the Q-system for id⊕ κ̄θκ. � We do not know if there exists a quadrilateral of the hyper�nite II1 fa tors in Class IV di�erent from the above example. 6. Classifi ation II Theorem 6.1. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors su h that the indi es of all the elementary subfa tors are less than or equal to 4. Then one of the following o urs: (1) The prin ipal graphs of P ⊂ M and Q ⊂ M are A3 and those of N ⊂ P and N ⊂ Q are A5. In this ase N is the �xed point algebra of an outer a tion of the symmetri group S3 on M . (2) The prin ipal graphs of P ⊂ M and Q ⊂ M are A4 and those of N ⊂ P and N ⊂ Q are D6. (3) The prin ipal graphs of all the elementary subfa tors are A7. (4) The prin ipal graphs of P ⊂ M and Q ⊂ M are A3 and those of N ⊂ P and N ⊂ Q are D(1)6 . In this ase N is the �xed point algebra of an outer a tion of the dihedral group D8 of order 8 on M . (5) The prin ipal graphs of P ⊂ M and Q ⊂ M are D4 and those of N ⊂ P and N ⊂ Q are E(1)6 . In this ase N is the �xed point algebra of an outer a tion of the alternating group A4 on M . (6) The prin ipal graphs of P ⊂ M and Q ⊂ M are A5 and those of N ⊂ P and N ⊂ Q are E(1)7 . In this ase the quadrilateral Q is given by the onstru tions in Example 2.6 with S{1,2,3,4}. (7) The prin ipal graphs of all the elementary subfa tors are E In ea h ase, su h a quadrilateral of the hyper�nite II1 fa tors is unique up to onju- ga y. We use the symbols ι = ιP,N , ιQ = ιQ,N , κ = ιM,P , κQ = ιM,Q. When N ⊂ P is 2-supertransitive, we also use the symbols ξ and ξ̂ as in Se tion 4. Lemma 6.2. Let Q = P ⊂ M N ⊂ Q be an irredu ible non ommuting quadrilateral of fa tors su h that the indi es of all the elementary subfa tors are less than or equal to 4. Then (1) The prin ipal graph of N ⊂ P is neither A3, D4 nor D(1)4 . CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 47 (2) If the prin ipal graph of N ⊂ P is E(1)6 , then that of P ⊂ M is D4. There exists an outer a tion a tion of the alternating group A4 on M su h that N is the �xed point algebra of the a tion. (3) If the prin ipal graph of N ⊂ P is D(1)n , then n = 6 and the prin ipal graph of P ⊂ M is A3. There exists an outer a tion a tion of the dihedral group D8 of order 8 on M su h that N is the �xed point algebra of the a tion. (4) If N ⊂ P and N ⊂ Q are 2-supertransitive, then the prin ipal graph of P ⊂ M is not D Proof. (1) follows from Remark 3.15. (2) Sin e the prin ipal graph of N ⊂ P is E(1)6 , there exists an outer automorphism α ∈ Aut(P ) of order 3 su h that [ξ2] = [idP ]⊕ [α]⊕ [α2]⊕ 2[ξ]. The proof of Lemma 4.1 shows that dim(κ̄κ, ξ) = 0 and 2 ≤ dim(κ̄κ, ξ2) = dim(κ̄κ, idP ⊕ α⊕ α2). Therefore κ̄κ ontains α and so the prin ipal graph of P ⊂ M is D4. The rest of the statement follows from [16℄, [21℄. (3) Thanks to (1), the number n is at least 5 and we have irredu ible de omposition [ιῑ] = [idP ]⊕ [α]⊕ [ξ], [ῑι]⊕ [τ ]⊕ [ξ̂], where d(ξ) = d(ξ̂) = 2 and α ∈ Aut(P ) and τ ∈ Aut(N) are outer automorphisms of order 2 satisfying [αξ] = [ξ], [τ ξ̂] = [ξ̂]. Sin e [κιτ ] = [κι], the rossed produ t N ⋊τ Z/2Z is identi�ed with the intermediate subfa tor of N ⊂ P generated by N and a unitary u ∈ M satisfying uxu∗ = τ(x) for all x ∈ N . If ῑQιQ ontained τ as well, the two in lusions N ⊂ P and N ⊂ Q would have a ommon intermediate subfa tor N ⋊ τZ/2Z, whi h is a ontradi tion as we have P ∩ Q = N . Sin e Q is non ommuting ῑQιQ must ontain ξ̂ and the prin ipal graph of N ⊂ Q is either A5 m with m ≥ 5. Assume that the prin ipal graph of N ⊂ Q is D(1)m . Then we have the irredu ible de omposition [ῑQιQ] = [idN ] ⊕ [τQ] ⊕ [ξ̂] su h that [τQ] 6= [τ ] and τQ is ontained in ξ̂2. If n were not equal to 6, the endomorphism ξ̂2 would ontain exa tly two automorphisms idP and τ , whi h would imply [τ ] = [τQ], a ontradi tion. Thus we get n = m = 6 and [ξ̂2] = [idN ]⊕ [τ ]⊕ [τQ]⊕ [τ ′], where τ ′ is an automorphisms of N . It is known that there are 4 di�erent subfa tors with the prin ipal graph D 6 [25℄. We show that the only one ase among them is possible. Sin e the dual prin ipal graph of N ⊂ P is also D(1)6 , we have [ξ2] = [id] ⊕ [α] ⊕ [α1]⊕ [α2] where α1 and α2 are automorphisms of P . The proof of Lemma 4.1 shows 2 ≤ dim(ῑκ̄κι, ξ̂) = 1 + dim(κ̄κ, α1 ⊕ α2). If κ̄κ ontained both α1 and α2, it would ontain α too, whi h ontradi ts irredu ibil- ity of N ⊂ M . Thus we may assume that κ̄κ ontains only α1, whose period must be two in onsequen e. Therefore either [M : P ] = 2 or the prin ipal graph of P ⊂ M is 48 PINHAS GROSSMAN AND MASAKI IZUMI k . Let L = P ⋊α1 Z/2Z, whi h is regarded as an intermediate subfa tor of P and M . Then it is known [26, Chapter 8℄ that N ⊂ L is given by the �xed point algebra of an outer a tion of either D8 or the Ka -Paljutkin algebra on L and [ῑσ̄σι] = [idN ]⊕ [τ ]⊕ [τQ]⊕ [τ ′]⊕ 2[ξ̂], where we set σ = ιL,P . We laim that Q is an intermediate subfa tor of N ⊂ L. Sin e d(ξ̂) = 2, Theorem 2.9,(1) shows that the multipli ity of ξ̂ in ῑκ̄κι is two and ῑκ̄κι ontains automorphisms with multipli ity at most one. On the other hand, the endomorphism ῑQιQ is ontained in ῑσ̄σι, and so [28, Theorem 3.9℄ shows the laim. Sin e Q is a quadrilateral we get M = L. Sin e [M : P ] = [M : Q] = 2, there exist period two automorphisms β, β1 ∈ Aut(M) su h that P = Mβ and Q = Mβ1 . If N ⊂ M were given by the Ka -Paljutkin algebra, we would have [κιῑκ̄] = [idM ]⊕ [β1]⊕ [β2]⊕ [β1β2]⊕ 2[ρ], with d(ρ) = 2, and the two automorphisms β1 and β2 would generate a group G of order 4 in Aut(M). However, this means P ∩Q = MG 6= N , whi h is a ontradi tion. Therefore N is the �xed point algebra of an outer a tion of D8. Now we assume that the prin ipal graph of N ⊂ P is A5 and we show that Q would be ommuting, whi h ontradi ts the assumption. Sin e [M : P ][P : N ] = [M : Q][Q : N ] we have [M : P ] = 3 and [M : Q] = 4. In this ase, we have [ῑQιQ] = [idN ]⊕ [ξ̂], [ιQῑQ] = [idQ]⊕ [ξQ], [ξ̂2] = [idN ]⊕ [τ ]⊕ [ξ̂], [τ 2] = [idM ], [τ ξ̂] = [ξ̂τ ] = [ξ̂], [ξ2Q] = [idQ]⊕ [αQ]⊕ [ξQ], [α2Q] = [idQ], [αQξQ] = [ξQαQ] = [ξQ]. As in [18, Lemma 3.2℄, we an hoose representatives of [αQ] and [τ ] su h that α idQ, τ 2 = idN , and αQιQ = ιQτ hold. We laim [κQαQ] = [κQ]. Indeed, sin e κι = κQιQ and [ιτ ] = [ι], we have [κQαQιQ] = [κQιQτ ] = [κQιQ] and 1 = dim(κQαQιQ, κQιQ) = dim(κQαQ, κQιQῑQ) = dim(κQαQ, κQ ⊕ κQξQ). Sin e we have dim(κQαQ, κQξQ) = dim(κQ, κQξQαQ) = dim(κQ, κQξQ) = dim(κ̄QκQ, ξQ) = 0, we get the laim. The laim implies that we an regard L := Q⋊αQZ/2Z as an intermediate subfa tor of Q ⊂ M . On the other hand, we an regard R := N ⋊τ Z/2Z as an intermediate subfa tor of N ⊂ P . Sin e we have αQιQ = ιQτ , we have the in lusion R ⊂ L. Moreover [18, Theorem 3.1℄ shows that N is the �xed point algebra of an outer a tion of S3 on L. In parti ular R ⊂ L N ⊂ Q is a ommuting square as we have [L : R] = 3 and [L : Q] = 2. By Galois orresponden e, the subfa tor R is the �xed point subalgebra of L under a subgroup of S3 isomorphi to Z/3Z, and by duality, there exists an order three outer automorphism β ∈ Aut(R) su h that L = R⋊βZ/3Z. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 49 Sin e [P : R] = 2, there exits an order two automorphism ϕ ∈ Aut(R) su h that P = R ⋊ϕ Z/2Z. Sin e ιM,RιM,R ontains ϕ and β and [M : R] = 6, we have M = N ⋊ Γ, where Γ is the group generated by [ϕ] and [β], whose order must be 6. In parti ular P ⊂ M R ⊂ L is a ommuting square. Therefore we have EQEP = EQELEP = EQER = EN , whi h shows that Q is ommuting, a ontradi tion. Thus the statement is proven. (4) Suppose that N ⊂ P and N ⊂ Q are 2-supertransitive and the prin ipal graph of P ⊂ M is D(1)n . If Q were o ommuting, Corollary 3.12 would imply 4 = [M : P ] ≤ [P : N ] − 1 whi h is a ontradi tion. If Q were non o ommuting, the statement (3) applied to the dual quadrilateral Q̂ would imply M = N ⋊ D8. However, this shows that Q is ommuting and we get a ontradi tion. Therefore the lemma is proven. � Proof of Theorem 6.1. Assume �rst that N ⊂ P and N ⊂ Q are 2-supertransitive. Then Lemma 6.2,(4) shows that P ⊂ M is either 2-supertransitive or M = P ⋊ G where G is a �nite abelian group of order 2,3, or 4. If the latter o urs, then the proof of Lemma 4.1 implies that ξ2 ontains a non-trivial automorphism, and so the prin ipal graph is either A5, E6, or E 6 . Applying Lemma 6.2,(1) to the dual quadrilateral Q̂, we see that Q is o ommuting. Sin e the E6 subfa tor has trivial se ond ohomology, Corollary 3.12 implies that E6 never o urs. For the same reason, we have [M : P ] = [P : N ]− 1 = 2 for A5 and we get ase (1) from Theorem 2.7. For 6 , we get ase (5) from Lemma 6.2,(2). If P ⊂ M andQ ⊂ M are 2-supertransitive, either Theorem 4.3 or Lemma 6.2,(2) is applied and we get ases (1),(2),(3),(6), and (7) from Theorem 5.2, 5.4, and 5.17. Assume now that N ⊂ P is not 2-supertransitive. Then Lemma 6.2,(1) implies that the prin ipal graph of N ⊂ P is D(1)n with n ≥ 5, and so Lemma 6.2,(4) implies the ase (4). � Remark 6.3. Ex ept for the ase (4), we have Θ(P,Q) = cos−1 1/([P : Q]−1) thanks to Lemma 2.5 and Corollary 3.11. In the ase (4), we have Θ(P,Q) = π/4 thanks to [54, Theorem 6.1℄. 7. α-indu tion and GHJ pairs 7.1. α-indu tion and angles. Let N be a properly in�nite fa tor and {λi}i∈I be a �nite system of irredu ible endomorphisms in End0(N ). We assume: (1) there exists 0 ∈ I su h that λ0 = idN , (2) for i 6= j ∈ I, we have [λi] 6= [λj ], (3) for ea h i ∈ I, there exists ī ∈ I su h that [λi] = [λī], (4) there exist non-negative integers Nkij su h that [λiλj] = Nkij [λk], 50 PINHAS GROSSMAN AND MASAKI IZUMI (5) the system {λi}i∈I has a braiding {ε(λi, λj)}i,j∈I (see [3, De�nition 2.2℄ for the de�nition), (6) N ⊂ M is an irredu ible in lusion of fa tors of �nite index su h that [ιM,N ιM,N ] = nj [λj], where J is a subset of I and nj is a positive integer. We naturally extend the braiding to the ategory generated by {λi}i∈I and use the same symbol ε(ρ, σ) for the extension. We set ε+(ρ, σ) = ε(ρ, σ) and ε−(ρ, σ) = ε(σ, ρ)∗. For simpli ity, we use the following notation: ν = ιM,N , γ = νν̄ ∈ End0(M), γ̂ = ν̄ν ∈ End0(N ). Then α-indu tion α±λi ∈ End0(M) is de�ned by α±λi = ν̄ −1 · Adε±(λi, γ̂) · λi · ν̄. (See [44℄, [58℄.) Note that for x ∈ N we have ν̄(ν(x)) = γ̂(x) and α±λi(x) = λi(x), whi h means α±λiν = νλi. Let Hj = (ν, νλj). Then thanks to Theorem 2.9, we have as a linear spa e. We hoose an orthonormal basis {t(j)k}njk=1 of Hj . Lemma 7.1. Let the notation be as above. Then for t ∈ Hj we have α±λi(t) = ε ±(λi, λj) Proof. Sin e ν̄(t) ∈ (γ̂, γ̂λj), the statement follows from the equation ε(λi, γ̂)λi(ν̄(t))ε(λi, γ̂) ∗ = ν̄(t) γ̂ λi γ̂ λi λj = ν̄(t) γ̂ λi γ̂ λi λj = γ̂(ε(λi, λj) ∗)ν̄(t). From the above lemma, it is easy to show (ρ, σ) ⊂ (α±ρ , α±σ ). We �x i0 ∈ I and set λ = λi0 , M = M, P = α+λ (M), Q = α−λ (M), N = P ∩ Q, and R = λ(N ). Sin e α±λ ν = νλ, we have R ⊂ N . In general these two fa tors do not oin ide and we give a des ription of N now. We set Ji0 = {j ∈ J ; ε(λj, λ)ε(λ, λj) is a s alar}, and set mj = ε(λj, λ)ε(λ, λj) for j ∈ Ji0. Re all that φρ = r∗ρρ̄(·)rρ is the standard left inverse of ρ. Sin e ε(λj, λ)ε(λ, λj) ∈ (λλj, λλj), we have φλ(ε(λj, λ)ε(λ, λj)) ∈ CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 51 (λj , λj), whi h is a s alar. It is easy to show that j ∈ Ji0 if and only if |φλ(ε(λj, λ)ε(λ, λj))| = 1. Lemma 7.2. Let the notation be as above. Then N = α+λ ( j∈Ji0 NHj) = α−λ ( j∈Ji0 NHj). In parti ular, the von Neumann algebra N is a fa tor. Proof. Let x, y ∈ M satisfy α+λ (x) = α−λ (y). Then Theorem 2.9 shows that we have expansion of x and y as a(j)kt(j)k, b(j)kt(j)k, where a(j)k and b(j)k are elements in N uniquely determined by x and y respe tively. Thus we get λ(a(j)k)ε(λ, λj) ∗t(j)k = α λ (x) = α λ (y) = λ(b(j)k)ε(λj, λ)t(j)k, whi h implies λ(a(j)k)ε(λ, λj) ∗ = λ(b(j)k)ε(λj, λ) for all j ∈ J and 1 ≤ k ≤ nj . Ap- plying φλ to both λ(a(j)k) = λ(b(j)k)ε(λj, λ)ε(λ, λj) and λ(a(j)k)(ε(λj, λ)ε(λ, λj)) b(j)k, we get a(j)k = b(j)kφλ(ε(λj, λ)ε(λ, λj)), a(j)kφλ(ε(λj, λ)ε(λ, λj)) = b(j)k, whi h shows that a(k)j 6= 0 only if |φλj (ε(λ, λj)ε(λj, λ))| = 1, whi h is equivalent to j ∈ Ji0 . On the other hand, when j ∈ Ji0 we have ε(λj, λ) = mjε(λ, λj)∗ and α+λ ( j∈Ji0 a(j)kt(j)k) = α j∈Ji0 m−1j a(j)kt(j)k), ∀a(j)k ∈ N , whi h shows the �rst statement. Sin e j∈Ji0 NHj is an intermediate von Neumann algebra between N and M, it must be a fa tor, and so is N . � Assume that P ⊂ M has no non-trivial intermediate subfa tor. Then Q = P ⊂ M N ⊂ Q is a quadrilateral though it is not irredu ible in general. Theorem 7.3. Let the notation be as above. Then Ang(P,Q) = {cos−1 |φλ(ε(λj, λ)ε(λ, λj))|; j ∈ J} \ {0, 52 PINHAS GROSSMAN AND MASAKI IZUMI In parti ular, when the braiding is non-degenerate, we have Ang(P,Q) = {cos−1 |S00Si0j ||S0i0S0j | ; j ∈ J} \ {0, π where Sij is as in [3, Proposition 2.4℄. Proof. Sin e M = j∈J NHj, we have P = α+λ (M) = λ(N )α+λ (Hj), Q = α−λ (M) = λ(N )α−λ (Hj). Thanks to Remark 2.12, to ompute the eigenvalues of EPEQEP it su� es to ompute EPEQ on α λ (Hj). We laim that for every t ∈ Hj the equalities λ (t)) = φλ(ε(λ, λj) ∗ε(λj, λ) ∗)α−λ (t), EP (α λ (t)) = φλ(ε(λj, λ)ε(λ, λj))α λ (t), hold. Note that we have EQ = α λ φα− and φα− = r∗λα (·)rλ. Thus (α+λ (t)) = r (α+λ (t))rλ = r (ε(λ, λj) ∗t)rλ = r λλ̄(ε(λ, λj) ∗)ε(λj, λ)trλ = r∗λλ̄(ε(λ, λj) ∗)ε(λj, λ)λj(rλ)t. Sin e r∗λλ̄(ε(λ, λj) ∗)ε(λj, λ)λj(rλ) = d(λj) d(λj) = φλ(ε(λ, λj) ∗ε(λj, λ) we get the �rst equation. The se ond one follows from a similar omputation. Using the laim, we get EPEQ(α λ (t)) = |φλ(ε(λj, λ)ε(λ, λj))|2α+λ (t), whi h proves the statement. � 7.2. GHJ pairs. We �rst re all the onstru tion of the GHJ subfa tors, whi h �rst appeared in the book of Goodman, de la Harpe and Jones [11℄. Our presentation is based on [9℄. Let G be one of the Dynkin diagrams of type A, D, or E, with a distinguished vertex ∗ and let Ãn be the string algebra String(n)G (see [9, p.554℄ for the de�nition). For i ≥ 1, we denote by ei the Jones proje tion in Ãi+1∩Ã′i−1 de�ned by [9, De�nition 11.5℄. Then the Jones proje tions {ek}∞k=1 satisfy the Temperley-Lieb relations: (1) eiei±1ei = τei (2) eiej = ejei if |i− j| ≥ 2 (3) tr(eiw) = τtr(w) if w is a word on e1, ..., ei−1. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 53 Here τ = 1/(4 cos2 π/m), where m is an integer asso iated to G alled the Coxeter number. For ea h n, the Coxeter number of An is n+1, the Coxeter number of Dn is 2n− 2, and the Coxeter numbers of E6, E7, and E8 are 12, 18, and 30 respe tively. Let B̃i be the subalgebra of Ãi generated by 1 and e1, ..., ei−1, for ea h i = 2, 3, .... Then the towers B̃i ⊂ Ãi form ommuting squares, and the von Neumann algebra B̃ generated by ∪∞i=1B̃i is a subfa tor of the fa tor à generated by ∪∞i=1Ãi in the GNS representation of the unique tra e on ∪∞i=1Ãi. This is an irredu ible subfa tor with �nite index, alled the GHJ subfa tor for (G, ∗). The prin ipal graphs of the GHJ subfa tors were omputed by Okamoto in [50℄. Be ause of the Temperley-Lieb relations, there is a unitary representation of the braid group inside the algebras B̃i, sending the usual braid group generators σi to gi = (t + 1)ei − 1, where t = e2πi/m, m again being the Coxeter number of G. Let vi = g1g2 · · · gi−1 and wi = g−11 g−12 · · · g−1i−1. We set C̃i = viÃi−1v∗i and D̃i = wiÃi−1w∗i . Then the towers C̃i ⊂ Ãi and D̃i ⊂ Ãi form ommuting squares, and the resulting subfa tor P̃ ⊂ M̃ and Q̃ ⊂ M̃ are irredu ible and [M̃ : P̃ ] = [M̃ : Q̃] = 4 cos2 π/k. For G = An the prin ipal graphs of P̃ ⊂ M̃ and Q̃ ⊂ M̃ are An, for G = D2n+1 they are A4n−1, for G = D2n they are D2n, for G = E7 they are A17 , and for G = En with n = 6, 8 they are En. The pair P̃ and Q̃ is alled the GHJ pair [13℄. (Though ∗ is assumed to be an endpoint in [13, De�nition 6.2.5℄, we don't pose this ondition here.) We set Ñ = P̃ ∩ Q̃ and set R̃ to be the subfa tor generated by {ei}∞i=2. By onstru tion, we have R̃ ⊂ Ñ though these two fa tors do not ne essarily oin ide in general. For a spe ial ase, the angles between P̃ and Q̃ are omputed in [13, Theorem 6.3.2℄. Now we show that P̃ ⊂ M̃ Ñ ⊂ Q̃ is essentially the same obje t as P ⊂ M N ⊂ Q ussed in the previous subse tion for an appropriate {λi}i∈I and N ⊂ M. Conse- quently, the angles between P̃ and Q̃ an be easily omputed by Theorem 7.3. Our argument below is inspired by [2, Appendix℄. Let N be the AFD type III1 fa tor and let {λi}ki=0 be a system of irredu ible endomorphisms isomorphi to the irredu ible se tors for the SU(2)k WZWmodel (see [2℄ for details). Su h a system an be obtained either by the loop group onstru tion [55℄ or by ombination of [57℄ and [14℄ using the quantum SU(2) at a root of unity. Then the system {λi}ki=0 satis�es the assumptions (1)-(5) of the previous subse tion. The prin ipal graph of λ1(N ) ⊂ N is Ak+1 and [λiλj ] = 0≤2l≤min{|i+j|,k}−|i−j| [λ|i−j|+2l], d(λi) = (i+1)π sin π Sij = k + 2 ((i+ 1)(j + 1)π k + 2 54 PINHAS GROSSMAN AND MASAKI IZUMI Assume that an in lusion of fa tors N ⊂ M satis�es the assumption (6). We set i0 = 1 and we use the same notation as in the previous subse tion. It is known that su h an in lusion is ompletely lassi�ed by a pair (G, ∗) (up to graph automorphism) where G is one of the Coxeter graphs of type A, D, or E [36, Theorem 2.1℄, so that we an identify Ãn with (νλ n, νλn) and B̃n with ν((λ n, λn)). Sin e νλ = α±λ ν, we have the in lusion relation α±λ (Ãn) ⊂ (α±λ νλn, α±λ νλn) = (νλn+1, νλn+1) = Ãn+1. Re all that the tra e tr on Ãn is given by φνλn . Lemma 7.4. Let the notation be as above. Then (1) For x ∈ Ãn, we have α±λ (x) = ε±(λ, λn)∗xε±(λ, λn). In onsequen e, the restri tion of α±λ to Ãn, as a map from Ãn to Ãn+1, is tra e preserving. (2) For x ∈ Ãn we have EP (x) ⊂ α+λ (Ãn) and EQ(x) ⊂ α−λ (Ãn). The restri tions of EP and EQ to Ãn is tra e preserving. Proof. (1) Let x ∈ Ãn. Then ν̄(x) ∈ (γ̂λn, γ̂λn) and ε(λ, γ̂)λ(ν̄(x))ε(λ, γ̂)∗ = ν̄(x) γ̂ λ λn = ν̄(x) γ̂ λ λ ν λ λ = ν̄(ε(λ, λn)∗xε(λ, λn)), whi h shows α+λ (x) = ε(λ, λ n)∗xε(λ, λn). The se ond equality an be shown in the same way. (2) Sin e EP (x) = α λ φα+ (x) = α+λ (r λ (x)rλ), it su� es to show r λ (x)rλ ∈ Ãn for x ∈ Ãn, whi h an be easily shown by using (1). This also shows tr(EP (x)) = tr(α λ φα+ (x)) = tr(φα+ (x)) = φνλnφα+ (x) = φα+ νλn(x) = φνλn+1(x) = tr(x). Sin e ε(λ, λn)∗ = ε(λ, λ)∗λ(ε(λ, λ)∗) · · ·λn−1(ε(λ, λ)∗), we an identify α+λ (Ãn−1) with C̃n and α λ (Ãn−1) with D̃n (for an appropriate hoi e of the braiding). We now show that P ⊂ M N ⊂ Q P̃ ⊂ M̃ Ñ ⊂ Q̃ are onjugate after tensor produ t with the AFD type III1 fa tor. Lemma 7.5. Let L be a fa tor and σ ∈ Mor0(N ,L). We set An = (σλn, σλn) and Bn = σ((λn, λn)). Let A and B be the II1 fa tors generated by ∪nAn and ∪nBn respe tively on the GNS Hilbert spa e of the unique tra e on ∪nAn. Then A ∩ B′ = (σ, σ). Proof. Sin e (λn, λn) is generated by Jones proje tions, the statement follows from the �atness of the Jones proje tions [9, Chapter 12℄. � CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 55 Theorem 7.6. Let the notation be as above. Then P̃ ⊗ R∞ ⊂ M̃ ⊗ R∞ Ñ ⊗ R∞ ⊂ Q̃⊗ R∞ P ⊂ M N ⊂ Q are onjugate where R∞ is the AFD type III1 fa tor. Proof. First we show that R ⊂ M and R̃ ⊂ M̃ have the same standard invariant. We �x a positive integer n larger than half of the depth of R ⊂ M , whi h is �nite now. Let ρ = νλ. Then R = ρ(N ) and (ρ̄ρ)n−1ρ̄((ρ, ρ)) ⊂ (ρ̄ρ)n−1((ρ̄ρ, ρ̄ρ)) ⊂ · · · ⊂ ((ρ̄ρ)n, (ρ̄ρ)n) ∪ ∪ ∪ C ⊂ (ρ̄ρ)n−1((ρ̄, ρ̄)) ⊂ · · · ⊂ ((ρ̄ρ)n−1ρ̄, (ρ̄ρ)n−1ρ̄) is the standard invariant of R ⊂ M (or rather (ρ̄ρ)n(N ) ⊂ (ρ̄ρ)n−1ρ̄(M)). We set A2j,m = (ρ̄ρ)n−j(((ρ̄ρ)jλm, (ρ̄ρ)jλm)), A2j+1,m = (ρ̄ρ)n−j−1ρ̄((ρ(ρ̄ρ)jλm, ρ(ρ̄ρ)jλm)). Note that ρ̄ρ is a dire t sum of irredu ibles in {λi}ki=0. When m is larger than the depth of λ(N ) ⊂ N , the in lusion graph for Aj,m ⊂ Aj+1,m is the re�e tion of that for Aj+1,m ⊂ Aj+2,m thanks to Frobenius re ipro ity. Therefore Aj,m ⊂ Aj+1,m ⊂ Aj+2,m is the basi onstru tion. Let Aj,∞ be the fa tor generated by ∪mAj,m in the GNS representation of the unique tra e of ∪mA2n,m. Then we may identify R̃ ⊂ M̃ with A0,∞ ⊂ A1,∞ and A0,∞ ⊂ A1,∞ ⊂ A2,∞ ⊂ · · · ⊂ A2n,∞ is the Jones tower. Lemma 7.5 implies A2j,∞ ∩ A′0,∞ = (ρ̄ρ)n−j(((ρ̄ρ)j , (ρ̄ρ)j)), A2j+1,∞ ∩ A′0,∞ = (ρ̄ρ)n−j−1ρ̄((ρ(ρ̄ρ)j, ρ(ρ̄ρ)j)), whi h shows that R ⊂ M and R̃ ⊂ M̃ have the same prin ipal graph. On the other hand, we have A2j,∞ ∩ A′1,∞ ⊃ (ρ̄ρ)n−j(((ρ̄ρ)j−1ρ̄, (ρ̄ρ)j−1ρ̄)), A2j+1,∞ ∩ A′1,∞ ⊃ (ρ̄ρ)n−j−1ρ̄((ρρ̄)j, (ρρ̄)j)). Sin e [M : R] = [M̃ : R̃], the equality holds in the above in lusions and the two subfa tors R ⊂ M and R̃ ⊂ M̃ have the same standard invariant. Therefore we on lude that the two in lusions R ⊂ M and R̃ ⊗ R∞ ⊂ M̃ ⊗ R∞ are onjugate thanks to Popa's result [53℄ with [19℄ (see [47℄ too). To �nish the proof, it su� es to show that eP = eP̃ and eQ = eQ̃ hold in the above identi� ation of the standard invariants, whi h an be done by using Lemma 7.4,(2). � 56 PINHAS GROSSMAN AND MASAKI IZUMI Remark 7.7. Sin e |φλ(ε(λj, λ)ε(λ, λj))| = |S1j | d(λ1)d(λj)|S00| | sin π 2(j+1)π | sin 2π (j+1)π | cos (j+1)π | cos π the operator ε(λj, λ)ε(λ, λj) is a s alar if and only if j = 0, k. Thus we have Ji0 = J ∩ {0, k}. It is known that k ∈ J o urs only if k is even. In this ase, we may and do assume λ2k = idN by hoosing a representative λk satisfying λkλk/2 = λk/2, whi h implies that there exists a unitary uk ∈ Hk satisfying u2k = 1. Therefore we may regard N +NHk as the rossed produ t N ⋊λk Z/2Z. We make this assumption and identi� ation in what follows. The following orollary answers the question raised in [13, omment after Propo- sition 6.2.4℄ in the negative. (The subfa tor R̃ is denoted by TL2 in [13℄.) Corollary 7.8. Let the notation be as above. Then R̃ = P̃ ∩ Q̃ if and only if k /∈ J . If k ∈ J , the index of R̃ ⊂ P̃ ∩ Q̃ is 2. Proof. Thanks to Theorem 7.6, we may work on R and N instead of R̃ and Ñ , whi h together with Lemma 7.2 show the �rst statement. When k ∈ J , Lemma 7.2 shows N = α±λ (N +NHk) and the se ond statement follows from Remark 7.7. � The following orollary is a generalization of [13, Theorem 6.3.2℄. Corollary 7.9. Let the notation be as above. Then Ang(P̃ , Q̃) = {cos−1 | cos (j+1)π | cos π | ; j ∈ J} \ {0, Example 7.10. Let N ⊂ M be the in lusion of the AFD type III1 fa tors asso iated with the onformal embedding SU(2)10 ⊂ SO(5)1 (see [58, p.381℄). Then the orre- sponding GHJ subfa tor is given by E6 with ∗ equal to the vertex of the minimum value of the Perron-Frobenius eigenve tor, and so its prin ipal graph is as in Theorem 5.4,(2) (see [9, p.726℄,[50℄). Sin e [γ̂] = [λ0]⊕ [λ6], we have R̃ = Ñ , and so Ñ ⊂ P̃ is onjugate to the GHJ subfa tor, while P̃ ⊂ M̃ is the E6 subfa tor. Sin e dim(νλ, νλ) = dim(γ̂, λ2) = dim(λ0 ⊕ λ6, λ0 ⊕ λ2) = 1, the in lusion Ñ ⊂ M̃ is irredu ible. The angle between P̃ and Q̃ is omputed as Θ(P̃ , Q̃) = cos−1(2 − 3) by using either of [13, Theorem 6.3.2℄, Theorem 4.3, or Corollary 7.9. It is observed in [13, Proposition 6.2.6℄ that R̃ ⊂ M̃ is irredu ible if and only if ∗ is an endpoint. However, even when ∗ is not an endpoint, the in lusion Ñ ⊂ M̃ may be irredu ible (though it is very subtle in general to de ide whether Ñ ⊂ M̃ is irredu ible or not as we will see below). We use this observation to ompute the opposite angles of quadrilaterals onstru ted in [12℄. The following theorem is shown in [12℄. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 57 Theorem 7.11. Let B ⊂ A D ⊂ C be an irredu ible non ommuting quadrilateral of the hyper�nite II1 fa tors su h that D ⊂ B and D ⊂ C are supertransitive and [B : D] and [C : D] are less than 4. Then the prin ipal graphs of D ⊂ B and D ⊂ C are A2n+1 with an integer n ≥ 2. For ea h integer n ≥ 2, su h a quadrilateral exists and is unique up to onjuga y. We now ompute the �opposite angles� of the above quadrilateral. Theorem 7.12. Let Ĉ ⊂ D̂ A ⊂ B̂ be the dual quadrilateral of the quadrilateral in the previous theorem. Then Ang(B̂, Ĉ) = {cos−1 (2j+1)π cos π ; j = 1, 2, · · · , [n− 1 where [(n− 1)/2] is the integer part of (n− 1)/2. Proof. We show that the above quadrilateral is given by the GHJ pair of A2n+1 with ∗ equal to the midpoint. Thanks to Theorem 7.6, we an use α-indu tion instead of the GHJ onstru tion for this purpose. Let N and {λi}ki=0 be as before with k = 2n. Let (M, {λ̃i}ki=0) be a opy of (N , {λi}ki=0). We embed N into M by ν : N ∋ x 7→ λn(x) ∈ M. Then we have ν̄ = λ̃n regarded as a map from M into N , and so γ̂ = λ2n and γ = λ̃2n. Sin e [λ2n] = [λ2i], we have J = {0, 2, 4, · · · , k} and Ji0 = {0, k}. Therefore thanks to Corollary 7.8, N = α±λ (N ⋊λk Z/2Z) = α±λ (ν(N ) + ν(N )uk), where uk ∈ (ν, νλk) = (λ̃n, λ̃nλ̃k). As before, we set M = M, P = α+λ (M), Q = α−λ (M), N = P ∩ Q, and Q = P ⊂ M N ⊂ Q . Then the prin ipal graphs of P ⊂ M and Q ⊂ M are A2n+1. To prove the theorem, it su� es to show that N ⊂ M is irredu ible and Q is not o ommuting thanks to Corollary 7.9 and Theorem 7.11. We laim α±λ = Adε ±(λ, λn)λ̃. Indeed, sin e ε ±(λ, γ̂) = λn(ε ±(λ, λn))ε ±(λ, λn) and ε±(λ, λn) ∈ (λλn, λnλ) = (λν̄, ν̄λ̃), we have Ad(ε±(λ, γ̂))λν̄(x) = Adλn(ε ±(λ, λn))Adε ±(λ, λn)λν̄(x) = Adν̄(ε ±(λ, λn))ν̄(λ̃(x)) for all x ∈ M, whi h shows the laim. In parti ular, the two subfa tors P and Q are inner onjugate in M . Therefore if N ⊂ M is irredu ible, Theorem 3.10 applied to the dual quadrilateral Q̂ shows that Q is not o ommuting. 58 PINHAS GROSSMAN AND MASAKI IZUMI Now our only task is to show M ∩N ′ = C. Note that N is generated by ν(λ(N )) and ν(ε(λ, λk) ∗)uk with uk ∈ (ν, νλk), and so M ∩N ′ = (νλ, νλ) ∩ {ν(ε(λ, λk)∗)uk}′. Therefore it su� es to show (λnλ, λnλ) ∩ {λn(ε(λ, λk)∗)u}′ = C. where u is a unitary in (λn, λnλk). By Frobenius re ipro ity, we have dim(λnλ, λnλ) = dim(λ 2) = dim(λ2n, λ0 ⊕ λ2) = 2, and we an hoose a basis {1, s} of (λnλ, λnλ) with where v ∈ (λn, λnλ2) and w ∈ (λ2λ, λ). On one hand, we have sλn(ε(λ, λk) ∗)u = v where ṽ = ε(λ2, λn) ∗v. On the other hand, λn(ε(λ, λk) ∗)us = Sin e dim(λn, λ2λnλk) = 1, there exists a s alar c su h that λ2 λn λn = c λ2 λn CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 59 whi h implies sλn(ε(λ, λk) ∗)u = cλn(ε(λ, λk) ∗)us. The number c is given by Kirillov- Reshetikhin's quantum 6j-symbol with q = eπi/(n+1) (see [37, (5.4)℄). Using the notation in [37℄ (irredu ible representations are parameterized by half integers there), we get Note that although an expli it formula of the 6j-symbols depends on the hoi e of intertwiners in general, the above one does not be ause the same intertwiners u and ṽ appear in the both sides. This an be omputed by [37, Theorem 5.1℄ and it turns out that we have c = −1. Therefore N ⊂ M is irredu ible. � 7.3. Asymptoti in lusions and angles. Another ri h sour e of non ommuting quadrilaterals is asymptoti in lusions. We show that a te hnique similar to the one we used in the previous subse tion works in this ase too. Let N ⊂ M be an irredu ible in lusion of the hyper�nite II1 fa tors with �nite index and of �nite depth and let N ⊂ M ⊂ M1 ⊂ · · · ⊂ Mn ⊂ · · · be the Jones tower. The algebra ∪nMn has a unique tra e and we denote by M∞ the fa tor generated by ∪nMn in the GNS representation of the unique tra e of ∪nMn. Then the in lusion M ∨ (M∞ ∩ M′) ⊂ M∞ is alled the asymptoti in lusion of N ⊂ M (see [9, Chapter 12.6℄ for details). It is easy to show that N ∨ (M∞ ∩ N ′) ⊂ M∞ N ∨ (M∞ ∩M′) ⊂ M∨ (M∞ ∩M′) is an irredu ible ommuting quadrilateral of fa tors. We show how to ompute the �opposite angles" of this quadrilateral. Example 7.13. When the prin ipal graph of N ⊂ M is An, there exists a sequen e of Jones proje tion {ei}∞i=1 su h that N = {ei}∞i=2 ′′ ⊂ M = {ei}∞i=1 . We set e0 = eN , e−1 = eM and e−(i+1) = eMi for i ≥ 1. Then M∞ = {ei}∞i=−∞ , M∨ (M∞ ∩M′) = {ei}′′i 6=0, N ∨ (M∞ ∩ N ′) = {ei}′′i 6=1, and N ∨ (M∞ ∩M′) = {ei}′′i 6=0,1. The index of the asymptoti in lusion in this ase is omputed in [8℄ and the prin ipal graphs are obtained in [10℄. It is well-known that M∞ ∩M′ is naturally identi�ed with the opposite algebra Mopp of M and M ∨ (M∞ ∩ M′) is naturally identi�ed with M ⊗ Mopp, and so we identify N ∨ (M∞ ∩ M′) with N ⊗ Mopp too. However, we never identify N ∨ (M∞∩N ′) with N ⊗N opp as we annot take ommon fa torizations M⊗Mopp and N ⊗ N opp for M∨ (M∞ ∩M′) and N ∨ (M∞ ∩ N ′). By utting the Hilbert spa e L2(M∞) by an appropriate proje tion in M′∞, we may assume that M∞ a ts on L2(M⊗Mopp) = L2(M)⊗ L2(Mopp). Sin e eN may be onfused with eN ⊗ 1 in this representation, we use the symbol e for eN in M∞. Let J = JM ⊗ JMopp be the anoni al onjugation of M⊗Mopp. We set M := J N ∨ (M∞ ∩M′) J = J(N ⊗Mopp)′J = M1 ⊗Mopp, P := J M∨ (M∞ ∩M′) J = J(M⊗Mopp)′J = M⊗Mopp, 60 PINHAS GROSSMAN AND MASAKI IZUMI Q := J N ∨ (M∞ ∩ N ′) J, N := JM′∞J. Note that Q is not equal to M1 ⊗ N opp as we formally (just formally) broke the symmetry between M∨ (M∞ ∩M′) and N ∨ (M∞ ∩ N ′) by representing M∞ on L2(M⊗Mopp). However, we still have the following: Lemma 7.14. Let the notation be as above. Then Q and M1 ⊗ N opp are inner onjugate in M . In parti ular, we have M⊗QM ∼= M1⊗Mopp1 as M−M bimodules. Proof. Note that N ⊗N opp ⊂ N ⊗Mopp ⊂ N ∨ (M∞ ∩ N ′) = (N ⊗Mopp) ∨ {e} is the basi onstru tion with the Jones proje tion e. Thus Q ⊂ M1 ⊗Mopp ⊂ J(N ⊗N opp)′J = M1 ⊗Mopp1 is the basi onstru tion too, whi h shows that Q and M1⊗N opp are inner onjugate in M . � Our purpose of this subse tion is to show the following theorem: Theorem 7.15. Let the notation be as above. If the prin ipal graph of N ⊂ M is An, then Ang(P,Q) = {cos−1 (j+1)π cos π ; j = 1, 2, · · · [n− 2 We go ba k to the general (not ne essarily An) ase. Let Q = P ⊂ M N ⊂ Q be as above. To ompute Ang(P,Q) we may repla e M with M ⊗ B(ℓ2)⊗ B(ℓ2) ∼= M⊗ B(ℓ2)⊗ (M⊗ B(ℓ2))opp, and so we may and do assume that N ⊂ M is an in lusion of properly in�nite fa tors from now on. In this ase, the in lusion N ⊂ P is des ribed as the Longo- Rehren in lusion [46℄ whose stru ture is well-studied [22℄. Therefore the stru ture of N ⊂ P ⊂ M is ompletely understood. It is a little tri ky to de ide the position of Q, but we an handle it by using Lemma 7.14. We use the symbols ι, ιQ, κ, and κQ as in Se tion 4. We re all the notation in [22℄ to des ribe the Longo-Rehren in lusion N ⊂ P . Let ν : M →֒ M1 be the in lusion map and let ∆ = {ρξ}ξ∈∆0 be the set of irredu ibles ontained in ∪n(ν̄ν)n. We may assume that ρe = idM with e ∈ ∆0 and [ρξ] = [ρξ̄]. We identifyMopp withM′ = JMMJM and de�ne j : M → Mopp by j(x) = JMxJM for x ∈ M. For x ∈ M, the orresponding element xopp ∈ Mopp is identi�ed with j(x∗). We denote by ρ ξ the endomorphism of Mopp orresponding to ρξ. In our situation, we have ρ ξ = jρξj . It is known that we have [ιῑ] = [ρ̂ξ], where ρ̂ξ = ρξ⊗ρoppξ , and so d(ι)2 = d(ξ)2. We hoose an isometry Vξ ∈ (ρξ⊗ρoppξ , ιῑ) su h that rῑ = Ve, whi h will be denoted by V for simpli ity. We denote rι ∈ N by W , CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 61 whi h is des ribed as follows. Let {T (ζξ,η)i} i=1 be an orthonormal basis of (ρζ , ρξρη), and set ξ,η = ξ,η)i ⊗ j(T ( ξ,η)i), whi h does not depend on the hoi e of the basis. Then we have ξ,η,ζ d(ξ)d(η) Vξρ̂ξ(Vη)T To des ribe the onditional expe tation EQ, we use the following lemma: Lemma 7.16. Let B ⊂ A be an irredu ible in lusion of property in�nite fa tors with �nite index and let µ : B →֒ A be the in lusion map. Let [µµ̄] = i=0 ni[σi] be the irredu ible de omposition and let {s(i)l}nil=1 be an orthonormal basis of (µ, σiµ). Then EB(x) = [A : B] d(σi)s(i) l σi(x)s(i)l, ∀x ∈ A. Proof. Let s̃(i)l = d(σi)s(i) l σi(rµ̄). Then {s̃(i)l} l=1 is an orthonormal basis of (σi, µµ̄) thanks to Frobenius re ipro ity. Thus EB(x) = µ(r µµ̄(x)rµ) = µ(r µ)µµ̄(x)µ(rµ) = µ(r∗µ)s̃(i)lσi(x)s̃(i) l µ(rµ). On the other hand, µ(r∗µ)s̃(i)l = d(σi)µ(r µ)s(i) l σi(rµ̄) = d(σi)s(i) l σi(µ(r µ)rµ̄) = d(σi) s(i)∗l , whi h shows the statement. � Re all that to ompute Ang(P,Q), it su� es to determine the eigenvalues of EPEQ restri ted to P . Lemma 7.17. Let [ν̄ν] = nξ[ρξ] be the irredu ible de omposition. Then for x ∈ P , we have EPEQ(x) = d(ι)2 [M : N ] W ∗Vξρ̂ξ(x)V Proof. Thanks to Lemma 7.14, we have [κQκ̄Q] = nξ[idM1 ⊗ ρ ξ ]. Thus to ompute EQ using Lemma 7.16, it su� es to obtain (κQ, (idM1⊗ρoppξ )κQ) for ξ ∈ ∆1. Although it is not so easy to apture this spa e dire tly, we have (κQ, (idM1 ⊗ ρ ξ )κQ) ⊂ (κQιQ, (idM1 ⊗ ρ ξ )κQιQ) = ((ν ⊗ idMopp)ι, (ν ⊗ ρoppξ )ι) ∼= (ιῑ, ν̄ν ⊗ ρoppξ ). 62 PINHAS GROSSMAN AND MASAKI IZUMI Comparing the dimensions of the both sides above, we get equality for the above in lusion. Let {t(ξ)i} i=1 be an orthonormal basis of (ν, νρξ). We laim that { d(ι)√ (t(ξ)∗i ⊗ 1)V ∗ξ W} is an orthonormal basis of ((ν⊗ idMopp)ι, (ν⊗ ρoppξ )ι). Indeed, it is easy to show that (t(ξ)∗i ⊗ 1)V ∗ξ W belongs to the intertwiner spa e ((ν ⊗ idMopp)ι, (ν ⊗ ρ ξ )ι), and so the operator W ∗Vξ(t(ξ)lt(ξ) i ⊗ 1)V ∗ξ W is already a s alar, whi h is equal to ENEP (W ∗Vξ(t(ξ)lt(ξ) i ⊗ 1)V ∗ξ W ) = EN(W ∗VξEP (t(ξ)lt(ξ)∗i ⊗ 1)V ∗ξ W ) 〈t(ξ)l, t(ξ)i〉 EN (W ξ W ) = W ∗EN (VξV ξ )W = d(ξ)δi,l d(ι)2 d(ξ)δi,l d(ι)2 This shows the laim. Thus Lemma 7.16 shows that for x ∈ P we have EQ(x) = [M : N ] d(ι)2W ∗Vξ(t(ξ)i ⊗ 1)(idM ⊗ ρoppξ )(x)(t(ξ)∗i ⊗ 1)V ∗ξ W d(ι)2 [M : N ] W ∗Vξρ̂ξ(x)(t(ξ)it(ξ) i ⊗ 1)V ∗ξ W. Therefore we get EPEQ(x) = d(ι)2 [M : N ] W ∗Vξρ̂ξ(x)EP (t(ξ)it(ξ) i ⊗ 1)V ∗ξ W d(ι)2 [M : N ] W ∗Vξρ̂ξ(x)V To ompute the eigenvalues of EPEQ|P , we need the rossed produ t-like de om- position as in Theorem 2.9 based on the irredu ible de omposition of ῑι. It is known that every se tor ontained in a power of ῑι is of the form σ̃α, where σ is a �nite dire t sum of endomorphisms in ∆ and α is a parameter of the orresponding half-braiding εασ(ξ) ∈ (σρξ, ρξσ) (see [22, Se tion 4℄ for details). Let Uα(σ) = σ(ξ)⊗ 1)(σ ⊗ idMopp)(V ∗ξ ). Then σ̃α is the restri tion of AdUα(σ)(σ ⊗ idMopp) to N . Irredu ible σ̃α is ontained in ῑι if and only if σ ontains idM. In this ase, we have (ι, ισ̃α) = Uα(σ) (idM, σ)⊗ C Therefore to ompute Ang(P,Q), it su� es to ompute EPEQ on this spa e thanks to Remark 2.11. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 63 Lemma 7.18. Let σ̃α be an irredu ible endomorphism of N ontained in ῑι. Then for v ∈ (idM, σ), EPEQ(U α(σ)(v ⊗ 1)) = Uα(σ)(φν̄ν(εασ(ν̄ν)v)⊗ 1). The map (idM, σ) ∋ v 7→ φν̄ν(εασ(ν̄ν)v) depends only on the se tor of ν̄ν and φν̄ν(ε σ(ν̄ν)v) = nξd(ξ) [M : N ]φρξ(ε σ(ξ)v). Proof. It is routine work to show the se ond statement. Sin e we have Uα(σ)(v ⊗ 1) = σ(η)v ⊗ 1)V ∗η , Lemma 7.17 implies EPEQ(U α(σ)(v ⊗ 1)) d(ι)2 [M : N ] W ∗Vξρ̂ξ(Vη(ε σ(η)v ⊗ 1)V ∗η )V ∗ξ W [M : N ] η,ζ∈∆0 nξd(η) iρξ(ε σ(η)v)T ( ξ,η)i ⊗ 1 V ∗ζ . Thanks to the half-braiding relation, we have i ρξ(ε σ(η)v) = ε σ(ζ)σ(T ( ξ,η)) ∗ρξ(v). Thus Frobenius re ipro ity implies d(η)T ( iρξ(ε σ(η)v)T ( ξ,η)i = εασ(ζ) d(η)σ(T ( ξ,η)) ∗ρξ(v)T ( ξ,η)i = d(ξ)d(ζ)εασ(ζ) σ(r∗ρξ̄ρξ(T ( ))i))ε ∗ρξ(vT ( )∗i )rρξ̄ = d(ξ)d(ζ)εασ(ζ) σ(r∗ρξ̄ρξ(T ( ))i))ε ∗ρξ(σ(T ( )∗i )v)rρξ̄ = d(ξ)d(ζ)εασ(ζ) σ(r∗ρξ̄ρξ(T ( )iT ( )∗i ))ε ∗ρξ(v)rρξ̄ = d(ξ)d(ζ)εασ(ζ)σ(r )εασ(ξ) ∗ρξ(v)rρξ̄ . 64 PINHAS GROSSMAN AND MASAKI IZUMI Using the half-braiding relation again, we see that this is equal to d(ξ)d(ζ)εασ(ζ)r σ(ξ̄)v)rρξ̄ = d(ξ)d(ζ)ε σ(ζ)φρξ̄(ε σ(ξ̄)v), whi h �nishes the proof. � Proof of Theorem 7.15. Assume that the prin ipal graph of N ⊂ M is An. The stru ture of N −N se tors asso iated with the in lusion N ⊂ P is des ribed in [22, Se tion 7℄. We may assume that there exists a system of endomorphism {λi}n−1i=0 of M isomorphi to the irredu ible se tors of the SU(2)n−1 WZW model su h that N = λ1(M). Then we have [ν̄ν] = [λ21] and ∆ = {λ2j} [(n−1)/2] j=0 . When i − j is even, we set σi,j = λiλj and εσi,j(λl) = ε +(λi, λl)λi(ε −(λj, λl)). Then {εσi,j (λl)}l is a half-braiding for σi,j and we denote by σ̃i,j the orresponding endomorphism of N . Assume �rst that n is even. In this ase, the endomorphism σ̃2i,2i is irredu ible [ῑι] = n/2−1⊕ [σ̃2i,2i]. Sin e (idM, σ2i,2i) = Crλ2i , we get Ang(P,Q) = {cos−1 r∗λ2iφλ21(εσ2i,2i(λ 1)rλ2i); i = 0, 1, · · · , } \ {0, π It is easy to show r∗λ2iφλ21(εσ2i,2i(λ 1)rλ2i) = φλ1(ε(λ2i, λ1)ε(λ1, λ2i)) and we get the statement. Assume now that n = 2s+1 is odd. Then σ̃i,i is irredu ible for i = 0, 1, 2, · · ·s− 1 while σ̃s,s is de omposed into two se tors, say µ0 and µ1 su h that [ῑι] = [σ̃i,i]⊕ [µ0]. Then a similar omputation as above shows Ang(P,Q) = {cos−1 |φλ1(ε(λi, λ1)ε(λ1, λi))|; i = 1, 2, · · · , s}, (sin e EPEQ((ι, ισ̃s,s)) = 0 we have EPEQ((ι, ιµ0)) = 0 too) whi h shows the state- ment. � 8. Appendix In the proof of Theorem 5.19 we use the fa t that there exists a unique Q-system for idP ⊕ κ̄θκ up to equivalen e. We give a proof of this statement here. The next lemma follows from Frobenius re ipro ity [20, Se tion 2.3℄, whi h says that the right normalization of the norm of an element v ∈ (τ, ρσ) is d(ρ)d(σ) Lemma 8.1. Let L, M, and N be properly in�nite fa tors and ρ ∈ Mor0(N ,M), σ ∈ Mor0(L,N ), and τ ∈ (L,M) be irredu ible morphisms. If v ∈ (τ, ρσ) satis�es CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 65 ||v|| = d(ρ)d(σ) , then (d(τ)d(σ) (d(τ)d(ρ) We re all the onstru tion of the Haagerup subfa tor in [23, Se tion 7℄. Let O4 be the Cuntz algebra generated by isometries {S0, T0, T1, T2} and let d = (3 + 13)/2 (whi h will be d(ρ)). We introdu e an endomorphism ρ and a period 3 automorphism α of O4 by setting (8.1) α(S0) = S0, α(Ti) = Ti+2, (8.2) ρ(S0) = i∈Z/3Z TiTi, (8.3) ρ(Ti) = −i + T−iS0S j,k∈Z/3Z A(i+ j, i+ k)TjTi+j+kT where i is understood as an element of Z/3Z and A(0, 0) = 1− 1 d− 1 , A(0, 1) = A(0, 2) = A(1, 0) = A(2, 0) = A(1, 1) = A(2, 2) = d− 1 , A(1, 2) = A(2, 1) = 4d− 1 2(d− 1) . Then ρ and α extend to an irredu ible endomorphism and an outer automorphism respe tively of the weak losure M of O4 in the GNS representation of some KMS state, whi h will be denoted by the same symbols. Moreover we have S0 ∈ (idM , ρ2), T0 ∈ (ρ, ρ2), T1 ∈ (αρ, ρ2), and T2 ∈ (α2ρ, ρ2) and αρ = ρα2. It is easy to show that ρ satis�es the ondition of Lemma 3.5, namely (8.4) ρ(T0)S0 = T0S0, (8.5) dS0 + (d− 1)T 20 = dρ(S0) + (d− 1)ρ(T0)T0. Therefore there exist a Q-system for idM ⊕ ρ and a subfa tor P ⊂ M su h that [κκ̄] = [idM ] ⊕ [ρ], where κ : P →֒ M is the in lusion map. Note that ρ = η̂ 66 PINHAS GROSSMAN AND MASAKI IZUMI and α = θ in the notation of Theorem 5.19. Lemma 3.5 shows that we an hoose isometries v = rκ̄, w = κ(rκ), and v1 ∈ (ρ, κκ̄) su h that (3.3) holds: (8.6) w = v1ρ(v)v v1ρ(v1)S0v v1ρ(v1)T0v To �nish the proof of Theorem 5.19, it su� es to show that there exists a unique Q-system for idP ⊕ κ̄ακ up to equivalen e. For this purpose we solve the equations (3.1) and (3.2) with σ := κ̄ακ. For this, we use the following notation: (d(κ)2 v∗1 = (d+ 1 v∗1 , (d+ 1 κ̄ ρ ρκ̄ Every vertex expressing an element in (αρ, ρα2), (ρ, αρα) et . su h as will always mean 1. We hoose S0 as rρ = r̄ρ and set := d1/4T0, = d1/4T ∗0 . = d1/2S∗0ρ(d 1/4T0) = d 1/4T ∗0 = = ρ(d1/2S∗0)d 1/4T0 = d 1/4T ∗0 = CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 67 In the same way, we have Equation (8.4) means ρ ρ ρ ρ ρ ρ We use a similar expression for d1/4T1 ∈ (αρ, ρ2) = (ρα2, ρ2) and d1/4T2 ∈ (α2ρ, ρ2) = (ρα, ρ2). Sin e we have κρκ̄ κρκ̄ κ̄ ρ κ we simply express this intertwiner by . In a similar way, we have κ̄ κα κ̄ κα and we simply express it by κ̄ κα . The diagram κ̄ κα is also interpreted in the same way. We set κ̄ α κ κ̄ α κ ρ ∈ (idP , σ2). Then thanks to Lemma 8.1, we have ||R|| = d(κ)2 d(κ)d(ρ) d(κ)2 = d(κ) = d+ 1. All we have to show is the following: Theorem 8.2. Let the notation be as above. Then there exist exa tly two elements S ∈ (σ, σ2) satisfying ||S|| = d(σ)1/4 = (d+ 1)1/4 and (8.7) σ(S)R = SR, 68 PINHAS GROSSMAN AND MASAKI IZUMI (8.8) (R− σ(R)) = σ(S)S − S2. (Note that if S satis�es the above ondition, so does −S.) To prove the theorem, we hoose a basis of (σ, σ2). Note that we have dim(σ, σ2) = dim(κ̄ακ, κ̄ακκ̄ακ̄) = dim(κκ̄ακκ̄, ακκ̄α) = dim(α⊕ αρ⊕ α2ρ⊕ α2ρ2, α2 ⊕ ρ) = 2. We set α α κκ̄ ∈ (σ, σ2), α α κκ̄ ρ ∈ (σ, σ2). Then sin e rκ̄ and v1 are orthogonal, so are S1 and S2. Lemma 8.3. Let the notation be as above. Then ||S1|| = ||S2|| = d(σ)1/4 = (d+1)1/4. Proof. For S1, we have ||S1||2 = d(κ) κκ̄ α κκ̄ Sin e this is already a s alar, it is equal to κ = κκ̄ d(κ)√ = d(κ). For S2, ||S2||2 = d(κ)√ α κκ̄ = d(κ). CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 69 The above lemma shows that every element S ∈ (σ, σ2) satisfying ||S|| = d(σ)1/4 is uniquely expressed as S = aS1 + bS2, where a and b are omplex numbers satisfying |a|2 + |b|2 = 1. Therefore (8.7) and (8.8) are equivalent to the following equations respe tively: aS1R + bS2R = aσ(S1)R + bσ(S2)R, (R− σ(R)) = a2(σ(S1)S1 − S21) + ab(σ(S1)S2 − S1S2 + σ(S2)S1 − S2S1) + b2(σ(S2)S2 − S22). The following lemma will be frequently used in what follows. Lemma 8.4. Let the notation be as above. Then Proof. The �rst equality is easy. For the se ond, we have Thanks to Equation (8.6), we have (d(κ)2 d(κ)1/2v∗1ρ(v 1)wv1 = d2 − 1 d1/4T0, whi h shows the statement. � Lemma 8.5. Let the notation be as above. Then we have S1R = σ(S1)R, S2R = σ(S2)R. In parti ular, (8.7) always holds. Proof. We introdu e a linear isomorphism F : (idP , σ 3) → (κκ̄, ακκ̄ακκ̄α) by F (x) = α κ κ̄ κ κ̄α α Sin e every intertwiner above belongs to the spa e (idP , σ 3), it su� es to show the equalities after applying F to the intertwiners. 70 PINHAS GROSSMAN AND MASAKI IZUMI For F (S1R), we have F (S1R) = α α κ α α κ Using Lemma 8.4 with onsideration of [α2(idM ⊕ ρ)α] = [idM ]⊕ [αρ], we get F (S1R) = α α κ In a similar way, we have F (σ(S1)R) = Sin e we have = dα2(S∗0)S0 = d, = dα(S∗0)S0 = d, we get F (S1R) = F (σ(S1)R). CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 71 For F (S2R), we have F (S2R) = α α κ κ̄ α α κ Using Lemma 8.4 with onsideration of [ρ idMα] = [αρ] and [ρρα] = [α]⊕ [ρ]⊕ [αρ]⊕ [α2ρ], we get F (S2R) = α α κ In a similar way, F (σ(S2)R) = Sin e we have = d3/4ρ(T ∗0 )T2T0 = d 3/4A(2, 1)T1 = d 3/4A(1, 2)T1, = d3/4α(T ∗0 )αρ(T1)T0 = d 3/4T ∗2 ρ(T2)T0 = d 3/4A(1, 2)T1, we get F (S2R) = F (σ(S2)R). � Lemma 8.6. Let S = aS1 + bS2 with a, b ∈ C. Then Equation (8.8) is equivalent to (8.9) = a2 + d− 1ab, 72 PINHAS GROSSMAN AND MASAKI IZUMI (8.10) = ab− b (8.11) 0 = a2 − ab√ − A(1, 2)b2, (8.12) 0 = ab+ 1 + (d− 1)A(1, 2) (d− 1)3/2 b Proof. Let G : (σ, σ3) → (κκ̄ακκ̄, ακκ̄ακκ̄α) be the linear isomorphism de�ned in a similar way as in the proof of the previous lemma. We �rst ompute G(R), G(σ(R)), G(S21) et . For G(R), we have G(R) = α κ κ̄ α κ α κ κ̄ α κ α κ κ̄ α κ Applying Lemma 8.4 to the se ond term, we get G(R) = α κ κ̄ α κ α κ κ̄ α κ α κ κ̄ α κ The intertwiner G(σ(R)) is the mirror image of G(R). Sin e α α ρ αρ dS0 = α α ρ αρ CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 73 we get G(R− σ(R)) α κ κ̄ α κ α κ κ̄ ακ α κ κ̄ α κ α κ κ̄ α κ For G(S21), Lemma 8.4 implies G(S21) = α α κ κ̄ α κ κ̄κ κ̄ α α κ κ̄ α κ κ̄κ κ̄ α α κ κ̄ α α κ κ̄κ κ̄ α α κ κ̄ α α κ κ̄κ κ̄ α α κ κ̄ α α κ κ̄κ κ̄ α α κ κ̄ α α κ κ̄κ κ̄ 74 PINHAS GROSSMAN AND MASAKI IZUMI where we use α(S0) = S0. The intertwiner G(σ(S1)S1) is the mirror image of G(S A similar argument shows G(S2S1) = α α κ κ̄ α α κ κ̄κ κ̄ α α κ κ̄ α α κ κ̄κ κ̄ and G(σ(S2)S1) is its mirror image. For G(S1S2), we have G(S1S2) = α α κ κακ κ̄ α α κ κακ κ̄ α α κ κακ κ̄ α α κ κακ κ̄ α α κ κακ κ̄ α α κ κακ κ̄ CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 75 Sin e (α2α, αρ) = (α2α, ραρ) = 0, we get G(S1S2) = α α κ κακ κ̄ α α κ κακ κ̄ The intertwiner G(σ(S1)S2) is the mirror image of G(S1S2). A similar argument shows G(S22) = α α κ κακ κ̄ α α κ κακ κ̄ α α κ κακ κ̄ and G(σ(S2)S2) is the mirror image of G(S Now (8.8) is equivalent to α αα ρ α αα ρ α αα ρ αα α ρ αα α ρ 76 PINHAS GROSSMAN AND MASAKI IZUMI α α α α α α α α α ρ α ρ ρ α ρ ρ α ρ ρα α α ρ α αα ρ α ρ ρα α α ρ α ρ − d− 1 ρ α αα ρ α ρ It is easy to show that the �rst and the se ond equations are equivalent to (8.9), the third and fourth are equivalent to (8.10), and the �fth and sixth are equivalent CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 77 to (8.11). For the last equation, we have ρα α α ρ α αα ρ α ρ 2 = dT2ρα 2(S∗0)T0 − dα(T1)S∗0ρα(T0) = dT2ρ(S 0)T0 − dT0S0ρ(T2) = d(T2T 0 − T0T ∗1 ), ρα α α ρ α ρ ρ α αα ρ α ρ = dρ(T ∗0 )T2ρ(T 2 )T0 − dα(T ∗0 )αρ(T1)T ∗2 ρα(T0) = dρ(T ∗0 )T2ρ(T 2 )T0 − dT ∗2 ρ(T2)T ∗2 ρ(T2) l∈Z/3Z A(2, l)T2+lT )∗( ∑ k∈Z/3Z A(2, 2 + k)T2+kT l∈Z/3Z A(1, l + 2)T1+lT k∈Z/3Z A(1, k + 2)T1+kT l∈Z/3Z A(2, l)A(2, l + 1)TlT l+1 − d k∈Z/3Z A(1, k)A(1, k + 2)Tk−1T (d− 1)2 + A(1, 2) 0 − T0T ∗1 ). Therefore the last equation is equivalent to (8.12). � Proof of Theorem 8.2. Equation (8.11) shows b 6= 0 and so (8.12) implies (8.13) a = − B + 1 (d − 1)3/2 b. 78 PINHAS GROSSMAN AND MASAKI IZUMI where B = (d− 1)A(1, 2). Using the fa t that B satis�es B2−B+ d = 0, we an see that this is ompatible with (8.11). Iterating (8.13) into (8.9), we get (B + 1)2 (d− 1)3 b 2 − B + 1 d− 1 b (d− 1)3 B2 + 2B + 1− (d− 1)2(B + 1) (d− 1)3 3B − d+ 1− (d+ 2)(B + 1) (d− 1)3 (1− d)B + d− d2 = − 1 (d− 1)2 B + d)b2, whi h shows (8.14) b2 = − (d− 1) (B + d) We get the same equation from (8.10). Now all we have to show is that (8.13) and (8.14) are ompatible with |a|2+|b|2 = 1. The equation (8.13) and |a|2 + |b|2 = 1 imply |b|2 = 1 + (d− 1)3 |B + 1| 2 = 1 + (d− 1)3 = 1 + d− 1 = d− 1 . Sin e |B + d|2 = d2 + 2d = d(d− 1)2, this is ompatible with (8.14). � Referen es 1. Asaeda, M. and Haagerup, U., Exoti subfa tors of �nite depth with Jones indi es (5+ 13)/2 and (5 + 17)/2. Comm. Math. Phys. 202 (1999), 1�63. 2. Bö kenhauer, J., Evans, D. E., and Kawahigashi, Y. Chiral stru ture of modular invariants for subfa tors. Comm. Math. Phys. 210 (2000), 733�784. 3. Bö kenhauer, J., Evans, D. E., and Kawahigashi, Y. On α-indu tion, hiral generators and modular invariants for subfa tors. Comm. Math. Phys. 208 (1999), no. 429�487. 4. Bis h, D. A note on intermediate subfa tors. Pa i� Journal of Mathemati s, 163 (1994), 201�216. 5. Bis h, D., Bimodules, higher relative ommutants and the fusion algebra asso iated to a sub- fa tor. In Operator algebras and their appli ations. Fields Institute Communi ations, Vol. 13, Ameri an Math. So . (1997), 13�63. 6. Bis h, D. and Haagerup, U., Composition of subfa tors: new examples of in�nite depth sub- fa tors. Ann. S i. É ole Norm. Sup. (4) 29 (1996), 329�383. 7. Bis h, D. and Jones, V. F. R., Algebras asso iated to intermediate subfa tors. Inventiones Mathemati ae, 128 (1997), 89�157. 8. Choda, M. Index for fa tors generated by Jones' two sided sequen e of proje tions. Pa i� J. Math. 139 (1989), 1�16. 9. Evans, D. E. and Kawahigashi, Y., Quantum symmetries on operator algebras. Oxford Uni- versity Press (1998). 10. Evans, D. E. and Kawahigashi, Y., Orbifold subfa tors from He ke algebras. II. Quantum doubles and braiding. Comm. Math. Phys. 196 (1998), 331�361. 11. Goodman, F., de la Harpe, P. and Jones, V. F. R., Coxeter graphs and towers of algebras. MSRI Publi ations (Springer) (1989), 14. CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 79 12. Grossman, P. (in press), Forked Temperley-Lieb algebras and intermediate subfa tors. J. Fun t. Anal. (2007), doi:10.1016/j.jfa.2007.03.014 13. Grossman, P. and Jones, V. F. R., Intermediate subfa tors with no extra stru ture. J. Amer. Math. So . 20 (2007), 219�265. 14. Hayashi, T. and Yamagami, S., Amenable tensor ategories and their realizations as AFD bimodules. J. Fun t. Anal. 172 (2000), 19�75. 15. Hiai, F., Minimizing indi es of onditional expe tations onto a subfa tor. Publ. Res. Inst. Math. S i. 24 (1988), 673�678. 16. Hong, J. H., Subfa tors with prin ipal graph E 6 . A ta Appl. Math. 40 (1995), 255�264. 17. Izumi, M., Appli ation of fusion rules to lassi� ation of subfa tors. Publ. Res. Inst. Math. S i., 27 (1991), 953�994. 18. Izumi, M., Goldman's type theorem for index 3. Publ. Res. Inst. Math. S i. 28 (1992), 833�843. 19. Izumi, M., On type II and type III prin ipal graphs of subfa tors. Math. S and. 73 (1993), 307�319. 20. Izumi, M., Subalgebras of in�nite C -algebras with �nite Watatani indi es II. Cuntz-Krieger algebras. Duke J. Math. 91 (1998), 409-461. 21. Izumi, M., Goldman's type theorems in index theory. Operator algebras and quantum �eld theory (Rome, 1996), 249�269, Int. Press, Cambridge, MA, 1997. 22. Izumi, M., The stru ture of se tors asso iated with Longo-Rehren in lusions. I. General theory. Comm. Math. Phys. 213 (2000), 127�179. 23. Izumi, M., The stru ture of se tors asso iated with Longo-Rehren in lusions. II. Examples. Rev. Math. Phys. 13 (2001), 603�674. 24. Izumi, M., Chara terization of isomorphi group-subgroup subfa tors. Int. Math. Res. Not. 2002, no. 34, 1791�1803. 25. Izumi, M. and Kawahigashi, Y., Classi� ation of subfa tors with the prin ipal graph D n . J. Fun t. Anal. 112 (1993), 257�286. 26. Izumi, M. and Kosaki, H., Ka algebras arising from omposition of subfa tors: general theory and lassi� ation. Mem. Amer. Math. So . 158 (2002), no. 750. 27. Izumi, M. and Kosaki, H., On a subfa tor analogue of the se ond ohomology. Reviews in Mathemati al Physi s, 14 (2002), 733�757. 28. Izumi, M., Longo, R., and Popa, S. A Galois orresponden e for ompa t groups of automor- phisms of von Neumann algebras with a generalization to Ka algebras. J. Fun t. Anal. 155 (1998), 25�63. 29. Jones, V. F. R. (1980). A tions of �nite groups on the hyper�nite type II1 fa tor. Memoirs of the Ameri an Mathemati al So iety, 237. 30. Jones, V. F. R., Index for subfa tors. Inventiones Mathemati ae, 72 (1983), 1�25. 31. Jones, V. F. R. (in press). Planar algebras I. New Zealand Journal of Mathemati s. QA/9909027 32. Jones, V. F. R., Quadrati tangles in planar algebras. In preparation, (2003), http://math.berkeley.edu/∼vfr 33. Jones, V. F. R. and Xu, F., Interse tions of �nite families of �nite index subfa tors. Interna- tional Journal of Mathemati s, 15 (2004), 717�733. 34. Kawahigashi, Y., On �atness of O neanu's onne tions on the Dynkin diagrams and lassi�- ation of subfa tors. J. Fun t. Anal. 127 (1995), 63�107. 35. Kawahigashi, Y. and Longo, R., Classi� ation of two-dimensional lo al onformal nets with c < 1 and 2- ohomology vanishing for tensor ategories. Comm. Math. Phys. 244 (2004), 63�97. 36. Kawahigashi, Y. and Longo, R., Pennig, U., and Rehren, K.-H., The lassi� ation of non-lo al hiral CFT with c < 1. math.OA/0505130, to appear in Comm. Math. Phys. 37. Kirillov, A. N. and Reshetikhin, N. Yu., Representations of the algebra Uq(sl(2)), q-orthogonal polynomials and invariants of links. In�nite-dimensional Lie algebras and groups (Luminy- Marseille, 1988), 285�339, Adv. Ser. Math. Phys., 7, World S i. Publ., Teane k, NJ, 1989. http://math.berkeley.edu/~vfr http://arxiv.org/abs/math/0505130 80 PINHAS GROSSMAN AND MASAKI IZUMI 38. Kosaki, H. Extension of Jones' theory on index to arbitrary fa tors. J. Fun t. Anal. 66 (1986), 123�140. 39. Kosaki, H. and Yamagami, S., Irredu ible bimodules asso iated with rossed produ t algebras. Internat. J. Math. 3 (1992), 661�676. 40. Landau, Z., Fuss-Catalan algebras and hains of intermediate subfa tors. Pa i� J. Math., 197 (2001), 325�367. 41. Landau, Z., Ex hange relation planar algebras. J. of Fun t. Anal., 195 (2002), 71�88. 42. Longo, R., Index of subfa tors and statisti s of quantum �elds. II. Corresponden es, braid group statisti s and Jones polynomial. Comm. Math. Phys. 130 (1990), 285�309. 43. Longo, R., A duality for Hopf algebras and for subfa tors. I. Comm. Math. Phys. 159 (1994), 133�150. 44. Longo, R. and Rehren, K.-H. (1995). Nets of subfa tors. Reviews in Mathemati al Physi s, 7, 567�597. 45. Longo, R. and Roberts, J. E. (1997). A theory of dimension. K-theory, 11, 103�159. 46. Masuda, T., An analogue of Longo's anoni al endomorphism for bimodule theory and its appli ation to asymptoti in lusions. Internat. J. Math. 8 (1997), 249�265. 47. Masuda, T., An analogue of Connes-Haagerup approa h for lassi� ation of subfa tors of type III1. J. Math. So . Japan, 57 (2005), 959�1001. 48. Nakamura, M. and Takeda, Z., A Galois theory for �nite fa tors. Pro . Japan A ad. 36 (1960), 258�260. 49. O neanu, A. Quantum symmetry, di�erential geometry of �nite graphs and lassi� ation of subfa tors, University of Tokyo Seminary Notes 45, (1991), (Notes re orded by Kawahigashi, 50. Okamoto, S., Invariants for subfa tors arising from Coxeter graphs. Current Topi s in Operator Algebras, World S ienti� Publishing, (1991), 84�103. 51. Pimsner, M. and Popa, S., Entropy and index for subfa tors. Annales S ienti�ques de l'É ole Normale Superieur, 19 (1986), 57�106. 52. Popa, S., Classi� ation of amenable subfa tors of type II. A ta Math. 172 (1994), 163�255. 53. Popa, S., Classi� ation of subfa tors and their endomorphisms. CBMS Regional Conferen e Series in Mathemati s, 86. Published for the Conferen e Board of the Mathemati al S ien es, Washington, DC; by the Ameri an Mathemati al So iety, Providen e, RI, 1995. 54. Sano, T. and Watatani, Y., Angles between two subfa tors. Journal of Operator Theory, 32 (1994), 209�241. 55. Wassermann, A., Operator algebras and onformal �eld theory. III. Fusion of positive energy representations of LSU(N) using bounded operators. Invent. Math. 133 (1998), 467�538. 56. Watatani, Y., Latti es of intermediate subfa tors. J. Fun t. Anal., 140 (1996), 312�334. 57. Wenzl, H., C∗ tensor ategories from quantum groups. J. Amer. Math. So . 11 (1998), 261�282. 58. Xu, Feng, New braided endomorphisms from onformal in lusions. Comm. Math. Phys. 192 (1998), 349�403. Pinhas Grossman, Department of Mathemati s, 1326 Stevenson Center, Vanderbilt University, Nashville, TN, 37203 e-mail: pinhas.grossman�vanderbilt.edu Masaki Izumi, Department of Mathemati s, Graduate S hool of S ien e, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan e-mail: izumi�math.kyoto-u.a .jp 1. Introduction 2. Preliminaries 2.1. Quadrilaterals 2.2. Sectors 3. Q-systems and angles 4. (2,2)-supertransitive quadrilaterals 5. Classification I 5.1. Class I 5.2. Class II 5.3. Class III 5.4. Class IV 6. Classification II 7. -induction and GHJ pairs 7.1. -induction and angles 7.2. GHJ pairs 7.3. Asymptotic inclusions and angles 8. Appendix References

0704.1122

Investigation of Energy Spectrum of EGRET Gamma-ray Sources by an Extensive Air Shower Experiment

Investigation of Energy Spectrum of EGRET Gamma-ray Sources by an Extensive Air Shower Experiment M. Khakian Ghomi2, M. Bahmanabadi1,2, F. Sheidaei1, J. Samimi1,2, A. Anvari1,2 1Department of physics, Sharif University of Technology, P.O. box 11365 - 9161, Tehran, Iran. 2ALBORZ Observatory(http://sina.sharif.edu/∼observatory) E-mail: [email protected] abstract Ultra-High-Energy (UHE) (E > 100 TeV) Extensive Air Showers (EASs) have been monitored for a period of five years (1997 − 2003), using a small array of scintillation detectors in Tehran, Iran. The data have been analyzed to take in to account of the dependence of source counts on zenith angle. Because of varying thickness of the overlaying atmosphere, the shower count rate is extremely dependent on zenith angle. During a calendar year different sources come in the field of view of the array at varying zenith angles and have different effective observation time equivalent to zenith in a day. High energy gamma-ray sources from the EGRET third catalogue where observed and the data were analyzed using an excess method. Upper limits were obtained for 10 EGRET sources [1]. Then we investigated the EAS event rates for these 10 sources and obtained a flux for each of them using parameters of our experiment results and simulations. Finally we investigated the gamma-ray spectrum in the UHE range using these fluxes with reported fluxes of the EGRET sources. keywords:The EGRET sources, Extensive Air Showers (EASs), Gamma-ray sources, Gamma-ray spectrum http://arxiv.org/abs/0704.1122v1 http://sina.sharif.edu/~observatory Investigation of Energy Spectrum of EGRET Gamma-ray... 2 1. Introduction The EGRET instrument on-board Compton gamma-ray Observatory (CGRO) has detected about 271 high energy (> 100 MeV) gamma-ray sources [2]. Effective sensitivity of EGRET is in the energy range from 100 MeV to 30 GeV. The EGRET gamma-ray sources in many aspects like characteristics, different energy ranges and etc. have been investigated [3, 4, 5]. Whether the EGRET sources emit gamma-ray at still higher energies or not, is an interesting question. Gamma-rays with energies of about 100 TeV and more, entering the earth atmosphere, produce Extensive Air Shower (EAS) events which could be observed by the detection of the secondary particles of the EAS events on the ground level [6]. This gamma-ray induced EAS events are investigated for diffuse Galactic [7] and extragalactic sources, also Galactic [8], [9] and extragalactic [10] gamma-ray point sources have been investigated too. In this work we have investigated 10 of these point sources. Our small particle detector array is located at the Sharif University of Technology in Tehran, Iran at about 1200 m above sea level (≡ 890 g cm−2). This small array is a prototype for a large EAS array to be built at an altitude of 2600 m (≡ 756 g cm−2) at ALBORZ Observatory (AstrophysicaL oBservatory for cOsmic Radiation on alborZ) (http://sina.sharif.edu/ ∼observatory/) near Tehran. The results of the experiments with this prototype observatory with about 1.7×105 recorded EAS events was reported earlier [1] and has been shown that some of the EGRET GeV point sources are gamma-ray emitters at energies about 100TeV. Here we report the results of our recent investigation using the data of our earlier experiments to extend the energy spectrum of the observed EGRET sources up to 100TeV and more. In this investigation we present the observational results for 10 EGRET third catalogue sources. Then we investigate the effective area and effective time of observation of each source. Finally we compare the obtained fluxes and spectral indices with the presented fluxes and spectral indices of the EGRET third catalogue sources at the 3rd EGRET catalogue. 2. Experimental arrangement Our array is constructed of four slab of plastic scintillation detectors (100×100×2 cm3). They are housed in white painted pyramidal boxes [11] which arranged in a square; at 51◦ 20 E and 35◦ 43 N, elevation 1200 m (≡ 890 g cm−2). Two different experimental configurations were used in the experimental set up. The first (E1) and the second (E2) experimental configurations are identical except the size of the square array. In E1 the size is 8.75 m × 8.75 m and in E2 the size is 11.30 m × 11.30 m. More details of the experimental setup is given in the reference [1]. http://sina.sharif.edu/ Investigation of Energy Spectrum of EGRET Gamma-ray... 3 3. Data Analysis The logged time lags between the scintillation detectors and Greenwich Mean Time (GMT) of each EAS event were recorded as raw data. We synchronized our computer to GMT (http://www.timeanddate.com). Our electronic system has a recording capability of 18.2 times per second. If an EAS event occurs, its three time lags will be recorded and if it does not occur ’zero’ will be recorded. Therefore the starting time of each experiment and the count of records gives us the GMT of each EAS event. We estimated the energy threshold and the mean energy of our experiment and also we calculated the statistical significance of 98 of the 3rd EGRET sources which were in the Field Of View (FOV) of our array. The complete analysis procedure [1] is itemized as follows: • The local coordinates: zenith and azimuth angles of each EAS event (z, ϕ) were calculated using a least-square method based on the logged time lags and coordinates of the scintillators. A zenith angle cut off of 60◦ is implemented to increase the significance [12]. • The distributions of the local angles of the EAS events were investigated to understand the general behavior of these events. We fitted these distributions with the two functions as follows [13]: dN = Az sin z cos n zdz (1) where Az = 95358 and n = 5.00, and also f(ϕ) = Aϕ +Bϕ cos(ϕ− ϕ1) + Cϕ cos(2ϕ− ϕ2) (2) where Aϕ = 1, Bϕ = 0.081 , Cϕ = 0.069, ϕ1 = 95 ◦ and ϕ2 = −193 • Celestial coordinates (RA,Dec) of each EAS event were calculated using its local coordinates, the GMT of the event and geographical latitude of our array (http://tycho.usno.navy.mil/sidereal.html). Then we calculated galactic coordinates (l,b) of each EAS event from its equatorial coordinates for epoch J2000 [14]. • We estimated the errors in (l,b) of the investigated EGRET sources from the error factors in the array. In this stage we obtained r̄e = 4.35 ◦ ± 0.82◦ as the mean angular error of our experiment. • We investigated cosmic-ray initiated EAS events by simulations based on a homogeneous distribution of primary charged particles. These simulations incorporated all known parameters of the experiment. [1] • We investigated the statistical significance of 98000 random sources and also 98 sources of the 3rd EGRET catalogue, using the method of Li & Ma (Li & Ma 1983) we derived the best-known locations for the EGRET sources in the TeV range [1]. http://www.timeanddate.com http://tycho.usno.navy.mil/sidereal.html Investigation of Energy Spectrum of EGRET Gamma-ray... 4 3.1. Shadow of the moon Observing the shadow of the moon in EAS experiments which usually might have a much larger error circle than the disk of the moon is a very difficult task and requires a careful scrutinization of the data. The difficulty is compounded since a realistic radius for the error circle of the experiment is only obtained by the observation of the shadow which could be indicated as a deficit of shower counts falling in the error circle centered about the moving location of the moon as compared to the average shower counts falling in error circles centered at other positions in sky during the observation time. To carry out the scrutinization of our data, we have proceeded as follows.We have divide our data into sequential time segments and for each time segment we have used the mean values of the local coordinates of moon (θm, ϕm) moving over our observatory site. These coordinates were obtained using the information provided by the http://aa.usno.navy.mil. Now for an assumed radius of circle of error, angular radius ρerr ranging from 0.5 ◦ to 15◦ incremented by 0.5◦, we have calculated the number of showers falling in each circle. This has been done by calculating the angular separations Θs between the arrival direction of each shower event (θs, ϕs) and the direction of the moon at the time of recording of that event, using the following equation from spherical geometry: cosΘs = cos θm cos θs + sin θm sin θs cos(ϕm − ϕs) (3) Obviously if Θs < ρerr that shower is counted as falling in the moon’s error circle. In order to compare the obtained result with random sampling and scrutinize the difference for each value of ρerr we have chosen 1000 random locations in the sky denoted by local coordinates (θr, ϕr) and have calculated the number of showers falling in the error circles countered about each of the 1000 random locations. This was similarly done by calculating the angular separation of each shower arrival direction (θs, φs) with the direction of the center of the randomly chosen error circle (Θr) from above equation with (θm, ϕm) replaced by (θr, ϕr). If for any shower event Θr < ρerr that shower is counted as falling in the error circle of that random position. In these computations (both for the moon as well as for the random locations) a weight factor was used for each of the shower events in order to account for the site-specific effects in our data which depend on the arrival directions of shower events. These effects are : (1) The different thickness and density of the overlying atmosphere which are effective in shower development and hence its detectability at the height of the observatory. (2) The geomagnetic effect on the azimuthal arrival direction of the showers. These effects which are specific for each observation site, were separately and independently determined for our site and are reflected in the dependence of the number of shower events on zenithal and azimuthal angles which are given by equations 1 and 2, respectively [13]. To take account of these effects in our observed data, we have assigned a weight factor to each shower arrival direction which is the product of these two independent factors. The weight factor is thus: W (θs, ϕs) = cos n θs(Aϕ +Bϕ cos(ϕ− ϕ1) + Cϕ cos(2ϕ− ϕ2)) (4) http://aa.usno.navy.mil Investigation of Energy Spectrum of EGRET Gamma-ray... 5 where the constants n, Aϕ,Bϕ and Cϕ are given below Eqs.1 and 2. It should be remembered that in our earlier work, [1] reporting the observation of EGRET gamma-ray point sources in TeV data by excess method, both of these effects were carefully taken into account by determining the exposure map of our observations by simulations incorporating these effects along with other particularities of our observations and by correcting of our observed data by dividing it by the exposure map in galactic coordinates. 3.2. Distributions of number of EAS events in error circles Here we discuss the distribution of number of EAS events falling in the error circles of different radii as determined according to the aforementioned procedure. We first present the results for each of the sets of 1000 randomly chosen error circles. Fig.2 shows the histogram of frequency of occurrence of number of circles with respect to the number of EAS falling in the error circle. The histograms distributions corresponding to different radii of error circles are shown here as illustration. These for other radii show similar distributions and all these distributions nearly fit gaussian distributions with a mean and a variance which increase with increasing radius. This result is very assuring and shows the correctness of our sophisticated numerical procedure and the validity of using the mean number of events, in these randomly chosen locations of error circles in the sky to compare with the number of events falling in the error circle countered about the moving moon. Fig.3 shows the variation of the mean number of events of these distribution as a function of the chosen radius of the error circles. It is seen that these calculated means show a nearly exact dependence on the square of the radius. This result is not surprising and is exactly what one would expect to get from a correct random sampling of statistical data. However, in view of the complexity and sophistication involved in our entire procedure (with inclusion of the weight factors), this result is again very assuring and shows that we can use these mean number of events to compare with that falling in the error circles centered about the moving moon and rule out the possibility of the deficit in the number of events falling in the moon centered circle as due to statistical fluctuation. In Fig.3 we have also shown the number of events falling in the moon-countered circles for comparison with the means of the random samplings. It is seen that in every case( for every chosen error circle radius) the deficit of number of EAS from the direction of the moving moon is quite significant as compared to the error bars of the mean of random sampling( Fig.3) which is taken as the variance of nearly gaussian distributions of Fig.2. As seen here the deficit which is from 1.6 to about 7.1 times the standard deviation of the mean of random distributions is quite significant and since it is definitely associated with the moving moon it must be associated with some moon- related phenomenon. We will not discuss this phenomenon here and rather simply call it the shadow of the moon in our EAS observed data. Investigation of Energy Spectrum of EGRET Gamma-ray... 6 3.3. Estimation of energy thresholds Our detected EAS events are a mixture of cosmic-ray and gamma-ray events. In E1 the total number of EAS events was 53,907 and the duration of the experiment was 501,460 seconds. So the mean event rate of the first experiment was 0.1075 events per second. The distribution of the time between successive events was investigated and found to be in good agreement with an exponential function, indicating that our event sampling is completely random [16]. In E2 the total number of events was 173,765 and the duration of the second experiment was 2,902,857 seconds, so its mean event rate was 0.05986 events per second. We refined the data to separate out the acceptable events. Events are acceptable if there is a good coincidence between the four scintillator pulses, also we omitted the events with zenith angles more than 60◦ because of their less accuracy. Therefore after these separations we obtained smaller data sets of 46,334 and 120,331 events for E1 and E2 respectively. Since we cannot determine the energy of the showers on an event by event basis, we estimate our lower energy threshold by comparing our event rate to the following cosmic-ray integral spectrum [17], J(E) = 2.78× 10−5E−2.22 + 9.66× 10−6E−1.62 − 1.94× 10−12 40 ≤ E ≤ 5000 TeV (5) The obtained lower energy limits are 39 TeV in E1 and 54 TeV in E2. The calculated mean energies with above energy spectrum are 94 and 132 TeV in E1 and E2, respectively. Since the distribution of cosmic-ray events within the array around these energy ranges is homogeneous and isotropic, we used an excess method [18] to find signatures of the EGRET 3rd catalogue gamma-ray sources. This method was used for both E1 and E2. 4. Calculation of effective area and time Number of secondary particles in the growth profile of EAS events increases in atmosphere until the shower maximum and then decreases after it. In energy of about 100 TeV the shower maximum height is at about 500g cm−2 and a fraction of these secondary particles arrive to the ground level particle detectors of our array at Tehran level (1200 m≡ 890 g cm−2). For calculating the effective surface of each experiment (E1 and E2) we used Greisen lateral distribution of electrons which is known the NKG formula [19] and CORSIKA simulation code [20] for the simulation of the two sets of proton showers with energy thresholds of 39 TeV and 54 TeV respectively for E1 and E2 at our array level. From our logged EAS events we obtained a zenith distribution function for E1 and E2, which the mean zenith angles are 26◦ for both of them. Also we calculated the z̄ which was obtained from the weight curve of the zenith distribution dN/dz ∝ cosn z sin z and we obtained the 26◦ too. This weight curve was obtained by fitting the function dN/dz to Investigation of Energy Spectrum of EGRET Gamma-ray... 7 our data in the E1 and E2 with n = 5.00. So in the first approximation we used the effective thickness of the passed atmosphere as 890g cm−2/ cos(26◦) = 980g cm−2. In the thickness, the average number of the secondary particles for the two experiments are NE1 = 5265 and NE2 = 8571, these two numbers obtained from 1000 simulated proton showers for each energy threshold. So based on the NKG formula the mean effective surfaces of EAS events at Tehran level are 965 m2 and 2173 m2 for E1 and E2 respectively. With these results we could obtain the mean effective surface of our array in the upper level of the atmosphere (The surface that if a primary particle passes through it, the array could detect its EAS events) 718m2 and 1751m2 for E1 and E2 respectively. For calculation of the effective time of observation of each source in every 24 hours, we used the spherical geometry and the track of each source in the local coordinates. Each source with its celestial coordinates right Ascention, Declination (RA,Dec) is introduced in the 3rd EGRET catalogue. Time duration of each source is calculated by reaching the source to the zenith angle of 60◦ from the direction of east to the same zenith angle from the west, and the distribution function dN/dz =∝ cosn z sin z which is related to the zenith distribution effect [1]. Finally we obtained the mean effective time of observation of our array for all 10 sources equivalent to 4h,28 (equivalent to existence of 4h,28 the source at zenith) for every 24 hours. The FOV of our array with the 60◦ zenith angle cutoff is π steradian. With these calculated factors we obtained fluxes(events cm−2s−1sr−1) for each of the 10 sources in E1 and E2 which are shown in Table 1. 5. Results Our results have been compared with the EGRET results. For each source we have fluxes and energies from EGRET, E1 and E2, so we extracted a spectral index for each source and compared it with the reported spectral index of EGRET. Some information about the 10 EGRET sources like Name, RA, Dec, and spectral index and its error, (γ ± ∆γ)EGT , are from the 3rd EGRET catalogue [2]. Other information like mean energy for E1 and E2 which are 94TeV and 132TeV respectively (are not shown in the table), fluxes of the two experiments and spectral indexes and their errors, (γ±∆γ)OUR, have been calculated in this analysis. The last column shows the agreement of our spectral indices to its in the 3rd EGRET catalogue. 5.1. Result of Energy Analysis with Simulated Showers For further energy analysis of our measured data of EAS events observed at our site we have used the CORSIKA code[20] to simulate showers with the inclusion of geomagnetic field pertinent to the location of our site from data provided by http://www.ngdc.noaa.gov. We have simulated a total of 7350 showers entering the top of atmosphere at various zenith angles (0◦, 15◦, 30◦, 45◦, 60◦) and for each zenith angle at 12 various azimuth angles ranging from 0◦ to 360◦ every 30◦. For each angle http://www.ngdc.noaa.gov Investigation of Energy Spectrum of EGRET Gamma-ray... 8 the simulations were repeated 10 times for energies less than 50 TeV and 5 times for energies greater than 50 TeV with separately 10 TeV intervals from 10 TeV to 100 TeV. These simulations were carried out similarly for entering protons as well as gamma- rays. For each simulation the number of secondary charged particles of the simulated shower at the height of observation of our site was determined from CORSIKA code. In order to make an energy analysis comparing gamma-ray initiated showers with those initiated by protons in our observations we have calculated a mean number of secondary shower particles for each energy by averaging over all the angles using the site-specific weight factor discussed in sec 3.1(Eq. 4). The ratio of angle-averaged mean of number of secondary charged particles in gamma-ray initiated showers to that of the mean for proton-initiated showers is shown in Fig.4 as a function of energy in the energy range of our simulations. The ratio shown here incorporates almost all of related particularities of our experiments. Since in our observations we have used exactly identical experimental procedure, equipment, thresholds and set-ups for all of the recorded showers irrespective of the nature of the radiation which initiates the shower the relative detection efficiency of our array and observations could only depend on the ratio of these means depicted in Fig.4. Now we use the relative detection efficiency calculated from simulated data and shown in Fig.4 to estimate the relative number of gamma-ray initiated showers to that initiated by protons, using the known energy spectrum of proton showers of the form = Np10( 10TeV )−Γp. (6) Where dNp is the number of proton showers in the energy interval E to E + dE, Np10 is a constant and Γp is the spectral index of EAS producing protons. Now denoting our array’s relative detection efficiency by η(E), we can write for the expected differential number of gamma-ray initiated showers: = η(E)Np10( 10TeV )−Γγ (7) where Γγ is the spectral index of EAS producing gamma- rays. Dividing Eq.7 by Eq.6 and upon integration from a threshold energy to infinity, the ratio of observed gamma- ray EAS events to that initiated by protons is obtained. Thus we write: NγEAS NpEAS η(E)E−Γγ dE E−Γp dE where Et is assumed threshold energy. For the energy range covered in this analysis, we use Γp = 2.7 and for gamma-ray spectral index we use the mean value of the spectral indices that we have estimated above (Table.1) for the sources with the highest statistical significance (those observed at a statistical significance level of higher than 1.5σ). Thus using our estimations of table 1 we have Γγ = 2.3. In order to carry out the integrations in Eq.8 we are forced to impose a truncation since with our rather limited computer shower simulations we only have η(E) for 10TeV < E < 100TeV . For the sake of numerical consistency, we have imposed this truncation to E=100 TeV in Investigation of Energy Spectrum of EGRET Gamma-ray... 9 Name(3EG J) RA,Dec ID (γ ±∆γ)EGT logFE1 logFE2 (γ ±∆γ)OUR |γOUR − γEGT | 0237+1635 39.3,16.5 A 1.85±0.12 -11.25 -11.53 1.90±0.27 0.05 0407+1710 61.8,17.1 2.93±0.37 -10.77 -11.45 4.20±0.40 1.27 0426+1333 66.6,13.5 2.17±0.25 -11.11 -11.28 1.15±0.48 1.02 0808+5114 122.1,51.2 a 2.76±0.34 -11.01 -11.58 3.87±0.42 1.11 1104+3809 126.1,38.1 A 1.57±0.15 -11.10 -11.72 4.21±0.50 2.64 1308+8744 197.0,87.7 2.17±0.66 -10.96 -11.53 3.87±0.45 1.70 1608+1055 242.1,10.9 A 2.63±0.24 -10.92 -11.72 4.34±0.50 1.71 1824+3441 276.2,34.6 2.03±0.50 -10.65 -11.25 4.07±0.35 2.04 2036+1132 309.1,11.5 A 2.83±0.26 -11.08 -11.73 4.41±0.53 1.58 2209+2401 332.4,24.0 A 2.48±0.50 -10.91 -11.31 2.71±0.42 0.23 Table 1. Comparison of our spectral indices of the 10 sources with the spectral indices of them which is introduced by the 3rd EGRET catalogue. The last column shows agreement of our spectral indices to EGRET spectral indices. the denominator of Eq.8 as well as its numerator. The result of these calculations obviously depend on the value of an assumed threshold energy, Et, therefore the numerical calculations were repeated for values of Et from 10 TeV to 100 TeV with 10 TeV intervals. The result of these calculations, that is, the ratio of the number of gamma-ray EAS events to that of proton EAS events expected to be detected in our experiments at our site is shown as a function of the threshold energy in Fig.5. Here we see that this ratio shows a maximum at a value of Et = 40 TeV which very nearly corresponds to the lower value of the two threshold energies of our two experiments. 6. Concluding remarks We believe the main reasons for success in these observations and investigations on EGRET gamma-ray point sources despite the low statistics of ∼ 3 × 105 EAS events from our small array of ALBORZ prototype observatory, relies on the following two favorite points of strength: (1) we have intensively studied the location- dependent factors which influence shower development and shower count from various angular bins in the sky. These factors are the mass of the overlying atmosphere and anisotropy in azimuth angles which it is attributed to the effect of the geomagnetic field [21], [13], [22]. For our particular site we investigated these effects which are in form of Eqs. 1 and 2. We carefully refined our data from these effects finally the investigation of the EGRET gamma-ray point sources are based on the corrected data. (2) Our site and the duration of our observations in the data set have been such that 10 of the EGRET gamma-ray point sources have crossed our site at small enough zenith angles to make their observation possible at least with a statistical significance of 1.5σ with Li and Ma criterion in the excess method which we have used[1]. In order to Investigation of Energy Spectrum of EGRET Gamma-ray... 10 investigate this rather fortunate point for our site and the time of these observations, we have calculated the average zenith all angle of passage of each of 271 EGRET gamma- ray point sources during our entire observation period, for each of these sources. We have also calculated the statistical significance(number of sigma) for excess counts from these sources. The result is shown here in Fig.1 and it convincingly shows an inverse correlation between the statistical significance of excess shower counts from the EGRET point source location and the average zenith angle of the transit of the source over our site. The computed coefficient of this correlation(correlation between inverse statistical significance and z) is 0.77. Obviously the EGRET sources investigated here are those with highest statistical significance corresponding to the lowest zenith angles of the passage of the source over our site. Our results shows that our spectral indices within error limits are in agreement with EGRET spectral indices for most of the considered here. For the calculation of the effective area of the EAS events at first , we calculated the primary particle energy, then we extracted the number of the secondary particles from a set of CORSIKA simulations, and finally we used the Greizen lateral distribution for these particles. In the near future with a larger array and larger set of logged EAS events at ALBORZ observatory we hope to calculate these results more accurately. It is worth remarking that due to increased probability of pair- production interaction of VHE and UHE gamma-rays with various low energy universal photons, the absorption of these gamma-rays becomes significant for sources located at cosmological distances. This absorption has been studied extensively [23, 24, 25]. The lack of observation of VHE and UHE gamma-rays from extragalactic EGRET gamma-ray sources has been generally attributed to the absorption of the gamma-rays from these sources which are located at cosmological distances. Stecker and De Jager (1997) gave a parametric relation for optical depth (τj) as a function of the source gamma energy (ETeV ) and red shift (z) for the two models (j=1,2) they have studied. For the 39TeV energy threshold of our experiments, we use their parametric relation with an optical depth of unity to calculate the red shift of the sources we have observed in our EAS experiments and have investigated here(Table 1). The result is z=0.0032 for their model 1 and z=0.0024 for their model 2. Ong,[26] have listed the red shifts of two classes of EGRET extragalactic sources and the calculated red shifts for the sources investigated here is smaller than red shifts listed for these classes of objects. However, we should remark that the recent discovery of the unexpected by hard spectra of the Blazar sources observed in HESS data [27] bear as the possibility of observation of EGRET gamma-ray point sources at higher energies. It would be open for future ground-base observations. The energy analysis carried out here using simulated shows with CORSIKA has shown that the ratio of number of gamma-ray generated EAS events expected to be detected at our site relative to the number of proton- generated EAS events show a maximum at a value of 40 TeV for the threshold energy of observation of EAS events at our site. The Investigation of Energy Spectrum of EGRET Gamma-ray... 11 fortunate fact that this value corresponds to the lower value of the threshold energies of our two experiments, provides further support for the fact that the data collected in our experiments at our site has had this extra advantageous attribute for the observation of gamma-ray initiated EAS events in the 10 TeV to 100 TeV energy range and thus the extra advantage for observing gamma-ray point sources in this range. 7. acknowledgements This research was supported by a grant from the national research council of Iran for basic sciences. The many useful and conductive comments by anonymous referee is very much appreciated. Also many thank from Prof. James Matthews for his comments and attentions to our works at ICRC 2005, Pune, India. References [1] Khakian Ghomi, M., Samimi, J., Bahmanabadi, M., 2005, A&A, 434, 459 [2] Hartman, R.C., et al., 1999, ApJ, 123, 79 [3] Bhattacharia, D., Akyuse A., Miyagi T., Samimi J., 2003, A&A, 404, 163 [4] Cillis, A.N., Hartman R.C., 2005 ApJ, 621, 291C [5] Zhang, S., Collmar, W. , Hermsen, W., Schonfelder, V., 2004, A&A, 421, 983Z [6] Gaisser, T.K., Cosmic Rays and Particle Physics, Cambridge Univ Press, New York, (1990) [7] Brezinskii, V.S., Kudriavtsev, V.A., 1990, ApJ 349, 620B [8] Atkins, R., Benbow, W., Berlay, D. et al., 2005, PhRvL 95y, 1103A [9] McKay, T.A., Borione, A., Catanese, M. et al., 1993, ApJ 417, 742 [10] Fidelis, V.V., Neshpor, Yu.I., Eliseev, V.S. et al., 2005, A&A 24, 53F [11] Bahmanabadi, M., et al. 1998, Experimental Astronomy, 8, 211 [12] Mitsui, K., et al., 1990, Nucl. Inst. Meth., A290, 565 [13] Bahmanabadi, M., et al. 2002, Experimental Astronomy, 13, 39 [14] Roy, A.E., Clarke, D., Astronomy : Principle and Practice, (Adant Hilger, Glasgow, 1991) [15] Li, T., Ma, Y., 1983, ApJ, 272, 317 [16] Bahmanabadi, M., et al. 2003, Experimental Astronomy, 15, 13 [17] Borione, A., et al. 1997, ApJ, 481,313 [18] Amenomori, M., et al., 2002, ApJ, 580, 887 [19] Kamata, K., Nishimura, J. (1958) Prog. Theor. Phys.(kyoto) suppl. 6, 93 [20] Heck, D., et al., Report FZKA 6019(1998), Forschungszentrum Karlsruhe http://www-ik.fzk.de/corsika/physics-description/corsika-phys.html [21] Ivanov, A.A. et al., 1999, JETP Let. 69, 288-293 [22] He, H.H., Sun, B.G., Zhou, Y., 2005, ICRC, India, 6, 5-8 [23] Stecker, F.W, Cosmic Gamma ray, Moro Book Corp., Baltimore (1971) [24] Fazio, G.G., Stecker, F.W., 1970, Nature, 226, 135 [25] Stecker, F.W., De Jager, O.C, 1997, ApJ, 476, 716 [26] Ong, R.A., et al., 2001, ICRC, Germany, 2593. [27] Aharonian , F., et al., Nature 440(2006) 1018-1021 http://www-ik.fzk.de/corsika/physics-description/corsika-phys.html Investigation of Energy Spectrum of EGRET Gamma-ray... 12 Figure 1. Distribution of statistical significance versus average zenith angle of transit of calculated for 271 EGRET gamma-ray sources over our site(only sources with significance > 1.5σ are shown.). Investigation of Energy Spectrum of EGRET Gamma-ray... 13 Figure 2. Histograms show the frequency of circles of error with radius ρ and random chosen centers in the sky vs. the number of EAS events falling in each error circle. Investigation of Energy Spectrum of EGRET Gamma-ray... 14 Figure 3. Dependence of the mean number of EAS events falling in the random error circles as a function of the radius of the error circle, ρ. Smooth curve shows a ρ2 dependence. Error bars are the standard deviation of the distribution shown in Fig.2, and statistical errors for the moon. Open circles (◦) are random error circles. Filled circles(•) are error circles centered about the location of the moving moon. Investigation of Energy Spectrum of EGRET Gamma-ray... 15 Figure 4. The ratio of the mean number of secondary charged particles at the location of our site produced in gamma-ray generated simulated Extensive Air Shower to that produced by proton generated showers as a function of the energy of the primary radiation( Gamma-ray and proton) entering the top of the atmosphere. Investigation of Energy Spectrum of EGRET Gamma-ray... 16 Figure 5. The ratio of the mean number of gamma-ray generated showers to that of proton generated showers expected to be observed at our site as a function of the threshold energy of our experiments. Introduction Experimental arrangement Data Analysis Shadow of the moon Distributions of number of EAS events in error circles Estimation of energy thresholds Calculation of effective area and time Results Result of Energy Analysis with Simulated Showers Concluding remarks acknowledgements

0704.1123

Conjugate field and fluctuation-dissipation relation for the dynamic phase transition in the two-dimensional kinetic Ising model

Conjugate field and fluctuation-dissipation relation for the dynamic phase transition in the two-dimensional kinetic Ising model D.T. Robb,1, 2, ∗ P.A. Rikvold,1, 3, 4 A. Berger,5 and M.A. Novotny6 1School of Computational Science, Florida State University, Tallahassee, Florida 32306, USA 2Department of Physics, Clarkson University, Potsdam, New York 13699, USA 3Center for Materials Research and Technology and Department of Physics, Florida State University, Tallahassee, Florida 32306-4350, USA 4National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA 5San Jose Research Center, Hitachi Global Storage Technologies, San Jose, California 95120, USA 6Department of Physics and Astronomy and HPC2 Center for Computational Sciences, Mississippi State University, Mississippi State, Mississippi 39762, USA (Dated: October 24, 2018) http://arxiv.org/abs/0704.1123v2 Abstract The two-dimensional kinetic Ising model, when exposed to an oscillating applied magnetic field, has been shown to exhibit a nonequilibrium, second-order dynamic phase transition (DPT), whose order parameter Q is the period-averaged magnetization. It has been established that this DPT falls in the same universality class as the equilibrium phase transition in the two-dimensional Ising model in zero applied field. Here we study for the first time the scaling of the dynamic order parameter with respect to a nonzero, period-averaged, magnetic ‘bias’ field, Hb, for a DPT produced by a square-wave applied field. We find evidence that the scaling exponent, δd, of Hb at the critical period of the DPT is equal to the exponent for the critical isotherm, δe, in the equilibrium Ising model. This implies that Hb is a significant component of the field conjugate to Q. A finite- size scaling analysis of the dynamic order parameter above the critical period provides further support for this result. We also demonstrate numerically that, for a range of periods and values of Hb in the critical region, a fluctuation-dissipation relation (FDR), with an effective temperature Teff (T, P,H0) depending on the period, and possibly the temperature and field amplitude, holds for the variables Q and Hb. This FDR justifies the use of the scaled variance of Q as a proxy for the nonequilibrium susceptibility, ∂〈Q〉/∂Hb, in the critical region. PACS numbers: 05.70.Ln, 64.60.Ht, 89.75.Da, 75.70.Cn ∗Corresponding author: [email protected] mailto:[email protected] I. INTRODUCTION The dynamic phase transition (DPT) in a ferromagnetic system below its critical temper- ature was first observed in numerical solutions of a mean-field model exposed to an oscillating magnetic field [1, 2]. It was then studied further, both in mean-field models [3, 4, 5, 6] and in kinetic Monte Carlo (KMC) simulations [5, 6, 7, 8, 9, 10]. A review of this early work can be found in Ref. [11]. More recently, the study of the DPT has expanded to include varying (and often more physical) model geometries. These include mean-field studies of domain- wall motion in an anisotropic XY model in one dimension [12, 13, 14], KMC simulations of a three-dimensional Ising system [15], and KMC simulations of a uniaxially anisotropic Heisenberg system in an off-axial field [16], an elliptically polarized applied field [17], and with the effect of a thin-film surface energy [18, 19, 20]. The phenomenon has also been observed in simulations of CO oxidation under oscillating CO pressure [21, 22]. Further simulation studies of the DPT in the two-dimensional kinetic Ising model have appeared [23, 24, 25, 26, 27], as well as analytical studies of the DPT [28, 29, 30, 31]. Here, we concentrate on the DPT in the two-dimensional kinetic Ising model. It was ob- served in simulations of this model that there exists a singularity at a critical period of the applied oscillating field [9, 10], and that the critical exponents β and γ (and, with less accu- racy, ν) are consistent with the universality class of the equilibrium two-dimensional Ising transition in zero field [23] (β = 1/8, γ = 7/4, and ν = 1). In those studies, the techniques of finite-size scaling were extended to the study of the dynamic order parameter (Q, defined in Sec. II) in the non-equilibrium steady state. This provided evidence for a diverging cor- relation length at a critical value of the period. In particular, because the field conjugate to Q and a fluctuation-dissipation relation were not known, a susceptibility could not be measured directly, and the scaled variance X L = L 2 (〈Q2〉 − 〈Q〉2), where L is the linear system size, was used as a proxy. An analytical argument, based on the correspondence of the two-dimensional kinetic Ising model and the continuous, two-dimensional Ginzburg- Landau model at the equilibrium critical point, provided an effective Hamiltonian for the non-equilibrium system and confirmed that the DPT is in the Ising universality class [28]. These findings are consistent with earlier symmetry arguments that any continous phase transition in a stochastic cellular automaton that preserves the Ising up-down symmetry should be in the equilibrium Ising universality class [32, 33]. Recently, experiments were performed on a [Co(0.4nm)/Pt(0.7nm)]3 multilayer film with strong uniaxial anisotropy [34], whose equilibrium behavior is known to be Ising-like [35, 36]. The film was exposed to an oscillating (sawtooth) applied field with varying period, in the presence of constant ‘bias’ magnetic fields Hb of varying strength and sign. (The bias field is defined explicitly in Sec. II.) The behaviors of the dynamic order parameter and its variance, as functions of the applied field period and the bias field, provided strong evidence for the presence of the DPT in this experimental system. The observed behavior of the order parameter with respect to the bias field supported previous conjectures that the conjugate field could include the period-averaged magnetic field as an important component, and stimulated the numerical investigations in this paper. This paper is organized as follows. In Sec. II, we describe the two-dimensional kinetic Ising model and our computational methods. In Sec. III, we verify directly the scaling of the dynamic order parameter with respect to the period-averaged magnetic field at the critical period, with scaling exponent δd ≈ δe = 15, in agreement with the equilibrium Ising transition. In Secs. IV and V, we derive the expected asymptotic scaling functions in a finite- size scaling analysis of the dynamic phase transition with non-zero period-averaged bias field, and then compare the expected scaling of the dynamic order parameter to our numerical results. In Sec. VI, we present numerical data assessing the applicability of a fluctuation- dissipation relation (FDR) to this far-from-equilibrium system. We then in Sec. VII compare the expected scaling of the susceptibility (of the dynamic order parameter) to our numerical data, using the results of Sec. VI to reconcile our findings with previous results on the scaling of the fluctuations of the dynamic order parameter. Finally, we present a summary of our results in Sec. VIII. II. COMPUTATIONAL MODEL In order to facilitate comparison with previous results, we employ the same model and computational method as in Ref. [23]. Specifically, we perform kinetic Monte Carlo (KMC) simulations of a two-dimensional periodic square lattice of Ising spins Si, which can take only the values Si = ±1. The Hamiltonian of the model is H = −J 〈i,j〉 SiSj −H(t) Si, (1) where J > 0 is the ferromagnetic exchange interaction, 〈i,j〉 runs over all nearest-neighbor pairs, i runs over all L 2 lattice sites, and H(t) is an oscillating, spatially uniform applied magnetic field. The form of H(t) is here taken as a square wave with amplitude H0 = 0.3J and period P , measured in Monte Carlo steps per spin (MCSS). The square-wave form not only allows for more efficient KMC simulation, but also reduces the critical period and the finite-size effects for the DPT [23]. Other symmetric field shapes, such as sinusoidal [9, 10] and sawtooth [34], yield essentially the same results, but with a larger critical period and with stronger finite-size effects. The Glauber single-spin-flip MC algorithm with updates at randomly chosen sites is used, in which each attempted spin flip is accepted with probability W (Si → −Si) = 1 + exp (∆E/T ) , (2) where ∆E is the energy change that would result from acceptance of the spin flip, and T is the absolute temperature in energy units (i.e., with Boltzmann’s constant set to unity). All simulations were performed at T = 0.8Tc, where Tc = 2.269J is the equilibrium critical temperature of the square-lattice Ising ferromagnet in zero applied field [37]. The system responds to the oscillating field via the time-dependent magnetization per site, m(t) = Si(t). (3) The dynamic order parameter is defined as the average of m(t) over a given field cycle i [1] : (i−1)P m(t)dt. (4) We define the bias field, so named because it measures the shift (or ‘bias’) of the periodic field toward either negative or positive field values, as the period-averaged magnetic field, H(t)dt. (5) This definition applies generally to any periodic magnetic field H(t). In this paper, the applied field consists of a square wave with period P superposed with a constant magnetic field. Applying (5), since the period-average of the square-wave field is zero, the bias field Hb in this case is simply equal to the superposed constant magnetic field. III. SCALING WITH RESPECT TO THE BIAS FIELD In the two-dimensional equilibrium Ising model, the critical isotherm is given (in the thermodynamic limit, i.e., as L → ∞) as m (T = Tc, H → 0) ∝ H 1/δe , (6) where the critical exponent δe = 15 [38]. For finite systems, this relationship breaks down when the infinite-system correlation length, ξ∞ (T = Tc, H), which diverges as H → 0, becomes comparable to the linear system size L. The relationship also naturally breaks down at larger fields away from the critical region. Therefore, a plot of m vs H for a given system size L will follow the power law (6) for a range of H near the critical value H = 0, with this range extending to smaller H as L is increased [39]. We can determine directly whether the non-equilibrium system exhibits a similar rela- tionship, 〈Q〉 (P = Pc, Hb → 0) ∝ H b (7) in an analogous way. In Fig. 1, we plot 〈Q〉 vs Hb at the critical value of the period, P = Pc. In previous work, the reversal time for the magnetization, following instantaneous reversal of the uniform magnetic field H at (H = 0.3J, T = 0.8Tc), was found as τ = 74.5977 MCSS [9, 10]. The critical scaled half-period for the square waveform was determined to be Θc = Pc/ (2τ) = 0.918± 0.005 [23]. This yields Pc = 136.96± 0.75 MCSS, and in our simulations and analysis in this paper we use Pc = 136.96 MCSS. A power-law dependence is indeed seen to hold in Fig. 1, within a range which extends to lower values of Hb as L is increased. We fit the L = 256 data between the points labeled A and B in Fig. 1, finding a statistically significant fit with power-law exponent δd = 14.85 ± 0.18. As including points with Hb ≥ 0.01J was found to greatly reduce the statistical significance of the fit, the value Hb = 0.01J serves as a boundary of the scaling region at P = Pc. This result is consistent with an exponent δd = δe = 15, suggesting that the bias field Hb, for these parameters and the square waveform, is the dominant component of a conjugate field which exhibits the same scaling exponent in the DPT as does the applied magnetic field in the equilibrium Ising transition. IV. FINITE-SIZE SCALING ANALYSIS WITH BIAS FIELD To provide more complete evidence that Hb is the dominant component of the field conjugate to 〈Q〉, in the next several sections we will demonstrate data collapse onto a two- parameter finite-size scaling function for the system-size dependent quantity 〈Q〉L at points (P ≥ Pc, Hb > 0), for lattice sizes L = 90, 128, 180, and (in several cases) L = 256, using the critical exponents for the equilibrium Ising system. In this section, we briefly review the theory of finite-size scaling as it applies to this system. We then determine the expected asymptotic forms of the scaling functions, which are compared in later sections of the paper to our computational data. The theory of finite-size scaling [40, 41] states that near a continuous phase transition, the singular part of the free-energy density for a d-dimensional system of linear size L can be written as fL ≈ L |ǫ|L1/ν , HLβδ/ν , (8) where ǫ = (T − Tc)/Tc, H (in units of kBT ) is the field conjugate to the order parameter, ν is the critical exponent for the correlation length, β is the exponent for the order parameter, δ is the exponent for the critical isotherm, and Y± are scaling functions above (+) and below (−) the critical point. This yields for the order parameter at finite L = L−β/νF0± |ǫ|L1/ν , HLβδ/ν , (9) where the exponent for L in the prefactor is obtained by using the hyperscaling relation dν = 2 − α and the exponent equality α = 2 − β(δ + 1). Further differentiation yields the susceptibility, = Lγ/νG0± |ǫ|L1/ν , HLβδ/ν , (10) where the exponent for L in the prefactor is obtained by using the exponent equality γ = β(δ − 1). It has previously been shown analytically that the DPT for a sinusoidal applied field, which is symmetric under H(t) → −H(t+P/2) and so which can safely be assumed to have Hc = 0, has an effective Ginzburg-Landau free-energy density in the same universality class as the equilibrium Ising model [28]. It therefore appears reasonable to write corresponding scaling functions for the dynamic order parameter 〈Q〉 and its associated susceptibility χ̂, 〈Q〉L = L −β/νF± |θ|L1/ν , (Hc/J)L , (11) χ̂L = L γ/νG± |θ|L1/ν , (Hc/J)L , (12) where θ = (P − Pc)/Pc, and Hc is the (as yet unknown) field conjugate to 〈Q〉. In this paper we express Hc (and Hb) in units of the exchange constant, J , so that the second scaling parameter in Eqs. 11 and 12 is dimensionless. The specific form, Hc/J , with which Hc is assumed to enter the second scaling parameter needs more theoretical investigation, and could conceivably change as the theory of the DPT is further developed. However, this should not affect our conclusions [42]. Computational results for sinusoidal and square-wave fields, which both are symmetric underH(t) → −H(t+P/2) and so presumably haveHc = 0, have previously confirmed the scaling behavior with respect to θ. The exponent values were determined as γ/ν = 1.74± 0.05, β/ν = 0.126± 0.005, and ν = 0.95± 0.15 [23], consistent with the exact values for the two-dimensional equilibrium Ising model, γ = 7/4 = 1.75, β = 1/8 = 0.125, and ν = 1. We now determine the expected asymptotic forms of the scaling functions F+(y1, y2) and G+(y1, y2), where we emphasize that the + subscript indicates that the scaling functions refer to the range P ≥ Pc, and where the scaling parameters are y1 ≡ θL 1/ν and y2 ≡ (Hc/J)L βδ/ν . y1 ≫ y2. We expect χ̂L ∼ θ −γ = Lγ/νy 1 (independent of y2) and 〈Q〉L = χ̂LHc ∼ θ −γHc ∼ L−β/νy 1 y2, where γ = β(δ − 1) was used to obtain the exponent for L in 〈Q〉L. y1 ≪ y2. We expect that 〈Q〉L ∼ H c ∼ L −β/νy 2 and χ̂L = ∂〈Q〉L/∂Hc ∼ L (1−δ)/δ (both independent of y1), where γ = β(δ − 1) was used to obtain the exponent for L in χ̂L. Thus, the asymptotic forms of the scaling functions are expected to be F+(y1, y2) ≡ L β/ν〈Q〉L ∼ 1 y2 for y1 ≫ y2 2 for y1 ≪ y2 G+(y1, y2) ≡ L −γ/ν χ̂L ∼ 1 for y1 ≫ y2 (1−δ)/δ 2 for y1 ≪ y2 . (14) V. COMPARISON OF FIRST SCALING FUNCTION TO COMPUTATIONAL RESULTS In Fig. 2(a), using the equilibrium values βe and νe in calculating F+(y1, y2) ≡ L β/ν〈Q〉L, we present a plot of the scaling function F+ vs y1 for different values of y2, for lattice sizes L = 90, 128, and 180. Here and for the remainder of the paper, exponents with the subscripts ‘d’ and ‘e’ refer to the behavior of the nonequilibrium system (with a dynamic phase transition) and the equilibrium system, respectively. The scaling function exhibits a power-law dependence in the regime y1 ≫ y2, which is consistent with Eq. (13). At progressively larger values of the constant y2, the power-law scaling can be seen to begin at increasing values of y1, as would be expected. A best-fit line to the final five points of the L = 180 data at y2 = 3.39 yields an estimate of the scaling exponent −γd = −1.76 ± 0.07 in Eq. (13). This is consistent with the previous results for Hc = 0 cited above [23], and it supports the hypothesis that γd = γe = 7/4 = 1.75. In Fig. 2(b), we present just the data for y2 = 3.39, including additional data points at y1 = 280 and 477. The data deviate from the power-law behavior for L = 90 at y1 > 149, for L = 128 at y1 > 280, and for L = 180 at y1 > 477. This locates the boundary of the scaling regime (for y2 = 3.39) at θ = y1/L 1/ν ≈ 2.65 . In Fig. 3, again using βe and νe in calculating F+, we plot the scaling function F+ vs y2 at different values of y1, in order to examine the scaling behavior for y1 ≫ y2. For the constant values y1 = 43.4, 69.7, and 149, power-law scaling can be observed in the regime y1 ≫ y2. A best-fit line to the y1 = 149 data for the five points from y2 = 3.39 to 84.6 yields a scaling exponent of 1.01± 0.01, which is consistent with the value of 1 expected from Eq. (13). In order to investigate the scaling of F+ in the asymptotic limit y1 ≪ y2, we plot in Fig. 4 the scaling function F+(y1, y2) vs y2 at the critical period P = Pc (i.e., y1 = 0), at lattice sizes L = 128, 180, and 256. In the range 20 < y2 < 50, power-law scaling is observed for all three lattice sizes. For y2 > 50, the data deviate from power-law scaling, with the smallest lattice size deviating first, as expected in a finite-size scaling plot. A fit of the L = 256 data in the range from y2 = 8.46 to 84.6 produces a scaling exponent 0.0673± 0.0008. Since the constant factors Lβδ/ν and Lβ/ν do not affect the fit of the scaling exponent, this is the same exponent found in the fit of 〈Q〉 vs Hb in Fig. 1. The reciprocal of this scaling exponent is thus δd = 14.85 ± 0.18, which is consistent with the exponent of the critical isotherm, δe = 15, in the equilibrium Ising model. The comparison of the second scaling function, G+(y1, y2), to numerical data is more clearly presented after the relationship of the susceptibility χ̂L and the scaled variance X has been examined. Therefore, we present in the next section numerical results on the extent of applicability of an FDR between χ̂L and X L , before turning in Sec. VII to the second scaling function. VI. APPLICABILITY OF A FLUCTUATION-DISSIPATION RELATION FDRs, such as the Einstein relation, Green-Kubo relations, etc., hold a central place in equilibrium statistical mechanics. This is essentially a consequence of detailed balance and the role of the partition function as a moment-generating function, and thus such relations cannot be readily extended to nonequilibrium steady states. However, it has recently been shown that certain FDRs can be extended to far-from equilibrium steady states by use of an effective temperature [43, 44]. Here we will therefore consider whether the nonequilibrium susceptibility and the scaled variance of the dynamic order parameter can be related as χ̂L ≡ ∂〈Q〉L L2 (〈Q2〉L − 〈Q〉 , (15) with an effective temperature Teff , in a way analogous to the equilibrium FDR, ∂〈m〉L L2 (〈m2〉L − 〈m〉 , (16) in which T is the temperature. As mentioned in Secs. I and IV, this conjecture motivated the use in previous work of the scaled variance X L as a proxy for χ̂L in investigating the scaling behavior of the nonequilibrium system near its critical period. To test the extent to which Eq. (15) holds, we computed values of χ̂L and X L for a range of periods from P = 140 to 250 MCSS and a range of bias fields from Hb = 0 to an upper limit between 0.005J and 0.2J . (The bias field necessary to ‘saturate’ the nonequilibrium system, i.e., to produce values of χ̂L and X L near zero, increases as the period is increased.) The computations were perfomed at L = 180. The quantity χ̂L was computed directly as a numerical derivative: χ̂L(P,Hb) ≈ (〈Q〉(P,Hb +∆Hb)− 〈Q〉(P,Hb −∆Hb)) /2∆Hb. (17) The choice of ∆Hb = 0.1Hb was found to produce sufficiently accurate values of the numerical derivative across the range of bias fields studied. The results for periods P = 140 through 190 MCSS are shown in Fig. 5. A linear relationship is seen to exist between χ̂L and X L , for each value of P , over a wide range of χ̂L values. At each period, the dependence becomes nonlinear below a certain value of χ̂L, as illustrated for periods P = 150, 170, and 190 MCSS in Fig. 6. Since low values of the susceptibility χ̂L correspond to large values of the bias field Hb, we interpret this breakdown of linearity as an indication that the FDR in Eq. (15) holds only in a scaling regime around the critical point, i.e., for a limited range of Hb around Hb = 0. The relationship between X L and χ̂L at the higher periods, P = 220 and P = 250 MCSS, is more complicated, as shown in Fig. 7. At P = 220 MCSS, following the nonlinear regime at low χ̂L, there is a linear relationship with slope Teff = (6.27± 0.11)J up to χ̂L ≈ 13 J followed by a second distinct linear dependence with slope Teff ≈ (4.16 ± 0.29)J above χ̂L = 13 J −1. At P = 250 MCSS, the initial nonlinear dependence is again present. Then the first linear regime has Teff = (6.49 ± 0.07)J up to χ̂L ≈ 13 J −1, and is followed by a regime which can be characterized as either linear with very gentle slope (0.27± 0.22)J , or as an effective ‘saturation’ of X L past χ̂L = 13 J In Fig. 8 we plot the best-fit slopes from Figs. 5 and 7, which according to Eq. (15) represent estimates of Teff , vs the scaling parameter θ = (P − Pc) /Pc. We have included in the plot the slopes of both linear regimes for the values θ = 0.606 and 0.825 (P = 220 and 250 MCSS). For θ below 0.4 (P ≈ 190 MCSS), Teff increases with θ in a way not inconsistent with a linear relationship (with slope 2.97J). It may be interesting to note that an extrapolation of the linear relationship to θ = 0 (P = Pc) yields the value Teff = 3.39J , which is significantly higher than the critical temperature, Tc = 2.2619J , of the equilibrium Ising system. However, one should not put too much emphasis on the numerical values of Teff [45], as they could easily be changed. For instance, if Hb is only proportional to the full conjugate field Hc, with a proportionality constant different from unity, this would trivially change Teff in Eq. (15). The important result, which we have demonstrated to hold in the critical region, is the linear relationship between X L and χ̂L. We can thus characterize the extent of applicability of an FDR to the DPT above the critical period as follows. For θ < 0.4, an FDR holds outside of a small nonlinear regime at low χ̂L (high Hb), with an effective temperature Teff which increases approximately linearly with θ. For θ above 0.4, two linear relationships appear to exist between X L and χ̂L in separate regimes, making it impossible to define a unique Teff at a given value of θ. An understanding of the nonlinear regime, which is present at low χ̂L for all periods examined, as well as of the complication of the FDR above θ = 0.4, would be highly desirable. We hope that these numerical results can stimulate the development of, as well as test the accuracy of, a theoretical description of the non-equilibrium steady states produced in the presence of non-zero Hb for this DPT. VII. COMPARISON OF SECOND SCALING FUNCTION TO COMPUTA- TIONAL RESULTS We now test the asymptotic scaling forms for G+ in Eq. (14). First, we note that in performing least-squares fits, one normally requires the goodness-of-fit parameter q, i.e., the probability that (assuming the fit relationship were true) random error alone could produce the observed data, to be greater than 10−3 to consider the fit reasonable. Within this section, however, and in the captions to Figs. 9 through 11, it will be useful for descriptive purposes to refer to scaling exponents resulting from attempts at least-squares fits with q values below this acceptable range. We will refer to the results of such unsuccessful fitting attempts as ‘nominal’ scaling exponents, and for clarity will report the value of the parameter q for each scaling exponent presented in this section. In Fig. 9, we plot the scaling function G+ vs y1 for y2 = 8.46 at L = 180, using γe and νe to calculate G+, and evaluating χ̂L numerically according to Eq. (17). In addition, we plot in the same figure the scaling function GX+ (y1, y2) ≡ X −γ/ν vs y1, for the same values of y2 and L. A fit to all four G+ data points yields a scaling exponent −1.60± 0.03 (q = 0.02), while a fit to the last three G+ data points yields a scaling exponent −1.71±0.05 (q = 0.25). We will provide evidence in the next paragraph that only the last three G+ data points, and not the first, satisfy the asymptotic condition y1 ≫ y2. Thus, these data are consistent with power-law scaling of χ̂L with exponent −γd = −γe = −7/4 = −1.75. Attempts to fit the GX+ data to all four and the last three data points yield nominal scaling exponents −1.73 ± 0.01 (q < 10−15) and −1.81 ± 0.02 (q = 2.3 × 10−14), respectively. Thus, while in Fig. 9 it appears that the power-law relationships with these nominal scaling exponents give respectable visual fits to the GX+ data, there are variations in the data which, while small, are larger than the statistical error bars, and which prevent a statistically significant fit. We will describe the causes of these variations later in this section. We present in Fig. 10 a plot of G+ and G + vs y1 at y2 = 0, again for L = 180, for a larger range from y1 = 30.3 to 477. With y2 = 0, we expect that y1 = 30.3 (and indeed, essentially any nonzero value of y1) should satisfy the asymptotic scaling condition y1 ≫ y2. An attempt to fit to all six G+ data points in Fig. 10 gives a nominal scaling exponent −1.65 ± 0.03 (q = 3.8 × 10−7), while a fit to the first five points (y1 = 30.3 through 280) gives a scaling exponent −1.74±0.03 (q = 0.07). Excluding the first data point at y1 = 30.3 has little effect on either fit. This supports the assumption that with y2 = 0, the asymptotic scaling condition y1 ≫ y2 holds for y1 = 30.3, while for y2 = 8.46, as used in Fig. 9, the asymptotic scaling condition does not hold for y1 = 30.3. Attempts to fit all of, and the first five of, the GX+ data points to power-law scaling again yield only nominal scaling exponents −2.01± 0.01 (q < 10−15) and −2.10± 0.01 (q < 10−15), respectively. We considered that the low statistical significance of the fits to the GX+ data could be caused by underestimation of the error bars on X L . These error bars were calculated by (i) finding the correlation time in the numerical data series Qi from the simulation, and sampling data at intervals of twice the correlation time; (ii) dividing this sampled data into k > 16 groups and calculating the value of X L within each group; (iii) finding the mean and standard error of this collection of X L values. As a check on self-consistency, we performed several independent calculations of X L by this method, and found that the standard error of these values (corrected for small sample size) was comparable to the standard error found within each calculation. Thus, we have strong evidence that the error bars for X L (and G are accurate. These scaling results can be understood in light of the observations in Sec. VI on the relationship between X L and χ̂L. We can reasonably assume that the breakdown in scaling of G+ past y1 = 280 (P = 350 MCSS, θ = 1.56) in Fig. 10 occurs because this is the boundary of the critical region. The small variations of the GX+ data for P < 350 MCSS in Fig. 10 from a scaling relationship with exponent −δd = −1.75 then have three main causes. The first cause is the multiplication of the accurately scaling function G+ by the θ- and y1-dependent value Teff , according to Eq. (15). However, such a variation would also occur in an analogous plot for the scaling of XML ≡ L 2 (〈m2〉 − 〈m〉2) vs y1,e = ǫL 1/ν = ((T − Tc) /Tc)L 1/ν in the equilibrium Ising model, since the susceptibility χML ≡ ∂〈m〉/∂H scales with exponent −γe, and XML is related to χ L by the ǫ-dependent temperature T according to Eq. (16). This effect is small enough to be neglected in equilibrium critical scaling, and, since the change in Teff from θ = 0.02 to θ = 0.4 is comparable to the change in T from ǫ = 0.02 to ǫ = 0.4 in the equilibrium transition, it can also be neglected here. The second cause is the presence of the nonlinear regimes in the plots of X L vs χ̂L at low χ̂L, resulting in non-zero X L -intercepts in the application of Eq. (15) to Figs. 5 and 7. Because of this, division of the X L data by the appropriate Teff values (given in the caption of Fig. 5) does not quite reproduce the corresponding χ̂L data, and (even below θ = 0.4) does not quite result in scaling consistent with δd = 1.75 with statistical signficance. The third and most signficant cause of the variations of the GX+ data is the ‘doubly linear’ behavior observed in Fig. 7 for θ > 0.4, which prevents identification of a unique Teff in this range. The assumption that X L can be used as a proxy for χ̂L is thus fairly well justified close to the critical period, where Teff varies over a limited range and the more complicated effects observed at θ > 0.4 are not relevant. This is supported by Fig. 9, where the data points cluster closely around the line corresponding to power-law scaling with exponent −1.73 ≈ −γe. However, because of the first two causes just described, there are small systematic variations in the GX+ data which prevent a statistically significant fit to a pure power-law relationship as a function of y1. In order to clarify the relationship of these scaling results to those in previous work, we also plot in Fig. 11 data of G + (y1, y2) ≡ X −γ/ν ≡ (〈Q2〉 − 〈|Q|〉2)L−γ/ν vs y1 at y2 = 0, again using the equilibrium values γe and νe to calculate G + . This can be directly compared to Fig. 11(d) in Ref. [23], in which the quantity we call X L was called X L . Attempted fits to all five data points and to the last four data points of G + in Fig. 10 produce nominal scaling exponents −1.60 ± 0.02 and −1.69± 0.02 (both with q < 10−15). The agreement in Fig. 11(d) of Ref. [23] of the line with slope −7/4 with the data for θ > θc must therefore be viewed as qualitative. The method used in Ref. [23] to numerically estimate γd, however, which involves finite-size scaling at the critical period, is fully consistent with the results of this paper, since at each period with θ < 0.4 we have found that the FDR in Eq. (15) holds to a very good approximation. Finally, in Fig. 12, we plot GX+ vs y2 at P = Pc, to study its scaling in the regime y1 ≪ y2. As just noted, the use of X L as a proxy for χ̂L is well justified at P = Pc by our results. Power-law scaling is perhaps suggested in the range 20 < y2 < 50 for L = 180, and it is clearly obeyed from y2 = 8.42 to 84.2 for L = 256. The scaling exponent was determined as (1− δd)/δd = −0.914± 0.030, which is consistent with the corresponding equilibrium value (1− δe)/δe = −14/15 ≈ −0.933. VIII. CONCLUSIONS AND OUTLOOK In this article, we have continued the computational study of the dynamic phase transition (DPT) in the two-dimensional kinetic Ising model exposed to a periodically oscillating field, which was begun in Refs. [9, 10, 23]. We have established two distinct but related results about the field conjugate to the dynamic order parameter. First, we have identified the period-averaged magnetic field, or ‘bias field’, Hb as an important component of the full conjugate field. This claim is supported by numerical evidence that the dynamic order parameter and its susceptibility follow critical scaling with respect to Hb. In particular, the scaling exponent δd of the conjugate field was determined for the first time, and found by finite-size scaling analysis of large-scale kinetic Monte Carlo simulations to be equal to the critical-isotherm exponent for the equilibrium Ising transition, δe = 15. Furthermore, in agreement with previous results [23], the dynamic scaling exponents γd, βd, and νd were also found to equal their equilibrium Ising counterparts, γe = 7/8, βe = 1/8, and νd = 1. These results further strengthen previous numerical [9, 10, 23] and analytical [28, 32, 33] claims that the DPT in a periodically driven two-dimensional kinetic Ising model belongs to the universality class of the equilibrium two-dimensional Ising model. However, with respect to the direct applicability of the symmetry arguments of Refs. [32, 33], we caution the reader that what is claimed in the present paper (as well as in Ref. [28]) is only equivalence of the phase transitions in the driven kinetic Ising model and the equilibrium Ising model. Outside the critical region, it is neither clear how closely P −Pc and Hb play the roles of T −Tc and the ordinary magnetic field, respectively, nor how closely the dynamic order parameter, Q, corresponds to the average equilibrium magnetization. From our discussion of the FDR in Sec. VI, it appears likely that one or more of these relations break down outside the critical region. Much theoretical work remains to be done in this area. The second main result of this article is that a fluctuation-dissipation relation (FDR), that is, a proportionality relation between the scaled fluctuations X L ≡ L 2 (〈Q2〉 − 〈Q〉2) and the susceptibility χ̂L with a slope we have called Teff , holds for a range of periods above Pc and for a range of bias fields around Hb = 0. We stress again that we have found the FDR of Eq. (15) to hold only in the critical region in this nonequilibrium system, in contrast to the equilibrium FDR of Eq. (16) which follows directly from the partition function, and which thus holds everywhere. We note that, for the parameters used in our computation at least, the critical region in which the nonequilibrium FDR holds (P < 190 MCSS) is somewhat smaller than the critical region in which power-law scaling is obeyed (P < 350 MCSS). In previous work, when the conjugate field had not been identified, the scaled fluctuations L were used as a proxy for the (then unknown) quantity χ̂L. The evidence for the FDR presented here shows this assumption to be fully justified at the critical period (see Fig. 12), and to be a very good approximation – nearly as good as the use of the scaled fluctuations as a proxy for the susceptibility in the equilibrium Ising model – in the critical region where the FDR holds. There are at least three further computational projects suggested by the progress re- ported here. The first is to investigate whether the field Hb functions as the conjugate field, with scaling exponents consistent with the equilibrium Ising transition for periods P < Pc, below the critical period. In the equilibrium system, the study of critical scaling in nonzero field for T < Tc is complicated by the long time correlations and strong finite-size effects which accompany the bimodal distributions of magnetization below Tc. Similar effects would complicate the investigation of scaling with respect to Hb in the DPT for P < Pc, but the advanced techniques [46, 47] which make the equilibrium simulations tractable do not ex- tend obviously to the nonequilibrium case. The second computational project suggested is to determine the nature of the full conjugate field Hc. The third project would be to study the FDR at different values of the temperature, T , and the amplitude, H0, of the driving field. Finally, we remark that it would be very desirable to extend the current understanding of the theory of nonequilibrium steady states to include the conjugate field Hb, the FDR found in the critical region, and the scaling of Hb. Acknowledgments Research at Florida State and Mississippi State Universities was supported by NSF Grant No. DMR-0444051, and at Clarkson University by NSF Grant No. DMR-0509104. This research also used resources of the National Center for Computational Sciences at Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. [1] T. Tomé and M. J. de Oliveira, Phys. Rev. A 41, 4251 (1990). [2] J. F. F. Mendes and E. J. S. Lage, J. Stat. Phys. 64, 653 (1991). [3] M. F. Zimmer, Phys. Rev. E 47, 3950 (1993). [4] M. Acharyya and B. K. Chakrabarti, Phys. Rev. B 52, 6550 (1995). [5] M. Acharyya, Phys. Rev. E 56, 2407 (1997). [6] M. Acharyya, Phys. Rev. E 58, 179 (1998). [7] W. S. Lo and R. A. Pelcovits, Phys. Rev. A 42, 7471 (1990). [8] M. Acharyya, Phys. Rev. E 56, 1234 (1997). [9] S. W. Sides, P. A. Rikvold, and M. A. Novotny, Phys. Rev. Lett. 81, 834 (1998). [10] S. W. Sides, P. A. Rikvold, and M. A. Novotny, Phys. Rev. E 59, 2710 (1999). [11] B. K. Chakrabarti and M. Acharyya, Rev. Mod. Phys. 71, 847 (1999). [12] T. Yasui, H. Tutu, M. Yamamoto, and H. Fujisaka, Phys. Rev. E 66, 036123 (2002). [13] N. Fujiwara, H. Tutu, and H. Fujisaka, Phys. Rev. E 70, 066132 (2004). [14] N. Fujiwara, H. Tutu, and H. Fujisaka, Prog. Theor. Phys. Suppl. 161, 181 (2006). [15] Y. Z. Shao, W. R. Zhong, and G. M. Lin, Acta Physica Sinica 53, 3165 (2004). [16] M. Acharyya, Int. J. Mod. Phys. C 14, 49 (2003). [17] M. Acharyya, Phys. Rev. E 69, 027105 (2004). [18] H. Jang and M. J. Grimson, Phys. Rev. E 63, 066119 (2003). [19] H. Jang, M. J. Grimson, and C. K. Hall, Phys. Rev. B 67, 094411 (2003). [20] H. Jang, M. J. Grimson, and C. K. Hall, Phys. Rev. E 68, 046115 (2003). [21] E. Machado, G. M. Buend́ıa, P. A. Rikvold, and R. M. Ziff, Phys. Rev. E 71, 016120 (2005). [22] G. M. Buend́ıa, E. Machado, and P. A. Rikvold, J. Mol. Struct.: THEOCHEM 769, 189 (2006). [23] G. Korniss, C. J. White, P. A. Rikvold, and M. A. Novotny, Phys. Rev. E 63, 016120 (2001). [24] G. Korniss, P. A. Rikvold, and M. A. Novotny, Phys. Rev. E 66, 056127 (2002). [25] A. Chatterjee and B. K. Chakrabarti, Phys. Rev. E 67, 046113 (2003). [26] A. Chatterjee and B. K. Chakrabarti, Phase Transitions 77, 581 (2004). [27] M. Acharyya, Phys. Rev. E 59, 218 (1999). [28] H. Fujisaka, H. Tutu, and P. A. Rikvold, Phys. Rev. E 63, 036109 (2001); 63, 059903(E) (2001). [29] H. Tutu and N. Fujiwara, J. Phys. Soc. Jpn. 73, 2680 (2004). [30] E. Z. Meilikhov, JETP Letters 79, 620 (2004). [31] S. B. Dutta, Phys. Rev. E 69, 066115 (2004). [32] G. Grinstein, C. Jayaprakash, and Y. He, Phys. Rev. Lett. 55, 2527 (1985). [33] K. E. Bassler and B. Schmittmann, Phys. Rev. Lett. 73, 3343 (1994). [34] D. T. Robb, Y. H. Xu, O. Hellwig, J. McCord, A. Berger, M. A. Novotny, and P. A. Rikvold, Evidence for a dynamic phase transition in [Co/Pt]3 magnetic multilayers (2007), to be sub- mitted to Phys. Rev. B. [35] M. Bander and D. L. Mills, Phys. Rev. B 38, R12015 (1988). [36] C. Back, C. Würsch, A. Vaterlaus, U. Ramsperger, U. Maier, and D. Pescia, Nature 378, 597 (1995). [37] L. Onsager, Phys. Rev. 65, 117 (1944). [38] D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, UK, 2000). [39] K. Binder and D. P. Landau, Phys. Rev. B 30, 1477 (1984). [40] V. Privman and M. E. Fisher, Phys. Rev. B 30, 322 (1984). [41] V. Privman, in V. Privman, ed., Finite Size Scaling and Numerical Simulation of Statistical Systems (World Scientific, Singapore, 1990), pp. 4–98. [42] Until the theory of the DPT, e.g. in Ref. [28], is extended to include Hc, it is unclear in what form it should enter the dimensionless scaling parameter, i.e. as Hc/J , Hc/kBT , or otherwise. Here we have chosen the form Hc/J . Since our numerical data is all taken at a single temperature, T = 0.8Tc = 1.8152J , a change in this form would at most multiply values of y2 by a constant factor, and would not affect the scaling relations reported in this paper. [43] K. Hayashi and S.-I. Sasa, Phys. Rev. E 69, 066119 (2004). [44] K. Hayashi and S.-I. Sasa, Phys. Rev. E 71, 046143 (2005). [45] To avoid misunderstandings, note that it is certainly possible to define other sensible effective temperatures for an Ising-like model, so that one should be clear about the definition being used in a given context. For example, the nonequilibrium FDR proposed in Refs. [48, 49] connects two-time response and correlation functions, and applies to critical ageing in a two- parameter family of Ising-like models. This family includes many models without detailed balance (e.g., the noisy voter and majority models), and one model – the equilibrium Ising model with Glauber dynamics – with detailed balance. In contrast, our Eq. (15) concerns single-time response and correlations, and the kinetic Ising model driven by an oscillating field is not included in the two-parameter family of models considered in Refs. [48, 49]. [46] B. A. Berg and T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992). [47] B. A. Berg and T. Neuhaus, Phys. Lett. B 267, 249 (1991). [48] C. Godrèche and J. M. Luck, J. Phys. A: Math. Gen. 33, 9141 (2000). [49] F. Sastre, I. Dornic, and H. Chaté, Phys. Rev. Lett. 91, 267205 (2003). 0.0001 0.001 0.01 0.1 H (units of ) L = 90 L = 128 L = 180 L = 256 1/ = 0.0673 (fit) 1/ = 0.0666 (ref.) FIG. 1: (Color online.) Log-log plot of the dynamic order parameter, 〈Q〉, vs bias field, Hb, at P = Pc for L = 90, 128, 180, and 256. A least-squares fit to power-law scaling of the L = 256 data, in the range between the labels A and B above, produced a statistically significant fit with scaling exponent 1/δd = 0.0673±0.0008 (corresponding to δd = 14.85±0.18). The dotted line corresponds to the scaling exponent 1/δd = 0.0673. A reference line representing scaling with the equilibrium Ising exponent, δe = 15 (1/δe = 0.0666), is shown as the dashed line. 1 10 100 1000 y = L 0.001 L = 90 L = 128 L = 180 Slope = -1.76 (fit) Slope = -1.75 (ref.) y = 3.39 y = 8.46 y = 16.9 y = 33.9 y = 84.6 y = 169 1 10 100 1000 y = L 0.001 L = 90 L = 128 L = 180 Slope = -1.76 (fit) Slope = -1.75 (ref.) FIG. 2: (Color online.) Log-log plots of the scaling function F+(y1, y2) vs y1 for lattice sizes L = 90, 128, and 180. The values of y1 plotted are 4.00, 17.2, 30.3, 43.4, 69.7, 147, and (in (b)) 280 and 477. (a) The data are shown for the values of y2 listed on the plot. The best-fit line for the last five points of the L = 180 data at y2 = 3.39, with slope −1.76 ± 0.07, is included along with a reference line with the slope −γe = −1.75. (b) The data for y2 = 3.39 with two additional y1 values illustrates the boundary of the regime of power-law scaling. 1 10 100 0.001 L = 90 L= 128 L = 180 Slope = 1.01 (fit) Slope = 1.00 (ref.) ν y = 0.00 y = 4.00 y = 17.1 y = 30.3 y = 43.4 y = 69.7 y = 149 FIG. 3: (Color online.) Log-log plot of the scaling function F+(y1, y2) vs y2 = (Hb/J)L βδ/ν for lattice sizes L = 90, 128, and 180, for the constant values of y1 labeled in the plot. The values of y2 used are y2 = 3.39, 8.46, 16.9, 33.9, 84.6, and 169. The dotted line represents the best fit to the first five points of the L = 180 data at y1 = 149, and has a slope of 1.01 ± 0.01. The dashed line shows the slope value of 1, expected from Eq. (13). 10 100 L = 128 L = 180 L = 256 Slope = 0.0673 (fit) Slope = 0.0666 (ref.) FIG. 4: (Color online.) Log-log plot of the scaling function F+(y1, y2) vs y2 = (Hb/J)L βδ/ν for lattice sizes L = 128, 180, and 256, at the critical period Pc, where y1 = 0. In the L = 256 data, near the values y1 = 16, 32, 165, and 335, two closely spaced data points are actually plotted. The best-fit line to the L = 256 data in the range 8.46 < y2 < 84.6, shown as a dotted line in the plot, corresponds to a scaling exponent 1/δd = 0.0673 ± 0.0008. A reference line corresponding to scaling exponent 1/δe = 1/15 = 0.0666 is also shown. These results are in complete agreement with those shown in Fig. 1. 0 100 200 P = 140 MCSS P = 150 MCSS P = 160 MCSS P = 170 MCSS P = 190 MCSS χ̂ (units of )-1 FIG. 5: (Color online.) The scaled fluctuations X L of the dynamic order paramater plotted vs its susceptibility χ̂L to the bias field Hb, calculated at L = 180, for periods P = 140, 150, 160, 170 and 190 MCSS. The quantity χ̂L was calculated using the numerical derivative in Eq. (17). The best-fit lines shown, whose slopes increase monotically with the period P of the data to which they were fit, were calculated as (3.239J)χ̂L+10.42, (3.557J)χ̂L+8.735, (3.980J)χ̂L+3.889, (4.232J)χ̂L+5.868 and (4.497J)χ̂L + 5.371, respectively. 0 10 20 30 P = 140 MCSS P = 150 MCSS P = 160 MCSS P = 170 MCSS P = 190 MCSS χ̂ (units of )-1 FIG. 6: (Color online.) Closeup of Fig. 5, showing the relationship of X L and χ̂L at low values of χ̂L, which correspond to large values of the bias field Hb. For P = 150, 170 and 190 MCSS, data have been taken (and are shown) down to very low values of χ̂L, where the breakdown of the linear relationship between X L and χ̂L can be clearly seen. The dashed lines are the same best-fit lines shown in Fig. 5. 0 10 20 30 40 P = 220 MCSS P = 250 MCSS χ̂ (units of )-1 FIG. 7: (Color online.) The scaled fluctuations X L of the dynamic order paramater plotted vs its susceptibility χ̂L to the bias fieldHb, calculated for lattice size L = 180, at periods P = 220 and 250 MCSS. At each period, the data were fit (purely phenomenologically) to two linear relationships. For P = 220 MCSS, the fits were calculated as (6.265J)χ̂L−7.497 at low χ̂L, and (4.161J)χ̂L+19.94 at high χ̂L. For P = 250 MCSS, the fits were (6.485J)χ̂L−9.240 at low χ̂L, and (0.2726J)χ̂L+67.92 at high χ̂L. 0 0.2 0.4 0.6 0.8 1 FIG. 8: (Color online.) The effective temperature Teff , obtained as the slopes of the linear fits to the data in Figs. 5 and 7, plotted vs θ = (P − Pc) /Pc. For the values θ = 0.606 and 0.825 (P = 220 and 250 MCSS), the slopes of both linear regimes fit in Fig. 7 are plotted as Teff values, using filled squares and diamonds rather than filled circles. The straight line is a weighted least-squares fit to the data below θ ≈ 0.4 (P < 190 MCSS), and has a slope of 2.97J . 20 50 100 200 y = L 0.001 Slope = -1.73 (nom.) Slope = -1.81 (nom.) Slope = -1.75 (ref.) Slope = -1.60 (fit) Slope = -1.71 (fit) FIG. 9: (Color online.) Log-log plots of G+(y1, y2) and G + (y1, y2) vs y1, over the range y1 = 30.3 through 149, for y2 = 8.46 at L = 180. The relatively small error bars on each data point can be seen inside the larger symbols. The solid and dash-dash-dotted lines are fits to all four and the last three G+ data points, respectively, and correspond to scaling exponents −1.60 ± 0.03 and −1.71±0.05. The dotted and dash-dotted lines are the result of attempts to fit all four and the last three GX+ data points, respectively. They correspond to nominal scaling exponents −1.73 ± 0.01 and −1.81± 0.02. The dashed line is a reference line corresponding to scaling exponent −1.75. 20 50 100 200 500 y = L 0.0001 0.001 Slope = -2.01 (nom.) Slope = -2.10 (nom.) Slope = -1.75 (ref.) Slope = -1.65 (nom.) Slope = -1.74 (fit) FIG. 10: (Color online.) Log-log plots of G+(y1, y2) and G + (y1, y2) vs y1, over the range y1 = 30.3 through 477, for y2 = 0 at lattice size L = 180. The relatively small error bars on each data point can be seen inside the larger symbols. The solid and dash-dash-dotted lines show the result of attempts to fit all six and the first five G+ data points, and correspond to a nominal scaling exponent −1.65 ± 0.03, and a statistically significant scaling exponent −1.74 ± 0.03, respectively. The dotted and dash-dotted lines are the result of attempts to fit all six and the first five GX+ data points, and correspond to nominal scaling exponents −2.01 ± 0.01 and −2.10± 0.01, respectively. The dashed line is a reference line corresponding to scaling exponent −1.75. 20 50 100 200 y = L 0.001 Slope = -1.59 (nom.) Slope = -1.70 (nom.) Slope = -1.75 (ref.) FIG. 11: (Color online.) Log-log plot of G + (y1, y2) vs y1, over the range y1 = 30.3 through 149, for y2 = 0 at lattice size L = 180. The relatively small error bars on each data point can be seen inside the larger symbols. The solid and dotted lines are the results of attempts to fit all five and the last four data points with a power-law relationship, and correspond to nominal scaling exponents of −1.59 ± 0.02 and −1.70 ± 0.03, respectively. The dashed line is a reference line corresponding to a scaling exponent of −1.75. 10 100 0.001 L = 128 L = 180 L = 256 Slope = -0.914 (fit) Slope = -0.933 (ref.) FIG. 12: (Color online.) Log-log plot of GX+ (y1, y2) vs y2 = (Hb/J)L βδ/ν for lattice sizes L = 128, 180, and 256, at the critical period Pc, where y1 = 0. The best-fit line to the L = 256 data in the range 8.46 < y2 < 84.6, shown as a dotted line in the plot, corresponds to a scaling exponent (1 − δd)/δd = −0.914 ± 0.029. A reference line (dashed), corresponding to a scaling exponent (1− δe)/δe = −14/15 ≈ −0.933, is also shown. Introduction Computational model Scaling with respect to the bias field Finite-size scaling analysis with bias field Comparison of first scaling function to computational results Applicability of a fluctuation-dissipation relation Comparison of second scaling function to computational results Conclusions and outlook Acknowledgments References

0704.1124

Non-commutativity and Open Strings Dynamics in Melvin Universes

MAD-TH-07-02 Non-commutativity and Open Strings Dynamics in Melvin Universes Danny Dhokarh, Akikazu Hashimoto, and Sheikh Shajidul Haque Department of Physics, University of Wisconsin, Madison, WI 53706 Abstract We compute the Moyal phase factor for open strings ending on D3-branes wrapping a NSNS Melvin universe in a decoupling limit explicitly using world sheet formalism in cylindrical coordinates. http://arxiv.org/abs/0704.1124v2 Melvin Universe is an exact axially symmetric solution of Einstein gravity in a background with magnetic flux [1]. It arises naturally as a Kaluza-Klein reduction of twisted flat space ds2 = −dt2 + d~x2 + dr2 + r2(dϕ+ ηdz)2 + dz2 , (1) along the coordinate z. The twist is parameterized by variable η. The fact that z ∼ z+2πR is periodic makes the twist deformation physical. Melvin universes has a natural embedding in string theory [2–4]. Simply embed (1) in 11-dimensional supergravity. Reducing along z gives rise to a background in type IIA string theory with a background of magnetic RR 2-form field strength. Along similar lines, one can embed (1) in type IIA supergravity and T-dualize along z. This gives rise to a background in type IIB string theory ds2 = −dt2 + d~x2 + dr2 + r2dϕ2 1 + η2r2 1 + η2r2 1 + η2r2 dϕ ∧ dz̃ 1 + η2r2 z̃ = z̃ + 2πR̃, R̃ = , (2) with an axially symmetric magnetic NSNS 3-form field strength in the background. String theories in backgrounds like (2) are very special in that the world sheet theory is exactly solvable [5–10]. Quantization of open strings in Melvin backgrounds have also been studied and was shown to be exactly solvable [11, 12] as well. Embedding D-branes in Melvin universes can give rise to interesting field theories in the decoupling limit. A D3-brane extended along t, z̃, and two of the ~x coordinates gives rise to a non-local field theory known as the “dipole” theory [13, 14]. Orienting the D3- brane to be extended along the t, r, ϕ, and z̃ coordinates, on the other hand, gives rise to a non-commutative gauge theory with a non-constant non-commutativity parameter1 [16,17]. These are field theories, whose Lagrangian [17] is expressed most naturally using the deformation quantization formula of Kontsevich2 [19]. Field theories arising as a decoupling limits of various orientations of D-branes in Melvin and related closed string backgrounds along these lines3 were tabulated and classified in Table 1 of [16].4 1The first explicit construction of models of this type is [15]. 2General construction of non-commutative field theory on curved space-time with non-constant non- commutativity parameter, arising from D-branes in non-vanishing H field background, and their relation to the deformation quantization formula of Kontsevich, was first discussed in [18]. 3The S-dual NCOS theories with non-constant non-commutativity parameter was studied in [20, 21]. 4More recently, a novel non-local field theory, not included in the classification of [16], was discovered [22, 23]. To show that the decoupled field theory is a non-commutative field theory, the authors of [16] presented the following arguments: • The application of Seiberg-Witten formula5 [24] )µν = [(g +B)µν ] −1 (3) to the closed string background (2) gives the following open string metric and the non-commutativity parameter Gµνdx ν = −dt2 + dr2 + r2dϕ2 + dz2 θϕz = 2πα′η (4) which are finite if α′ is scaled to zero keeping ∆ = α′η fixed. • Solution of the classical equations of motion of an open string traveling freely on the D3-brane with angular momentum J has a dipole structure whose size is given by [16] L = θϕzJ . (5) Another suggestive argument is the similarity between α′ → 0 limit of critical string theory and the boundary Poisson sigma-model [25] as was pointed out, e.g., in [26]. As was emphasized in [26], however, the two theories are not to be understood as being equivalent or continuously connected. This apparent similarity therefore does not constitute a proof that the decoupled theory is a non-commutative field theory. A physical criteria for non-commutativity is the Moyal-like phase factor in scattering amplitudes. Scattering amplitudes of open strings ending on a D-brane can be computed along the lines reviewed in [27]. In the case of the constant non-commutativity parameter, one can show very explicitly that 1x(τ1)eip 2x(τ2) . . . eip nx(τn)〉G,θ = e θijpm ǫ(τn−τm)〈eip 1x(τ1)eip 2x(τ2) . . . eip nx(τn)〉G,θ=0 which implies that the scattering amplitudes receive corrections in the form of the Moyal phase factor [24, 28, 29]. The goal of this article is to derive the analogous statement (60) for the model of [16, 17]. Once (60) is established in polar coordinates, the connection to Kontsevich formula follows from performing a change of coordinates to the rectangular coordinate system and a non-local field redefinition as is described in [17, 30]. 5The normalization of B field is such that BHashimoto−Thomas = 2πα BSeiberg−Witten. A useful first step in this exercise is to reproduce the master relation (6) in a slightly different formalism than what was used in [24]. Let us begin by constructing the closed string background as follows. Start with flat space ds2 = dy′2 + dz̃2 , (7) where y and z̃ are compactified with period L = 2πR. Then, I T-dualize along the z direction so that the metric becomes ds2 = dy′2 + dz2 . (8) II Twist by shifting the coordinates y′ = y + ηz ds2 = (dy + ηdz)2 + dz2 . (9) III T-dualize on z so that 1 + η2 (dy2 + dz̃2), B = 1 + η2 dy ∧ dz̃ . (10) The open string metric associated to this background is Gµνdx ν = dy2 + dz̃2, θyz̃ = 2π∆2 (11) if we scale ∆2 = α′η . (12) The transformation of the coordinates and the orientation of the branes are illustrated in figure 1. This sequence of dualities was referred to as the “Melvin shift twist” in [16]. The approach of [24] was to work directly in the duality frame III, but one can just as easily work in a framework which makes the T-duality between duality frame II and III manifest, by working with a sigma model of the form dσ1dσ2 ∂ay∂by + 2η∂ayVb + (1 + η 2)VaVb + iǫab∂az̃Vb where we have chosen to work in conformal gauge in Eucledian signature. This action utilizes the Bushar’s formulation of T-duality [31]. To see this more explicitly, consider integrating out the field z̃. This imposes the constraint dV = 0 → Va = ∂az (14) Figure 1: In I and II, the thick line denotes a D2-brane, and the dotted line is the minimum energy configuration of the open strings ending on the D2-branes. The I and II are related by coordinate transformation y′ = y + ηz. III is the T-dual of II, and the shaded region in III denotes a D3-brane. which brings the action (13) into the form dσ1dσ2 ∂ay∂by + 2η∂ay∂bz + (1 + η 2)∂az∂bz which is the sigma model for the duality frame II. On the other hand, integrating out V first gives rise to a sigma model of the form dσ1dσ2 1 + η2 δab (∂ay∂by + ∂az̃∂bz̃) + i 1 + η2 ǫab∂ay∂bz̃ which is the string action for the duality frame III. In extracting non-commutative gauge theory as a decoupling limit, we are interested in embedding a D-brane extended along the y and z̃ coordinates in the duality frame III. We must therefore take the sigma model to be defined on a Riemann surface with one boundary, which we take to be the upper half plane. It is also necessary to impose the appropriate boundary condition for all of the world sheet fields. We impose the boundary condition which is free along the y direction and Dirichlet along the z direction: ∂ny(σ, σ̄) + ηVn(σ, σ̄)|∂Σ = 0 , (17) Vt|∂Σ = ∂tz|∂Σ = 0 . (18) Using the equation of motion from the variation of Va field η∂by + (1 + η 2)Vb + iǫab∂az̃ = 0 (19) and (18), we infer ∂nz̃ − iη∂ty = 0 . (20) The boundary conditions (17) and (20) are precisely the boundary condition imposed in the analysis of [24]. In order to complete the derivation of (6), we add a source term e−Ssource = eikyny(σn,σ̄n)+ikznz̃(σn,σ̄n) = e n(ikyny(σn,σ̄n)+ikznz̃(σn,σ̄n)) (21) to the action (13). Integrating out the V fields and bringing the sigma model (13) into duality frame III would lead to identical computation as what was described in [24] to derive (6). We will show below that the same conclusion can be reached using a slightly different argument which turns out to easily generalize to the case of Melvin deformed theories [16, 17]. The approach we take here is to go to the duality frame I. This brings the sigma model (13) to a simpler form dσ1dσ2 δab (∂ay ′ + ∂az∂bz) . (22) The z̃ field in the vertex operator now plays the role of a disorder operator of the dual field z. It has the effect of shifting the Dirichlet boundary condition, incorporating the fact that strings are stretched along the z direction in frames I and II. Also, the fact that the periodicity in (y′, z) coordinate system are twisted (y′, z) = (y′ + ηLn, z + Ln) (23) requires an insertion of a disorder operator for the y′(σ, σ̄) field as well. We therefore find that the source term has the form e−Ssource = eikyny ′(σn,σ̄n)+iηkznỹ ′(σn,σ̄n)−iηkynz(σn,σ̄n)+ikznz̃(σn) . (24) The boundary condition is now simply Neumann for y′ ′(σ, σ̄) = 0| ∂Σ , (25) and Dirichlet for z ∂tz(σ, σ̄) = 0|∂Σ . (26) In this form, y′ and the z sector decouple, allowing their correlators to be computed sepa- rately. In order to compute the correlation functions involving order and disorder operators with boundary conditions (25) and (26), it is convenient to decompose the fields into holo- morphic and anti holomorphic parts y′(σ, σ̄) = y′(σ) + ȳ′(σ̄) , ỹ′(σ, σ̄) = y′(σ)− ȳ′(σ̄) , (27) z(σ, σ̄) = z(σ) + z̄(σ̄) , z̃′(σ, σ̄) = z(σ)− z̄(σ̄) . (28) Their correlation functions are given by 〈y′(σ1)y ′(σ2)〉 = − α′ log(σ1 − σ2) (29) 〈ȳ′(σ1)ȳ ′(σ2)〉 = − α′ log(σ̄1 − σ̄2) (30) 〈ȳ′(σ̄1)y ′(σ2)〉 = − α′ log(σ̄1 − σ2) (31) 〈z(σ1)z(σ2)〉 = − α′ log(σ1 − σ2) (32) 〈z̄(σ̄1)z̄(σ̄2)〉 = − α′ log(σ̄1 − σ̄2) (33) 〈z̄(σ̄1)z(σ2)〉 = α′ log(σ̄1 − σ2), (34) from which we infer 〈y′(σ1, σ̄1)y ′(σ2, σ̄2)〉 = − α′(log(σ1 − σ2) + log(σ1 − σ̄2) + log(σ̄1 − σ2) + log(σ̄1− σ̄2)) (35) 〈ỹ′(σ1, σ̄1)y ′(σ2, σ̄2)〉 = − α′(log(σ1 −σ2) + log(σ1 − σ̄2)− log(σ̄1 −σ2)− log(σ̄1 − σ̄2)) (36) 〈ỹ′(σ1, σ̄1)ỹ ′(σ2, σ̄2)〉 = − α′(log(σ1 −σ2)− log(σ1 − σ̄2)− log(σ̄1− σ2)+ log(σ̄1 − σ̄2)) (37) 〈z̃(σ1, σ̄1)z̃(σ2, σ̄2)〉 = − α′(log(σ1 − σ2) + log(σ1 − σ̄2) + log(σ̄1 − σ2) + log(σ̄1 − σ̄2)) (38) 〈z(σ1, σ̄1)z̃(σ2, σ̄2)〉 = − ′(log(σ1 − σ2) + log(σ1 − σ̄2)− log(σ̄1 − σ2)− log(σ̄1 − σ̄2)) (39) 〈z(σ1, σ̄1)z(σ2, σ̄2)〉 = − α′(log(σ1−σ2)− log(σ1− σ̄2)− log(σ̄1−σ2)+ log(σ̄1− σ̄2)) . (40) In terms of these correlation functions, one can easily show that 〈O(σ1, σ̄1)O(σ2, σ̄2)〉 (41) ′(ky1ky2 + kz1kz2)(log(σ1 − σ2) + log(σ1 − σ̄2) + log(σ̄1 − σ2) + log(σ̄1 − σ̄2)) −ηα′(ky1kz2 − ky2kz1)(log(σ1 − σ̄2)− log(σ̄1 − σ2)) η2α′(ky1ky2 + kz1kz2)(log(σ1 − σ2)− log(σ1 − σ̄2)− log(σ̄1 − σ2) + log(σ̄1 − σ̄2)) On(σn, σ̄n) = ikyny ′(σn, σ̄n) + iηkznỹ ′(σn, σ̄n)− iηkynz(σn, σ̄n) + ikznz̃(σn, σ̄n) . (42) When these operators are pushed toward the boundary σ → τ + 0+i , (43) the correlation function (41) reduces to 〈O(τ1)O(τ2)〉 = 2α ′(ky1ky2 + kz1kz2) log(τ1 − τ2)− πiηα ′(ky1kz2 − ky2kz1)ǫ(τ2 − τ1) (44) where ǫ(τ), following the notation of [24], is a function which takes the values ±1 depending on the sign of τ . The term of order η2 vanishes in this limit. From these results, we conclude eOn(τn)〉 = e m<n〈Om(τm)On(τn)〉 (45) from which the main statement (6) follows immediately. Finally, let us discuss the generalization of (6) to D3-brane embedded into Melvin universe background (2) along the lines of [16, 17]. We will consider the simplest case of embedding (2) into bosonic string theory. For the Melvin universe background (2), it is convenient to prepare a vertex operator that corresponds to tachyons in cylindrical basis V (ν,m,~k) = dk1 dk2 δ(ν 2 − k21 − k imθeik1x1(σ,σ̄)+k2x2(σ,σ̄)+ ~k~x(σ,σ̄) ~k~x(σ,σ̄)Jν(r(σ, σ̄))e imϕ(σ,σ̄) (46) where r2 = x21 + x 2, ϕ = arg(x1 + ix2), θ = arg(k1 + ik2) . (47) As long as ~k2 + ν2 are taken to satisfy the on-shell condition of the tachyon, (46) is linear combination of operators of boundary conformal dimension 1, and must itself be an operator of boundary conformal dimension one. Such construction of vertex operator as a linear superposition is similar in spirit to what was considered in [32, 33]. dσ1dσ2 ∂ar∂br + r ∂aϕ∂bϕ+ 2ηr ∂aϕVb + (1 + η 2)VaVb + iǫab∂az̃Vb on the upper half plane. Integrating out z̃ brings this action to the form appropriate for the analogue of II dσ1dσ2 ∂ar∂br + r 2∂aϕ∂bϕ+ 2ηr 2∂aϕ∂bz + (1 + η 2r2)∂az∂bz . (49) The vertex operators can be represented as a source term e−Ssource = Jvn(r(σn, σ̄n))e imnϕ(σn,σ̄n)+ikznz̃(σn,σ̄n) (50) where z̃ is a disorder operator. Now, if we let ϕ′(σ, σ̄) = ϕ(σ, σ̄) + ηz(σ, σ̄) , (51) the action becomes dσ1dσ2 ∂ar∂br + r ′ + ∂az∂bz e−Ssource = Jvn(r(σn, σ̄n))e On (53) On = imnϕ ′(σn, σ̄n) + iηkznϕ̃ ′(σn, σ̄n)− iηmnz(σn, σ̄n) + ikznz̃(σn, σ̄n) (54) where ϕ̃′(σ, σ̄) (55) is the disorder field for ϕ′ satisfying the relation ∂aϕ̃′ = iǫabr2∂bϕ ′ (56) which follows naturally from the Busher rule applied to the ϕ fields. This time, the problem is slightly complicated by the fact that (r, ϕ′) sector is interacting. It is still the case that (ϕ′, z) sector, for some fixed r(σ, σ̄), is non-interacting. We will exploit this fact and do the path integral in the order where we integrate over ϕ′ and z first. The two point function of ϕ′ formally has the form 〈ϕ′(σ1, σ̄1)ϕ ′(σ2, σ̄2)〉 = (∂r 2(σ, σ̄)∂)−1 . (57) Then, it follows that 〈ϕ′(σ1, σ̄1)∂ aϕ̃′(σ2, σ̄2)〉 = iǫ ab(∂b)−1 (58) from which it follows 〈ϕ̃′(σ1, σ̄1)ϕ ′(σ2, σ̄2)〉 = − α′(log(σ1−σ2)+ log(σ1− σ̄2)− log(σ̄1−σ2)− log(σ̄1− σ̄2)) (59) in complete analogy with (36). The correlator (59) tells us that while the field-field correlator 〈ϕ′ϕ′〉 is complicated and r dependent, the field/disorder field correlator 〈ϕ̃′ϕ′〉 stays simple and topological. We can then proceed to compute the analogue of (44) and (45) for the operator (54) in the (ϕ′, z) sector. While we do not explicitly compute the 〈ϕ̃′ϕ̃′〉 correlator which appear at order η2 in (44), it is clear that the boundary condition forces this term to vanish as was the case in the earlier example. The term of order η in the exponential can be made to take the Moyal-like form (2π∆)(makzb−kzamb)ǫ(τb−τa) (60) which is finite in the scaling limit α′ → 0 with keeping ∆ finite. This is precisely the scaling considered in [16, 17]. The dependence on r(σ, σ̄) drops out for this term of order η, allowing us to further path integrate over this field trivially, with the only effect of η being the overall phase factor (60). This establishes that the decoupled theory of D-branes in Melvin universes considered in [16,17] has an effec- tive dynamics which includes the Moyal-like phase factor involving the angular momentum quantum number m and the momentum kz. In Cartesian coordinates, this Moyal phase cor- responds to a position dependent non-commutativity [16,17]. This analysis extends straight forwardly to other simple models of position dependent non-commutativity, such as6 the “Melvin Null Twist” [15] and “Null Melvin Twist” [34]. It would be interesting to extend this analysis to superstrings and to consider the scattering of states other than the open string tachyon. Acknowledgements We would like to thank I. Ellwood and O. Ganor for discussions. This work was supported in part by the DOE grant DE-FG02-95ER40896 and funds from the University of Wisconsin. References [1] M. A. Melvin, “Pure magnetic and electric geons,” Phys. Lett. 8 (1964) 65–70. [2] F. Dowker, J. P. Gauntlett, D. A. Kastor, and J. H. Traschen, “Pair creation of dilaton black holes,” Phys. Rev. D49 (1994) 2909–2917, hep-th/9309075. [3] F. Dowker, J. P. Gauntlett, S. B. Giddings, and G. T. Horowitz, “On pair creation of extremal black holes and Kaluza-Klein monopoles,” Phys. Rev. D50 (1994) 2662–2679, hep-th/9312172. 6Using the terminology of [16]. http://www.arXiv.org/abs/hep-th/9309075 http://www.arXiv.org/abs/hep-th/9312172 [4] K. Behrndt, E. Bergshoeff, and B. Janssen, “Type II Duality Symmetries in Six Dimensions,” Nucl. Phys. B467 (1996) 100–126, hep-th/9512152. [5] J. G. Russo and A. A. Tseytlin, “Constant magnetic field in closed string theory: An Exactly solvable model,” Nucl. Phys. B448 (1995) 293–330, hep-th/9411099. [6] A. A. Tseytlin, “Melvin solution in string theory,” Phys. Lett. B346 (1995) 55–62, hep-th/9411198. [7] J. G. Russo and A. A. Tseytlin, “Exactly solvable string models of curved space-time backgrounds,” Nucl. Phys. B449 (1995) 91–145, hep-th/9502038. [8] A. A. Tseytlin, “Exact solutions of closed string theory,” Class. Quant. Grav. 12 (1995) 2365–2410, hep-th/9505052. [9] J. G. Russo and A. A. Tseytlin, “Heterotic strings in uniform magnetic field,” Nucl. Phys. B454 (1995) 164–184, hep-th/9506071. [10] J. G. Russo and A. A. Tseytlin, “Magnetic flux tube models in superstring theory,” Nucl. Phys. B461 (1996) 131–154, hep-th/9508068. [11] E. Dudas and J. Mourad, “D-branes in string theory Melvin backgrounds,” Nucl. Phys. B622 (2002) 46–72, hep-th/0110186. [12] T. Takayanagi and T. Uesugi, “D-branes in Melvin background,” JHEP 11 (2001) 036, hep-th/0110200. [13] A. Bergman and O. J. Ganor, “Dipoles, twists and noncommutative gauge theory,” JHEP 10 (2000) 018, hep-th/0008030. [14] A. Bergman, K. Dasgupta, O. J. Ganor, J. L. Karczmarek, and G. Rajesh, “Nonlocal field theories and their gravity duals,” Phys. Rev. D65 (2002) 066005, hep-th/0103090. [15] A. Hashimoto and S. Sethi, “Holography and string dynamics in time-dependent backgrounds,” Phys. Rev. Lett. 89 (2002) 261601, hep-th/0208126. [16] A. Hashimoto and K. Thomas, “Dualities, twists, and gauge theories with non-constant non-commutativity,” JHEP 01 (2005) 033, hep-th/0410123. [17] A. Hashimoto and K. Thomas, “Non-commutative gauge theory on D-branes in Melvin universes,” JHEP 01 (2006) 083, hep-th/0511197. http://www.arXiv.org/abs/hep-th/9512152 http://www.arXiv.org/abs/hep-th/9411099 http://www.arXiv.org/abs/hep-th/9411198 http://www.arXiv.org/abs/hep-th/9502038 http://www.arXiv.org/abs/hep-th/9505052 http://www.arXiv.org/abs/hep-th/9506071 http://www.arXiv.org/abs/hep-th/9508068 http://www.arXiv.org/abs/hep-th/0110186 http://www.arXiv.org/abs/hep-th/0110200 http://www.arXiv.org/abs/hep-th/0008030 http://www.arXiv.org/abs/hep-th/0103090 http://www.arXiv.org/abs/hep-th/0208126 http://www.arXiv.org/abs/hep-th/0410123 http://www.arXiv.org/abs/hep-th/0511197 [18] L. Cornalba and R. Schiappa, “Nonassociative star product deformations for D-brane worldvolumes in curved backgrounds,” Commun. Math. Phys. 225 (2002) 33–66, hep-th/0101219. [19] M. Kontsevich, “Deformation quantization of Poisson manifolds, I,” Lett. Math. Phys. 66 (2003) 157–216, q-alg/9709040. [20] R.-G. Cai, J.-X. Lu, and N. Ohta, “NCOS and D-branes in time-dependent backgrounds,” Phys. Lett. B551 (2003) 178–186, hep-th/0210206. [21] R.-G. Cai and N. Ohta, “Holography and D3-branes in Melvin universes,” Phys. Rev. D73 (2006) 106009, hep-th/0601044. [22] O. J. Ganor, “A new Lorentz violating nonlocal field theory from string theory,” hep-th/0609107. [23] O. J. Ganor, A. Hashimoto, S. Jue, B. S. Kim, and A. Ndirango, “Aspects of Puff Field Theory,” hep-th/0702030. [24] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP 09 (1999) 032, hep-th/9908142. [25] A. S. Cattaneo and G. Felder, “A path integral approach to the Kontsevich quantization formula,” Commun. Math. Phys. 212 (2000) 591–611, math.qa/9902090. [26] L. Baulieu, A. S. Losev, and N. A. Nekrasov, “Target space symmetries in topological theories. I,” JHEP 02 (2002) 021, hep-th/0106042. [27] A. Hashimoto and I. R. Klebanov, “Scattering of strings from D-branes,” Nucl. Phys. Proc. Suppl. 55B (1997) 118–133, hep-th/9611214. [28] C.-S. Chu and P.-M. Ho, “Noncommutative open string and D-brane,” Nucl. Phys. B550 (1999) 151–168, hep-th/9812219. [29] V. Schomerus, “D-branes and deformation quantization,” JHEP 06 (1999) 030, hep-th/9903205. [30] B. L. Cerchiai, “The Seiberg-Witten map for a time-dependent background,” JHEP 06 (2003) 056, hep-th/0304030. [31] T. H. Buscher, “Path integral derivation of quantum duality in nonlinear sigma models,” Phys. Lett. B201 (1988) 466. http://www.arXiv.org/abs/hep-th/0101219 http://www.arXiv.org/abs/q-alg/9709040 http://www.arXiv.org/abs/hep-th/0210206 http://www.arXiv.org/abs/hep-th/0601044 http://www.arXiv.org/abs/hep-th/0609107 http://www.arXiv.org/abs/hep-th/0702030 http://www.arXiv.org/abs/hep-th/9908142 http://www.arXiv.org/abs/math.qa/9902090 http://www.arXiv.org/abs/hep-th/0106042 http://www.arXiv.org/abs/hep-th/9611214 http://www.arXiv.org/abs/hep-th/9812219 http://www.arXiv.org/abs/hep-th/9903205 http://www.arXiv.org/abs/hep-th/0304030 [32] H. Liu, G. W. Moore, and N. Seiberg, “Strings in a time-dependent orbifold,” JHEP 06 (2002) 045, hep-th/0204168. [33] H. Liu, G. W. Moore, and N. Seiberg, “Strings in time-dependent orbifolds,” JHEP 10 (2002) 031, hep-th/0206182. [34] L. Dolan and C. R. Nappi, “Noncommutativity in a time-dependent background,” Phys. Lett. B551 (2003) 369–377, hep-th/0210030. http://www.arXiv.org/abs/hep-th/0204168 http://www.arXiv.org/abs/hep-th/0206182 http://www.arXiv.org/abs/hep-th/0210030

0704.1125

High Galactic Latitude Interstellar Neutral Hydrogen Structure and Associated (WMAP) High Frequency Continuum Emission

Revised, July 15, 2007 High Galactic Latitude Interstellar Neutral Hydrogen Structure and Associated (WMAP) High Frequency Continuum Emission Gerrit L. Verschuur Physics Department, University of Memphis, Memphis, TN 38152 [email protected] ABSTRACT Spatial associations have been found between interstellar neutral hydrogen (HI) emission morphology and small-scale structure observed by the Wilkinson Microwave Anisotropy Probe (WMAP) in an area bounded by l = 60◦ & 180◦, b = 30◦ & 70◦, which was the primary target for this study. This area is marked by the presence of highly disturbed local HI and a preponderance of intermediate- and high-velocity gas. The HI distribution toward the brightest peaks in the WMAP Internal Linear Combination (ILC) map for this area is examined and by comparing with a second area on the sky it is demonstrated that the associations do not appear to be the result of chance coincidence. Close examination of several of the associations reveals important new properties of diffuse interstellar neutral hydrogen structure. In the case of high-velocity cloud MI, the HI andWMAP ILC morphologies are similar and an excess of soft X-ray emission and Hα emission have been reported for this feature. It is suggested that the small angular-scale, high frequency continuum emission observed by WMAP may be produced at the surfaces of HI features interacting one another, or at the interface between moving HI structures and regions of enhanced plasma density in the surrounding interstellar medium. It is possible that dust grains play a role in producing the emission. However, the primary purpose of this report is to draw attention to these apparent associations without offering an unambiguous explanation as to the relevant emission mechanism(s). Subject headings: Interstellar matter, neutral hydrogen, cosmology, WMAP 1. Introduction Examination of the Wilkinson Microwave Anisotropy Probe (WMAP) images immedi- ately fires the imagination, especially the summary image that has received so much publicity, http://arxiv.org/abs/0704.1125v2 – 2 – the so-called Internal Linear Combination (ILC) map (Hinshaw et al, 2006). It is readily accessed at http://map.gsfc.nasa.gov and was produced after subtraction of suspected galac- tic components from the 5-channel observations carried out by WMAP and then combining the data. But do the remaining structures in the ILC map truly reveal the fingerprints of processes that took place shortly after the universe was born? Upon close inspection, certain features in the WMAP ILC map (hereafter the ILC map) look hauntingly familiar to those who have spent their careers studying Galactic, interstellar neutral hydrogen (HI) structure. This can be recognized in a visual comparison of the above-referenced ILC image and the all-sky HI column density map found at http://www.astro.uni-bonn.de. Several extended areas of excess emisison at high galactic latitudes (b >30◦) are present in both maps; for example in the areas around galactic longitude and latitude (l,b) = (170◦,80◦), which exhibits extensive intermediate- and high-velocity HI, and around (l,b) = (160◦,-35◦). Also, a striking area of extended ILC structure that reaches from (l,b) = (230◦,0◦) through (l,b) = (315◦,- 35◦) has a counterpart in the HI data where a tongue of emission follows roughly the same axis, emerging from the galactic disk HI at (l,b) = (255◦,-15◦) and stretching through (l,b) = (310◦,-35◦). These apparent associations initiated a closer look at the HI structure and ILC data for several areas of sky. If the ILC small-scale structures correspond to cosmological signals, absolutely no associations with HI structure, other than those due to pure chance, should be found. The thrust of this paper is to highlight the fact that a close relationship between ILC and HI structures appears evident in the data. This deserves closer attention. Lagache (2003) has examined the relationship between WMAP structure and interstellar HI by considering the HI emission integrated over all velocities as a function of position and suggests that excess WMAP emission may be associated with small, transiently heated, dust particles. In the data presented below, the apparent associations are found by examining HI area maps produced by integrating over limited velocity ranges as opposed to the total HI content. Small-scale HI features due to relatively weak and narrow emission profile structure tend to become lost when maps of integrated HI content are plotted. In §2 the data and analysis are described. In §3 several of the most informative exam- ples of close correspondences and morphological associations between HI and ILC structure are presented. In the discussion in §4 reference is made to possible, dust-related emission mechanisms that may be invoked to account for the associations and a geometric model for producing the apparent morphological associations is outlined. In the conclusions in §5 the hope is expressed that others will be motivated to seek confirmation (or otherwise) of the claims made here. http://map.gsfc.nasa.gov – 3 – 2. Data and Analysis The goal of the present study was to determine if parallels exist between small-angular- scale (1◦ to 2◦) galactic HI structure and high-radio-frequency continuum emission features observed by WMAP. Neutral hydrogen emission profile data with an angular resolution of 0.6◦ were obtained from the Leiden-Dwingeloo, sidelobe-corrected HI Survey (Hartmann & Burton, 1997) as well as the more extensive Leiden-Argentina-Bonn (LAB) All-Sky Survey (Kalberla et al. 2005). In our study the distribution of the HI brightness was initially plotted as a function of galactic longitude and latitude (l,b maps) every 10 km/s in velocity integrating over a 10 km/s range. The focus of the first phase of this study is Region A, bounded by l = 60◦ & 180◦, b = 40◦ & 70◦ for which 25 maps covering the velocity range of -200 to +50 km/s with respect to the local-standard-of-rest were produced. The HI and ILC data in the form of l,b maps were visually examined in a manner akin to blink comparison using a transparent overlay of the ILC data to search for apparent associations between small–scale structures. The galactic HI sky is filled with small-scale structure evident in such l,b maps (e.g., see Hartmann & Burton, 1997) so that any attempt to compare with small-scale structure found in the ILC data, which also consists of many small peaks of order 1 to 2◦ across, will lead to chance agreements in position. This was tested (to first-order) by also comparing the ILC map for the Region A data with the HI l,b maps for Region B, chosen as l = 180◦ to 300◦, b = 40◦ to 70◦. This is located symmetrically opposite the galactic anti-center with respect to the Region A. To this end, 20 HI maps from -120 to +80 km/s at 10 km/s intervals were produced to cover the extent of the galactic HI emission in Region B. For Regions A & B, lists of the brightest ILC peaks were produced. For Region A this included all 51 ILC peaks with amplitudes ≧0.100 mK. For Region B, 51 peaks were also used whose amplitudes are ≧0.117 mK. The centers and amplitudes of the ILC and the apparently associated HI peaks were recorded and the angular separations between the ILC and apparently associated HI peaks calculated. The same was done for two lists of 51 negative amplitude ILC peaks for Regions A & B whose amplitude limits are ≦-0.131 & ≦-0.144 mK respectively. 2.1. Statistics of the Apparent Associations The morphologies of both HI and ILC structures are complex yet distinct brightness features of a small angular extent of order 1◦ to 2◦ are common in both classes of data. To derive a rough estimate of the likelihood of finding chance associations between ILC and – 4 – HI peaks it is possible to calculate how likely it is that one of NWpeaks in the ILC map will coincide by chance with one of NHpeaks in the HI maps for the the same area, A, assuming both to be randomly distributed on the sky. The number, N, of HI peaks expected to lie by chance within a distance r of any ILC peak, that is within an area of order π r2 square degrees centered on the ILC peak, is given N = πr2A−1NWpeaks NHpeaks. (1) This is a first-order approach to the problem and ignores an edge-effect that will be introduced for large separations, r, which reduces the effective area, A, to be entered in Eqtn. 1. At small separations this is a minor correction. This estimate is in any case limited for several other reasons. For example, the two classes of structure are treated as being random on the sky, which is not so for HI, because it is known to be filamentary. Also, the apparent associations that are found (see Table 1 below) are spread over all velocities with associations at anomalous velocities relatively over-represented in Region A. In addition, no account is taken of morphological similarities between the two types of structure other than to search for sets of closed contours on a scale of 1◦ to 2◦. Morphological similarities reported in §3.3 below are difficult to quantify in a formal manner but in any event would reduce the likelihood of chance associations. For each of the HI maps about 10 to 15 contour levels were plotted and the number of peaks, defined as sets of closed contours with dimensions of order 1◦ to 2◦, was counted on each of the HI maps. A given HI peak on one contour map can usually be followed over at least two or three adjacent maps for HI at velocities outside ± 20 km/s with respect to the local-standard-of-rest. In one case the relevant HI peak could be followed over as many as 8 maps (i.e., 80 km/s). Furthermore, low velocity HI (−20 < v < 20 km/s), narrow emission line components of order 6 km/s wide tend to appear in only one contour map integrating over a 10 km/s width. On average, each HI peak is identified in 2 adjacent velocity maps so that the value of NHpeaks is this half of the number found in all the HI l,b maps used in the analysis. In the case of Region A, about 452 obvious HI peaks in the 25 area maps were counted, which sets NHpeaks = 226. For Region B the HI is more highly concentrated to low latitudes and low velocities and fewer HI peaks were counted in the 10 km/s wide HI maps at b≧40◦. Using the same criterion that on average an HI peak can be followed over 2 adjacent HI maps, NHpeaks = 130 is used in the calculations. – 5 – 2.2. The Angular Distribution of Apparent Associations In Fig. 1 the distribution of the angular offsets in 0.◦2 bins between ILC and HI features for a number of data comparisons are shown. Fig. 1(a) presents the histogram of angular separations between ILC and HI peaks when the ILC peaks for Region A are compared to the HI data for Region A. It shows a clear excess at small angular separations with the peak occurring at about 0.◦8, only slightly larger than the beamwidth. Beyond 1◦ separation, the values fall below expectations for chance associations indicated by a dashed line that represents Eqtn. 1. Of the 51 ILC peaks in Region A, some 60 associations make up the histogram. As will be discussed below, this carries a clue to the nature of the phenomenon producing the associations. A similar excess of offsets between ILC and HI peaks with small angular separations is seen in Fig. 1(b) for Region B, which shows the results obtained by overlaying the ILC data on the HI maps for the same area. The numbers involved are smaller than for Region A with the bulk of the HI concentrated to low velocities. An estimate of the significance of the apparent associations found in Figs. 1(a) and (b) can be obtained by overlaying the ILC data for Region B on the HI peaks for Region A, and vice versa. Figure 1(c) shows what is found when the Region B ILC peaks are overlain on Region A HI maps. Figure 1(d) shows the converse. Neither plot shows evidence for systematic small-scale angular offsets other than might be expected from chance. Finally, negative ILC peaks and HI structure were compared. Figure 1(e) shows the results found by overlaying Region A negative amplitude peaks on Region A HI maps and Fig. 1(d) shows the results derived by overlaying Region B negative ILC peaks on HI data for the Region B. Fig. 1(g) plots the results obtained when comparing the negative ILC peaks for Region B with the HI for Region A and Fig. 1 (h) shows the converse. None of these four plot show convincing evidence for systematic small-scale angular offsets other than might be expected from chance. Another way to summarize these results is to compare the total number of cases of close associations at small angular separations. For example, for those offset by 1.◦1 or less, Region A contains 41 cases, compared to the 23 estimated from chance according to Eqtn. (1), which compares to 20, 17, 21 cases for the plots in Figs. 1 (c),( e) & (g), which are independent estimates of what is to be expected due to chance associations. Similarly, for Region B the direct comparison produces 22 cases of small offsets compared to 13 predicted by chance, with the other three comparisons summarized in Figs. 1 (d), (f) &(h) producing 10, 9 & 6 cases respectively. This suggests that the close associations found in Region B are also significant. – 6 – These plots point to the possibility that the ILC continuum peaks are located at the boundaries of galactic HI emission features, which implies at the interface between HI struc- tures and surrounding plasma, or at the interface between colliding HI structures. 3. Case Studies of Associated HI and Continuum Structures If the close associations between ILC and HI structures found especially in Region A are indeed significant then they should be expected to reveal underlying aspects of interstellar gas dynamics that may allow the cause of the relationships to be understood. That would then remove the study from the realm of a statistics to that of interstellar physics. The cases outlined below emerged from the study of an extended version of Region A with the latitude boundary set to 30◦. Table 1 lists the apparent associations with an identifying number for the ILC peak given in column 1 and its galactic coordinates (in degrees) in columns 2 & 3. The peak ILC brightness temperature in mK is given in column 4. Column 5 gives the center velocity in km/s with respect to the local standard of rest of the HI map used to determine the positions of the relevant HI feature; hence -95 km/s refers to the HI map integrating from -100 to -90 km/s. Columns 6 and 7 give the galactic coordinates for the HI peak with the amplitude in units of K.km/s in Column 8. The angular offset in degrees between associated peaks is given in Column 9. Table 1 includes 64 ILC peaks with amplitudes ≧0.100 mK. All but two of these appear to have associated HI peaks yet the table includes 83 entries for HI associated with the remaining 62 ILC peaks. This is consistent with multiple HI structures being involved in creating some of the continuum emission peaks. In those directions where the most dramatic associations between ILC and HI structure were noted, the HI data were examined in more detail by using HI maps made by integrating over 5, 2 or 1 km/s and latitude-velocity b,v plots with a view to obtaining further insights into the nature of the possible relationship between the two forms of emission. In most of the plots below, the ILC data are shown as contours overlain on the HI morphology plotted using inverted gray-scale shading and these examples are offered be- cause they each reveal something interesting and unexpected about the nature of interstellar matter. This is regarded as highly significant because these directions would never have been chosen for closer study if it were not for the apparent associations with the continuum emission peaks highlighted in the ILC map. – 7 – 3.1. Source 25 at (l,b) = (112.◦3,57◦.8): Directly overlapping HI and continuum features Figure 2(a) shows the brightness of a high-velocity (v ≦-100 km/s) HI feature # 25 from Table 1 centered at -118 km/s integrated over 5 km/s (peak value 51 K.km/s). Fig. 2(b) shows the brightness of an intermediate-velocity (between -100 and -30 km/s) HI feature centered at -87 km/s integrated over 5 km/s (peak value 47 K.km/s). By using Gaussian analysis of the HI profiles every 0.◦5 in latitude and longitude across the peaks, the coordi- nates of the centers of the HI structures were accurately derived and found to be identical in latitude (to < 0.◦1) and also identical to the latitude center of the ILC peak. The centers of two HI features are identical in longitude but offset from the ILC peak by 0.◦3, half the the beamwidth used in the HI studies. Examination of the latitude-velocity b,v plot at l = 112◦ for this feature revealed no significant HI emission at -100 km/s between the two peaks seen in Figs. 2(a) and (b). However, a marked lack of low-velocity HI emission around zero velocity is revealed. (Low- velocity is defined as between + 30 and -30 km/s.) Fig. 2(c) shows the integrated HI content over the velocity interval -8 to +2 km/s with the continuum contours overlain. The peak in the ILC structure is clearly co-located with a lack of low-velocity HI. Also at low velocities an HI peak at (l,b) = (115◦, 57◦) is associated with a secondary peak in the continuum emission. Further relationships emerge when the integrated HI emission between -130 & -120 km/s is examined, Fig. 2(d). Two HI maximum at (l,b) = (113◦,55◦) and (l,b) = (115◦,54◦) overlie HI minima seen in Fig. 2(c). These plots suggest a direct relationship between high- and intermediate-velocity HI and a distinct minimum in low-velocity HI, with all of them related to the presence of an ILC continuum emission peak. This contrasts with those models that would place the high- and intermediate-velocity HI at very different distances well removed from local HI. For example, Wakker (2001) places the high-velocity gas at distances of several kiloparsecs with the intermediate-velocity HI at about 1 kpc, both of which contrast with the realm of high- latitude, low-velocity H, which is local at distances of order 50 to 100 pc. Blaauw & Tolbert (1966) originally noted that intermediate-velocity HI and a relative lack of low-velocity gas is a hallmark of that area of sky encompassed by Region A, which places them both within about 100 pc of the Sun. Here we find striking evidence in Fig. 2 that HI structure in all three velocity regimes is related, something that would not have been noticed but for the apparent association with a significant peak in the ILC structure. This implies that HI gas in all three velocity regimes in this direction is local. (The possibility that small-scale, intermediate-velocity structure is associated with a lack of low-velocity gas has been briefly noted by Burton et al. (1992) as well as Kuntz & Danly (1996) without their following up – 8 – on the implications.) 3.2. Sources 11 & 31 centered at (l,b) = (119.◦5, 57◦) and association between high- and low-velocity HI Figure 3(a) shows the HI structure of high-velocity gas integrated between -140 and -110 km/s associated with two ILC continuum peaks, 11 & 31 (Table 1) identified in Fig. 3(b). The ILC peaks are linked by a ridge of emission straddled by two HI features (peak amplitudes 22 K.km/s) whose morphologies closely follow the continuum radiation contour lines. Gaussian analysis of the HI profiles in a 0.◦5 grid for the entire area covered in Fig. 3(a) was carried out and allowed the center velocities of the two HI components to be determined. They are -127 and -118 km/s for the northern and southern peaks, respectively. The Gaussian analysis also allowed the HI column density for two other features to be mapped, shown in Figs. 3 (b) and (c). Their presence was recognized in a set of HI emission profiles at l=120◦ shown in Fig. 3(d). Most striking is a component at -8 km/s, that is seen only at (l,b) = (120◦,57◦). It has a peak column density of 7×1018 cm−2 and its morphology is plotted in Fig 3(b), which shows that it is located precisely on the saddle in the continuum contours between two peaks. Even more striking is the fact that it is unresolved in angle. Fig. 3(c) shows the column density plot for the -17 km/s feature evident in Fig. 3(d), again based on the results of the Gaussian analysis of the area profiles (peak column density 19×1018 cm−2). The location of this component also appears related to the presence of the continuum ridge with the southern peak located at the position of the HV peak that nestles in the indentation (or pinch) in the continuum emission ridge. At this stage of the analysis, no clear relationship between intermediate-velocity HI structure and other features in the area of Fig. 3 has been noted. However, very weak positive velocity HI emission is also found toward this structure but it will require confirming observations to determine if it is real. The plots in Fig. 3 represent the second example of a close relationship between HI structures at low- and high-velocities, with each of them related to the presence of an ILC feature. In addition, the discovery of the angularly unresolved, low-velocity, high-galactic- latitude emission peak is unprecedented and deserving of observations at much higher angular resolution. – 9 – 3.3. Continuum source 33 at (l,b) = (89.3◦, 34.5◦) and associated high-velocity HI Figure 4 illustrates the dramatic association between ILC emission peak 33 (Table 1) and HI. Here nine inverted gray-scale images of the HI at different velocities are overlain by a contour map of the ILC structure. All the HI plots are single channel maps (1 km/s wide) and the peak brightness temperatures are of order 0.8 K, except for the map at -200 km/s where the peak is 0.3 K and at -150 km/s where it is 1.1 K. The limits of the associated HI emission are -135 km/s at the same location as the peak at -140 km/s, and -205 km/s. The HI centroid of emission shifts in velocity as it follows the ridge of the ILC emission feature with a bifurcation of the HI peak starting at -170 km/s to produce two components that ”move” along opposite sides of the continuum ridge as the velocity is increased. A preliminary attempt to sketch the structure revealed in these and HI l,b maps at smaller velocity intervals suggests a twisted pattern around an axis defined by the contin- uum radiation, possibly related to helical magnetic field structure around this axis. Higher resolution HI data are desirable to untangle the HI structure in this fascinating area. Note that the map of total HI in these directions carries no hint of the existence of the structure seen in Fig. 4 because it is very faint compared to low-velocity gas in this area of sky. Examination of b,v plots and emission profile data for this area reveal further details that are relevant to understanding the apparent relationship between HI and ILC features. This is illustrated in Fig. 5(a) in which several HI emission profiles cutting across the continuum feature seen in Fig. 4 are shown. In a manner similar to that reported in §3.2 above, a distinct additional component, in this case at -40 km/s, emerges in one of the profiles. Its morphology integrated between -42 to -38 km/s is shown in Fig. 5(b) as an inverted gray scale image (peak value 6.7 K.km/s) with the same ILC contours shown in Fig. 4 overlain. Fig. 5(c) illustrates in contour map form how closely this intermediate-velocity HI structure (contours from 3 K.km/s in steps of 0.05 K.km/s) mimics the continuum contours seen in (b). Their axes are aligned to better than 5◦. This HI component is clearly associated with the continuum as well as the HI structure at high velocities seen in Fig. 4. Further examination of the b,v contour maps reveals a phenomenon also found for the HI- continuum association noted in §4.1 (Fig. 2) above, a dearth of low-velocity emission where the anomalous velocity HI structure shows a peak. This is illustrated in Fig 5(d) where the integrated HI emission from -5 to +15 km/s (peak 36 K.km/s) is shown. A distinct minimum (36 K/km/s) compared to the maximum value in this plot (112 K.km/s) in the low velocity integrated HI emission is found at the location of the peak in the intermediate-veloicity HI emission at -40 km/s. This, in turn, coincides with the location of continuum source 32 and the HV structure seen in Fig. 4. – 10 – 3.4. Close-up view of HVC MI Figure 6 shows the inverted gray-scale image of the HI brightness associated with high- velocity cloud MI integrated over the velocity range -140 to -100 km/s. The double HI peaks are offset from and parallel to a pair of ILC peaks, sources 19 and 57. A third ILC peak 43 is also associated with a weak HI feature, see Table 1. This main double HI feature is well known and corresponds to high-velocity cloud HVC MI. A crucial clue as to the likely cause for associations between the small-scale HI and high- frequency continuum structures is found in a report of excess soft X-ray emission toward HVC MI found by Herbstmeier et al. (1995). Their Figure 7a has been adapted to correspond to the data in Figs. 6(a) & (b) and is shown as Fig. 6(c) where the contours correspond to the HI emission from HVC MI similar to the data in (a). The shaded pixels overlain correspond to areas of excess soft X-ray emission. Furthermore, Tufte, Reynolds & Haffner (1998) have reported excess Hα emission at several positions toward HVC MI. The close spatial relationship seen in Fig. 6(c) between the X-ray hot spots and the HI and the high-frequency continuum emission observed by WMAP, as well as the presence of Hα emission contains critical clues that should lead to understanding the nature of the associations between HI and weak, high-frequency continuum emission. It is predicted that high-velocity cloud MII, a companion to MI but located outside Region A, will show a similar relationship between HI and ILC structures. (The outcome of this prediction will be discussed in a future report.) In Fig. 6(d) the ILC contours are overlain on the low-velocity HI data. Again a relative lack of LV emission, here integrated between 5 & +5 km/s, is seen to be closely associated with the continuum and HI structure at high velocities. The HI column density is 1.8 K.km/s for the minimum compared to 6.2 K.km/s for the maximum at the top left of (d). 4. Discussion Several authors have considered the possibility that excess WMAP emission may be produced by spinning dust grains, in turn possibly associated with HI. For example, Davies et al. (2006) note that the spectrum of the low WMAP frequency data is consistent with such a model. In contrast, Banday et al. (2003) find evidence in earlier COBE data of a component in the continuum emission with a dust-like morphology but with a synchrotron- like spectrum. Larson & Wandelt (2004) have suggested that both the ILC data positive and negative peaks do not have sufficient amplitude to be accounted for by a cosmological interpretation of the data, which implies some other mechanism may be playing a role in generating the observed signals. In a specific case involving the detection of anomalous – 11 – microwave emission toward the Perseus molecular cloud, Watson et al. (2005) present an hypothesis that dipole emission from spinning dust grains can account for the spectrum of the continuum emission in the range 10 to 50 GHz. An extensive literature exists concerning the possible role of spinning dust grains as the cause of continuum emission in the frequency range of the WMAP experiment. This includes Draine & Lazarian (1998), Finkbeiner at al. (2002), Schlegel et al. (1998), and de Oliviera et al. (2005). An alternative mechanism for producing low density electrons possibly capable of generating high-frequency radio emission involves a plasma physical phenomenon suggested by Verschuur (2007). In general, many high latitude ILC peaks listed in Table 1 for extended Region A have a corresponding, closely-spaced HI peak found in two or more l,b maps made at 10 km/s intervals. The typical angular offset is approximately 0.◦8 (Fig. 1a). Including the cases below b = 40◦ in the statistics makes no difference to the distribution seen in Fig. 1a. Angular offsets of order 1◦ between parallel HI, dust and Hα filaments have been reported by Verschuur et al. (1992). Based on the above results it is suggested that the ILC peaks are associated with HI structures that are interacting (probably colliding) with other HI structures, or interacting with regions of enhanced plasma density in surrounding interstellar space through which the HI is moving. In one case illustrated above, two HI features at distinctly different velocities are coincident in position at the location of the ILC peak, Fig.2. In other cases, two HI features at (nearly) the same velocity straddle a continuum emission peak in position, for example as shown in Fig. 3. Depending on the geometry of the situation, positional coincidence will only be observed in those directions where a collision between two HI features (clouds) occurs along the line-of-sight. However, where the HI features are interacting while moving along an axis oriented at some angle to the line-of-sight, the continuum emission peak will be observed as offset from the HI peak(s). A number of clues that may help account for the physics underlying the production of continuum emission at the surface of moving HI structures exist in the literature. For example, as was described in §3.4, Herbstmeier et al. (1995) report excess soft X-ray emission at the boundary of HVC MI where the ILC continuum structure is prominent (Fig. 6). On a larger scale, Kerp et al. (1999) find evidence for widespread soft X-ray emission over much of Region A toward the high-velocity structures but the angular resolution of their data do not allow for closer comparison with our results. Furthermore, Tufte et al. (1998) found Hα emission high-velocity clouds MI & MII as well as other high-velocity features. This raises tantalizing questions as to the emission mechanism that could produce the continuum radiation. If it involves the formation of dust at the (shocked) interface between the HI and surroundings by a mechanism involving spinning dust grains such as has been proposed by – 12 – Draine & Lazarian (1998), the presence of both excess Hα and soft X-ray emission will also have to be taken into account. Lagache (2003) has searched for associations between HI and WMAP peaks but confined his study to the total HI content as a function of position. In the examples above the amplitudes of the HI peaks show a relationship to ILC peaks that varies widely and in many cases, such as for the HI features shown in Figs. 4 or 5, are invisible in maps of total HI content over the relevant areas. Land & Slosar (2007) also studied the relationship between WMAP peaks and HI structure but considered only direct, point-to-point associations that are in fact relatively rare. Far more likely are small angular offsets between peaks in the two forms of emission. Another interesting, possibly related phenomenon, has been found by Liu & Zhang (2006) who studied the cross-correlation betweenWMAP and Egret γ ray data and concluded that an unknown source of radiation, most likely of galactic origin, is implied by their analysis. Such a source would produce foreground residuals that need to be removed in order to minimize their role in confusing the cosmological interpretation of the WMAP data. Perhaps the source of this unknown radiation of galactic origin is to be found in processes occurring at the surfaces of Galactic HI structures moving through interstellar space and/or interacting with one another. Based on what is found in these examples, it is possible that where two HI features are interacting (colliding) current sheets are created in which particle acceleration may underlie the production of the continuum emission observed by WMAP. Excess electrons at these interfaces may initially be introduced through the process described by Verschuur (2007) and references therein. To pursue this study this further, ILC structure should be compared with l,b HI maps integrated over smaller velocity ranges that the 10 km/s used in this overview and higher angular resolution HI observations of some of the most interesting HI features shown in Figs. 2 to 6 are desirable. It is hoped that research into the relationships between interstellar HI morphology and high-frequency radio emission observed by WMAP will be stimulated by this work. 5. Conclusions The goal of this study was to determine if evidence exists to suggest that small-scale structure in the WMAP ILC data and HI are related. To that end, the map of ILC structure guided the study and led not only to the discovery of what appear to be highly significant – 13 – spatial relationships between the two data sets but also drew attention to unexpected proper- ties of the galactic HI that turned out to be especially interesting and relevant to accounting for the associations. In Region A clear associations have been found between small-scale structure observed by WMAP and galactic HI features identified in maps in which the HI is integrated over 10 km/s. In Region B a similar excess of associated features was noted. In contrast, no significant associations are found when comparing the ILC data for one Region with the HI data for the other Region, nor when the minima in the ILC data are treated as peaks. This argues that the apparent associations found in Region A are not due to chance. While studying the data for Region B in which the HI is more concentrated to low velocities and latitudes it became clear that HI maps at smaller velocity intervals should be used in the search for associations. A future report will describe such work together with extensive studies of structures at high latitudes in the southern galactic hemisphere. Taken as an ensemble, the spatial associations between HI and ILC emission peaks point to the existence of one or more processes occurring in interstellar space capable of generating weak continuum radiation observed by WMAP. The radiation appears to originate at the surfaces of dynamic and interacting HI structures. Where this interaction is viewed along the direction in which the HI is moving the radio continuum structure will overlap in position but where the HI has a transverse component of motion (much more common) the two forms of emission will appear closely offset on the sky. Of particular significance is the fact that the associations between ILC and HI structure discussed here led to the discovery that HI structures in the three velocity regimes (high, intermediate and low) are physically related to one another, which places them at a common distance from the Sun, probably of order 100 pc. Rigorous studies of the apparent associations between ILC and HI structures should make use of the full LAB survey spectral resolution of 1 km/s to produce HI maps integrating over 2 km/s velocity intervals, a limit set by what is known about low-velocity HI emission line structure. I am grateful to Wayne Landsman and Gary Hinshaw for the first-year ILC continuum emission data in digital form and in rectangular coordinates. The encouragement and advice offered by an anonymous referee as well as Richard Lieu as this paper evolved to completion was essential to its progress. I am particularly grateful to Tom Dame for providing me with the necessary software for a Mac computer to allow the production of the vast majority of the final area maps of HI emission presented in this paper. I also greatly appreciate en- couragement from and discussions with Joan Schmelz and Butler Burton and the important – 14 – feedback offered by Dave Hogg, Mort Roberts, Ed Fomalont, Eric Priest, Petrus Martens, Tony Peratt, Max Bonamente, Shuang-Nan Zhang & Eric Feigelson as well as numerous audience members who attended seminars I gave in the early phases of this project. REFERENCES Albert, C. E. & Danly, L. 2004, p. 73, High-Velocity Clouds, ed. van Woerden, H, Wakker, B.P., Schwarz, U. J. & de Boer, K.S. (Kluwer Academic Publishers: Dordrecht) Banday, A.J., Dickinson, C., Davies. R.D., Davis, R.J., & Gorski, K.M. 2003, MNRAS, 345, Blaauw, A. & Tolbert, C.R. 1966, Bull. Astr. Inst. Neth., 18, 405 Burton, W.B., Hartmann, D. & West, S.R.D., 1996, Unsolved Problems of the Milky Way, Eds. Blitz, L. & Teuben, P. (Springer: New York) Burton, W.B., Bania, T.M., Hartmann, D., & Yuan, T., 1992, ”Evolution of Interstellar Matter and Dynamics of Galaxies”, p25, Eds. Palous, J., Burton, W.B. & Lindblad, P.O. (Cambridge University Press: Cambridge) Davies, R.D.,Dickinson,C., Banday,A.J., Jaffe, T.R., & Gorski, K. M. 2006, MNRAS, 370, Draine, B.T. & Lazarian, A. 1998, ApJ, 494, L19 de Oliviera-Costa, Tegmark, M., Davies, R.D., Gutiérrez, C,M., Lasenby, A.N., Rebolo, R., & Watson, R.A. 2004, ApJ, 606, L89 Finkbeiner, D.P., Schlegel, D.J., Frank, C, & Heiles, C. ApJ, 566, 898 Hartmann, D. & Burton, W.B. 1997, Atlas of Galactic Neutral Hydrogen, Cambridge Univ. Press. Herbstmeier, U., Mebold, U., Snowden, S.L., Hartmann,D., Burton, W.B., Moritz, P., Kalberla, P.M.W., & Egger, R. 1995, A&A, 298, 606 Hinshaw, G. et al. 2006. Available in electronic form at http:lambda.gsfc.nasa.gov. Kalberla, P.M.W., Burton, W.B., Hartmann, D., Arnal, E.M., Bajaja, E., Morras, R., & Pöppel, W.G.L. 2005, A&A, 440, 775 – 15 – Kerp, J., Lesch, H., & Mack, K.-H. A&A, 286, L13 Kuntz, K.D., & Danly, L. 1996, ApJ, 457, 701 Larson, D. L. & Wandelt, B. D. 2004, Ap.J, 613, L85 Lagache, G. 2003, A&A, 405, 813 Land, K & Slosar, A. 2007, Astro-ph 0706.1703v1 Liu, X., & Zhang, S-N. 2006, ApJ, 636, L1 Tufte, S.L., Reynolds, R.J., & Haffner, L.M. 1998, ApJ, 504, 773 Schlegel, D.J., Finkbeiner, D.P., & Davies, M. 1998, ApJ, 500, 525 Verschuur, G. L. 2007, IEEE Trans Plasma Sc., August, in press Verschuur, G.L., Rickard, L. J., Verter, F., Pound, M.W., & Leisawitz, D., 1992, ApJ, 390, Wakker, B.P. 2001, ApJS, 136, 463 Watson, R.A., Rebolo, R., Rubiño-Mart́ın, J.A., Hildebrandt, S., Gutiérrez, C,M., Fernández-Cerezo, S., Hoyland, R.J., & Battistelli, C.M. 2005, ApJ, 624, L89 This preprint was prepared with the AAS LATEX macros v5.2. – 16 – Table 1. TARGET AREA ASSOCIATIONS No. l b ILC Temp. HI Velocity l b HI amplitude Angular offset (1) (2) (3) (4) (5) (6) (7) (8) (9) 1 139.6 40.1 0.227 -55 139.2 40.5 37 0.50 1 139.6 40.1 0.227 5 139.6 41.0 139 0.90 2 174.4 56.8 0.223 -95 175.0 55.6 8 1.24 3 160.3 62.9 0.218 -115 163.0 64.0 6 1.65 3 160.3 62.9 0.218 -15 160.0 61.5 34 1.41 4 176.1 53.0 0.212 5 176.5 53.5 16 0.55 5 140.3 41.5 0.202 -45 140.3 41.0 24 0.50 6 172.3 59.5 0.199 -65 168.2 58.5 27 2.31 7 174.0 50.5 0.182 -125 171.5 51.4 4 1.83 7 174.0 50.5 0.182 -15 173.1 50.1 29 0.70 8 177.2 57.6 0.182 -105 175.5 55.5 8 2.29 9 168.1 67.4 0.180 -125 167.0 65.1 30 2.34 9 168.1 67.4 0.180 -25 168.2 66.5 26 0.90 9 168.1 67.4 0.180 -45 168.3 68.5 44 1.10 10 171.6 56.3 0.179 -65 170.8 56.0 16 0.54 10 171.6 56.3 0.179 -55 172.0 56.0 13 0.37 11 118.1 57.0 0.175 (-121) 120.0 58.0 1.1 1.44 11 118.1 57.0 0.175 (-115) 120.0 56.5 1.0 1.15 11 118.1 57.0 0.175 (-8) 120.0 57.1 1.8 1.04 12 159.6 50.0 0.172 none none none · · · · · · 13 169.5 46.5 0.172 -15 167.5 47.6 42 1.76 13 169.5 46.5 0.172 -5 170.0 47.2 47 0.78 14 174.7 48.2 0.172 -15 175.5 46.5 28 1.78 15 174.4 31.8 0.172 -5 174.0 31.0 110 0.87 16 101.8 59.5 0.169 -5 100.8 58.5 32 1.12 17 78.2 55.9 0.168 5 78.5 56.0 62 0.20 17 78.2 55.9 0.168 -25 79.5 56.5 26 0.94 18 82.3 55.7 0.168 -15 80.0 56.7 35 1.64 19 165.7 64.8 0.168 -125 167.0 65.0 30 0.59 19 165.7 64.8 0.168 -15 165.9 65.4 18 0.61 – 17 – Table 1—Continued No. l b ILC Temp. HI Velocity l b HI amplitude Angular offset 20 91.5 36.7 0.166 -135 91.0 38.1 15 1.46 20 91.5 36.7 0.166 -35 91.1 37.9 17 1.24 20 91.5 36.7 0.166 -15 91.9 37.0 36 0.44 21 170.2 54.5 0.165 -55 170.0 55.0 9 0.51 21 170.2 54.5 0.165 5 169.6 55.9 17 1.44 22 135.4 51.1 0.164 -45 135.9 51.1 98 0.31 23 151.2 48.8 0.162 -165 152.3 48.5 4 0.78 23 151.2 48.8 0.162 -15 152.1 49.1 11 0.66 24 172.6 46.1 0.161 -15 173.3 46.0 22 0.50 25 112.3 57.8 0.159 (-87) 112.8 57.6 4.8 0.33 25 112.3 57.8 0.159 (-125) 112.1 57.9 9 0.15 26 62.6 42.9 0.157 -15 63.5 42.7 19 0.69 27 165.6 48.2 0.155 -15 166.2 48.0 26 0.45 28 65.0 37.1 0.155 -115 64.5 37.5 5 0.56 29 68.6 37.1 0.150 -145 66.6 37.0 5 1.60 30 93.2 66.6 0.150 -5 94.0 65.0 32 1.63 31 121.6 56.8 0.150 -140 to -110 see text see text see text 1.00 32 113.4 66.4 0.149 -15 115.4 66.0 22 0.90 33 89.3 34.5 0.146 (-170) 89.0 34.5 0.8 0.25 33 89.3 34.5 0.146 -42 to -38 89.4 34.5 7 0.08 34 158.8 32.0 0.145 -55 157.5 32.0 30 1.10 35 94.6 38.6 0.143 -165 94.7 38.0 4 0.61 36 77.0 46.9 0.143 -25 79.0 47.0 16 1.37 37 117.0 48.0 0.138 -85 117.5 47.3 19 0.78 38 78.2 62.5 0.138 -25 77.0 62.0 22 0.75 39 62.6 38.7 0.137 -15 61.1 39.5 22 1.42 40 167.3 55.7 0.136 -135 167.0 55.5 16 0.26 41 172.8 35.9 0.132 -5 172.0 36.0 25 0.66 42 174.7 63.6 0.131 -55 172.7 65.0 32 1.66 43 163.8 63.8 0.130 –125 162.1 63.6 4.5 0.78 44 131.8 57.6 0.128 -75 131.0 57.5 30 0.44 – 18 – Table 1—Continued No. l b ILC Temp. HI Velocity l b HI amplitude Angular offset 44 131.8 57.6 0.128 -115 133.0 57.0 33 0.44 45 125.2 40.0 0.127 -65 123.9 40.5 23 1.11 46 138.7 52.7 0.126 -55 139.3 52.5 16 0.41 47 166.3 34.6 0.125 -45 165.8 35.0 36 0.57 48 118.5 68.3 0.124 -25 119.0 68.0 19 0.35 49 132.9 50.0 0.122 none none none · · · · · · 50 153.6 68.5 0.122 -85 155.1 67.5 20 1.14 51 159.3 56.1 0.120 -35 159.9 56.0 16 0.35 52 95.4 46.9 0.120 -115 95.0 46.0 7 0.94 52 95.4 46.9 0.120 -65 96.0 47.5 7 0.73 53 149.4 67.2 0.118 -5 149.6 67.5 56 0.31 54 160.7 57.4 0.115 -85 160.0 58.0 5 0.71 55 142.7 35.2 0.115 -25 143.7 34.5 12 1.20 56 167.3 61.6 0.114 -5 167.9 59.8 18 1.82 57 163.3 66.2 0.114 -125 165.0 66.4 25 0.71 58 140.7 67.0 0.109 -95 138.6 67.0 22 0.82 59 145.7 37.1 0.109 15 145.1 37.9 20 0.93 60 143.4 66.6 0.106 -95 146.3 67.3 14 1.35 61 146.3 36.5 0.105 -55 146.3 36.5 30 0.00 62 158.6 67.2 0.104 -95 160.6 66.0 14 1.43 63 164.2 69.1 0.103 -45 164.0 69.0 28 0.12 64 148.2 67.9 0.100 -85 146.5 67.5 23 0.75 – 19 – Fig. 1.— The distribution of angular separations between ILC peaks and closely spaced HI peaks for b ≧40◦for various combinations of ILC and HI data for the two Regions A &B, see text. The dashed line represents a model calculation for a random distribution of peaks according to Eqtn. 1, and the systematic excess of associations with low angular separations with respect to the model in the top two plots supports the hypothesis that the continuum emission arises at the surfaces of HI features. – 20 – Fig. 2.— ILC peak 25 from Table 1 at (l,b) = (112.◦3, 57.◦8) at the center of the map (contours from +0.04 in steps of 0.02 mK) overlain on HI (inverted gray scale) structure at the velocities indicated. In (a) the ILC peak is associated with a high-velocity HI feature and in (b) with an intermediate-velocity HI feature. In (c) the ILC peak is compared to low velocity HI and overlies a clear HI deficit. It is striking that the integrated high-velocity HI emission displayed in (d) shows two other peaks that are co-located with minima in (c), see text. These structures strongly suggest that the anomalous velocity gas is interacting with low-velocity HI, producing the ILC continuum emission in the process. – 21 – Fig. 3.— (a) The ILC peaks 11 & 31 from Table 1 labeled in (b) with contour levels from 0.02 mK in steps of 0.03 mK centered at (l,b) = (120◦,57◦). A ridge of ILC structures bridges the gap between the two peaks. In (a) the integrated HI brightness over the velocity range indicated shows two high-velocity features with center velocities determined to be -127 & -118 km/s (for the north and south components) straddling the ILC ridge, see text. Frame (d) shows several HI emission profiles at the latitudes indicated. The HI l,b map of the striking, unresolved, component at -8 km/s at b = 57◦ is shown in (b). In (c) the area map of the HI component at -17 km/s is compared to the ILC structure. – 22 – Fig. 4.— The elongated ILC structure, peak 33, Table 1, at (l,b) = (89◦, 34.◦5) (contour levels from 0.03 mK in steps of 0.02 mK) showing a direct relationship to the high-velocity HI structure in 1 km/s bands at the velocities indicated. The HI structure follows the ILC structure closely as the center velocity shifts. At -135 and -205 km/s no significant HI emission is detectable. – 23 – Fig. 5.— Further associations between the ILC continuum source 33, Table 1, and HI. (a) HI profiles toward the ILC peak at the latitude and longitudes indicated. The profile at longitude 89.◦5 shows excess emission at -40 km/s. The gray-scale representation of this feature is shown in (b) with the ILC contours overlain. Frame (c) shows the HI data from (b) in contour form to highlight the similarity to the ILC contours in (b). In (d) the ILC peak is overlain on the HI emission at low velocities showing the coincidence with a lack of low velocity emission. – 24 – Fig. 6.— ILC peaks 19, 43 & 57 around (l,b) = (165◦, 65.◦5) indicated as contours (from 0.06 to 0.20 mK in intervals of 0.02 mK) in (a) (b) & (d) overlain on inverted gray-scale images of the HI emission toward high-velocity cloud MI at the velocities indicated. (c) The location of a soft X-ray excess emission (pixels) with respect to high-velocity cloud MI contours adapted from Herbstmeier et al. (1995), see text. Introduction Data and Analysis Statistics of the Apparent Associations The Angular Distribution of Apparent Associations Case Studies of Associated HI and Continuum Structures Source 25 at (l,b) = (112.3,57.8): Directly overlapping HI and continuum features Sources 11 & 31 centered at (l,b) = (119.5, 57) and association between high- and low-velocity HI Continuum source 33 at (l,b) = (89.3, 34.5) and associated high-velocity HI Close-up view of HVC MI Discussion Conclusions

0704.1126

Negative- and positive-phase-velocity propagation in an isotropic chiral medium moving at constant velocity

arXiv:0704.1126v1 [physics.optics] 9 Apr 2007 Negative– and positive–phase–velocity propagation in an isotropic chiral medium moving at constant velocity Tom G. Mackaya and Akhlesh Lakhtakiab a School of Mathematics James Clerk Maxwell Building University of Edinburgh Edinburgh EH9 3JZ, United Kingdom email: [email protected] b CATMAS — Computational & Theoretical Materials Sciences Group Department of Engineering Science & Mechanics 212 Earth & Engineering Sciences Building Pennsylvania State University, University Park, PA 16802–6812 email: [email protected] Abstract Analysis of electromagnetic planewave propagation in a medium which is a spatiotemporally homoge- neous, temporally nonlocal, isotropic, chiral medium in a co–moving frame of reference shows that the medium is both spatially and temporally nonlocal with respect to all non–co–moving inertial frames of ref- erence. Using the Lorentz transformations of electric and magnetic fields, we show that plane waves which have positive phase velocity in the co–moving frame of reference can have negative phase velocity in cer- tain non–co–moving frames of reference. Similarly, plane waves which have negative phase velocity in the co–moving frame can have positive phase velocity in certain non–co–moving frames. Keywords: Isotropic chiral medium, Lorentz transformation, negative phase velocity, nonlocality 1 Introduction Analysis of planewave propagation in a frame of reference that is uniformly moving with respect to the medium of propagation can lead to the emergence and understanding of new phenomenons. In this respect, the Minkowski constitutive relations, as widely described in standard books [1, 2], are strictly appropriate to instantaneously responding mediums only [3]. For realistic material mediums, recourse should be taken to the Lorentz transformation of electromagnetic field phasors [4, 5]. In an earlier study on plane waves in a medium that is spatiotemporally homogeneous, temporally nonlocal, isotropic and chiral in a co–moving frame of reference, we reported that planewave propagation with negative phase velocity (NPV) is possible with respect to a non–co–moving frame of reference, even though the medium does not support NPV propagation in the co–moving frame [6]. That study applies strictly only at low translational speeds. In this paper, we demonstrate by means of an analysis based on the Lorentz–transformed electromagnetic fields in the non–co-moving frame, that our conclusion remains qualitatively valid for realistic mediums even at high translational speeds. http://arxiv.org/abs/0704.1126v1 2 Planewave analysis We consider a spatiotemporally homogeneous, spatially local, temporally nonlocal, isotropic chiral medium, characterized in the frequency domain by the Tellegen constitutive relations [7] ′ = ǫ0ǫ ′ + i ǫ0µ0ξ B′ = −i√ǫ0µ0ξ′E′ + µ0µ′rH′ , (1) in an inertial frame of reference Σ′. The relative permittivity ǫ′r, relative permeability µ r and chirality parameter ξ′ are complex–valued functions of the angular frequency ω′ if the medium is dissipative, and real–valued if it is nondissipative [1, p. 71]; ǫ0 and µ0 are the permittivity and permeability of free space, respectively. The electromagnetic field phasors are related by the Maxwell curl postulates as ∇×H′ + iω′D′ = 0 ∇×E′ − iω′B′ = 0 . (2) Our attention is focused on a plane wave, described by the Σ′ phasors E′ = E′0 exp [i (k • r′ − ω′t′)] H′ = H′ exp [i (k′ • r′ − ω′t′)] , (3) which propagates in the medium characterized by (1), with wavevector k′ = k′ k̂′ and wavenumber k′ = k0k There are four possibilities for the relative wavenumber: k′r ∈ {k′r1, k′r2, k′r3, k′r4} where k′r1 = r + ξ k′r2 = r − ξ′ k′r3 = −k′r1 k′r4 = −k′r4 . (4) With respect to frame Σ′, the plane wave is assumed to be uniform; i.e., k̂′ ∈ R3. Suppose that the inertial frame Σ′ is moving at constant velocity v = vv̂ relative to another inertial frame Σ. The electromagnetic field phasors in Σ are related to those in Σ′ by the Lorentz transformations [4, 5] E = (E′ • v̂) v̂ + γ I − v̂v̂ ′ − v ×B′ B = (B′ • v̂) v̂ + γ I − v̂v̂ v ×E′ H = (H′ • v̂) v̂ + γ I − v̂v̂ ′ + v ×D′ D = (D′ • v̂) v̂ + γ I − v̂v̂ ′ − v ×H , (5) where I = x̂ x̂ + ŷ ŷ + ẑ ẑ is the 3×3 identity dyadic, γ = 1/ 1− β2 and the relative translational speed β = v/c0, with c0 = 1/ ǫ0µ0 being the speed of light in free space. In terms of the Σ phasors, the plane wave is described by E = E0 exp [i (k • r− ωt)] H = H0 exp [i (k • r− ωt)] . (6) The phasor amplitude vectors {E0,H0} and {E′0,H′0} are related via the transformations (5), whereas [4] I + (γ − 1) v̂ v̂ ′ + γ vt′, (7) t = γ v • r′ , (8) k = γ • v̂ + I − v̂ v̂ ′, (9) ω = γ (ω′ + k′ • v) . (10) Since k′ ∈ C for a dissipative medium, we have from (9) that k = kRk̂R + ikI k̂I with kR ∈ R, kI ∈ R, k̂R ∈ R3, and k̂I ∈ R3, but k̂R 6= k̂I in general; i.e., the plane wave is generally nonuniform with respect to Σ. Similarly, from (10) we have that ω = ωR + iωI with ωR ∈ R and ωI ∈ R. Expressing the Σ phasors as E = {E0 exp [− (kI • r− ωIt)]} exp [i (kR • r− ωRt)] H = {H0 exp [− (kI • r− ωIt)]} exp [i (kR • r− ωRt)] , (11) we note that that the periodic propagation of phase is governed by kR and ωR, whereas attenuation or growth of the wave amplitude is governed by kI and ωI . By writing the phasor amplitudes as E0 = E0R + iE0I and H0 = H0R + iH0I , the corresponding cycle–averaged Poynting vector may be expressed as P = exp (−2kI • r) E0R ×H0R ∫ t0+ cos2 (kR • r− ωRt) exp (2ωIt) dt − (E0I ×H0R +E0R ×H0I)× ∫ t0+ cos (kR • r− ωRt) sin (kR • r− ωRt) exp (2ωIt) dt +E0I ×H0I ∫ t0+ sin2 (kR • r− ωRt) exp (2ωIt) dt for a cycle beginning at time t = t0. The phase velocity is given by k̂R. (13) Whether the plane wave has positive phase velocity (PPV) or negative phase velocity (NPV) in the refer- ence frame Σ is determined by the sign of vp • P: positive for PPV and negative for NPV. Criterions for determining whether the phase velocity is positive or negative with respect to the reference frame Σ′ are presented elsewhere [8, 9]. 3 Numerical results and discussion For the sake of illustration, let us consider the cycle–averaged Poynting vector evaluated at the point r = 0 with the temporal averaging starting from t0 = 0; i.e., 8πωI |ω|2 |ω|2 + ω2I E0R ×H0R −ωRωI (E0I ×H0R +E0R ×H0I) + ω2R E0I ×H0I . (14) Without loss of generality, let us assume that the plane wave propagates along the z′ Cartesian axis; i.e., k̂′ = ẑ′. It follows then from the Maxwell curl postulates (2) that E′ lies in the x′y′ plane with ŷ′ • E′ = ix̂′ • E′ for k′r = k r1, k ŷ′ • E′ = −ix̂′ • E′ for k′r = k r2, k . (15) Further, we take the velocity v to lie in the x′z′ Cartesian plane as per v̂ = x̂′ sin θ + ẑ′ cos θ . (16) In Figure 1 the distributions of PPV and NPV in the reference frame Σ for the dissipative scenario wherein ǫ′r = 6.5 + 1.5i, ξ ′ = 1 + 0.2i and µ′r = 3 + 0.5i are displayed for k r ∈ {k′r1, k′r2, k′r3, k′r4}. Clearly, for all four values of k′r, propagation is of the PPV type for β = 0 (i.e., with respect to the Σ ′ frame). As the relative translational speed increases, the phase velocity of the plane waves corresponding to k′r1 and k eventually becomes negative provided that π/2 < θ < π. In contrast the plane waves corresponding to k′r3 and k′r4 have NPV at sufficiently large values of β provided that 0 < θ < π/2. Now, let us look at the scenario where the isotropic chiral medium supports NPV propagation in the co–moving reference frame. This situation arises when |Re {ξ′} | > Re , for example [8, 9]. We take ǫ′r = 6.5 + 1.5i, ξ ′ = 10 + 2i and µ′r = 3 + 0.5i. The corresponding distributions of NPV and PPV are mapped against β and θ in Figure 2. We see that the plane waves corresponding to relative wavenumbers k′r1 and k r3 have NPV when β is small but have PPV when β is sufficiently large. On the other hand, the plane waves corresponding to relative wavenumbers k′r2 and k r4 have PPV when β is small but have NPV when β is sufficiently large. To conclude, by means of the Lorentz–transformed electromagnetic fields, we have demonstrated that a plane wave with PPV in an isotropic chiral medium can have NPV when observed from a no–co–moving inertial reference frame. Similarly a NPV plane wave in the co–moving frame can be PPV from a non–co– moving frame. Acknowledgements The authors are most grateful to Professor I.M. Besieris (Virginia Polytechnic Institute and State University) for a discussion on the Minkowski constitutive relations for realistic mediums. TGM is supported by a Royal Society of Edinburgh/Scottish Executive Support Research Fellowship. References [1] H.C. Chen, Theory of electromagnetic waves, McGraw–Hill, New York, NY, USA, 1983. [2] J.A. Kong, Electromagnetic wave theory, Wiley, New York, NY, USA 1986. [3] I.M. Besieris and R.T. Compton Jr., Time–dependent Green’s function for electromagnetic waves in moving conducting media, J. Math. Phys. 8 (1967), 2445–2451. [4] C.H. Pappas, Theory of electromagnetic wave propagation, McGraw–Hill, New York, NY, USA, 1965. [5] A. Lakhtakia and W.S. Weiglhofer, Lorentz covariance, Occam’s razor, and a constraint on linear constitutive relations, Phys. Lett. A 213 (1996), 107–111; correction 222 (1996), 459. [6] T.G. Mackay and A. Lakhtakia, On electromagnetics of an isotropic chiral medium moving at constant velocity, Proc. R. Soc. A 463 (2007), 397–418; corrections (submitted). [7] A. Lakhtakia, Beltrami fields in chiral media, World Scientfic, Singapore, 1994. [8] T.G. Mackay, Plane waves with negative phase velocity in isotropic chiral mediums, Microwave Opt Technol Lett 45 (2005), 120–121; corrections: 47 (2005), 406. [9] T.G. Mackay and A. Lakhtakia, Simultaneous negative– and positive–phase–elocity propagation in an isotropic chiral medium, Microwave Opt Technol Lett 49 (2007), 1245–1246. 0 90 ° 180 ° 0 90 ° 180 ° 0 90 ° 180 ° 0 90 ° 180 ° Figure 1: The distribution of positive phase velocity (PPV) and negative phase velocity (NPV) in the Σ reference frame, in relation to β ∈ [0, 1) and θ ∈ [0◦, 180◦), for k′r ∈ {k′r1, k′r2, k′r3, k′r4}. Here, ǫ′r = 6.5+ 1.5i, ξ′ = 1 + 0.2i and µ′r = 3 + 0.5i. 0 90 ° 180 ° 0 90 ° 180 ° 0 90 ° 180 ° 0 90 ° 180 ° Figure 2: As Figure 1 but with ξ′ = 10 + 2i.

0704.1127

Sensitivity of solar off-limb line profiles to electron density stratification and the velocity distribution anisotropy

Astronomy & Astrophysics manuscript no. 2568˙AA November 30, 2018 (DOI: will be inserted by hand later) Sensitivity of solar off-limb line profiles to electron density stratification and the velocity distribution anisotropy N.-E. Raouafi1,2 and S. K. Solanki1 1 Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Straße 2, 37191 Katlenburg-Lindau, Germany 2 National Solar Observatory, 950 North Cherry Avenue, Tucson, AZ 85726, USA e-mail: [email protected] ; [email protected] Received ; accepted Abstract. The effect of the electron density stratification on the intensity profiles of the H I Ly-α line and the O VI and Mg X doublets formed in solar coronal holes is investigated. We employ an analytical 2-D model of the large scale coronal magnetic field that provides a good representation of the corona at the minimum of solar activity. We use the mass-flux conservation equation to determine the outflow speed of the solar wind at any location in the solar corona and take into account the integration along the line of sight (LOS). The main assumption we make is that no anisotropy in the kinetic temperature of the coronal species is considered. We find that at distances greater than 1 R⊙ from the solar surface the widths of the emitted lines of O VI and Mg X are sensitive to the details of the adopted electron density stratification. However, Ly-α, which is a pure radiative line, is hardly affected. The calculated total intensities of Ly-α and the O VI doublet depend to a lesser degree on the density stratification and are comparable to the observed ones for most of the considered density models. The widths of the observed profiles of Ly-α and Mg X are well reproduced by most of the considered electron density stratifications, while for the O VI doublet only few stratifications give satisfying results. The densities deduced from SOHO data result in O VI profiles whose widths and intensity ratio are relatively close to the values observed by UVCS although only isotropic velocity distributions are employed. These density profiles also reproduce the other considered observables with good accuracy. Thus the need for a strong anisotropy of the velocity distribution (i.e. a temperature anisotropy) is not so clear cut as previous investigations of UVCS data suggested. However, these results do not rule completely out the existence of some degree of anisotropy in the corona. The results of the present computations also suggest that the data can also be reproduced if protons, heavy ions and electrons have a common temperature, if the hydrogen and heavy-ion spectral lines are also non-thermally broadened by a roughly equal amount. Key words. Line: profiles – Scattering – Sun: corona – Magnetic fields – Sun: solar wind – Sun: UV radiation 1. Introduction Extreme-ultraviolet (EUV) spectral lines provide valuable in- formation on the plasma conditions in the solar wind accel- eration zone. Information on the densities, the temperatures, the bulk velocities and the velocity distributions of different species, can be obtained from the total intensity, Doppler width and shift of multiple line profiles with the appropriate instru- ment. Basically three techniques are used to deduce both bulk velocity and the velocity distribution. The first is the usual Doppler effect that provides infor- mation on the LOS velocity of the emitting source. The sec- ond is the Doppler dimming which yields information on (microscopic and macroscopic) velocity components not di- rected along the LOS, on the basis of the technique proposed by Rompolt (1967 & 1969) and Hyder & Lites (1970). See, e.g., Beckers & Chipman (1974) for a description. Note that Doppler dimming acts only on the radiative component of a Send offprint requests to: N.-E. Raouafi line. Rompolt (1967, 1969 & 1980), Heinzel & Rompolt (1987) and Gontikakis et al. (1997a,b) used the Doppler dimming of Ly-α to study moving prominences. It has later been considered to diagnose the solar wind (Kohl & Withbroe 1982; Withbroe et al. 1982; Strachan et al. 1989 & 1993; etc.). Optical pumping is a third process. It happens when, due to the Doppler effect, the absorption profile of the moving atom overlaps with a neighboring incident spectral line. This is the case for the coronal O VI 1037.61 Å line that can be excited by the chromospheric C II doublet at 1036.3367 and 1037.0182 Å when the solar wind speed reaches values between ∼ 100 and ∼ 500 km s−1 (Noci et al. 1987; Kohl et al. 1998; Li et al. 1998). Finally, Cranmer et al. (1999a) used optical pumping by Fe III 1035.77 Å to diagnose wind speeds around ∼ 530 km s−1. UVCS (the UltraViolet Coronagraph Spectrometer; Kohl et al. 1995 & 1997) on the SOHO (the SOlar and Heliospheric Observatory; Domingo et al. 1995) mission yielded valuable data on the solar corona, including the polar coronal holes up http://arxiv.org/abs/0704.1127v1 2 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles to more than 3.5R⊙ (Kohl et al. 1997 & 1998; Noci et al. 1997; Habbal et al. 1997; Li et al. 1998; etc.). One of the most excit- ing results obtained from data recorded by UVCS concerns the very broad lines emitted by heavy ions (namely O VI and Mg X) in polar coronal holes during the minimum of the solar activ- ity cycle. From the analysis of profiles of coronal lines (mainly the Ly-α, O VI and Mg X doublets) Kohl et al. (1997 & 1998), Noci et al. (1997) and others concluded that the evidence for highly anisotropic velocity distributions of the cited species is strong. For O VI and Mg X the ratio of kinetic temperatures in the directions perpendicular and parallel to the coronal mag- netic field, respectively, is found to range from 10 to more than 100 (Kohl et al. 1997 & 1998; Cranmer et al. 1999b; etc.). In addition, the heavy ions are deduced to be significantly hotter and faster than the protons (Li et al. 1997). In the present paper we extend the work of Raouafi & Solanki (2004; hereafter abbreviated as RS04) in different ways. Firstly, we consider a more complete set of published density profiles. Secondly, we study the effect of the density stratification on spectral lines having very different formation mechanisms: Ly-α is purely radiative; Mg X lines are exclu- sively collisional and the O VI doublet results from both mech- anisms. In addition, we also calculate the LOS integrated total intensities of the Ly-α and O VI lines. 2. Line formation in the corona In an optically thin medium, the general form of the contribu- tion to the emissivity of a given point on the LOS, assuming a Maxwellian velocity distribution and a Gaussian shaped inci- dent profile, is given by I(ν) = NXn+(θ)Ne(θ)αlu NXn+(θ)Blu IR (4 π) dΩ [f ×D × P ×M] (Ω, θ) where NXn+(θ) and Ne(θ) are densities of scattering ions Xn+ and electrons (cm−3), respectively. Blu and αlu are the Einstein coefficient for absorption between the two atomic lev- els l and u1 and the coefficient of electronic collisions, respec- tively. The width of the microscopic velocity distribution αs is assumed to be isotropic (no anisotropy in the kinetic tem- perature of scattering atoms/ions is considered throughout this paper)). αs includes the contribution due to thermal motions. δν = ν − ν0, where ν0 is the rest frequency of the reemitted line. The geometry of the scattering process is given by Fig. 1 in RS04. The other terms entering Eq. (1) are defined in the same paper. The different values of the coefficient Wul in the expres- sion of M(Ω, θ) and the corresponding spectral lines studied here are given in Table 1. Eq. (1) is general and applies to any spectral line irrespective of whether it is formed by collisions and/or by radiation. An exception is when the incident profiles deviates from a Gaussian (see Sect. 5.1 on how such a case can be dealt with). 1 In the present paper, we consider the assumption of a two level atom. Table 1. Values of Wul for the spectral lines considered here. Jl Ju Wul Examples of spectral lines Mg X 624.941 Å, O VI 1037.61 Å H I 1215.6736 Å Mg X 609.793 Å, O VI 1031.92 Å H I 1215.6682 Å Fig. 1. Electron density plotted as a function of the distance to Sun center for different empirical models (solid curve: DKL (Doyle et al. 1999a,b; Kohl et al. 1998; and Lamy et al. 1997) dotted: Guhathakurta et al. 1999; Dashes: Guhathakurta & Holzer 1994; Dot-dashes: Cranmer et al. 1999c; Triple-dot- dashes: Esser et al. 1999). According to Eq. (1), the profiles created by collisions and by resonant scattering at a given point along the LOS all have Gaussian shapes. However, there are significant differences be- tween the two components. The main differences are: 1) The intensity of the collisional component is proportional to the square of the electron density, where as the radiative com- ponent is only proportional to the density. Thus, the collisional component suffers the radial density drop more than the radia- tive one. 2) The radiative component is affected by the outflow speed of the scattering ions through the Doppler dimming effect, while the collisional component is not. Thus, the contributions to the collisional component from sections along the LOS supporting large outflow speeds are more important than the contributions to the radiative one, since the latter gets progressively out of resonance due to the increase in outflow speed of the solar wind with the distance from Sun center. 3) For a given point on the LOS, the Doppler shift of the col- lisional component reflects the real LOS speed uZ of the scat- tering atoms/ions. The shift of the radiative component, how- ever, corresponds to only a fraction of this speed. This point is very important for the broadening of the line profile. Note that in frame of the complete redistribution approximation, there is no shift difference between collisionally and radiatively excited profiles (see Sahal-Bréchot & Raouafi 2005). Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles 3 3. Atmospheric parameters The realistic synthesis of spectral line profiles requires the modeling of all the parameters entering the formation process of a given spectral line. According to Eq. (1), this includes the magnetic field, the solar wind outflow velocity, the den- sity stratifications of electrons and of atoms/ions, etc., at every point along the LOS. For the large scale magnetic field of the corona and the solar wind velocity we adopt the same descrip- tions as RS04, while for the density stratifications we consider a wider range of models. The coronal magnetic field is described by the model of Banaszkiewicz et al. (1998; see Fig. 1 of RS04). Cranmer et al. (1999c) used the same model to constrain the strength and su- perradial expansion of the magnetic field in the corona. The so- lar wind outflow speed is obtained through the mass-flux con- servation equation V (r, θ) = Ne(R⊙) Ne(r, θ) B(r, θ) B(R⊙, θ⊙) V (R⊙, θ⊙), (2) where Ne(R⊙), V (R⊙, θ⊙) and B(R⊙, θ⊙) are the electron density, the outflow speed of the ions and the coronal magnetic field at the base of the solar corona (solar surface), respectively. Ne(r, θ), V (r, θ) and B(r, θ) are the same quantities at coor- dinates (r, θ), with r > R⊙. The angles θ and θ⊙ are related to each other by the requirement that Eq. (2) is valid along a field line. This also implies that the velocity vector is par- allel to the magnetic field vector (needed for the calculation of the dimming rate of the radiative component and also for the determination of the Doppler shift of the reemitted spec- tral line). In order to be consistent with Ulysses observations that the fast solar wind varies by less than a few percent with latitude (Neugebauer et al. 1998; McComas et al. 2000; Zhang et al. 2002), we adopt V (R⊙, θ⊙) ∝ B(R⊙, θ⊙) as bound- ary condition on the outflow speed on the solar surface (see Table 2). We consider different values of V⊙ for the different species. We also consider different V⊙ for the different density stratification models in order to get as close as possible to the observed values of widths, total intensities and intensity ratios. RS04 have shown that the density stratification is a key quantity for determining the off-limb profiles of collision- dominated lines, in particular the line width. Here we con- sider a more complete set of empirical density stratifications than in that paper. In addition to the electron density profiles from (Doyle et al. 1999a,b; hereafter abbreviated as DKL) determined empirically by combining the results obtained by three instruments on SOHO (SUMER: Doyle et al. 1999a,b; UVCS: Kohl et al. 1999; LASCO: Lamy et al. 1997) and from Guhathakurta & Holzer (1994; SKYLAB) which were consid- ered by RS04, we also employ the models of Guhathakurta et al. (1999; SPARTAN), Cranmer et al. (1999c) and Esser et al. (1999; SPARTAN and Mauna Loa; Fisher & Guhathakurta 1995). The absolute values of the density in all these models are not very different. They are generally within the error bars of the coronal densities which are typically 20 - 30 % of the measured values (the main exception is the Esser et al. model). Except for the DKL model, the data for the other models has been fitted by power series functions based on the fact that the Fig. 2. Outflow speed of the O VI ions along the polar axis and field line arising from 70 degree latitude, respectively, for the different density models considered. The Boundary condition on the outflow speed of the coronal ions is chosen to be pro- portional to the solar surface magnetic field strength (see text). In order to get as close as possible to the measured values, we adopt different values of the proportionality coefficient of V (R⊙, θ⊙) with respect to B(R⊙, θ⊙) for the different den- sity models. The outflow speed of other species can be obtained by multiplying the present ones by a constant. density drops very fast close to the Sun and follows an ∼ r−2 function sufficiently far away. The description given by Doyle et al. for the DKL electron density stratification (Eq. (13) of RS04) assumes that the coronal gas is isothermal and in hydro- static equilibrium (see Guhathakurta & Fisher 1995 & 1998). The absolute value of the electron density directly affects the intensity of the emitted line at any point along the LOS. However, the importance of the density stratification resides therein that it also influences the LOS-integrated profile indi- rectly through the solar wind speed, which is determined via mass-flux conservation and thus depends on both the density stratification and the magnetic field. Even somewhat different density stratifications give quite different solar wind speeds and therefore very different line profiles. A more thorough discus- sion is given in the following Sects. The effect of the density stratification on the outflow speed and LOS velocity has been illustrated by RS04; see their Fig. 5. 4 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles Table 2. Proportionality coefficient, κ, of V (R⊙, θ⊙) toB(R⊙, θ⊙) V (R⊙, θ⊙) = κ×B(R⊙, θ⊙) used for the calculations of the different spectral lines considered in the present paper and corresponding to the different density models. Guhathakurta & Holzer Guhathakurta et al. Cranmer et al. Esser et al. (1994) (1999) (1999c) (1999) H I 0.63 0.85 0.35 0.30 0.13 O VI 0.85 0.88 0.50 0.35 0.13 Mg X 0.85 0.88 0.50 0.35 0.13 The solar wind speed profiles obtained for the combination of the chosen magnetic structure and the density stratifications are plotted in Fig. 2 for two extreme cases. In Fig. 2(a) the wind speed along the polar axis is displayed, while Fig. 2(b) shows the speed along the field line arising from 70 degree latitude on the solar surface. The LOS at 3.5R⊙ intersects this field line at 6 R⊙, which is the distance to which we integrate the profiles. The small decrease in the solar wind speed at high altitudes (solid line in panel a) indicates some mismatch between the DKL model and the Banaszkiewicz et al. magnetic structure right above the pole. This decrease does not affect our con- clusions since it occurs at altitudes above 4 R⊙, whereas the furthest LOS we consider passes at 3.5 R⊙ above the pole. The scaling factors of the solar wind speed at the solar sur- face are listed in Table 2. Clearly although the scaling factors of the heavy elements are not the same as for hydrogen, they differ by less than a factor of 1.5, with the hydrogen atoms moving more slowly. In order to compute the total intensities of the lines emit- ted in the polar coronal holes, we need the absolute density of each ion NXn+ , a parameter not considered by RS04. It can be written as follows NXn+ = Ne, (3) where (NXn+/NX) is obtained from the ionization balance of the X species, (NX/NH) is the abundance of X relative to hy- drogen and (NH/Ne) is the abundance of Hydrogen relative to electrons. Here we adopt a plasma with 10% of helium (the cosmic value) so that NH/Ne = 0.83. However, the coronal helium abundance could be considerably larger and could vary significantly through the region considered here (see Joselyn & Holzer 1978; Hansteen, Holzer & Leer 1993, Hansteen, Leer & Holzer 1994a,b & 1997). The considered fraction of neutral hy- drogen is 2.667 10−7 (John Raymond, private communication). NO5+/NO = 4.78 10 −3 (Nahar 1999) and the abundance of oxygen relative to hydrogen is taken to be 8.7 (Asplund et al. 2004). 4. Effect of the integration along the LOS on the collisional and radiative components The results that will be presented in the present Sect. are ob- tained by using the DKL density stratification. The use of any of the other density models does not change the conclusions. UVCS samples three types of lines: the Mg X lines at 609.793 Å and 624.941 Å are only excited by electron colli- sions, while the Ly-α line is exclusively created by resonant scattering of the solar disk radiation. The contribution of the collisional component is negligible (less than 1% according to Raymond et al. 1997). The O VI lines at 1031.92 Å and 1037.61 Å are excited by the combination of collisions and scattering of the transition region radiation. Here we consider separately the collisional and radiative components of the considered coronal spectral lines in order to illustrate better the effects of the solar wind outflow veloc- ity, density stratification and the integration along the LOS on each of them. The behavior of a particular line depends on the relative strengths of these two components. Figs. 3 and 4 display the profiles emitted from different sections along the LOS for two different projected heliocen- tric distances 1.5 and 3.5 R⊙ (i.e. different rays) for spectral lines created solely by electron collisions and purely from reso- nant scattering of the solar disk radiation, respectively. For the sake of clarity, we used constant αs to illustrate the effect of the LOS integration on the profiles in Figs. 3, 4 & 5. When comparing with observed profiles in Sect. 5, however, we only consider computations with height-dependent αs. At low altitudes (< 2 R⊙) the main contribution is given by the central part of the LOS (i.e. near the polar axis). This is due to the fast drop of the densities as a function of the helio- centric distance (Fig.1). However, when further from the solar disk, significant contributions are obtained from larger sections (larger |Z|) along the LOS, due to the relatively slow decrease of the densities far from the solar limb ( Fig. 1). Doppler dim- ming, which is largest at greater heights, concentrates the con- tribution of the radiative component to the observed line profile toward small |Z| (compare the bottom panels of Figs. 3 & 4). Another striking difference is seen in the Doppler shift behav- ior of the two components. The Doppler shift of the collisional profiles increases when moving away from the polar axis along the LOS (Fig. 3). The Doppler sift of the radiative component also increases at small |Z|, but ever more slowly at larger |Z|. Depending on the profile of the LOS velocity, the line shift can reach a limiting value characteristic of that LOS, or can even start to decrease again at greater |Z| (Fig. 4). The behavior seen in Figs. 3 & 4 can be understood as fol- lows. At a given Z , the Doppler shift of the collisional compo- nent is equal to the LOS speed uZ of the scattering atoms/ions (see first term of the RHS of Eq. (1)). However, for the radiative component the Doppler shift is given by (uZ − αsi nZ u · n). For small |Z| (close to the pole where the field lines are only slightly inclined with respect to the polar axis), nZ is very small and the Doppler shift is approximately equal to uZ , so that collisional and radiative line profiles arising at small |Z| Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles 5 Fig. 3. Contributions to the final line profile emanating from different sections of the LOS (different values of Z) for a spec- tral line excited only by electron collisions (namely a Mg X doublet line) for LOSs coming to within 0.5 R⊙ and 2.5 R⊙ of the solar surface (top and bottom frames, respectively). The LOS integrated profile is the sum of all the individual profiles emitted by the different sections along the LOS. The marked Z values are in units of R⊙. Blue Doppler shifts correspond to positive LOS speeds (toward the observer) that are obtained for positive values of Z . are similar. However, further away from the polar axis, nZ in- creases (in absolute value) so that, depending on nZ and αsi (= α2s/(α i + α s)), the expression (uZ − αsi nZ u · n) either saturates or even decreases again (αi is the Doppler width of the incident line profile). This effect is more marked at higher altitudes where the contribution from LOS sections with large |Z| are important (see bottom panel of Fig. 4). In Fig. 5 profiles for a purely collisionally (a) and purely radiatively (b) excited line are displayed. The plotted lines are obtained by integrating out to different distances along the LOS for a projected heliocentric distance of 3.5R⊙. Profiles of col- lisional lines turn out to be sensitive to the atmospheric parame- Fig. 4. Equivalent results as in Fig.3 but for a purely radiatively excited line. ters, in particular LOS velocity, out to |Z| = 6R⊙. The profile shape of the radiatively excited line is hardly altered by LOS integration, although the total intensity is significantly affected (not visible in Fig. 5 due to the normalization). 5. Application to coronal lines Among the spectral lines observed by UVCS in polar coronal holes three sets are of particular interest. The first is the most intense line emitted in the solar corona, Ly-α. This line is pro- duced by the resonant scattering of the radiation emitted by the solar disk. The few electron collisions are negligible for this line (Raymond et al. 1997). The second group of lines is the O VI doublet. These lines are excited by both, the radia- tion coming from the underlying chromosphere-corona transi- tion zone and by isotropic electron collisions. The third class is composed of the Mg X doublet, whose members are emitted following excitations produced almost exclusively by electron collisions (Wilhelm et al. 2004). 6 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles Fig. 5. Line profiles obtained by integrating over different distances between −Z0 and Z0 along the LOS for a spectral line excited only by electron collisions (a) and only by radiation (b). These profiles are obtained for a point of closest approach of 3.5 R⊙ from Sun center and for αs = 200 km s−1 (αi = 100 km s−1 for the radiative component). The effect of the LOS integration on the radiatively excited line is rather small compared to the collisional line. Fig. 6. Disk emission (+ signs) of Ly-α during the minimum of solar activity (Lemaire et al. 1998). The solid curve is a four- Gaussian fit to the data. The individual Gaussians contributing to the fit are given by the dotted, dashed, dot-dashed and triple- dot-dashed lines. The dependence of αs on r and θ is computed consistently. We assume that αs depends only on r, so that its value changes along the LOS. In the next subsections, we describe calcula- tions of the profiles of these three sets of lines and their com- parison with the observed ones. 5.1. The Ly-α line of hydrogen To calculate the coronal emission H Ly-α, it is necessary to know the shape of the line profile coming from the so- lar disk (Fig. 6). A four Gaussian fit plus a constant back- ground is applied to the complex shape of the observed pro- file. Numerically, we assume that coronal hydrogen atoms are illuminated by the photons of four incident Gaussian profiles Table 3. Parameters of the different Gaussians used to fit the Ly-α line. The Gaussians’ centers are given in (km s−1) with respect to the line center as obtained from the data (+ signs in Fig. 6). The amplitudes are in (1012 photons/cm2/s/(km s−1) and the widths are in (km s−1). Gaussian Amplitude Position Width Dotted 0.22 -94.01 16.81 Dashed 2.24 -54.79 34.39 Dot-dashed 2.70 7.41 125.28 Triple-dot-dashed 2.13 65.13 34.91 emitted by the solar disk, with of course different parame- ters, as listed in Table 3. We do not consider any center-to- limb variation in the intensity emitted by the solar disk in Ly-α (f(Ω, θ) = 1), which is in accordance with the filter images of Bonnet et al. (1980). A single Gaussian is sufficient to fit the calculated off- limb Ly-α profiles with good accuracy. Fig. 7(a) displays the e-folding (Doppler) widths of the LOS integrated Ly-α profiles as a function of the projected heliocentric distance on the plane of the sky (the widths of the best-fit Gaussian are plotted). The values of αs used to reproduce these profiles are also given in the same Figure (dot-dashed line) together with the best fit to the observational data (solid line) by Cranmer et al. (1999a) (see also Kohl et al. 1998). The computed Ly-α profiles are relatively in-sensitive to the details of the electron density stratification. For most of the density stratifications given in Fig. 1, the observed line width of Ly-α in polar coronal holes at different heights above the so- lar limb are comparable to the calculated ones in the presence of an isotropic velocity distribution and a gradual rise in αs with projected distance to the limb. The displayed results are all located in the 1-σ error bars of measured widths except for the widths obtained at low altitudes by using densities given by Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles 7 Fig. 7. Widths (a) and total intensities (b) of the LOS-integrated Ly-α line profile as a function of the projected heliocentric distance obtained for the different density stratification mod- els presented in Fig. 1. The correspondence symbols to density models are given in the top panel. The solid lines is the best fits to the observations recorded by UVCS and the dot-dashed line (top panel) is the turbulence velocity distribution of the hydro- gen atoms. The vertical lines are the statistic error bars of the measurements. Guhathakurta et al. (1999) or between 2.3 and 2.7 R⊙ based on the Esser et al. densities. The reasons for these departures are visible in Fig. 2; note the high/low outflow speeds obtained through these density models at these altitudes. Note that the outflow speed of the hydrogen atoms at the solar surface is a free parameter. By lowering the outflow speed at 1 R⊙ (solar surface) for Guhathakurta et al. (1999) we can obtain a bet- ter correspondence with the measured widths a low altitudes. However, this occurs only at the cost of lower widths also at higher altitudes, so that the overall correspondence with the data is not improved. Fig. 7(a) shows how the contribution to the line width from the small-scale velocity distribution, given by αs, which domi- nates at small r is replaced by contributions from the outflow- ing solar wind at larger r, as the line width significantly exceeds the αs value. Fig. 7(a) also demonstrates that a more rapid drop in den- sity with height leads to broader line profiles (compare profiles resulting from the DKL stratification with that of Guhathakurta et al. 1999). This means that of the two oppositely directed ef- fects introduced by a steeper density gradient, a more rapidly increasing wind speed and a smaller contribution to the mea- sured line profile from large |Z| values, the former dominates. Fig. 7(b) displays the total intensity of Ly-α as a function of the projected heliocentric distance for the different density stratifications considered in the present paper. The best fit to the observed intensities (solid curve) and the error bars of the mea- surements (vertical bars) are also plotted (see Cranmer et al. 1999a). All the density models considered give total intensities that are close to the observed ones (within a factor of 2− 3). 5.2. O VI doublet We compute the intensity profiles of the lines of the O VI doublet at 1031.92 Å 2s 2S1/2 − 2p 2P3/2 and 1037.61 Å 2s 2S1/2 − 2p 2P1/2 . In the solar corona, the O VI ions in- teract with electrons (with isotropic velocity distribution) and with the radiation coming from the underlying transition re- gion. The effects of the ions’ motion are taken into account via the Doppler dimming, the Doppler shift and the optical pump- ing of the O VI 1037.61 Å line by the chromospheric C II dou- blet (1036.3367 Å & 1037.0182 Å). This last effect is impor- tant above ∼ 2R⊙ where the solar wind outflow speed reaches values that enable optical pumping to occur (the exact radius at which this occurs depends on the adopted solar wind profile). The incident solar disk spectrum considered for the calcula- tions of the coronal O VI lines is shown in Fig. 2 of RS04. All the line profiles are assumed to be Gaussians. In the present computations, the radiances of the individual members of the O VI doublet on the disk spectrum are equated to 305 and 152.5 erg cm−2 s−1 sr−1, respectively. Both O VI incident lines are assumed to have the same widths, corresponding to 35 km s−1. We consider also two identical profiles for the C II doublet with widths of 25 km s−1 and radiances of 52 erg cm−2 s−1 sr−1 each. We use the limb-brightening measured by Raouafi et al. (2002) for the O VI line and no limb-brightening for the C II lines (see Warren et al. 1998). For comparison, Table 1 of RS04 presents different values of these parameters measured from different observations. The top panel of Fig. 8 displays the widths of the O VI 1031.92 Å line calculated for rays (LOSs) corresponding to dif- ferent projected heliocentric distances and for different density stratification models. These widths are obtained by applying a Gaussian fit to each calculated profile. All profiles are well rep- resented by a Gaussian. The profiles at the furthest considered ray (3.5 R⊙) show the strongest departure from a Gaussian shape. They are plotted for all considered density models in Fig. 10. The solid line represents the best fit to the UVCS data (Cranmer et al. 1999a). Note that no anisotropy in the kinetic temperature of the emitting ions is considered. 8 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles Fig. 8. The symbols represent the variation of the width (top panel) and intensity ratio (bottom panel) of the LOS integrated synthetic line profiles of O VI as a function of the heliocentric distance. The correspondence between symbols and models is indicated in the grey frame. The dot-dashed curve displays the used values of αs. The observed dependence is given by the solid curves (best fits to the observations; see Cranmer et al. 1999c), with the vertical lines being error bars. At small heights (< 2 R⊙) the widths of the calculated profile are comparable for most of the density models except for the one by Guhathakurta et al. (1999). For this model the boundary condition on the solar wind speed at the solar surface is chosen to fit the widths at high latitudes. This gives higher solar wind speeds already at 1.5 R⊙ and explains the broader profiles resulting from this model at low altitudes. At larger heights (> 2.0 R⊙) the line widths are very sensitive to the details of the electron density stratification. This can be easily seen by the difference in the width of the different profiles ob- tained through slightly different density stratification models. The model by DKL gives line widths comparable to the ones obtained from the data, except between 2 and 2.7 R⊙, where the obtained widths are slightly smaller than the observed ones. The bottom panel of Fig. 8 displays the ratio of total in- tensities of the O VI doublet lines (I1032/I1037) as a function a the projected heliocentric distance. This ratio exhibits a marked dependence on radial distance being well over 2 close to the Fig. 9. Total intensities of the O VI lines as a function of the projected heliocentric distance. The correspondence to the dif- ferent density stratifications is given by the symbols in the top panel. The solid curves are the best fits to the observed inten- sities by UVCS and the vertical bars represent estimates of the accuracy of the observations. Sun, then dropping rapidly (except for the density models by Esser et al. 1999, which produces larger ratios compared to the ones given by the other density models and which do not fit the observations in particular at distances between 2.3 and 3 R⊙). All the other models lead to a minimum in the ratio at r ≈ 2.5−3R⊙, which then increases again at larger r (a num- ber of the considered models exhibit a slightly different behav- ior, showing a slight decrease in the ratio out to r = 3.5 R⊙). Generally, most of the calculated intensity ratios (in particular those obtained from the DKL density stratification) are within the error bars of the ones observed by UVCS. Fig. 9 displays the computed total intensities of the O VI lines as a function of the projected heliocentric distance. The density model of DKL, Cranmer et al., Guhathakurta & Holzer, and for r ≥ 2 R⊙, Esser et al. give comparable intensities of the O VI doublet that agree relatively reasonably with respect to the observations. All models produce a slightly less steep drop in intensity with altitude than suggested by the observations, however. The density models by Guhathakurta et al. (1999) and Esser et al. (1999) give low intensities at low altitudes. This is due the fast drop of the electron density at low altitudes for these two models. Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles 9 Fig. 10. LOS integrated profiles of the O VI 1031.92 Å line obtained at a projected heliocentric distance of 3.5 R⊙ for the different electron density stratifications considered in the present paper. The obtained line profiles are all approxima- tively Gaussian shaped, although they differ strongly in width. 5.3. Mg X doublet The coronal Mg X ion emits a doublet at 609.793 Å and 624.941 Å that corresponds to the atomic transitions 2s 2S1/2− 2p 2P3/2 and 2s 2S1/2 − 2p 2P1/2, respectively. They are formed in the solar corona almost exclusively by electron col- lisions (the disk emission is almost non existent; see Curdt et al. 2001). Due to the weakness of the coronal emission in these lines, UVCS was only able to record data in the polar coronal holes up to a height of 2 R⊙ from Sun center. The members of the doublet have almost identical widths, so that a single value suffices to describe both of them. The widths of the observed profiles are displayed in Fig. 11 together with their error bars. Note the small, but systematic difference between the values determined by Esser et al. (1999) and Kohl et al. (1999). The widths of the calculated profiles and the values of microscopic velocities αs used to reproduce these profiles are plotted in the same Figure. The obtained synthetic profile widths are compa- rable with the observed widths within the accuracy of the data, in particular if we consider the scatter of the data point. Note that we consider only the widths obtained from single Gaussian fits to the data, i.e. assuming no anisotropy of the velocity dis- tribution. The line profiles obtained at 3.5 R⊙ for the differ- ent density models present a slight flattening at the central fre- quency which makes their one-Gaussian-fitting less accurate. The corresponding line widths in Fig. 11 are therefore more uncertain. At heights above 2R⊙ the variation of the width of the syn- thetic profiles of Mg X as a function of the projected heliocen- tric distance shows how sensitive the collisional line profile is to the details of the electron density stratification. This explains why the profiles emitted by heavy ions in the corona, namely O VI and Mg X, which are partially or totally collisional, are larger compared to the ones of Ly-α (pure radiative lines). Fig. 11. Widths of the LOS integrated line profiles of the Mg X doublet plotted as a function of the projected heliocentric dis- tance for the different density stratification models considered in the present paper. The dot-dashed line gives αs as a func- tion of the heliocentric distance. Also plotted are the measured widths of the Mg X lines together with the error bars. For the sake of clarity they are plotted slightly to the right (full circles; data of Kohl et al. 1999) and to the left (full diamonds; data of Esser et al. 1999) from their respective x-coordinates. A comparison with Fig. 8 shows that the line widths ob- tained for the Mg X lines are only slightly larger than for the O VI lines. This suggests that already for O VI the excitation through electron collisions dominates over resonant scattering. 6. Discussion We have considered the effect of the density stratification on the strongest lines observed by UVCS in the polar holes of the solar corona in the presence of a consistent model of the large scale magnetic field and solar wind structure. We consider lines with different formation processes (pure scattering lines, purely col- lisionally excited lines, and lines for which both processes are important; represented by Ly-α, the Mg X and O VI doublets, respectively). We calculate the line profiles, total intensities and (for the O VI lines) intensity ratio of the spectral lines emitted at different altitudes in the polar coronal holes. Integration along the LOS is taken into account. The velocity distributions of the reemitting atoms/ions are considered to be simple Maxwellians with a drift macroscopic velocity vector that is identical to the solar wind outflow velocity, which is calculated according to mass-flux conservation. We note that the main assumption un- derlying this investigation is that no anisotropy in the kinetic temperature of the scattering atoms/ions is considered. 10 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles It is found that whereas the width of the Ly-α profile reacts little to the choice of density stratification, the widths of the O VI and Mg X lines are strongly dependent. The Ly-α total intensities obtained from the different density stratifications are comparable and reasonably fit the observed intensities by most models at most heights. The line widths of the O VI and Mg X doublets obtained at different heights in the polar coronal holes are found to be very sensitive to details of the electron density stratifica- tion. At low altitudes in the polar coronal holes, the calcu- lated profiles of the lines of a given doublet all have compa- rable widths, which is due to the fact that the main contribu- tion comes from the area around the polar axis, due to the fast drop of the electron density with height. At greater heights, the density drops more slowly and contributions to the LOS inte- grated profile from sections of the LOS with large |Z| are in- creasingly important. At these relatively large heights, where the solar wind speed reaches values that leave the O VI line out of resonance, the reemitted lines behave like pure collisional lines. This is confirmed by the fact that the calculated O VI and Mg X lines exhibit similar, large line widths at these altitudes. Their widths are very sensitive to the LOS solar wind speed, which is strongly dependent on the density stratification, so that the differences between the profiles obtained through different density stratifications start to show up clearly at greater heights. The intensity ratios of the O VI doublet drops from values above 2 at heights of 1.5 R⊙ to reach a minimal value that is very close to unity at a height that depends on the density model. For most of the considered density stratification mod- els, the obtained intensity ratios are within the error bars out- lining the scatter of the observations. The LOS integrated total intensities of the O VI lines also show only a weak dependence on the density stratification. Generally, the obtained values are reasonable compared to the observations for both lines of the O VI doublet. Note that for the total intensity the absolute val- ues of the electron densities are as important as their stratifica- tion, which may explain the small dependence of the obtained intensities on the density stratification. For the DKL density stratification the calculated parame- ters are close to those obtained from observations carried out by UVCS, in spite of the fact that we only consider an isotropic ki- netic temperature of the scattering atoms and ions. Our analysis does not aim to decide between density models, since the exact results depend on, e.g, the details of the magnetic structure, or on how well the contribution of the polar plumes or foreground coronal material has been separated from the part of the coronal hole giving rise to the fast wind. However, it shows that a rea- sonable combination of the magnetic, solar wind and density structures exist that reproduce a wide variety of the observed parameters. Our analysis confirms and strengthens the conclu- sion of RS04 that the need for anisotropic velocity distributions (i.e. anisotropy of kinetic temperature of the heavy ions) in the solar corona may not be so pressing as previously concluded, although we stress that the current results do not rule out such anisotropies. It is beyond the scope of the present paper to determine the implications of this result for the mechanisms of heating and acceleration of different species in the polar coronal holes. In particular, it remains to be shown to what extend the ion cy- clotron waves proposed to heat the outer corona in numerous studies (Cranmer et al. 1999c; Isenberg et al. 2000 & 2001; Isenberg 2001 & 2004; Markovskii & Hollweg 2004; etc.) are consistent with an isotropic non-thermal broadening velocity (which is known to be present in all layers of the solar atmo- sphere). We note that the difference between the kinetic temper- atures of heavy ions and protons, found earlier by Li et al. (1997), is also present in our analysis. If, however, we consider the more realistic case that the αs values obtain a contribution both from thermal and non-thermal broadening, then the pro- tons and heavy ions give more consistent results. If we further assume both protons and heavy ions to be affected by a temper- ature of 0.9 106 K, which is typical of the electron temperature in the polar coronal holes (David et al. 1998), we find that the non-thermal broadening for the O VI lines varies between ∼ 85 km s−1 at 1.5 R⊙ to ∼ 210 km s−1 at 3.5 R⊙. For Ly-α these values are ∼ 165 km s−1 and ∼ 185 km s−1, respectively; i.e. they lie between the values deduced from the O VI lines. This relative consistency between the non-thermal broadening of protons and heavy ions is all the more surprising since we did not in any way optimize the computations with such an aim. We note that an extended model including the influence of polar plumes (Raouafi & Solanki, in preparation) suggests that the difference in the non-thermal broadening at low altitudes is at least partly due to the neglect of polar plumes in the present computations. Once more, we stress that the current work in no way rules out alternative interpretations; it simply opens up the possibility of considering models in which protons, heavy ions and electrons all have the same temperature. In agreement with earlier results we find a somewhat lower outflow speed for protons than heavy ions, however. Acknowledgements. The authors would like to thank Dr. T. Holzer for the very constructive and encouraging report on the manuscript. We are grateful to Steven Cranmer, Bernhard Fleck, Bernd Inhester, Eckart Marsch, Klaus Wilhelm, Giannina Poletto and Jean-Claude Vial for helpful discussions and critical comments that greatly im- proved the paper. References Asplund, M., Grevesse, N., Sauval, A. J., Allende Prieto, C., & Kiselman, D. 2004, A&A, 417, 751 Banaszkiewicz, M., Axford, W. I., & McKenzie, J. F. 1998, A&A, 337, 940 Beckers, J. M., & Chipman, E. 1974, Sol. Phys., 34, 151 Bonnet, R. M., Decaudin, M., Bruner, E. C., Jr., Acton, L. W., & Brown, W. A 1980, ApJ, 237, L47 Cranmer, S. R., Kohl, J. L., Noci, G., et al. 1999a, ApJ, 511, 481 Cranmer, S. R., Field, G. B., & Kohl, J. L. 1999b, ApJ, 518, 937 Cranmer, S. R., Field, G. B., & Kohl, J. L. 1999c, Space Sci. Rev., 87, Curdt, W., Brekke, P., Feldman, U., et al. 2001, A&A, 375, 591 David, C., Gabriel, A. H., Bely-Dubau, F., Fludra, A., Lemaire, P., & Wilhelm, K. 1998, A&A, 336, L90 Domingo, V., Fleck, B., & Poland, A. I. 1995, Sol. Phys., 162, 1 Doyle, J. G., Keenan, F. P., Ryans, R. S. I., Aggarwal, K. M., & Fludra, A. 1999a, Sol. Phys., 188, 73 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles 11 Doyle, J. G., Teriaca, L., & Banerjee, D. 1999b, A&A, 349, 956 Esser, R., Fineschi, S., Dobrzycka, D., et al. 1999, ApJ, 510, L63 Fisher, R. A., & Guhathakurta, M. 1995, ApJ, 447, L139 Gontikakis, C., Vial, J.-C., & Gouttebroze, P. 1997a, Sol. Phys., 172, Gontikakis, C., Vial, J.-C., & Gouttebroze, P. 1997b, A&A, 325, 803 Guhathakurta, M., & Holzer, T. E. 1994, ApJ, 426, 782 Guhathakurta, M., & Fisher, R. A. 1995, Geophys. Res. Lett., 22, 1841 Guhathakurta, M., & Fisher, R. A. 1998, ApJ, 499, L215 Guhathakurta, M., Fludra, A., Gibson, S. E., Biesecker, D., & Fisher, R. 1999, J. Geophys. Res., 104, 9801 Habbal, S. R., Woo, R., Fineschi, S., et al. 1997, ApJ, 489, L103 Hansteen, V. H., Holzer, T. E., & Leer, E. 1993, ApJ, 402, 334 Hansteen, V. H., Leer, E., & Holzer, T. E. 1994a, ApJ, 428, 843 Hansteen, V. H., Leer, E., & Holzer, T. E. 1994b, Space Sci. Rev., 70, Hansteen, V. H., Leer, E., & Holzer, T. E. 1997, ApJ, 482, 498 Heinzel, P., & Rompolt, B. 1987, Sol. Phys., 110, 171 Hyder, C. L., & Lites, B. W. 1970, Sol. Phys., 14, 147 Isenberg, P. A., Lee, M. A., & Hollweg, J. V. 2000, Sol. Phys., 193, Isenberg, P. A. 2001, J. Geophys. Res., 106, 29249 Isenberg, P. A., Lee, M. A., & Hollweg, J. V. 2001, J. Geophys. Res., 106, 5649 Isenberg, P. A. 2004, J. Geophys. Res., 109, 3101 Joselyn, J., & Holzer, T. E. 1978, J. Geophys. Res., 83, 1019 Kohl, J. L., & Withbroe, G. L. 1982, ApJ, 256, 263 Kohl, J. L., Esser, R., Gardner, L. D., et al. 1995, Sol. Phys., 162, 313 Kohl, J. L., Noci, G., Antonucci, E., et al. 1997, Sol. Phys., 175, 613 Kohl, J. L., Noci, G., Antonucci, E., et al. 1998, ApJ, 501, L127 Kohl, J. L., Esser, R., Cranmer, S. R., et al. 1999, ApJ, 510, L59 Lamy P., Quemerais E., Liebaria A., et al., 1997, Fifth SOHO Workshop, ESA SP-404, p. 491 Lemaire, P., Emerich, C., Curdt, W., Schuehle, U. & Wilhelm, K. 1998, A&A, 334, 1095L Li, X., Esser, R., Habbal, S. R. & Hu, Y.-Q. 1997, J. Geophys. Res., 102, 17419 Li, X., Habbal, S. R., Kohl, J. L., & Noci, G. 1998, ApJ, 501, L133 Markovskii, S. A., & Hollweg, J. V. 2004, ApJ, 609, 1112 McComas, D. J., Barraclough, B. L., Funsten, H. O., et al. 2000, J. Geophys. Res., 105, 10419 Nahar, S. N. 1999, ApJS, 120, 131 Neugebauer, M., Forsyth, R. J., Galvin, A. B., et al. 1998, J. Geophys. Res., 103, 14587 Noci, G., Kohl, J. L., & Withbroe, G. L. 1987, ApJ, 315, 706 Noci, G., Kohl, J. L., Antonucci, E., et al. 1997, Adv. Space Res., 20, Raouafi, N.-E., Sahal-Bréchot, S., Lemaire, P., & Bommier, V. 2002, A&A, 390, 691 Raouafi, N.-E., & Solanki S. K. 2004, A&A, 427, 725 Raymond, J. C., Kohl, J. L., Noci, G., et al. 1997, Sol. Phys., 175, 645 Rompolt, B. 1967, Acta Astronomica, 17, 329 Rompolt, B. 1969, IV Consultation on Solar Physics and Hydromagnetics. Edited by Jan Megentaler. A Universitatis Wratislaviensis No. 77. Matematyka, Fizyka, Astronomia VIII. Wroclaw, p.117 Rompolt, B. 1980, Hvar Observatory Bulletin, 4, 55 Sahal-Bréchot, S., & Raouafi, N.-E. 2005, A&A, in press Strachan, L., Kohl, J. L., Munro, R. H., & Withbroe, G. L. 1989, BAAS, 21, 840 Strachan, L., Kohl, J. L., Weiser, H., Withbroe, G. L., & Munro, R. H. 1993, ApJ, 412, 410 Warren, H. P., Mariska, J. T., & Wilhelm, K. 1998, ApJS, 119, 105 Wilhelm, K., Dwivedi, B. N., & Teriaca, L. 2004, A&A, 415, 1133 Withbroe, G. L., Kohl, J. L., Weiser, H., & Munro, R. H. 1982, Space Sci. Rev., 33, 17 Zhang, J., Woch, J., Solanki, S. K., & von Steiger, R. 2002, Geophys. Res. Lett., 29, 77 Introduction Line formation in the corona Atmospheric parameters Effect of the integration along the LOS on the collisional and radiative components Application to coronal lines The Ly- line of hydrogen Ovi doublet Mgx doublet Discussion

0704.1128

Subfactors and Hadamard Matrices

7 SUBFACTORS AND HADAMARD MATRICES WES CAMP AND REMUS NICOARA Abstract. To any complex Hadamard matrix H one associates a spin model commuting square, and therefore a hyperfinite subfactor. The stan- dard invariant of this subfactor captures certain ”group-like” symmetries of H . To gain some insight, we compute the first few relative commutants of such subfactors for Hadamard matrices of small dimensions. Also, we show that subfactors arising from Dita type matrices have intermediate subfac- tors, and thus their standard invariants have some extra structure besides the Jones projections. 1. Introduction A complex Hadamard matrix is a matrix H ∈ Mn(C) having all entries of absolute value 1 and all rows mutually orthogonal. Equivalently, 1√ H is a unitary matrix with all entries of the same absolute value. For example, the Fourier matrix Fn = (ω ij)1≤i,j≤n, ω = e 2πi/n, is a Hadamard matrix. In the recent years, complex Hadamard matrices have found applications in various topics of mathematics and physics, such as quantum information theory, error correcting codes, cyclic n-roots, spectral sets and Fuglede’s con- jecture. A general classification of real or complex Hadamard matrices is not available. A catalogue of most known complex Hadamard matrices can be found in [TZ]. The complete classification is known for n ≤ 5 ([H]) and for self-adjoint matrices of order 6 ([BeN]). The connection between Hadamard matrices and von Neumann algebras arose from an observation of Popa ([Po2]): a unitary matrix U is of the form H , H Hadamard matrix, if and only if the algebra of n×n diagonal matrices Dn is orthogonal onto UDnU∗, with respect to the inner product given by the trace on Mn(C). Equivalently, the square of inclusions: C(H) = Dn ⊂ Mn(C) C ⊂ UDnU∗ is a commuting square, in the sense of [Po1],[Po2]. Here τ denotes the trace on Mn(C), normalized such that τ(1) = 1. http://arxiv.org/abs/0704.1128v1 2 WES CAMP AND REMUS NICOARA Such commuting squares are called spin models, the name coming from sta- tistical mechanical considerations (see [JS]). By iterating Jones’ basic con- struction, one can construct a hyperfinite, index n subfactor from H (see for instance [JS]). The subfactor associated to H can be used to capture some of the symmetries of H , and thus to classify H to a certain extent (see [BHJ],[Jo2],[BaN]). Let N ⊂ M be an inclusion of II1 factors of finite index, and let N ⊂ M e2⊂ M2 ⊂ ... be the tower of factors constructed by iterating Jones’ basic construction (see [Jo1]), where e1, e2, ... denote the Jones projections. The standard invariant GN,M is then defined as the trace preserving isomorphism class of the following sequence of commuting squares of inclusions of finite dimensional ∗-algebras: C = N ′ ∩N ⊂ N ′ ∩M ⊂ N ′ ∩M1 ⊂ N ′ ∩M2 ⊂ ... ∪ ∪ ∪ M ′ ∩M ⊂ M ′ ∩M1 ⊂ M ′ ∩M2 ⊂ ... The Jones projections e1, e2, ..., en are always contained in N ′ ∩Mn. If the index of the subfactor N ⊂ M is at least 4, they generate the Temperley-Lieb algebra of order n, denoted TLn. In a lot of situations the relative commutant N ′ ∩Mn has some interesting extra structure, besides TLn. For instance, the five non-equivalent real Hadamard matrices of order 16 yield different dimen- sions for the second relative commutant N ′ ∩ M1, and thus are classified by these dimensions ([BHJ]). In this paper we investigate the relation between Hadamard matrices and their subfactors. We look at Hadamard matrices of small dimensions or of special types. The paper is organized as follows: in the first section we recall, in our present framework, several results of [Jo2],[JS] regarding computations of standard invariants for spin models. In section 2 we study the subfactors associated to Hadamard matrices of Dita type. These are matrices that arise from a construction of [Di], which is a generalization of a construction of Haagerup ([H]). Most known parametric families of Hadamard matrices are of Dita type. We show that the associated subfactors have intermediate subfactors. In the last section we present a list of computations of the second and third relative commutants N ′ ∩ M1, N ′ ∩ M2, for complex Hadamard matrices of small dimensions. We make several remarks and conjectures regarding the structure of the standard invariant. Most of the computations included were done using computers, with the help of the Mathematica and GAP softwares. We would like to thank Teodor Banica, Kyle Beauchamp and Dietmar Bisch for fruitful discussions and correspondence. Wes Camp was supported in part SUBFACTORS AND HADAMARD MATRICES 3 by NSF under Grant No. DMS 0353640 (REU Grant), and Remus Nicoara was supported in part by NSF under Grant No. DMS 0500933. 2. Subfactors associated to Hadamard matrices Let H be a complex n × n Hadamard matrix and let U = 1√ H . U is a unitary matrix, with all entries of the same absolute value. One associates to U the square of inclusions: C(H) = Dn ⊂ Mn(C) C ⊂ UDnU∗ where Dn is the algebra of diagonal n×n matrices and τ is the trace onMn(C), normalized such that τ(1) = 1. Since H is a Hadamard matrix, C(H) is a commuting square in the sense of [Po1],[Po2], i.e. EDnEUDnU∗ = EC. The notation EA refers to the τ -invariant conditional expectation from Mn(C) onto the ∗-subalgebra A. Recall that two complex Hadamard matrices are said to be equivalent if there exist unitary diagonal matrices D1, D2 and permutation matrices P1, P2 such that H2 = P1D1H1D2P2. It is easy to see that H1, H2 are equivalent if and only if C(H1),C(H2) are isomorphic as commuting squares, i.e. conjugate by a unitary from Mn(C). We denote by Ct(H) the commuting square obtained by flipping the upper left and lower right corners of C(H): t(H) = UDnU∗ ⊂ Mn(C) C ⊂ Dn We have: Ct(H) = Ad(U)C(H∗). Thus, Ct(H) and C(H) are isomorphic as commuting squares if and only if H,H∗ are equivalent as Hadamard matrices. We now recall the construction of a subfactor from a commuting square. By iterating Jones’ basic construction ([Jo1]), one obtains from Ct(H) a tower of commuting squares of finite dimensional ∗-algebras: UDnU∗ ⊂ Mn(C) ⊂ ... ∪ ∪ ∪ ∪ C ⊂ Dn ⊂ ... together with the extension of the trace, which we will still denote by τ , and Jones projections gi+2 ∈ Yi, i = 1, 2, .... 4 WES CAMP AND REMUS NICOARA Let MH be the weak closure of ∪iXi, with respect to the trace τ , and let NH be the weak closure of ∪iYi. NH ,MH are hyperfinite II1 factors, and the trace τ extends continuously to the trace of MH , which we will still denote by τ . It is well known that NH ⊂ MH is a subfactor of index n, which we will call the subfactor associated to the Hadamard matrix H . The standard invariant of NH ⊂ MH can be expressed in terms of commu- tants of finite dimensional algebras, by using Ocneanu’s compactness argument (5.7 in [JS]). Consider the basic construction for the commuting square C(H): Dn ⊂ Mn(C) e3⊂ P1 e4⊂ P2 e5⊂ ... ∪ ∪ ∪ ∪ C ⊂ UDnU∗ e3⊂ Q1 e4⊂ Q2 e5⊂ ... Ocneanu’s compactness theorem asserts that the first row of the standard invariant of NH ⊂ MH is the row of inclusions: D′n ∩ UDnU∗ ⊂ D′n ∩ Q1 ⊂ D′n ∩ Q2 ⊂ D′n ∩ Q3 ⊂ ... More precisely, if NH ⊂ MH e3⊂ MH,1 e4⊂ MH,2 e5⊂ ... is the Jones tower obtained from iterating the basic construction for the inclu- sion NH ⊂ MH , then: D′n ∩ Qi = N ′H ∩MH,i, for all i ≥ 1. Thus, the problem of computing the standard invariant of the subfactor associated to H is the same as the computation of D′n ∩ Qi. However, such computations seem very hard, and even for small i and for matrices H of small dimensions they seem to require computer use. Jones ([Jo2]) provided a diagrammatic description of the relative commutants D′n ∩ Qi (see also [JS]), which we express below in the framework of this paper. Let P0 = Mn(C) and let (ei,j)1≤i,j≤n be its canonical matrix units. Let i,j=1 ei,j . It is easy to check that e2 is a projection. Moreover: < Dn, e2 >= Mn(C) and e2xe2 = EDn(x)e2 for all x ∈ Mn(C). Thus, e2 is realizing the basic construction C ⊂ Dn e2⊂ Mn(C) Let ek,l ⊗ ei,j denote the n2 × n2 matrix having only one non-zero entry, equal to 1, at the intersection of row (i − 1)n + k and column (j − 1)n + l. Thus, ek,l ⊗ ei,j are matrix units of Mn(C) ⊗ Mn(C). In what follows, we SUBFACTORS AND HADAMARD MATRICES 5 will assume that the embedding of Mn(C) into Mn(C)⊗Mn(C) is realized as ek,l → ek,l ⊗ In, where ek,l ⊗ In = i=1 ek,l ⊗ ei,i. Lemma 2.1. Let P1 = Mn(C)⊗Dn, P2 = Mn(C)⊗Mn(C), e3 = i=1 ei⊗ei ∈ P1 and e4 = In ⊗ e2 ∈ P2. Then Dn ⊂ Mn(C) e3⊂ P1 is a basic construction with Jones projection e3 and Mn(C) ⊂ P1 e4⊂ P2 is a basic construction with Jones projection e4. Proof. To show that Dn ⊂ Mn(C) e3⊂ P1 is a basic construction it is enough to check that < Mn(C), e3 >= P1 and e3 is implementing EP1Mn(C). First part is clear, since ek,ie3ei,l = ek,l ⊗ ei,i are a basis for P1 = Mn(C) ⊗ Dn. To check that e3 implements the conditional expectation, let X = (xi,j) ∈ Mn(C). We have: e3(X ⊗ In)e3 = i,j=1 (Di ⊗Di)(X ⊗ In)(Dj ⊗Dj) DiXDi ⊗Di (DiXDi ⊗ In)e3 = EDn⊗In(X)e3 Since C ⊂ Dn e2⊂ Mn(C) is a basic construction, after tensoring to the left by Mn(C) it follows that Mn(C) ⊂ P1 e4⊂ P2 is a basic construction, with e4 = In ⊗ e2. Proposition 2.1. The algebras P1,P2,P3, ... constructed in (2) are given by P2k = ⊗k+1i=1Mn(C), P2k+1 = P2k ⊗Dn with the Jones projections e2k+2 = ⊗ki=1In ⊗ e2, e2k+3 = ⊗ki=1In ⊗ e3 Proof. Follows from the previous lemma, by tensoring successively by Mn(C). 6 WES CAMP AND REMUS NICOARA Proposition 2.2. Let H be a complex n×n Hadamard matrix, let U = 1√ i,j=1 ūi,jej,j ⊗ ei,i, U1 = UDU . Then the algebras Q1,Q2,Q3, ... constructed in (2) are given by Qk = UkPk−1U∗k , k ≥ 1 where Uk ∈ Pk are the unitary elements: U2k+1 = Π i=0(⊗iIn ⊗ U1 ⊗k−i In), U2k = U2k−1(⊗kIn ⊗ U), k ≥ 1. Proof. The unitary U1 satisfies: (AdU1)(Dn) = (AdU)(Dn) since U∗U1 = DU ∈ Dn. Moreover, we have: (AdU1)(e2) = (AdU)Ad( i,j=1 ūi,jej,j ⊗ ei,i)( k,l=1 ek,l) = (AdU)( i,k,l=1 ūi,kui,lek,l ⊗ ei,i) = (AdU)(AdU∗(e3)) It follows that AdU1 takes the basic construction C ⊂ Dn e2⊂ Mn(C) onto the inclusion C ⊂ UDnU∗ e3⊂ U1Mn(C)U∗1 . Thus this is also a basic construction, which shows thatQ1 = U1Mn(C)U∗1 . Moreover, it follows that each AdUi takes the basic construction Pi−1 ⊂ Pi ⊂ Pi+1 onto Qi ⊂ Qi+1 ⊂ Qi+2, which ends the proof. The first relative commutant D′n∩UDnU∗ is equal to C, since the commuting square condition implies Dn ∩ UDnU∗ = C. Thus the subfactor NH ⊂ MH is irreducible. In the following proposition we describe the higher relative commutants of the subfactor NH ⊂ MH as the commutants of some matrices Pi, i ≥ 1. Proposition 2.3. With the previous notations, let Pi denote the projection Uiei+3U i ∈ Pi+1, i ≥ 1. Then we have the following formula for the (i+ 1)-th relative commutant: D′n ∩ Qi = P ′i ∩ D′n ∩ Pi. SUBFACTORS AND HADAMARD MATRICES 7 Proof. We have: D′n ∩Qi = D′n ∩AdUi(Pi−1) = D′n ∩AdUi(e′i+3 ∩ Pi) = D′n ∩ P ′i ∩ AdUi(Pi) = D′n ∩ P ′i ∩ Pi We used the fact that Pi−1 ⊂ Pi ⊂ Pi+1 is a basic construction, and thus e′i+3 ∩ Pi = Pi−1. � Remark 2.1. The n2 × n2 matrix P1 = U1e4U∗1 can be written as a,b,c,d=1 a,bea,b ⊗ ec,d, where p a,b = ua,iūb,iūc,iud,i. This matrix is used in the theory of Hadamard matrices and it is called the profile of H. It is a result of Jones ([Jo2]) that the matrices P2i+1, i ≥ 1, depend only on P1. Indeed, one can check that P2i+1 = k1,l1,...,ki,li=1 k1,l1 a,b p k2,l2 k1,l1 ki,li ea,b ⊗ ek1,l1 ⊗ ek2,l2 ⊗ ...⊗ eki,li ⊗ ec,d. Thus, all higher relative commutants of even orders are determined by P1. Let ΓH denote the graph of vertices {1, 2, ..., n} × {1, 2, ..., n}, in which the distinct vertices (a, c) and (b, d) are connected if and only if p a,b 6= 0. The second relative commutant can be easily described in terms of ΓH . We recall this in the following Proposition, which is a reformulation of a result in [Jo2] (see also [JS]). Proposition 2.4. The second relative commutant of the subfactor NH ⊂ MH is abelian, its minimal projections are in bijection with the connected compo- nents of ΓH , and their traces are proportional to the sizes of the connected components. Proof. Let i,j=1 λ iei,i⊗ej,j, λ i ∈ {0, 1}, be a projection in the second relative commutant P ′1 ∩ (Dn ⊗Dn). We have: a,b,c,d=1 a,bea,b⊗ec,d)( i,j=1 iei,i⊗ej,j) = ( i,j=1 iei,i⊗ej,j)( a,b,c,d=1 a,bea,b⊗ec,d) Equivalently: a,c,i,j=1 q,iea,i ⊗ ec,j = b,d,i,j=1 i,b ei,b ⊗ ej,d. 8 WES CAMP AND REMUS NICOARA By relabeling and identifying the set of indices, it follows: (λca − λ a,i = 0. Thus, if the vertices (a, c) and (i, j) are connected then λca = λ i . This ends the proof. � 3. Matrices of Dita type In this section we investigate the standard invariant of subfactors associ- ated to a particular class of Hadamard matrices, obtained by a construction of P.Dita ([Di]), which is a generalization of an idea of U.Haagerup ([H]). These matrices have a lot of symmetries, and we show that for such matrices the sec- ond relative commutant has some extra structure besides the Jones projection. Let n be non-prime, n = kl with k, l ≥ 2. Let A = (ai,j) ∈ Mk(C) and B1, ..., Bk ∈ Mm(C) be complex Hadamard matrices. It is possible to construct an n × n Hadamard matrix from A,B1, ..., Bk by using an idea of [Di] (see also[H],[Pe]). This construction is a generalization of the tensor product of two Hadamard matrices: (6) H = a1,1B1 a1,2B2 ... a1,kBk a2,1B1 a2,2B2 ... a2,kBk ak,1B1 ak,2B2 ... ak,kBk Let (fi,j)1≤i,j≤k be the matrix units of Mk(C). We identify Mn(C) with the tensor product Mm(C)⊗Mk(C), with the same conventions as before. Thus: i,j=1 ai,jBj ⊗ fi,j One can use construct multi-parametric families of non-equivalent Hadamard matrices, by replacing B1, ..., Bk by B1D1, ...BkDk, where D1, ..., Dk are diag- onal unitaries. Some of the families of Hadamard matrices of small orders considered in the next section arise from this construction. Recall that the second relative commutant always contains the Jones projec- tion e3 = eii ⊗ eii. In the next proposition we show that the second relative commutant of a Dita type subfactor contains another projection f ≥ e3, so it has dimension at least 3. Proposition 3.1. Let H = (ai,jBj)1≤i,j≤k ∈ Mn(C) be a Dita type ma- trix, where A = (ai,j)1≤i,j≤k ∈ Mk(C) and B1, ..., Bk ∈ Mm(C) are complex SUBFACTORS AND HADAMARD MATRICES 9 Hadamard matrices, n = mk. Then the second relative commutant of the subfactor associated to H contains the projection: 1≤i,j≤n, i≡j(mod m) ei,i ⊗ ej,j ∈ Mn2(C). Proof. For 1 ≤ i ≤ n, let i0 = (i− 1)(mod m) + 1 and i1 = i−i0m + 1. We will use similar notations for 1 ≤ j ≤ n. Thus, the (i, j) entry of H is: hi,j = ai1,j1b i0,j0 where btr,s is the (r, s) entry of Bt, for all 1 ≤ t ≤ k, 1 ≤ r, s ≤ m. With these notations, the projection f can be written as i,j=1 iei,i ⊗ ej,j where λ i = 1 if i0 = j0 and λ i = 0 for all other i, j. According to Proposition 2.4, showing that f is in the second relative com- mutant is equivalent to showing that p i,c = 0 whenever c0 6= d0. Using the formula for the entries of P1 and the fact that i0 = j0 we obtain: i,c = ui,xūc,xūj,xud,x hi,xh̄c,xh̄j,xhd,x ai1,x1b i0,x0 āc1,x1 b̄ c0,x0 āj1,x1 b̄ j0,x0 ad1,x1b d0,x0 ai1,x1āc1,x1 b̄ c0,x0 āj1,x1ad1,x1b d0,x0 (ai1,x1āc1,x1āj1,x1ad1,x1( b̄x1c0,x0b d0,x0 ai1,x1āc1,x1āj1,x1ad1,x1δ whenever c0 6= d0. � 10 WES CAMP AND REMUS NICOARA We show that in fact the subfactor NH ⊂ MH associated to the Dita matrix H has an intermediate subfactor NH ⊂ RH ⊂ MH , and the projection f is the Bisch projection (in the sense of [Bi]) corresponding to RH . Proposition 3.2. Let H = 1≤i,j≤k ai,jBj ⊗ fi,j ∈ Mn(C) be a Dita type matrix, where A = (ai,j)1≤i,j≤k ∈ Mk(C) and B1, ..., Bk ∈ Mm(C) are complex Hadamard matrices, n = mk. Then: (a). The commuting square C(H) can be decomposed into two adjacent sym- metric commuting squares: Dm ⊗Dk ⊂ Mm(C)⊗Mk(C) Dm ⊗ Ik ⊂ U(Mm(C)⊗Dk)U∗ C ⊂ U(Dm ⊗Dk)U∗ (b). The commuting square Ct(H) can be decomposed into two adjacent symmetric commuting squares: U(Dm ⊗Dk)U∗ ⊂ Mm(C)⊗Mk(C) U(Im ⊗Dk)U∗ ⊂ Dm ⊗Mk(C) C ⊂ Dm ⊗Dk SUBFACTORS AND HADAMARD MATRICES 11 Proof. (a). We first show that Dm ⊗ Ik ⊂ U(Mm(C) ⊗ Dk)U∗. Equivalently, we check that U∗(Dm ⊗ Ik)U ⊂ (Mm(C)⊗Dk). Indeed, for D ∈ Dm we have: U∗(D ⊗ Ik)U = 1≤i′,j′≤k āi′,j′B j′ ⊗ fj′,i′)(D ⊗ Ik)( 1≤i,j≤k ai,jBj ⊗ fi,j) 1≤i,j,j′≤k āi,j′ai,jB j′DBj ⊗ fj′,j 1≤j,j′≤k āi,j′ai,j)B j′DBj ⊗ fj′,j 1≤j,j′≤k j′DBj ⊗ fj′,j 1≤j≤k B∗jDBj ⊗ fj,j ∈ (Mm(C)⊗Dk) The lower square of inclusions is clearly a commuting square, since C(H) is a commuting square. We check that Dm ⊗Dk ⊂ Mm(C)⊗Mk(C) Dm ⊗ Ik ⊂ U(Mm(C)⊗Dk)U∗ is a commuting square. For X ∈ Mm(C) and D ∈ Dk we have: U(X ⊗D)U∗ = 1 1≤i,j≤k ai,jBj ⊗ fi,j)(X ⊗D)( 1≤i′,j′≤k āi′,j′B j′ ⊗ fj′,i′) 1≤i,i′,j≤k āi′,jai,jBjXB j ⊗Dj,jfi,i′ 12 WES CAMP AND REMUS NICOARA Hence: EDn(U(X ⊗D)U∗) = EDn( 1≤i,i′,j≤k āi′,jai,jBjXB j ⊗Dj,jfi,i′) 1≤i,i′,j≤k EDm(āi′,jai,jBjXB j )⊗Dj,jδi i fi,i 1≤i,j≤k Dj,jEDm(BjXB j )⊗ fi,i 1≤j≤k Dj,jEDm(BjXB j )⊗ Ik ∈ Dm ⊗ Ik The lower commuting square is symmetric, since the product of the dimensions of its upper left and lower right corners equals the dimension of its upper right corner. This also implies that the upper commuting square is symmetric, since C(H) is symmetric. (b). The proof is similar to the proof of part (a). � Corollary 3.1. The subfactors associated to Dita matrices have intermediate subfactors. Proof. By iterating the basic construction for the decomposition of Ct(H) in commuting squares, we obtain the towers of algebras: U(Dm ⊗Dk)U∗ ⊂ Mm(C)⊗Mk(C) e3⊂ X1 e4⊂ X2 e5⊂ ... ∪ ∪ ∪ ∪ U(Im ⊗Dk)U∗ ⊂ Dm ⊗Mk(C) e3⊂ R1 e4⊂ R2 e5⊂ ... ∪ ∪ ∪ ∪ C ⊂ Dm ⊗Dk e3⊂ Y1 e4⊂ Y2 e5⊂ ... where Ri =< Ri−1, ei+2 >⊂ Xi. Let RH be the weak closure of ∪iRi. We have NH ⊂ RH ⊂ MH and RH is a II1 factor since the subfactor NH ⊂ MH is irreducible. Remark 3.1. It is immediate to check that the projection f ∈ Mn(C)⊗Mn(C) from Proposition 3.1 implements the conditional expectation from Mn(C)⊗In = Mn(C) onto Dm ⊗ Mk(C). It follows that f is the Bisch projection for the intermediate subfactor NH ⊂ RH ⊂ MH . SUBFACTORS AND HADAMARD MATRICES 13 4. Matrices of small order In this section we compute the second relative commutants of the subfactors associated to Hadamard matrices of small dimensions. For some of the matrices considered we also specify the dimension of the third relative commutant. Most computations included were done with the help of computers, using GAP and Mathematica. It is well known in subfactor theory that the dimension of the second relative commutant D′∩Q1 is at most n, with equality if and only if H is equivalent to a tensor product of Fourier matrices. In this case the subfactor NH ⊂ MH is well understood, being a cross-product subfactor. For this reason, we exclude from our analysis tensor products of Fourier matrices. Some of the matrices we present are parameterized and they yield contin- uous families of complex Hadamard matrices. In such cases, the strategy for computing the second relative commutant will be to determine which entries of the profile matrix P1 depend on the parameters, and for what values of the parameters are these entries 0. According to Proposition 2.4, the second rela- tive commutant will not change as long as the 0 entries of P1 do not change. Thus, to compute the second relative commutant for any other value of the parameters, it is enough to compute it for some random value. We will describe the second relative commutant by specifying its minimal projections. Each such projection p corresponds to a subset S ⊂ {1, 2, ..., n2}: p is the n2 × n2 diagonal matrix having 1 on diagonal positions i ∈ S and 0 on all other positions. Since the Jones projection e3 is always in the second relative commutant, one of the subsets of our partitions will always be {1, n+ 2, 2n+ 3, ..., kn+ k + 1, ..., n2}. Complex Hadamard matrices of dimension 4. There exists, up to equivalence, only one family of complex Hadamard matrices of dimension 4: F4(a) = 1 1 1 1 1 a −1 −a 1 −1 1 −1 1 −a −1 a , |a| = 1 The entries of P1 that depend on the parameter a are . Thus, the second relative commutant is the same for all values of a that are not roots of these equations. The roots a = 1, a = −1 yield matrices that are tensor products of 2 × 2 Fourier matrices. Thus the dimension of the second relative commutant 14 WES CAMP AND REMUS NICOARA is 4, and its minimal projections are given by the partition {1, 6, 11, 16}, {2, 5, 12, 15}, {3, 8, 9, 14}, {4, 7, 10, 13}. The roots a = i, a = −i yield the 4 × 4 Fourier matrix, thus the minimal projections are {1, 6, 11, 16}, {2, 7, 12, 13}, {3, 8, 9, 14}, {4, 5, 10, 15}. Any other values of a, |a| = 1, yield relative commutants of dimension 3: {1, 6, 11, 16}, {2, 4, 5, 7, 10, 12, 13, 15}, {3, 8, 9, 14}. This is not surprising, since this matrix is of Dita type (see Proposition 3.1). The dimension of the third relative commutant is 10, and the dimension of the fourth relative commutant is 35 unless a is a primitive root of order 8 of unity, in which case the dimension is 36. Based on this evidence, we conjecture that the principal graph of the subfactor associated to F4(a) is D 2k if a is a primitive root of order 2k of unity, and D ∞ otherwise. Complex Hadamard matrices of dimension 6. The Fourier matrix F6 is part of an affine 2-parameter family of Dita matrices: F6(a, b) = 1 1 1 1 1 1 1 a e π b e π −1 a π 1 e 1 −a b −1 a −b π 1 e −1 a e 2 i3 π b e i3 π The entries of P1 that depend on a, b are: 2 (1 + a −2 + b−2), 2 + 2 (−1) 2 (−1) , 2 − 2 (−1) 2 (−1) , 2 (1 + a2 + b2), 2 + 2 (−1) 3 a2 − 2 (−1) 3 b2, 2 − 2 (−1) 3 a2 + 2 (−1) 3 b2. Making one of these entries 0 yields the following possibilities: a = −1 3, b = −1 3 or a = −1 3, b = −1 3 or a = 1 3, b = 3 or a = 1 3, b = 1 3 or a = −1 3, b = 1 or a = −1 3, b = 1 3 or a = 1 3, b = −1 3 or a = 3, b = −1 3 or a = −1, b = −1 or a = 1, b = 1 or a = −1, b = 1 or a = 1, b = −1. In each of these cases the matrix F6(a, b) is a tensor product of Fourier matrices. For all other pairs (a, b) satisfying |a| = |b| = 1, the second relative commu- tant has dimension 4: {1, 8, 15, 22, 29, 36}, {2, 4, 6, 7, 9, 11, 14, 16, 18, 19, 21, 23, 26, 28, 30, 31, 33, 35}, {3, 10, 17, 24, 25, 32}, {5, 12, 13, 20, 27, 34}. SUBFACTORS AND HADAMARD MATRICES 15 The following family of self-adjoint, non-affine, complex Hadamard matrices was obtained in [BeN], one of the motivations being the search for Hadamard matrices of small dimensions that might yield subfactors with no extra struc- ture in their relative commutants, besides the Jones projections. BN6(θ) = 1 1 1 1 1 1 1 −1 x̄ −y −x̄ y 1 x −1 t −t −x 1 −ȳ t̄ −1 ȳ −t̄ 1 −x −t̄ y 1 z̄ 1 ȳ −x̄ −t z 1 where θ ∈ [−π,−arcos(−1+ )] ∪ [arcos(−1+ ), π] and the variables x, y, z, t are given by: y = exp(iθ), z = 1 + 2y − y2 y(−1 + 2y + y2) 1 + 2y + y2 − 1 + 2y + 2y3 + y4 1 + 2y − y2 1 + 2y + y2 − 1 + 2y + 2y3 + y4 −1 + 2y + y2 The entries of BN6 do not depend linearly on the parameters, thus this is not a Dita-type family. The corresponding subfactors have the second relative commutant generated by the Jones projection. We conjecture thatBN6(θ) give supertransitive subfactors, i.e. all the relative commutants of higher orders are generated by the Jones projections. There are other interesting complex Hadamard matrices of order 6, such as the one found by Tao in connection to Fuglede’s conjecture ([T]), or the Haagerup matrix ([H],TZ). We computed the second relative commutant for these matrices, and it is generated by the Jones projection. Complex Hadamard matrices of dimension 7. The following one- parameter family was found in [Pe], providing a counterexample to a conjecture of Popa regarding the finiteness of the number of complex Hadamard matrices of prime dimension. 16 WES CAMP AND REMUS NICOARA P7(a) = 1 1 1 1 1 1 1 1 a e π −1 −1 e i3 π π −1 e−i3 π −1 e i3 π π −1 e 1 −1 e−i3 π 1 1 −1 −1 e i3 π e i3 π e−2 i3 π e−i3 π π −1 −1 e−i3 π e−2 i3 π The second relative commutant of the associated subfactors is generated by the Jones projection, for all |a| = 1. For a = 1 we also computed the third relative commutant, and it is just the Temperley-Lieb algebra TL2. We conjecture that P7(a) yield subfactors with no extra structure in their higher order relative commutants, besides the Jones projections. Complex Hadamard matrices of dimension 8. The following 5-parameter family of Hadamard matrices contains the Fourier matrix and is of Dita type: F8(a, b, c, d, z) = 1 1 1 1 1 1 1 1 1 a e π i b c e π −1 a −i b c 1 i d −1 −i d 1 i d −1 −i d π z −i b c e i b c z 1 −1 1 −1 1 −1 1 −1 i b c −1 a e i4 π −i b c e 3 i4 π 1 −i d −1 i d 1 −i d −1 i d −i b c z −1 e 3 i4 π z i b c e The list of possible values of a, b, c, d, z that yield 0 entries for P1 is very long and we do not include it here. Outside these values, the second rela- tive commutant has dimension 4 and it is given by {1, 10, 19, 28, 37, 46, 55, 64}, {2, 4, 6, 8, 9, 11, 13, 15, 18, 20, 22, 24, 25, 27, 29, 31, 34, 36, 38, 40, 41, 43, 45, 47, 50, 52, 54, 56, 57, 59, 61, 63}, {3, 7, 12, 16, 17, 21, 26, 30, 35, 39, 44, 48, 49, 53, 58, 62}, {5, 14, 23, 32, 33, 42, 51, 60}. We analysed several other complex Hadamard matrices besides those in- cluded in this paper, such as those found by [MRS],[Sz]. We tried to cover SUBFACTORS AND HADAMARD MATRICES 17 most known examples of complex Hadamard matrices of dimensions 2, 3, ..., 11. We draw some conclusions: (1) Matrices of Dita type yield subfactors with intermediate subfactors, and thus the second relative commutant has some extra structure besides the Jones projection. We note that parametric families of Dita type exist for every n non-prime, and they contain the Fourier matrix Fn. (2) All non-Dita, non-Fourier matrices we tested have the second relative commutant generated by the Jones projection. The third relative com- mutant is also generated by the first two Jones projections for all cases we could compute. It remains an open problem whether there exist such complex Hadamard matrices that admit symmetries of higher or- References [BHJ] R. Bacher, P. de la Harpe, V. Jones, Carres commutatifs et invariants de structures combinatoires, Comptes rendus Acad. Sci. Math., 1995, vol. 320, no9, 1049-1054 [Bi] D. Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (1994), no. 2, 201-216. [BaN] T. Banica and R. Nicoara, Quantum groups and Hadamard matrices, preprint, math.OA/0610529 [BeN] K. Beauchamp and R. Nicoara, Orthogonal maximal abelian ∗-subalgebras of the 6× 6 matrices, preprint, math.OA/0609076. [Di] P. Dita, Some results on the parametrization of complex Hadamard matrices, J. Phys. A, 37 (2004) no. 20, 5355-5374 [GHJ] F. M. Goodman, P.de la Harpe and V.F.R.Jones,Coxeter Graphs and Towers of Algebras, Math. Sciences Res. Inst. Publ. Springer Verlag 1989 [H] U. Haagerup, Orthogonal maximal abelian *-subalgebras of the n × n matrices and cyclic n-roots, Operator Algebras and Quantum Field Theory (ed. S.Doplicher et al.), International Press (1997),296-322 [HJ] P. de la Harpe and V. F. R. Jones, Paires de sous-algebres semi-simples et graphes fortement reguliers, C.A. Acad. Sci. Paris 311, serie I (1990), 147-150 [Jo1] V. F. R. Jones, Index for subfactors, Invent. Math 72 (1983), 1–25. [Jo2] V. F. R. Jones, Planar Algebras I, math.QA/9909027 [JS] V. F. R. Jones and V. S. Sunder, Introduction to subfactors, Cambridge University press, (1997). [MRS] M. Matolcsi, J. Reffy, F. Szollosi, Constructions of complex Hadamard matrices via tiling abelian groups, preprint quant-ph/0607073. [Ni] R. Nicoara, A finiteness result for commuting squares of matrix algebras, J. Operator Theory 55 (2006), no. 2, 295–310. [Pe] M. Petrescu, Existence of continuous families of complex Hadamard matrices of certain prime dimensions and related results, PhD thesis, University of California Los Angeles, 1997. [Po1] S. Popa, Classification of subfactors : the reduction to commuting squares, Invent. Math., 101(1990),19-43 http://arxiv.org/abs/math/0610529 http://arxiv.org/abs/math/0609076 http://arxiv.org/abs/math/9909027 http://arxiv.org/abs/quant-ph/0607073 18 WES CAMP AND REMUS NICOARA [Po2] S. Popa, Orthogonal pairs of *-subalgebras in finite von Neumann algebras, J. Operator Theory 9, 253-268 (1983) [Sz] F. Szöllősi , Parametrizing Complex Hadamard Matrices, preprint, math.CO/0610297 [TZ] W. Tadej and K. Zyczkowski, A concise guide to complex Hadamard matrices Open Systems & Infor. Dyn.,13(2006), 133-177. [T] T. Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Letters 11 (2004), 251 W.C.: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA E-mail address : [email protected] R.N.: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA E-mail address : [email protected] http://arxiv.org/abs/math/0610297 1. Introduction 2. Subfactors associated to Hadamard matrices 3. Matrices of Dita type 4. Matrices of small order References

0704.1129

SW Sextantis stars: the dominant population of CVs with orbital periods between 3-4 hours

Mon. Not. R. Astron. Soc. 000, (2007) Printed 27 October 2018 (MN LATEX style file v2.2) SWSextantis stars: the dominant population of CVs with orbital periods between 3–4 hours P. Rodŕıguez-Gil1,2⋆, B. T. Gänsicke2, H.-J. Hagen3, S. Araujo-Betancor1, A. Aungwerojwit2,4, C. Allende Prieto5, D. Boyd6, J. Casares1, D. Engels3, O. Giannakis7, E. T. Harlaftis8, J. Kube9, H. Lehto10,11, I. G. Mart́ınez-Pais1,12, R. Schwarz13, W. Skidmore14, A. Staude13 and M. A. P. Torres15 1Instituto de Astrof́ısica de Canarias, Vı́a Láctea s/n, La Laguna, E-38205, Santa Cruz de Tenerife, Spain 2Department of Physics, University of Warwick, Coventry CV4 7AL, UK 3Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany 4Department of Physics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand 5McDonald Observatory and Department of Astronomy, University of Texas, Austin, TX 78712, USA 6British Astronomical Association, Variable Star Section, West Challow, OX12 9TX, England, UK 7Institute of Astronomy and Astrophysics, National Observatory of Athens, P.O. Box 20048, Athens 11810, Greece 8Institute of Space Applications and Remote Sensing, National Observatory of Athens, PO Box 20048, Athens 11810, Greece 9Alfred-Wegener-Institut für Polar- und Meeresforschung, Bürgermeister-Smidt-Straße 20, 27568 Bremerhaven, Germany 10Tuorla Observatory, University of Turku, FIN-21500 Piikkiö, Finland 11Department of Physics, FIN-20014 University of Turku, Finland 12Departamento de Astrof́ısica, Universidad de La Laguna, Tenerife, Spain 13Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany 14California Institute of Technology, Mail Code 105-24, Pasadena, CA 91125-24, USA 15Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138, USA Accepted 2007. Received 2007 ABSTRACT We present time-series optical photometry of five new CVs identified by the Ham- burg Quasar Survey. The deep eclipses observed in HS 0129+2933 (= TT Tri), HS 0220+0603, and HS 0455+8315 provided very accurate orbital periods of 3.35129827(65), 3.58098501(34), and 3.56937674(26) h, respectively. HS 0805+3822 shows grazing eclipses and has a likely orbital period of 3.2169(2) h. Time-resolved optical spectroscopy of the new CVs (with the exception of HS 0805+3822) is also pre- sented. Radial velocity studies of the Balmer emission lines provided an orbital period of 3.55 h for HS 1813+6122, which allowed us to identify the observed photometric signal at 3.39 h as a negative superhump wave. The spectroscopic behaviour exhibited by all the systems clearly identifies them as new SW Sextantis stars. HS 0220+0603 shows unusual N ii and Si ii emission lines suggesting that the donor star may have experienced nuclear evolution via the CNO cycle. These five new additions to the class increase the number of known SW Sex stars to 35. Almost 40 per cent of the total SW Sex population do not show eclipses, invalidating the requirement of eclipses as a defining characteristic of the class and the models based on a high orbital inclination geometry alone. On the other hand, as more SW Sex stars are identified, the predominance of orbital periods in the narrow 3–4.5 h range is becoming more pronounced. In fact, almost half the CVs which populate the 3–4.5 h period interval are definite members of the class. The dominance of SW Sex stars is even stronger in the 2–3 h period gap, where they make up 55 per cent of all known gap CVs. These statistics are confirmed by our results from the Hamburg Quasar Survey CVs. Remarkably, 54 per cent of the Hamburg nova-like variables have been identified as SW Sex stars with orbital periods in the 3–4.5 h range. The observation of this pile-up of systems close to the upper boundary of the period gap is difficult to reconcile with the standard theory of CV evolution, as the SW Sex stars are believed to have the highest mass transfer rates among CVs. Finally, we review the full range of common properties that the SW Sex stars exhibit. Only a comprehensive study of this rich phenomenology will prompt to a full understanding of the phenomenon and its impact on the evolution of CVs and the accretion processes in compact binaries in general. Key words: accretion, accretion discs – binaries: close – stars: individual: HS 0129+2933, HS 0220+0603, HS 0455+8315, HS 0805+3822, HS 1813+6122 – novae, cataclysmic variables ⋆ E-mail:[email protected] c© 2007 RAS http://arxiv.org/abs/0704.1129v2 2 P. Rodŕıguez-Gil et al. 1 INTRODUCTION The Palomar-Green (PG) survey (Green, Schmidt & Liebert 1986) led to the identification of several rela- tively bright (V ∼ 15), deeply eclipsing cataclysmic vari- ables (CVs) with orbital periods in the range 3 − 4 h, namely SWSex (Penning et al. 1984), DWUMa (Shafter, Hessman & Zhang 1988), BHLyn (Thorstensen et al. 1991; Dhillon et al. 1992), and PXAnd (Thorstensen et al. 1991). Szkody & Piché (1990) and Thorstensen et al. (1991) estab- lished a number of common traits among these systems, including unusually “V”-shaped eclipse profiles, the pres- ence of He II λ4686 emission, a substantial orbital phase lag (∼ 0.2 cycle) of the radial velocities of the Balmer lines with respect to the motion of the white dwarf, and single-peaked emission lines that display central absorp- tion dips around orbital phases ≃ 0.4 − 0.7. The “SW Sex phenomenon” was later extended to lower orbital in- clinations after the identification of several grazingly eclips- ing and non-eclipsing CVs which exhibit the spectroscopic properties characteristic of the SW Sex class (WXAri: Beuermann et al. 1992; Rodŕıguez-Gil et al. 2000; V795Her: Casares et al. 1996; LSPeg: Mart́ınez-Pais, Rodŕıguez-Gil & Casares 1999; Taylor, Thorstensen & Patterson 1999; V442Oph: Hoard, Thorstensen & Szkody 2000). The observational characteristics of the SWSex stars are not easily reconciled with the properties of a sim- ple, steady-state hot optically thick accretion disc which is expected to be found in intrinsically bright, weakly- magnetic CVs above the period gap. Nevertheless, a variety of mechanisms have been invoked to explain the behaviour of the SWSex stars, such as stream overflow (Hellier & Robinson 1994; Hellier 1996), mag- netic white dwarfs (Williams 1989; Casares et al. 1996; Rodŕıguez-Gil et al. 2001; Hameury & Lasota 2002), mag- netic propellers in the inner disc (Horne 1999), and self- obscuration of the inner disc (Knigge et al. 2000). While no unambiguous model for their accretion geometry has been found so far, it is becoming increasingly clear that the SWSex stars are not rare and unusual systems, but rep- resent an important, if not dominant fraction of all CVs in the orbital period range 3 − 4 h (Rodŕıguez-Gil 2005; Aungwerojwit et al. 2005). Hence, a thorough investigation of this class of systems is important in the context of under- standing CV evolution as a whole. In this paper, we present five new CVs identified in the Hamburg Quasar Survey (HQS; Hagen et al. 1995) which are classified as new SWSex stars on the basis of our follow- up photometry and spectroscopy. The new discoveries bring the total number of confirmed SWSex stars to 35, and we discuss the global properties of these objects as a class. 2 OBSERVATIONS AND DATA REDUCTION 2.1 Identification The five new CVs (Fig. 1) were selected for follow- up observations upon the detection of emission lines in their HQS spectra. Identification spectra of HS 0220+0603, HS 0805+3822, and HS1813+6122 were respectively ob- tained in October 1990 with the Calar Alto 3.5-m telescope, in December 1996 with the 1.5-m Tillinghast telescope at Table 2. Comparison stars used for the differential CCD pho- tometry (see Fig. 1). ID USNO-A2.0 R B C01 1125-00509642 13.3 14.0 C02 1125-00510117 14.8 15.2 C03 0900-00553937 15.8 17.7 C04 0900-00554015 16.4 18.9 C05 1725-00217755 13.8 15.1 C06 1725-00218117 15.4 15.9 C07 1275-07186372 15.0 15.0 C08 1275-07186091 15.7 17.2 C09 1500-06459684 14.6 15.1 C10 1500-06458763 15.8 16.5 Fred Lawrence Whipple Observatory, and in June 1991 with the Calar Alto 2.2-m telescope as part of QSO and galaxy searches. HS 0129+2933 and HS0455+8315 were both iden- tified as CVs in September 2000 using the Calar Alto 2.2-m telescope as part of a dedicated search for CVs in the HQS (Gänsicke et al. 2002). HS 0129+2933 (= TT Tri) was already identified as an eclipsing star by Romano (1978). At the time of writ- ing this paper we were aware of a multicolour photometric study by Warren, Shafter & Reed (2006), in which an eclipse ephemeris is derived for the first time. HS 0805+3822 was independently found as a CV in the Sloan Digital Sky Survey (SDSSJ080908.39+381406.2, Szkody et al. 2003) and was identified as a SWSex star. Flux-calibrated, low resolution identification spectra of the five new CVs are shown in Fig. 2. The spectra are dom- inated by strong, single-peaked Balmer and He i emission lines on top of blue continua. The high excitation emission lines of He ii λ4686 and the C iii/N iii Bowen blend near 4640 Å are also observed, and are especially intense in HS 0455+8315. Table 1 summarises some properties of the new 2.2 Optical photometry We obtained differential CCD photometry of the five new CVs at seven different telescopes: the 2.2-m telescope at Calar Alto Observatory, the 1.2-m telescope at Kryoneri Ob- servatory, the 0.82-m IAC80 and the 1-m Optical Ground Station at Observatorio del Teide, the 0.7-m telescope of the Astrophysikalisches Institut Potsdam, the 1.2-m tele- scope at Fred Lawrence Whipple Observatory, and the 0.7- m Schmidt-Väisälä telescope at Tuorla Observatory, and the 0.35-m telescope at West Challow Observatory. Details on the instrumentation are given in the notes to Table 3, which also contains the log of the photometric observations. The data obtained at Calar Alto, Kryoneri, and Tuorla were re- duced with the pipeline described by Gänsicke et al. (2004), which applies bias and flat-field corrections in MIDAS and then uses Sextractor (Bertin & Arnouts 1996) to extract aperture photometry for all objects in the field of view. The reduction of the AIP data was carried out completely us- c© 2007 RAS, MNRAS 000, The SW Sextantis stars 3 HS0220+0603 HS0805+3822 E HS1813+6122 Figure 1. 10′ × 10′ finding charts for HS 0129+2933, HS 0220+0603, HS 0455+8315, HS 0805+3822, and HS 1813+6122 obtained from the Digitized Sky Survey. See Table 2 for details on the comparison stars C1–C10. Table 1. Properties of the five new SWSex stars. The coordinates and the B and R magnitudes were taken from the USNO-B catalog (Monet et al. 2003). The J , H, and Ks magnitudes were taken from the 2MASS catalogue. The emission line parameters are measured from the public SDSS spectrum. HS0129+2933 HS 0220+0603 HS 0455+8315 HS 0805+3822 HS 1813+6122 Right ascension (J2000) 01h31m59.86s 02h23m01.65s 05h06m48.27s 08h09m08.40s 18h14m29.83s Declination (J2000) +29◦49′22.3′′ +06◦16′50.0′′ +83◦19′23.3′′ +38◦14′06.5′′ +61◦23′34.5′′ Orbital period (h) 3.35129827(65) 3.58098501(34) 3.56937674(26) ≃ 3.22 ≃ 3.38 B/R magnitudes 15.3 / 14.7 16.5 / 16.2 15.2 / 14.6 14.7 / 14.6 15.1 / 14.5 J/H/Ks magnitudes 14.6 / 14.4 / 14.3 15.4 / 15.1 / 14.8 14.4 / 14.3 / 14.1 15.3 / 15.2 / 14.9 14.8 / 14.7 / 14.6 Hα EW [Å] / FHWM [km s−1] 61 / 1160 73 / 1390 50 / 1180 25 / 1040 22 / 1050 Hβ EW [Å] / FHWM [kms−1] 25 / 1265 43 / 1560 29 / 1250 11 / 970 6 / 1020 Hγ EW [Å] / FHWM [km s−1] 16 / 1350 25 / 1545 31 / 1540 10 / 1010 5 / 930 He I λ5876 EW [Å] / FHWM [kms−1] 8 / 1120 15 / 1270 – / – 2 / 720 1 / 720 He I λ6678 EW [Å] / FHWM [kms−1] 5 / 1090 9 / 1240 10 / 1240 4 / 830 – / – He II λ4686 EW [Å] / FHWM [kms−1] – / – 6 / 2190 17 / 1900 4 / 700 – / – ing MIDAS. IRAF1 was used to correct the OGS, IAC80, and FLWO data for the bias level and flat-field variations, and to compute Point Spread Function (PSF) magnitudes of the target and comparison stars. Fig. 1 shows finding charts for the five new CVs and indicates the comparison stars used for differential photometry. The USNO R and B magnitudes IRAF is distributued by the National Optical Astronomy Ob- servatories. of the comparison stars are given in Table 2. Sample light curves are shown in Fig. 4. HS 0129+2933, HS 0220+0603, and HS0455+8315 are deeply eclipsing, HS 0805+3822 dis- plays evidence for grazing eclipses (similar to WXAri, Rodŕıguez-Gil et al. 2000), and all five systems show sub- stantial short time-scale variability. From the scatter in the comparison star light curves we estimate that the differential photometry is accurate to 1 per cent. c© 2007 RAS, MNRAS 000, 4 P. Rodŕıguez-Gil et al. Figure 2. Flux-calibrated optical spectra of the five new CVs. 2.3 Optical spectroscopy The spectroscopic data were obtained with five different tele- scopes: the 2.5-m Isaac Newton Telescope (INT) and the 2.56-m Nordic Optical Telescope (NOT) on La Palma, the 2.2-m and the 3.5-m telescopes at Calar Alto Observatory, and the 2.7-m telescope at McDonald Observatory. The log of spectroscopic observations can also be found in Table 3. Details on the different telescope/spectrograph setups are given in what follows: 1. At the INT we used the Intermediate Dispersion Spectro- graph (IDS) with the R632V grating, the 2048× 4100 pixel EEV10a CCD detector, and a 1.1′′ slit width. With this setup we sampled the wavelength region λλ4400− 6700 at a resolution (full width at half maximum, hereafter FWHM) of ∼ 2.5 Å. 2. The Andalućıa Faint Object Spectrograph and Camera (ALFOSC) was in place at the NOT. The spectra were im- aged on the 2048×2048 pixel EEV chip (CCD #8). A spec- tral resolution of ∼ 3.7 Å (FHWM) was achieved by using the grism #7 (plus the second-order blocking filter WG345) and a 1′′ slit width. The useful wavelength interval this con- figuration provides is λλ3800− 6800. 3. The spectroscopy at the 2.2-m Calar Alto telescope was performed with CAFOS. A 1.2′′ slit width and the G–100 grism granted access to the λλ4200 − 8300 range with a resolution of ∼ 4.5 Å (FWHM) on the standard SITe CCD (2048× 2048 pixel). 4. The double-armed TWIN spectrograph was used to carry out the observations at the 3.5-m telescope in Calar Alto. The blue arm was equipped with the T05 grating, while the T06 grating was in place in the red arm. The wavelength ranges λλ4070 − 5160 and λλ6080 − 7180 were sampled at 1.32 Å and 1.23 Å resolution (FWHM; 1.5′′ slit width) in the blue and the red, respectively. 5. The Large Cassegrain Spectrometer (LCS) on the 2.7-m telescope at McDonald Observatory was equipped with grat- ing #43 and the TI1 800× 800 pixel CCD detector. The use of a 1.0′′ slit width resulted in a resolution of 3 Å (FWHM) and a wavelength range of λλ3800− 5030. After the effects of bias and flat field structure were removed from the raw images, the sky background was subtracted. The one-dimensional target spectra were then obtained using the optimal extraction algorithm of Horne (1986). For wavelength calibration, a low-order polynomial was fitted to the arc data, the rms being always smaller than one tenth of the dispersion in all cases. The pixel-wavelength correspondence for each target spectrum was obtained by interpolating between the two nearest arc spectra. These re- duction steps were performed with the standard packages for long-slit spectra within IRAF. 2.4 HST/STIS far-ultraviolet spectroscopy A single far-ultraviolet (FUV) spectrum of HS 0455+8315 was obtained with the Hubble Space Telescope/Space Tele- scope Imaging Spectrograph (HST/STIS) as part of a large survey of the FUV emission of CVs (Gänsicke et al. 2003). The data were obtained using the G140L grating and the 52′′ × 0.2′′ aperture, resulting in a spectral resolution of λ/∆λ ≃ 1000 and a wavelength coverage of λλ1150 − 1710. The spectrum (Fig. 12) was obtained at an orbital phase of ≃ 0.75, well outside the eclipse (see Sect. 4.5 for a dis- cussion). The STIS acquisition image showed the system at c© 2007 RAS, MNRAS 000, The SW Sextantis stars 5 Table 3. Log of the observations. Date UT Telescope Filter/ Exp. Frames Grism (s) HS0129+2933 2002 Aug 29 03:22-04:35 INT R632V 600 7 2002 Aug 31 05:00-05:40 INT R632V 600 4 2002 Sep 02 04:20-05:01 INT R632V 600 4 2002 Sep 03 04:39-05:20 INT R632V 600 4 2003 Dec 17 19:33-01:11 NOT Grism #7 600 27 2002 Sep 22 23:42-03:35 KY R 10 1035 2003 Sep 29 05:29-06:00 IAC80 clear 15 75 2003 Dec 15 18:24-23:56 CA22 clear 15 641 2003 Dec 16 18:07-19:35 CA22 clear 20 151 2003 Dec 25 20:06-21:52 CA22 clear 15 176 2006 Nov 21 22:22-23:39 WCO clear 60 73 2006 Dec 11 18:49-19:11 WCO clear 60 22 2006 Dec 16 22:21-23:38 WCO clear 60 73 HS0220+0603 2002 Oct 15 23:13-02:47 KY R 25 328 2002 Oct 17 01:13-03:44 KY R 25 279 2002 Dec 08 23:27-00:13 CA22 G-200 600 4 2002 Dec 16 18:25-19:20 CA22 G-200 600 5 2002 Dec 29 20:39-23:41 CA22 V 30 217 2002 Dec 31 18:30-21:02 CA22 V 30 145 2003 Jan 26 19:41-23:01 IAC80 V,R 100 119 2003 Jan 28 19:24-23:24 IAC80 V 120 102 2003 Jul 11 04:19-05:10 OGS clear 50 54 2003 Jul 13 04:08-05:25 OGS clear 30 119 2003 Sep 22 03:07-04:16 IAC80 clear 30 100 2003 Sep 29 00:07-01:24 IAC80 clear 30 113 2003 Nov 09 00:50-02:03 IAC80 R 60 58 2003 Nov 17 23:41-00:27 OGS clear 17 108 2003 Dec 15 19:28-21:04 NOT Grism #7 600 9 2003 Dec 16 19:26-02:33 NOT Grism #7 600 38 2006 Nov 21 19:15-20:11 WCO clear 60 55 2006 Dec 11 19:16-19:49 WCO clear 60 33 2006 Dec 11 22:47-23:23 WCO clear 60 36 2006 Dec 16 21:04-22:16 WCO clear 60 70 HS0455+8315 2000 Oct 20 18:57-21:44 AIP R 60 142 2000 Nov 10 17:30-21:48 AIP R 30 364 2000 Nov 16 16:45-21:14 AIP R 30 425 2000 Dec 04 17:09-17:50 AIP R,B 30 64 2000 Dec 05 16:39-18:03 AIP R,B 30 135 2000 Jan 01 03:30-05:31 CA35 T05/T06 600 11 2001 Jan 01 23:03-23:45 CA35 T05/T06 300 5 2001 Jan 02 00:32-04:38 CA35 T05/T06 400 33 2002 Dec 09 09:06 HST G140L 730 1 2006 Nov 21 20:30-22:02 WCO clear 60 88 2006 Nov 23 23:05-00:09 WCO clear 60 62 2006 Dec 08 20:05-21:03 WCO clear 50 68 Date UT Telescope Filter/ Exp. Frames Grism (s) 2007 Jan 11 21:28-22:40 WCO clear 60 70 2007 Jan 13 23:33-00:15 WCO clear 50 50 2007 Jan 14 21:01-21:52 WCO clear 50 60 HS0805+3822 2005 Feb 11 19:28-04:32 Tuorla clear 90 325 2005 Feb 15 18:36-01:03 Tuorla clear 80 257 2005 Feb 28 04:39-10:00 FLWO1.2 clear 15-20 769 2005 Mar 01 03:12-09:56 FLWO1.2 clear 15 784 2005 Mar 04 05:59-09:33 FLWO1.2 clear 15-25 402 2005 Mar 24 20:03-00:04 Tuorla clear 30 373 2005 Mar 25 20:00-02:21 Tuorla clear 30 608 2005 Mar 27 03:07-08:18 FLWO1.2 clear 15-20 674 2005 Mar 28 04:06-07:12 FLWO1.2 clear 15-20 390 2005 Mar 29 04:48-08:04 FLWO1.2 clear 15-30 321 HS1813+6122 2000 Sep 24 20:23-20:54 CA22 R-100 600 2 2001 Aug 23 21:46-01:26 AIP R 30 346 2001 Aug 23 20:10-01:05 AIP R 60 251 2001 Aug 23 19:58-00:20 AIP R 60 199 2002 Jul 03 23:47-02:29 KY R 120 67 2002 Jul 05 19:54-22:31 KY R 120 69 2002 Aug 22 18:44-21:36 KY clear 10 660 2002 Aug 23 18:59-22:21 KY clear 10 800 2002 Sep 04 21:24-23:58 KY R 10 516 2002 Sep 04 00:23-00:54 INT R632V 600 4 2002 Sep 06 18:34-21:57 KY R 35 276 2003 Aug 20 18:11-21:43 KY clear 10 649 2002 Aug 27 22:38-00:11 INT R632V 600 9 2002 Aug 31 23:37-00:28 INT R632V 600 5 2002 Sep 01 21:55-22:37 INT R632V 600 4 2002 Sep 03 20:26-21:07 INT R632V 600 4 2003 Jun 28 22:49-00:35 KY R 60 93 2003 Jul 15 02:29-05:09 OGS clear 15 442 2003 Sep 23 20:23-00:05 IAC80 clear 7 690 2003 Sep 24 20:02-22:50 IAC80 clear 7 502 2003 Sep 25 21:18-23:48 IAC80 clear 10 400 2003 Sep 26 19:53-21:08 IAC80 clear 7 248 2003 Sep 27 19:36-23:19 IAC80 clear 7 680 2003 May 18 01:00-04:08 INT R632V 600 6 2003 May 19 02:48-04:13 INT R632V 600 8 2004 May 24 02:13-03:36 IAC80 clear 15 134 2004 May 25 01:44-05:19 IAC80 clear 15 411 2004 May 26 00:34-05:15 IAC80 clear 15 720 2003 Jun 29 00:52-03:58 CA22 G–100 600 15 2003 Jun 30 03:13-04:11 CA22 G–100 600 5 2004 Jul 17 04:17-04:50 McD #43 600 3 2004 Jul 18 04:11-05:19 McD #43 600 6 Notes on the instrumentation used for CCD photometry. CA22: 2.2-m telescope at Calar Alto Observatory, using CAFOS with a 2k × 2k pixel SITe CCD; KY: 1.2-m telescope at Kryoneri Observatory, using a Photometrics SI-502 516 × 516 pixel CCD camera; IAC80: 0.82-m telescope at Observatorio del Teide, equipped with Thomson 1k × 1k pixel CCD camera; OGS: 1-m Optical Ground Station at Observatorio del Teide, equipped with Thomson 1k × 1k pixel CCD camera; AIP: 0.7-m telescope of the Astrophysikalisches Institut Potsdam, with 1k × 1k pixel SITe CCD; FLWO: 1.2-m telescope at Fred Lawrence Whipple Observatory, equipped with MINICAM containing three 2048 × 4608 EEV CCDs; Tuorla: 0.7-m Schmidt-Väisälä telescope at Tuorla Observatory, equipped with a SBIG ST–8 CCD camera; WCO: 0.35-m telescope at West Challow Observatory with SXV-H9 CCD camera. c© 2007 RAS, MNRAS 000, 6 P. Rodŕıguez-Gil et al. Table 4. Eclipse timings (given in HJD − 2450000), cycle num- ber, and the difference in observed minus computed eclipse times using the ephemerides in Eqs. (1–4). See Warren et al. (2006) for additional eclipse timings of HS 0129+2933. T0 Cycle O − C (s) T0 Cycle O − C (s) HS0129+2933 2961.51100 2667 –1 2540.53210 0 –30 4061.32113 10038 10 2989.32739 3214 22 4081.31481 10172 –4 2989.46702 3215 21 4081.46397 10173 –8 2990.30418 3221 –36 4086.38789 10206 –2 2999.38161 3286 50 HS0455+8315 4061.46337 10892 5 1859.24683 0 –7 4081.29206 11034 20 1859.39549 1 –13 4086.45840 11071 –1 1865.34449 41 –12 HS0220+0603 1884.23301 168 8 2563.57452 0 42 1884.23333 168 24 2564.61852 7 2 4061.40138 14807 –28 2638.47610 502 –18 4063.48339 14821 –13 2666.37811 689 –3 4078.35639 14921 43 2666.37818 689 3 4112.41340 15150 –25 2668.31751 702 –29 4114.49583 15164 0 2668.46682 703 –20 4115.38843 15170 22 2831.70004 1797 –21 HS0805+3822 2834.68439 1817 –5 3413.42628 0 –9 2904.66298 2286 10 3455.38312 313 231 2911.52646 2332 4 3455.51361 314 –75 2952.55900 2607 40 3457.79144 331 –147 an approximate Rc magnitude of 15.4, close to the normal high-state brightness. 3 PHOTOMETRIC PERIODS 3.1 HS 0129+2933, HS 0220+0603, and HS 0455+8315 The deep eclipses detected in the light curves of HS 0129+2933, HS 0220+0603, and HS0455+8315 provide accurate information on the orbital periods of the systems. Determining the time of mid-eclipse2 in SWSex stars is notoriously difficult due to the asymmetric shape of the eclipse profiles. We have therefore employed the following method: the observed eclipse profile is mirrored in time around an estimate of the eclipse centre, and the mirrored profile is overplotted on the original eclipse data. The time of mid-eclipse is then varied until the central part of both eclipse profiles overlap as closely as possible. This empiri- cal method proved to be somewhat more robust compared to e.g. fitting a parabola to the eclipse profile. The mid- eclipse times are reported in Table 4. An initial estimate of the cycle count was then obtained by fitting eclipse phases (φobserved0 −φ −2 over a wide range of trial periods (Fig. 3). Once an unambiguous cycle count was established, a lin- ear eclipse ephemeris was fitted to the times of mid-eclipse. For HS 0129+2933, we also included the 22 eclipse timings 2 The time of mid-eclipse, T0, is the time of inferior conjunction of the donor star. reported by Warren et al. (2006) in our calculations. The resulting ephemerides are: T0(HJD) = 2 452 540.53244(21) + 0.139637428(27) × E (1) for HS 0129+2933, i.e. Porb = 3.35129827(65)) h, T0(HJD) = 2 452 563.574036(73) + 0.149207709(14) × E (2) for HS 0220+0603, i.e. Porb = 3.58098501(34) h, and T0(HJD) = 2 451 859.24463(12) + 0.148724030(11) × E (3) for HS 0455+8315, i.e. Porb = 3.56937674(26) h. 3.2 HS 0805+3822 Some of the light curves of HS 0805+3822 contain broad dips that we interpret as grazing eclipses, similar to those detected in WXAri (Rodŕıguez-Gil et al. 2000). The system displays a varying level of short time-scale variability, and we restrict the identification of the (presumed) eclipses to the nights with low flickering activity (Fig. 4, Table 4). No unique cycle count can be determined from the four eclipses detected (Fig. 3), and the eclipse ephemerides determined for the two most likely cycle count aliases are T0(HJD) = 2453413.4264(22) + 0.1267611(75) × E , (4) i.e. Porb = 3.0423(2) h, and T0(HJD) = 2453413.4264(23) + 0.1340385(84) × E , (5) i.e. Porb = 3.2169(2) h. The data of the nights during which grazing eclipses were detected folded over both periods are shown in Fig. 5. As an alternative approach, we have subjected the combined data from all nights to an analysis-of-variance (after subtracting the nightly mean magnitudes) us- ing Schwarzenberg-Czerny’s (1996) method, and find the strongest signal at 3.21 h (Fig. 3). Folding all data on that period gives a light curve which broadly resembles the “eclipse” light curves mentioned above. We conclude that the orbital period of HS 0805+3822 is ≃ 3.22 h. 3.3 HS 1813+6122 The light curve of HS 1813+6122 is characterized by rapid (10− 20min) oscillations with a typical amplitude of 0.1 − 0.2mag, superimposed by modulations with time-scales of hours and amplitudes of ≃ 0.1mag. We combined all data after subtracting the nightly means, and calculated a Scargle (1982) periodogram (Fig. 3, bottom panel). Two broad clus- ters of signals are found at 3.38 and 3.54 h, with the stronger signal at the shorter period. The same double-cluster struc- ture repeats with lower amplitudes at several ±1 cycle/day aliases. Fig. 6 shows the photometric data folded on either period, after subtracting a sine fit with the respective other period. Many of the known SWSex stars display posi- tive and/or negative superhumps in their light curves (e.g. Patterson & Skillman 1994; Rolfe, Haswell & Pat- terson 2000; Stanishev et al. 2002; Patterson et al. 2002; Patterson et al. 2005), and the pattern observed in c© 2007 RAS, MNRAS 000, The SW Sextantis stars 7 Figure 3. Time series analysis of the CCD photometry of the five new SWSex stars. The four top panels show the results of eclipse-phase fits for the deeply eclipsing systems HS 0129+2933, HS 0220+0603, and HS 0455+8315, as well as for the graz- ing eclipser HS0805+3822. For HS 0805+3822, an AOV peri- odogram (Schwarzenberg-Czerny 1996) is shown in the second- lowest panel, and a Scargle (1982) periodogram is shown for HS 1813+6122 in the bottom panel, with the likely orbital pe- riod and superhump period indicated. HS1813+6122 fits into that picture. On the basis of pho- tometry alone, it is usually not possible to unambiguously identify orbital and superhump periods, but based on the ra- dial velocity study described below in Sect. 4.3, we suggest that PSH = 3.38 h and Porb = 3.54 h. 4 SPECTROSCOPIC ANALYSIS 4.1 An overabundance of nitrogen in HS 0220+0603 HS0220+0603 shows emission lines not usually seen in CVs. In Fig. 7 we plot the average spectrum covering the region λλ4870−6500. We identify a group of lines around λ5250 as Fe ii transitions. They very likely are emission profiles plus an absorption component since a deep absorption trough is observed at the position of Fe ii λ5169. The He i λ5016 line is abnormally broad and has a strange profile with three peaks. This is probably the effect of blended He i and Fe ii emission (at λ5018). But the most notorious feature is the Figure 5. Phase-folded light curves of HS 0805+3822. Top and middle panels: data from the nights with low flickering activity folded over the eclipse ephemerides given by Eqs. (4,5). Bottom panel : all data folded over the period determined from an analysis- of-variance periodogram (ORT, Schwarzenberg-Czerny 1996). Figure 6. Phase-folded light curves of HS 1813+6122. Top panel : all data folded over PSH = 3.38 h, the strongest signal detected in the periodogram (Fig. 3), after subtracting a sine fit with Porb = 3.54 h, the period of the neighbouring signal. Bottom panel : all data folded over Porb = 3.54 h after subtracting a sine fit with PSH = 3.38 h. We interpret PSH and Porb as the superhump and orbital periods, respectively. c© 2007 RAS, MNRAS 000, 8 P. Rodŕıguez-Gil et al. Figure 4. Sample light curves of the five new SWSex stars. From top to bottom: HS0129+2933, HS 0220+0603, HS0455+8315, HS 0805+3822, and HS 1813+6122. HS 0805+3822 displays grazing eclipses. Those indicated by ticks above the light curves were used to calculate an ephemeris. broad N ii λ5680 emission. Another triple-peaked emission line profile is found at λ6350 which we identify as Si ii emis- sion. The N ii λ5680 transition has been observed in the in- termediate polar HS 0943+1404 (Rodŕıguez-Gil et al. 2005), and we interpret its presence as evidence of an anomalously large abundance of nitrogen. An observed overabundance of nitrogen in the accre- tion flow is directly linked to the chemical abundances of the donor star, and suggests that the donor has under- gone nuclear evolution via the CNO cycle. This implies that the donor had an initial mass of & 1.2M⊙ and the system evolved through a phase of thermal-time scale mass trans- fer (Schenker et al. 2002; Podsiadlowski, Han & Rappaport 2003). The theoretical models predict that up to 1/3 of all CVs may have undergone nuclear evolution. This seems to be confirmed by the substantial number of systems with evolved donor stars that have been found (e.g. Jameson, King & Sherrington 1980; Bonnet-Bidaud & Mouchet 1987; Thorstensen et al. 2002a, 2002b; Gänsicke et al. 2003). 4.2 Radial velocities Radial velocity curves of the Hα and He ii λ4686 emis- sion lines were computed for the eclipsing systems, except for HS 0129+2933 where He ii λ4686 is too weak and only Hα was measured. The individual velocity points were ob- tained by cross-correlating the individual profiles with single Gaussian templates matching the FWHMs of the respective average line profiles (see Table 1). Before measuring the ve- locities, the normalised spectra were re-binned to a uniform velocity scale centred on the rest wavelength of each line. The radial velocity curves of the three deeply-eclipsing sys- tems folded on their respective orbital periods are presented in Fig. 8. We fitted sinusoidal functions of the form: Vr = γ −K sin [2π (ϕ− ϕ0)] to the radial velocity curves. The fitting parameters are shown in Table 5. Note that the γ and K parameters given in the table are not the actual systemic velocity and radial velocity amplitude of the white dwarf (K1). The Hα line in HS 0129+2933, HS 0220+0603, and HS0455+8315 is de- layed with respect to the motion of the white dwarf by ϕ0 ∼ c© 2007 RAS, MNRAS 000, The SW Sextantis stars 9 Table 5. Radial velocity curves fitting parameters. Line γ (km s−1) K (km s−1) ϕ0 HS0129+2933 Hα 69± 1 282± 1 0.240± 0.001 HS0220+0603 Hα −37.0± 0.7 186± 1 0.170± 0.001 He II λ4686 −88± 16 295 ± 22 0.24± 0.01 HS0455+8513 Hα −27± 1 238± 1 0.155± 0.001 He II λ4686 5± 1 125± 7 0.14± 0.01 Figure 7. Orbitally-averaged spectrum of HS 0220+0603. Notice the unusual N ii λ5680 emission line. 0.2. The He ii λ4686 radial velocity curves of HS 0220+0603 and HS0455+8315 show the same phase offset. This phase lag indicates that the main emission site is at an angle (∼ 72◦) to the line of centres between the centre of mass of the binary and the white dwarf. The radial velocity curves are not sinusoidal in shape, showing significant distortion mainly at phases 0 (mid-eclipse) and 0.5. The spikes at ϕ ∼ 0 are due to a rotational disturbance, caused by the fact that the secondary first occults the disc material approaching us and then the receding material. This translates into a red velocity spike before mid-eclipse and a blue spike after it. 4.3 The orbital period of HS 1813+6122 Of the five new CVs, HS 1813+6122 is the only one not showing eclipses, but the photometry suggests a possible or- bital period of 3.54 h (Sect. 3.3). In an attempt to confirm this value we performed a period analysis on the Hα and Hβ radial velocities. The velocities were derived by using the double Gaussian method of Schneider & Young (1980) (which gave better results than the single Gaussian tech- nique), adopting a Gaussian FWHM of 200 km s−1 and a separation of 1600 km s−1. The 2002 September 1 and 2003 May INT data were obtained under poor observing conditions and were therefore excluded from the analysis. In Fig. 9 we show the resulting analysis-of-variance peri- odogram computed from the combined Hα and Hβ radial velocity curves. The periodogram shows its strongest peak Figure 8. Radial velocity curves of the deeply-eclipsing sys- tems HS0129+2933, HS0220+0603, and HS 0455+8315 folded on their respective orbital periods after averaging the data into 20 phase bins. The Hα velocities are delayed by ∼ 0.1 − 0.2 or- bital cycle with respect to the photometric ephemerides. The same is observed for the He ii λ4686 lines in HS 0220+0603 and HS 0455+8315. The orbital cycle has been plotted twice. at a period of 3.53 h, which confirms the value given by the photometric data as well as the presence of a negative su- perhump in the light curve of HS 1813+6122. A sine fit to the longest spectroscopic run (2003 Jun 29) with the pe- riod fixed at the above value yielded a tentative zero-phase time of T0(HJD) = 2 452 819.597(2). A preliminary trailed spectrum revealed the characteristic SWSex high-velocity S- wave reaching its maximum blue velocity at relative phase ∼ 0.3 (the phases are defined by using the above T0). By analogy with the eclipsing systems presented in this paper (Sect. 4.4), and with the eclipsing SW Sex stars in general (see e.g. Hellier 1996; Hoard & Szkody 1997; Hellier 2000; Rodŕıguez-Gil et al. 2001), this is expected to happen at ab- solute phase ϕ ∼ 0.5. Therefore, the Balmer radial velocities of HS 1813+6122 are delayed by ϕ0 ∼ 0.2 with respect to the white dwarf motion, a defining characteristic of the SWSex stars. Correcting for this we get a new time of zero phase of T0(HJD) = 2 452 819.568(2). Fig. 9 shows the combined Hα and Hβ velocities folded on the orbital period with the new phase definition. 4.4 Trailed spectra Trailed spectrograms of several emission lines of HS 0129+2933, HS 0220+0603, and HS0455+8315 were constructed after re-binning the spectra on to an uniform velocity scale centred on the rest wavelength of each line. They are shown in Fig. 10. The Hα and Hβ line emis- sion is dominated by a high-velocity emission component which neither follows the phasing of the primary nor the secondary. These S-waves reach their bluest velocity at ϕ ∼ 0.5 in the three eclipsing systems and have a velocity amplitude of & 600 km s−1. Weaker emission can also be seen underneath the dominant S-wave, possibly originating c© 2007 RAS, MNRAS 000, 10 P. Rodŕıguez-Gil et al. Figure 9. Top: Scargle periodogram of the Hα+Hβ radial veloc- ity curves of HS 1813+6122. Bottom: phase folded radial veloc- ity curve (no phase binning applied). See text for details on the adopted T0. A full cycle has been repeated. in the accretion disc. An absorption component is observed moving across the lines from red to blue, reaching maximum strength also at ϕ ∼ 0.5. The He i lines also display this high-velocity S-wave with the same phasing as the Balmer lines, as well as the absorp- tion component. The He i λ4472 line in HS0220+0603 shows it for approximately three quarters of an orbit, going well below the continuum. Only HS0220+0603 and HS 0455+8315 have a He ii λ4686 line strong enough to produce a clear trailed spectrogram. While in the former the emission seems to be entirely dominated by the high-velocity component, two components are observed in the latter: a low-velocity one (the strongest), and a weaker component which is probably the same S-wave we observe in the Balmer and He i lines. No significant absorption is detected in He ii λ4686. In Fig. 11 we present the Hα trailed spectrum of HS 1813+6122 after averaging the individual spectra into 20 orbital phase bins. The line shows a high-velocity S-wave with maximum blue velocity at ∼ −2000 km s−1. 4.5 The FUV spectrum of HS 0455+8315 The ultraviolet spectrum of HS 0455+8315 (obtained at ϕ ≃ 0.75) displays very strong emission lines of He, C, N, O, and Si, with line flux ratios compatible with those observed in the majority of CVs (Mauche, Lee & Kall- man 1997), indicating normal chemical abundances of the donor star. The slope of the FUV continuum is nearly flat as observed in a number of deeply eclipsing SWSex stars (e.g. DWUMa, Szkody 1987; Knigge et al. 2004; PXAnd, Thorstensen et al. 1991; BHLyn, Hoard & Szkody 1997), . Figure 11. Hα trailed spectra of HS 1813+6122. Black represents emission. The grey scale has been adjusted to enhance the high- velocity S-wave (left) and the line core (right). A whole cycle has been repeated. It has been argued that a relatively cold structure shields the inner disc and the white dwarf from view in high-inclination SWSex stars during the high state, specifically supported by the FUV detection of the hot white dwarf in DWUMa dur- ing a low state (Knigge et al. 2000; Araujo-Betancor et al. 2003). 5 SWSEX MEMBERSHIP The five new CVs presented in this paper have very much in common. In their average spectra the Balmer and He i lines display both single- or double-peaked profiles. The double- peaked profiles are likely a consequence of phase-dependent absorption components as the trailed spectra show. The lines are also characterised by highly asymmetric profiles with enhanced wings. The trailed spectra reveal the pres- ence of a high-velocity emission S-wave in all the systems with extended wings reaching a maximum velocity between ∼ ±2000 km s−1 (HS 1813+6122) and ∼ ±1000 km s−1 in the eclipsing systems. The tendency of non-eclipsing SW Sex stars to show broader S-waves may be evidence of emitting material with a vertical velocity gradient such as a mass outflow. The radial velocity curves also show a distinctive SW Sex feature. They are delayed with respect to the motion of the white dwarf, so that the red-to-blue crossing takes place at ϕ ∼ 0.2 instead of ϕ = 0. On the other hand, the eclipsing systems also display a discontinuity around mid- eclipse, probably a rotational disturbance, which indicates that part of the line emission comes from the accretion disc. All the features described above are defining charac- teristics of the SW Sex stars (see Thorstensen et al. 1991 and Rodŕıguez-Gil, Schmidtobreick & Gänsicke 2007). Even though each system exhibits its own peculiarities (e.g. un- usual spectral lines in HS 0220+0603 and strong He ii λ4686 emission in HS0455+8315), they all share the characteris- tic SWSex behaviour. We therefore classify them all as new SWSex stars. c© 2007 RAS, MNRAS 000, The SW Sextantis stars 11 Figure 10. Trailed spectra of (from top to bottom) HS0129+2933, HS 0220+0603, and HS 0455+8513. The individual spectra were averaged into 20 phase bins. Black represents emission. Unsampled phase bins are indicated by a blank spectrum. A full orbital cycle has been repeated for continuity. 6 THE SW SEX STARS IN THE CONTEXT OF CV EVOLUTION 6.1 How are the SW Sex stars discovered as CVs? CVs are found by a number of means. Many of them dis- play abrupt brightenings as a result of a dwarf nova eruption or a nova explosion. They also show a rich variety of pho- tometric variations like eclipses, orbital modulations, rapid oscillations, ellipsoidal modulations, etc. Others have been discovered by their blue colour, the emission of X-rays, or the presence of strong emission lines in their optical spectra. The total number of definite members of the SW Sex class so far known amounts to 35, out of which a remarkable number of 18 (51 per cent) has been found in UV-excess sur- c© 2007 RAS, MNRAS 000, 12 P. Rodŕıguez-Gil et al. Figure 12. HST/STIS far-ultraviolet spectrum of HS 0455+8315. veys (i.e. blue colour). This is not surprising, as the optical spectra of the SWSex stars in the high-state is characterised by a very blue continuum. On the other hand, 11 SWSex stars (31 per cent) have been identified as CVs from their emission line spectra, five of which were discovered in the HQS. Only four and two SWSex stars have been found be- cause of their brightness variability (including 3 novae) and X-ray emission, respectively. Any sort of CV search has its own selection effects, and the classification of a CV as a SW Sex star is no excep- tion. In fact, the deep eclipses that many of the SW Sex stars show initially made the sample clearly biased towards high inclination systems. This led many authors to link the SW Sex phenomenon to a mere inclination effect. At the last count, 13 out of a total of 35 SW Sex stars (37 per cent) do not display eclipses and are bona fide members of the class. Although an inclination effect may certainly be important (see the case of HL Aqr in Rodŕıguez-Gil et al. 2007), the increasing number of non-eclipsing systems poses serious difficulties to any model resting solely on a high or- bital inclination. In the following section we discuss on the impact of SW Sex stars in the (spectroscopic) HQS sample, which is unaffected by the high-inclination selection effect. 6.2 The role of the SW Sex stars in the big family of nova-like CVs In Table 6 we list the 35 known SW Sex stars along with their orbital periods. Before doing any statistics we want to stress the fact that Table 6 does not intend to present a definitive census of the SW Sex stars. This is because their defining characteristics are continuously evolving as we dig deeper into the understanding of this class of CVs. For comparison (and probably completeness) we also point the reader to Don Hoard’s Big List of SW Sex stars.3 Table 6 shows that a significant 37 per cent of the fam- ily does not show eclipses, confirming that the high incli- nation requirement is merely a selection effect. The orbital 3 http://spider.ipac.caltech.edu/staff/hoard/biglist.html Figure 13. Period distributions of confirmed SWSex stars (left panel) and nova-like variables and classical novae that are not known to exhibit SWSex behaviour. The orbital period gap and the 3–4 h orbital period range are indicated in light and dark grey, respectively. period distribution of the known sample of SW Sex stars is presented in the left panel of Fig. 13. The combined dis- tributions for non-SW Sex nova-likes and nova remnants are also plotted for comparison. The nova-likes and novae were selected from the Ritter (Ritter & Kolb 2003), CVcat (Kube et al. 2003), and Downes (Downes et al. 2005) cata- logues. Only systems with a robust orbital period determi- nation are included. These orbital period distributions reveal important fea- tures. About 40 per cent (35 out of 93) of the whole nova- like/classical nova population (which is preferentially found above the period gap; only 12 are below it) are indeed SW Sex stars. Even more remarkable is the fact that the SW Sex stars represent almost half (26 out of 53) the CVs in the narrow 3−4.5 h orbital period range. Above 4.5 h things radically change and only 14 per cent (4 out of 28) of the nova-like/nova population are known to be SWSex stars. The impact of SW Sex stars in the gap is also striking, as 55 per cent of all the nova-like/nova gap inhabitants are members of the class. The non-SW Sex nova-like/nova Porb distribution also shows a significant fraction of systems in the 3 − 4.5 h in- terval. This nicely depicts the tendency of nova-likes to ac- cumulate in the 3− 4.5 h period range. Therefore, it is very likely that more SW Sex stars are still to be found. In fact, time-resolved spectroscopic studies like the one reported in Rodŕıguez-Gil et al. (2007) are revealing the SWSex nature of many previously poorly studied nova-likes in this range. If the rate of detection and identification of SW Sex systems remains high the dominance of this class at the upper edge of the gap will eventually become even more pronounced. It is possible, however, that these numbers are the re- sult of a selection effect as the majority of SW Sex stars are bright, making them easily accesible to observations. There- fore, one can argue that a proper characterisation of the whole population of CVs above the gap needs to be made before addressing any conclusion. However, the fact the ma- jority of SW Sex stars have orbital periods between 3 and 4.5 h is a well established fact. 6.2.1 The impact of SW Sex stars on the HQS CV sample During the course of our spectroscopic search we have dis- covered 53 new CVs in the area/magnitude range covered c© 2007 RAS, MNRAS 000, The SW Sextantis stars 13 Table 6. The SW Sex stars. Object Porb (h) Eclipses References V348Pup 2.44 Yes 1 V795Her 2.60 No 2 RXJ1643.7+3402 2.89 No 3 V442Oph 2.98 No 4 AHMen 3.05 No 5 HS 0728+6738 3.21 Yes 6 HS 0805+3822 3.21 Grazing 7, this paper SWSex 3.24 Yes 8 HLAqr 3.25 No 5 DWUMa 3.28 Yes 8 SDSS J132723.39+652854.2 3.28 Yes 9 WXAri 3.34 Grazing 10 HS 0129+2933 3.35 Yes This paper V1315Aql 3.35 Yes 8 BOCet 3.36 No 5 AHPic 3.41 No 5 VZScl 3.47 Yes 11 LNUMa 3.47 No 5 RRPic 3.48 Grazing 12 PXAnd 3.51 Yes 8 V533Her 3.53 No 13 HS 1813+6122 3.54 No This paper HS 0455+8315 3.57 Yes This paper HS 0220+0603 3.58 Yes This paper HS 0357+0614 3.59 No 14 V380Oph 3.70 No 5 BHLyn 3.74 Yes 15 UUAqr 3.93 Yes 16 LXSer 3.95 Yes 17 V1776Cyg 3.95 Yes 18 LSPeg 4.19 No 19 V347Pup 5.57 Yes 20 RWTri 5.57 Yes 21 V363Aur 7.71 Yes 22 BTMon 8.01 Yes 23 References: 1 Rodŕıguez-Gil et al. (2001); 2 Casares et al. (1996); 3 Mickaelian et al. (2002); 4 Hoard et al. (2000); 5 Rodŕıguez-Gil et al. (2007); 6 Rodŕıguez-Gil et al. (2004); 7 Szkody et al. (2003); 8 Thorstensen et al. (1991); 9 Wolfe et al. (2003); 10 Beuermann et al. (1992); 11 Moustakas & Schlegel (1999); 12 Schmidtobreick, Tappert & Saviane (2003); 13 Rodŕıguez-Gil & Mart́ınez-Pais (2002); 14 Szkody et al. (2001); 15 Thorstensen, Davis & Ringwald (1991); 16 Hoard et al. (1998); 17 Young, Schneider & Shectman (1981); 18 Garnavich et al. (1990); 19 Taylor et al. (1999); 20 Thoroughgood et al. (2005); 21 Groot, Rutten & van Paradijs (2004); 22 Thoroughgood et al. (2004); 23 Smith, Dhillon & Marsh (1998) by the HQS (≃ 13600 square degrees/17.5 . B . 18.5; see Hagen et al. 1995). So far, we have determined the orbital period for 43 of them in a huge observational effort. Our preliminary results support the SW Sex excess within the 3 − 4.5 h range. Remarkably, 54 per cent (7 out of 13) of all the newly-discovered HQS nova-likes are indeed SW Sex stars, which is in agreement with the distribution discussed above. This gives further strength to the significancy of the observed (and still unexplained) pile-up of SW Sex stars in the 3− 4.5 h region. 6.3 CV evolution and the SW Sex stars This accumulation of systems just above the period gap seri- ously challenges our current understanding of CV evolution. The SW Sex stars are intrinsically very luminous, as the brightness of systems like DW UMa indicates. DW UMa is a SW Sex star likely located at a distance between ∼ 590−930 pc (Araujo-Betancor et al. 2003) that has an average mag- nitude of V ∼ 14.5, even though it is viewed at an orbital inclination of ∼ 82◦. Therefore, in order to show such high luminosities, either the SW Sex stars have an average mass transfer rate well above that of their CV cousins, or another source of luminosity exists. Neither the Rappaport, Joss & Verbunt (1983) nor the (empirical) Sills, Pinsonneault & Terndrup (2000) angular momentum loss prescriptions ac- count for a largely enhanced mass transfer rate (Ṁ) in the period interval where most of the SW Sex stars reside. In this regard, nuclear burning has been suggested as an extra luminosity source (Honeycutt 2001), but the necessary con- ditions for the burning to occur can only be found in the base of a magnetic accretion funnel, suggesting a magnetic nature (Honeycutt & Kafka 2004). Temporary cessation of nuclear burning would in principle explain the VY Scl low states that many SW Sex and other nova-likes undergo. If this is true, the majority of CVs above the period gap have to be magnetic, in stark contrast with a non-magnetic ma- jority below the gap. However, the fact that some dwarf novae (like HT Cas and RX And) also show low states (e.g. Robertson & Honeycutt 1996; Schreiber, Gänsicke & Mat- tei 2002) argues against this possibility, suggesting that the VY Scl states are likely the product of a decrease in the mass transfer rate from the donor star caused by starspots, as already proposed by Livio & Pringle (1994). The obser- vational results of Honeycutt & Kafka (2004) appear to sup- port this hypothesis (see also e.g. Howell et al. 2000). All of the above arguments point to accretion at a very high Ṁ as the most likely cause for the high luminosity observed in the SW Sex stars. One possibility could be en- hanced mass transfer due to irradiation of the inner face of the secondary star by a very hot white dwarf. In fact, a number of nova-like CVs in the ∼ 3 − 4 h orbital period range (including the SW Sex star DW UMa) have been ob- served to harbour the hottest white dwarfs found in any CV (see Araujo-Betancor et al. 2005), with effective tem- peratures peaking at ∼ 50 000 K. These high temperatures are most likely the result of accretion heating, as CVs are thought to spend on average ∼ 2 Gyr as detached binary systems (Schreiber & Gänsicke 2003), enough time for the white dwarfs to cool down to . 8000 K. Hence, the high effective temperatures measured in the 3 − 4 h range is a measure of a very high secular mass accretion rate of ∼ 5 × 10−9 M⊙ yr −1 (Townsley & Bildsten 2003), higher than predicted by angular momentum loss due to magnetic braking. Irradiation of the donor star has been observed in the non-eclipsing SW Sex stars HL Aqr, BO Cet, and AH Pic (Rodŕıguez-Gil et al. 2007), which supports the above hypothesis. Alas, the question of why the SW Sex stars have the highest mass transfer rates is still lacking a satisfactory explanation in the context of the current CV evolution the- c© 2007 RAS, MNRAS 000, 14 P. Rodŕıguez-Gil et al. 6.4 A rich phenomenology to explore It is becoming apparent that the SW Sex phenomenon is not restricted to the (mainly) spectroscopic properties ini- tially introduced by Thorstensen et al. 1991. These maver- ick CVs are now known to show a much more intricate be- haviour. Therefore, only a comprehensive study of this rich phenomenology will definitely lead to a full understanding of the SW Sex stars. In what follows we will review the range of common features exhibited by the SWSex stars and which implications can be derived from them. 6.4.1 Superhumps A significant one third of the SWSex stars is known to show apsidal (positive) or/and nodal (negative) super- humps, which are large-amplitude photometric waves mod- ulated at a period slightly longer or slightly shorter than the orbital period, respectively. Positive superhumps are be- lieved to be the effect of an eccentric disc which is forced to progradely precess by the tidal perturbation of the donor star (Whitehurst 1988). On the other hand, negative su- perhumps are likely linked to the retrograde precession of a warped accretion disc (Murray et al. 2002). Vertical changes in the structure of the disc may be triggered by the torque exerted on the disc by the tilted, dipolar magnetic field of the secondary star. Apparently, positive and negative super- humps are independent and can either coexist or alternate with a time scale of years. The detection of superhumps in the SWSex stars is of great importance as they exhibit the largest superhump pe- riod excesses. Therefore, they are fundamental in calibrat- ing the period excess–mass ratio relationship for CVs (see Patterson et al. 2005). 6.4.2 Variable eclipse depth The continuum light curves of some eclipsing members of the class reveal that the eclipse depth varies with a time-scale of several days. So far this variability has only been stud- ied in PXAnd (Stanishev et al. 2002; Boffin et al. 2003) and DWUMa (B́ıró 2000; Stanishev et al. 2004, not to be con- fused with the changing eclipse depth in different brightness states). In PX And, Stanishev et al. 2002 identify this long periodicity with the precession period of the accretion disc, which may also be true for DW UMa. The actual mechanism is not yet understood, although the eclipses of a retrogradely precessing, warped disc may account for what is observed in PXAnd (Boffin et al. 2003). 6.4.3 Low states Half of the nova-likes known to undergo VYScl faint states are SWSex stars. During these events the system brightness can drop by up to ∼ 4 − 5mag and can remain that low for months. As the disc warping effect mentioned above, these low states may be controlled by the strong magnetic activity of the donor star, and are believed to be driven by a sudden drop of the accretion rate from the secondary star due to large starspots located in the area around the inner Lagrangian point L1 (Livio & Pringle 1994). Interestingly, the disced VYScl stars concentrate in the 3 − 4.5 h orbital period stripe as the SWSex stars do. It would therefore not be a surprise to find that all nova-likes in that range are actually VYScl stars and even SWSex stars. Nevertheless, the presence of large starspots around the L1 point still have to be observationally confirmed. Us- ing Roche tomography techniques, Watson, Dhillon & Shah- baz (2006) discovered a heavily spotted secondary star in AEAqr (Porb = 9.88 h) with a spot distribution resembling that of other rapidly rotating, low-mass field stars. A stellar atmosphere plagued with spots has been also observed in the pre–CV V471Tau (Porb = 12.51 h, Hussain et al. 2006). Un- fortunately, the high spectral and time resolution required to image the M-dwarf donor in a CV with Porb ≃ 3.5 h at a distance of many hundred parsecs is currently beyond ob- servational reach. 6.4.4 Quasi-periodic oscillations On the grounds of short-term variability, the SWSex stars are also characterised for exhibiting quasi-coherent mod- ulations in their light curves. In a compilation of CVs displaying rapid oscillations, Warner (2004) lists 9 (only two deeply eclipsing) SWSex stars known to have quasi- periodic oscillations (QPOs) with a predominant time-scale of ∼ 1000 − 2000 s. For example, the non-eclipsing SWSex stars V442Oph and RXJ1643.7+3402 show strong QPO sig- nals dominating over other underlying higher-coherence os- cillations (Patterson et al. 2002). These results are based on hundreds of hours of photometric data, whose power spec- tra showed rapid frequency changes of the main signals in less than a day. Similar rapid variability was also detected in the optical light curve of the SWSex star HS 0728+6738 (Rodŕıguez-Gil et al. 2004), with a prominent signal (coher- ent for at least 20 cycles) at ∼ 600 s. 6.4.5 Emission-line flaring In connection with the QPO activity, the fluxes and EWs of the emission lines of some SWSex stars show modu- lations at the same time-scales (a phenomenon known as emission-line flaring): ∼ 1800 s in BTMon (Smith et al. 1998), ∼ 2000 s in LSPeg (Rodŕıguez-Gil et al. 2001), ∼ 1400 s in V533Her (Rodŕıguez-Gil & Mart́ınez-Pais 2002), ∼ 1800 s in DWUMa (V. Dhillon, private communication), ∼ 2400 s in RXJ1643.7+3402 (Mart́ınez-Pais, de la Cruz Rodŕıguez & Rodŕıguez-Gil 2007), and ∼ 1200 s in BOCet (Rodŕıguez-Gil et al. 2007). Remarkably, the radial veloci- ties measured in the last two objects are also modulated at the flux/EW periodicities, which suggests that emission- line flaring has to do with the dynamics of the line emit- ting source, and is not due to e.g. random fluctuations in the disc continuum emission. On the other hand, although DW UMa does exhibit emission-line flaring in the optical, such line variability was not detected in the far ultraviolet (Hoard et al. 2003). Since similar flaring in the optical is observed in the intermediate polar CVs (IPs; e.g. FOAqr Marsh & Duck 1996) caused by the rotation of the mag- netic white dwarf, all the described rapid variations seen in many SWSex stars have been associated to the pres- ence of magnetic white dwarfs (Rodŕıguez-Gil et al. 2001; Patterson et al. 2002). Nevertheless, the far ultraviolet data c© 2007 RAS, MNRAS 000, The SW Sextantis stars 15 of DW UMa presented by Hoard et al. (2003) can also be ex- plained with a stream overflow model, but do not necessary exclude a magnetic scenario either. 6.4.6 Variable circular polarisation Despite the fact that circular polarisation is not commonly detected in the majority of IPs, it is a sine qua non condition for IP membership (see further requirements in Patterson 1994). The cyclotron radiation emitted by the accretion columns (built up by disc plasma forced by the magnetic field to supersonically fall on to the white dwarf surface) is known to be circularly polarised, thus the detection of a sig- nificant level of circular polarisation is a unequivocal sign of magnetic accretion. This, and the possible magnetic nature of the QPO and line flaring activity prompted to the search for circular polarisation in the SWSex stars. Rodŕıguez-Gil et al. (2001) found circular polarisation modulated at 1776 s with a peak-to-peak amplitude of 0.3 per cent in LSPeg. Remarkably, the flaring observed in the Hβ high-velocity S-wave was modulated at 2010 s, which is just the synodic period between the polarisation period and the orbital period. V795Her also revealed variable circu- lar polarisation with a periodicity of 1170 s (or twice that) and showed an increasing polarisation level with wavelength (Rodŕıguez-Gil et al. 2002) as is expected for cyclotron emis- sion. In addition, RXJ1643.7+3402 shows circular polarisa- tion modulated at 1163 s (Mart́ınez-Pais et al. 2007). The 1776-s period of LSPeg was confirmed by Baskill, Wheatley & Osborne (2005) who detected a coherent modu- lation at 1854 s in ASCA X-ray light curves. The coincidence of both periods indicates a common origin, with the X-ray period reported by Baskill et al. likely being a more accurate measurement of the white dwarf spin period. Although polarimetric studies of many other SWSex stars have to be done, the results obtained so far suggest that magnetic accretion may play an important role in the SWSex phenomenon. However, with the few such studies so far at hand it is not possible to address any conclusion regarding the impact of magnetism in the whole class. Despite the broad implications in our understanding of accretion that the study of the SWSex stars have, the quest for a successful, global explanation of the phenomenon has been unfruitful so far. This, in addition to the (yet unex- plained) fact that the majority of SWSex stars (and many nova-likes) largely populate the narrow orbital period stripe between 3 and 4.5 hours, are seriously shaking the grounds on which CV evolution theory stands. Further study of these maverick systems will certainly provide fundamental clues to our understanding of CV evolution. ACKNOWLEDGMENTS To the memory of Emilios Harlaftis. AA thanks the Royal Thai Government for a stu- dentship. BTG was supported by a PPARC Advanced Fellowship, respectively. MAPT is supported by NASA LTSA grant NAG-5-10889. RS is supported by the Deutsches Zentrum für Luft und Raumfahrt (DLR) GmbH under contract No. FKZ 50 OR 0404. AS is supported by the Deutsche Forschungsgemeinschaft through grant Schw536/20-1. The HQS was supported by the Deutsche Forschungsgemeinschaft through grants Re 353/11 and Re353/22. This paper includes data taken at The McDonald Ob- servatory of The University of Texas at Austin. It is also based in part on observations obtained at the German- Spanish Astronomical Center, Calar Alto, operated by the Max-Planck-Institut für Astronomie, Heidelberg, jointly with the Spanish National Commission for Astronomy; on observations made at the 1.2m telescope, located at Kry- oneri Korinthias, and owned by the National Observatory of Athens, Greece; on observations made with the Isaac New- ton Telescope, which is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof́ısica de Canarias (IAC); on observations made with the 1.2m tele- scope at the Fred Lawrence Whipple Observatory, a facility of the Smithsonian Institution; on observations made with the IAC80 telescope, operated on the island of Tenerife by the IAC in the Spanish Observatorio del Teide; on observa- tions made with the OGS telescope, operated on the island of Tenerife by the European Space Agency, in the Spanish Observatorio del Teide of the IAC; on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the IAC; and on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Associ- ation of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. REFERENCES Araujo-Betancor S., Gänsicke B. T., Long K. S., Beuer- mann K., de Martino D., Sion E. M., Szkody P., 2005, ApJ, 622, 589 Araujo-Betancor S., et al., 2003, ApJ, 583, 437 Aungwerojwit A., et al., 2005, A&A, 443, 995 Baskill D. S., Wheatley P. J., Osborne J. P., 2005, MNRAS, 357, 626 Bertin E., Arnouts S., 1996, A&AS, 117, 393 Beuermann K., Thorstensen J. R., Schwope A. D., Ring- wald F. A., Sahin H., 1992, A&A, 256, 442 B́ıró I. B., 2000, A&A, 364, 573 Boffin H. M. J., Stanishev V., Kraicheva Z., Genkov V., 2003, in Sterken C., ed, ASP Conf. Ser. Vol. 292, Inter- play of Periodic, Cyclic and Stochastic Variability in Se- lected Areas of the H-R Diagram. Astron. Soc. Pac., San Francisco, p. 297 Bonnet-Bidaud J. M., Mouchet M., 1987, A&A, 188, 89 Casares J., Mart́ınez-Pais I. G., Marsh T. R., Charles P. A., Lázaro C., 1996, MNRAS, 278, 219 c© 2007 RAS, MNRAS 000, 16 P. Rodŕıguez-Gil et al. Dhillon V. S., Jones D. H. P., Marsh T. R., Smith R. C., 1992, MNRAS, 258, 225 Downes R. A., Webbink R. F., Shara M. M., Ritter H., Kolb U., Duerbeck H. W., 2005, VizieR Online Data Catalog, Gänsicke B. T., Araujo-Betancor S., Hagen H.-J., Harlaftis E. T., Kitsionas S., Dreizler S., Engels D., 2004, A&A, 418, 265 Gänsicke B. T., et al., 2003, ApJ, 594, 443 Gänsicke B. T., Hagen H. J., Engels D., 2002, in Gänsicke B. T., Beuermann K., Reinsch K., eds, ASP Conf. Ser. Vol. 261, The Physics of Cataclysmic Variables and Related Objects. Astron. Soc. Pac., San Francisco, p. 190 Garnavich P. M., et al., 1990, ApJ, 365, 696 Green R. F., Schmidt M., Liebert J., 1986, ApJS, 61, 305 Groot P. J., Rutten R. G. M., van Paradijs J., 2004, A&A, 417, 283 Hagen H. J., Groote D., Engels D., Reimers D., 1995, A&AS, 111, 195 Hameury J. M., Lasota J. P., 2002, A&A, 394, 231 Hellier C., 1996, ApJ, 471, 949 Hellier C., 2000, New Astronomy Reviews, 44, 131 Hellier C., Robinson E. L., 1994, ApJ, 431, L107 Hoard D. W., Szkody P., 1997, ApJ, 481, 433 Hoard D. W., Szkody P., Froning C. S., Long K. S., Knigge C., 2003, AJ, 126, 2473 Hoard D. W., Szkody P., Still M. D., Smith R. C., Buckley D. A. H., 1998, MNRAS, 294, 689 Hoard D. W., Thorstensen J. R., Szkody P., 2000, ApJ, 537, 936 Honeycutt R. K., 2001, PASP, 113, 473 Honeycutt R. K., Kafka S., 2004, AJ, 128, 1279 Horne K., 1986, PASP, 98, 609 Horne K., 1999, in Hellier C., Mukai K., eds, ASP Conf. Ser. Vol. 157, Annapolis Workshop on Magnetic Cataclysmic Variables. Astron. Soc. Pac., San Francisco, p. 349 Howell S. B., Ciardi D. R., Dhillon V. S., Skidmore W., 2000, ApJ, 530, 904 Hussain G. A. J., Allende Prieto C., Saar S. H., Still M., 2006, MNRAS, 367, 1699 Jameson R. F., King A. R., Sherrington M. R., 1980, MN- RAS, 191, 559 Knigge C., Araujo-Betancor S., Gänsicke B. T., Long K. S., Szkody P., Hoard D. W., Hynes R. I., Dhillon V. S., 2004, ApJ, 615, L129 Knigge C., Long K. S., Hoard D. W., Szkody P., Dhillon V. S., 2000, ApJ, 539, L49 Kube J., Gänsicke B. T., Euchner F., Hoffmann B., 2003, A&A, 404, 1159 Livio M., Pringle J. E., 1994, ApJ, 427, 956 Marsh T. R., Duck S. R., 1996, New Astronomy, 1, 97 Mart́ınez-Pais I. G., de la Cruz Rodŕıguez J., Rodŕıguez-Gil P., 2007, MNRAS, submitted Mart́ınez-Pais I. G., Rodŕıguez-Gil P., Casares J., 1999, MNRAS, 305, 661 Mauche C. W., Lee Y. P., Kallman T. R., 1997, ApJ, 477, Mickaelian A. M., Balayan S. K., Ilovaisky S. A., Chevalier C., Véron-Cetty M.-P., Véron P., 2002, A&A, 381, 894 Monet et al. D. G., 2003, AJ, 125, 984 Moustakas J., Schlegel E. M., 1999, BAAS, 31, 1421 Murray J. R., Chakrabarty D., Wynn G. A., Kramer L., 2002, MNRAS, 335, 247 Patterson J., 1994, PASP, 106, 209 Patterson J., et al., 2002, PASP, 114, 1364 Patterson J., et al., 2005, PASP, 117, 1204 Patterson J., Skillman D. R., 1994, PASP, 106, 1141 Penning W. R., Ferguson D. H., McGraw J. T., Liebert J., Green R. F., 1984, ApJ, 276, 233 Podsiadlowski P., Han Z., Rappaport S., 2003, MNRAS, 340, 1214 Rappaport S., Joss P. C., Verbunt F., 1983, ApJ, 275, 713 Ritter H., Kolb U., 2003, A&A, 404, 301 Robertson J. W., Honeycutt R. K., 1996, AJ, 112, 2248 Rodŕıguez-Gil P., 2005, in Hameury J.-M., Lasota J.-P., eds, ASP Conf. Ser. Vol. 330, The Astrophysics of Cata- clysmic Variables and Related Objects. Astron. Soc. Pac., San Francisco, p. 335 Rodŕıguez-Gil P., Casares J., Dhillon V. S., Mart́ınez-Pais I. G., 2000, A&A, 355, 181 Rodŕıguez-Gil P., Casares J., Mart́ınez-Pais I. G., Hakala P., Steeghs D., 2001, ApJ, 548, L49 Rodŕıguez-Gil P., Casares J., Mart́ınez-Pais I. G., Hakala P. J., 2002, in Gänsicke B. T., Beuermann K., Reinsch, K., eds, ASP Conf. Ser. Vol. 261, The Physics of Cata- clysmic Variables and Related Objects. Astron. Soc. Pac., San Francisco, p. 533 Rodŕıguez-Gil P., Gänsicke B. T., Barwig H., Hagen H.-J., Engels D., 2004, A&A, 424, 647 Rodŕıguez-Gil P., et al., 2005, A&A, 440, 701 Rodŕıguez-Gil P., Mart́ınez-Pais I. G., 2002, MNRAS, 337, Rodŕıguez-Gil P., Mart́ınez-Pais I. G., Casares J., Villada M., van Zyl L., 2001, MNRAS, 328, 903 Rodŕıguez-Gil P., Schmidtobreick L., Gänsicke B. T., 2007, MNRAS, in press Rolfe D. J., Haswell C. A., Patterson J., 2000, MNRAS, 317, 759 Romano G., 1978, Inf. Bull. Variable Stars, 1421, 1 Scargle J. D., 1982, ApJ, 263, 835 Schenker K., King A. R., Kolb U., Wynn G. A., Zhang Z., 2002, MNRAS, 337, 1105 Schmidtobreick L., Tappert C., Saviane I., 2003, MNRAS, 342, 145 Schneider D. P., Young P., 1980, ApJ, 238, 946 Schreiber M. R., Gänsicke B. T., 2003, A&A, 406, 305 Schreiber M. R., Gänsicke B. T., Mattei J. A., 2002, A&A, 384, L6 Schwarzenberg-Czerny A., 1996, ApJ, 460, L107 Shafter A. W., Hessman F. V., Zhang E.-H., 1988, ApJ, 327, 248 Sills A., Pinsonneault M. H., Terndrup D. M., 2000, ApJ, 534, 335 Smith D. A., Dhillon V. S., Marsh T. R., 1998, MNRAS, 296, 465 Stanishev V., Kraicheva Z., Boffin H. M. J., Genkov V., 2002, A&A, 394, 625 Stanishev V., Kraicheva Z., Boffin H. M. J., Genkov V., Papadaki C., Carpano S., 2004, A&A, 416, 1057 Szkody P., 1987, AJ, 94, 1055 Szkody P., Gänsicke B., Fried R. E., Heber U., Erb D. K., 2001, PASP, 113, 1215 Szkody P., Piché F., 1990, ApJ, 361, 235 c© 2007 RAS, MNRAS 000, The SW Sextantis stars 17 Szkody P., et al., 2003, AJ, 126, 1499 Taylor C. J., Thorstensen J. R., Patterson J., 1999, PASP, 111, 184 Thoroughgood T. D., et al., 2005, MNRAS, 357, 881 Thoroughgood T. D., Dhillon V. S., Watson C. A., Buckley D. A. H., Steeghs D., Stevenson M. J., 2004, MNRAS, 353, Thorstensen J. R., Davis M. K., Ringwald F. A., 1991, AJ, 102, 683 Thorstensen J. R., Fenton W. H., Patterson J., Kemp J., Halpern J., Baraffe I., 2002, PASP, 114, 1117 Thorstensen J. R., Fenton W. H., Patterson J. O., Kemp J., Krajci T., Baraffe I., 2002, ApJ, 567, L49 Thorstensen J. R., Ringwald F. A., Wade R. A., Schmidt G. D., Norsworthy J. E., 1991, AJ, 102, 272 Townsley D. M., Bildsten L., 2003, ApJ, 596, L227 Warner B., 2004, PASP, 116, 115 Warren S. R., Shafter A. W., Reed J. K., 2006, PASP, 118, Watson C. A., Dhillon V. S., Shahbaz T., 2006, MNRAS, 368, 637 Whitehurst R., 1988, MNRAS, 232, 35 Williams R. E., 1989, AJ, 97, 1752 Wolfe M. A., Szkody P., Fraser O. J., Homer L., Skinner S., Silvestri N. M., 2003, PASP, 115, 1118 Young P., Schneider D. P., Shectman S. A., 1981, ApJ, 244, c© 2007 RAS, MNRAS 000, Introduction Observations and data reduction Identification Optical photometry Optical spectroscopy HST/STIS far-ultraviolet spectroscopy Photometric periods HS0129+2933, HS0220+0603, and HS0455+8315 HS0805+3822 HS1813+6122 Spectroscopic analysis An overabundance of nitrogen in HS0220+0603 Radial velocities The orbital period of HS1813+6122 Trailed spectra The FUV spectrum of HS0455+8315 SWSex membership The SW Sex stars in the context of CV evolution How are the SW Sex stars discovered as CVs? The role of the SW Sex stars in the big family of nova-like CVs CV evolution and the SW Sex stars A rich phenomenology to explore

0704.1130

Relativeca Dopplera efekto \^ce unuforme akcelata movo -- II

Relativeca Dopplera efekto ĉe unuforme akcelata movo – II F.M. Paiva Departamento de F́ısica, U.E. Humaitá II, Colégio Pedro II Rua Humaitá 80, 22261-040 Rio de Janeiro-RJ, Brasil; [email protected] A.F.F. Teixeira Centro Brasileiro de Pesquisas F́ısicas 22290-180 Rio de Janeiro-RJ, Brasil; [email protected] 30–a de oktobro, 2018 Resumo Daŭrigante [1], luma fonto de unukolora radiado ĉe rekta movo ĉe konstanta propra akcelo pasas preter restanta observanto. Ĉe la special-relativeco, ni priskribas la obser- vatan Doppleran efikon. Ni ankaŭ priskribas la interesan nekontinuan efikon se trapaso okazas. - - - - - - - - - - - Extending [1], a light source of monochromatic radiation, in rectilinear motion under con- stant proper acceleration, passes near an observer at rest. In the context of special relativ- ity, we describe the observed Doppler effect. We describe also the interesting discontinuous effect when riding through occurs. An English version of this article is available. 1 Enkonduko Citâo [1] studis Doppleran efikon de lumo eligita el restanta fonto, vidata per akcelata observanto kiu pasas preter aŭ tra la fonto. Ĉi tie ni inversas tiun sistemon. Nun, fonto de unukolora radiado ĉe rekta movo ĉe konstanta propra akcelo pasas preter aŭ tra restanta observanto. La frekvenco ν de la lumo eligata estas konstanta; tamen, la observata frekvenco (t.e., la koloro) estas νobs, malsama ol ν. Tio estas nomata Dopplera efiko. La proporcio D = νobs/ν, nomata Dopplera faktoro, estas studata ĉi tie. Estiĝu inercia referenco sistemo S = {t, ~x}. El nefinia loko x = ∞, je nefinia estinta momento t = −∞, luma fonto ekvenis al origino x = 0, per la akso x. Ĝia komenca rapido estis −c, kaj ĝi estas malakcelata per konstanta propra akcelo g; tiel ĝi pasas la originon, poste momente restas ĉe x = −a je t = 0, kaj tuj reiras al nefinio x = ∞ kun sama akcelo. En x = 0, distance b el akso x, estas restanta observanto, kiel montras figuro (1.a). Ni konsideras la preterpason kun b 6= 0 kaj la trapason kun b = 0. Krome ni komparas niajn rezultojn al tiujn de [1]. http://arxiv.org/abs/0704.1130v1 Figuro 1: La sistemo. 1.a) Observanto (nigra sfereto) estas fiksa distance b de akso x. Luma fonto (blanka sfereto) estas iranta de x=−a al x=∞, kun konstanta propra akcelo g. Antaŭe ĝi venis el x=∞ (t=−∞), kun konstanta propra malakcelo g, ĝis x=−a (t=0). 1.b) La moviĝanta fonto eligas signalon en la momento t de la inercia sistemo S, kaj la restanta observanto ricevas tiun signalon en la momento τ , ekv. (6). Vidu la asimptotojn t=(τ + A)/2 kaj τ=−A. Vidu ankaŭ ke ne estas ricevo en tempo antaŭ τ=−A. Ĉi tie x estas la loko de fonto je momento t, ambaŭ mezurataj per la inercia referenco sistemo S. La tempo τ , ankaŭ en S, estas la propratempo de la restanta observanto kiam li ricevas signalon el x je t. Kiel figuro (1.a) evidentigas, la intertempo τ − t inter eligo kaj enigo de signalo estas x2 + b2/c. Por simpligi formulojn ni formale konsideros c = 1 kaj g = 1. Por aperigi la arbitrajn valorojn sufiĉas substitui a→ag/c2 , A→Ag/c2 , b→bg/c2 , x→gx/c2 , v→v/c , t →gt/c , τ→gτ/c , τ ′→gτ ′/c . (1) Farinte tiun simpligon kaj uzante la kondiĉojn x(0) = −a kaj v(0) = 0, la rapido v kaj la loko x de la fonto ĉe konstanta propra akcelo g estas [1] v(t) = 1 + t2 , x(t) = 1 + t2 −A , A := a + 1 . (2) Ĉar la fonto ne restas en S, tial la propratempo de eligo de signalo estas τ ′ 6= t. Vere dτ ′ = dt/γ(t) = 1− v2(t)/c2dt. La integro de dτ ′ fiksante τ ′ = 0 kiam t = 0 estas t=sinh τ ′ , (3) do la rapido kaj la loko de la fonto kiel funkcioj de τ ′ estas v(τ ′) = tanh τ ′ , x(τ ′) = cosh τ ′ −A . (4) Vidu ke la preterpasoj (x = 0) okazas kun rapido v0 := A2 − 1/A je ∓t0 (aŭ ∓τ ′0), kie t0 := A2 − 1 , τ ′ := cosh−1A . (5) Interesas kalkuli la rilaton inter la tempo t de eligo kaj la tempo τ de enigo. El figuro (1.a) τ= t+ x2 + b2 = t+ 1 + t2 −A)2 + b2 , (6) Figuro 2: Doppleraj faktoroj. 2.a) D kiel funkcio (8) de la propratempo τ ′ de moviĝanta fonto. Estas tri momentoj en kiuj la radiado ne havos kolor-ŝanĝon (D= 1); la unua (τ ′ ) antaŭas al momento −τ ′ de la unua preterpaso; la dua (τ ′=0) okazas kiam la fonto restas ĉe x=−a; kaj la tria (τ ′ ) antaŭas al momento τ ′ de la dua preterpaso. La radiadoj ĉe la momentoj ∓τ ′ preterpaso havas D=1/A, do ili estas ruĝ-delokigataj. La valoroj de τ ′ kaj τ ′ estas ĉe (9), kaj estas ĉe (5). 2.b) D kiel funkcio de la momento τ de ricevo de la lumo. La unuaj signaloj estas ricevataj kiam τ =−A, kvankam ili estis eligataj detempe de τ =−∞. Ne estas kolor-ŝanĝo je τ−, a2 + b2, kaj τ+. La radiado eligita en la preterpaso, ∓t0 en ekv. (5), estas observata je ∓t0 + b, ekv. (6), kun D = 1/A. uzante (2). Figuro (1.b) montras ke la enigo ne komencas kiam τ = −∞, sed jese kiam τ=−A. La fonto eligis tiujn komencajn signalojn je t=−∞ el x=∞, kiam ĝia rapido estis −c. Observu ankaŭ la asimptoton τ=2 t−A por t → ∞; la fonto eligos tiujn signalojn je t=∞, kaj ili enigos poste ‘duoble’ nefinia tempo al observanto. 2 Dopplera efiko Ĉar frekvenco estas la inverso de periodo, tial ni povas redifini la Doppleran faktoron kiel D(τ) := 1− v2(t) . (7) Ĉi tie dτ ′ estas la infinitezima propra intertempo per la fonto inter eligo de du lumaj signaloj, kaj dτ estas la infinitezima propra intertempo per la observanto inter enigo de tiuj du lumaj signaloj. La radiko rilatas al la tempa dilato pro la movado de la fonto. Kiam D < 1 la Dopplera efiko estas nomata ‘ruĝ-delokigo’, kaj kiam D > 1 ĝi estas nomata ‘viol-delokigo’. Ni emfazas ke estas aliaj difinoj de Dopplera faktoro [2]. Sekve ni kalkulas la Doppleran faktoron kiel funkcio de τ ′, τ , x kaj de t. Ni komencas kun D(τ ′). Uzante la difinon (7) en la formo D = (dτ/dτ ′)−1, derivante (6) kaj uzante t(τ ′) kaj x(τ ′) el (3) kaj (4), la faktoro estiĝas D(τ ′) = cosh τ ′ + (cosh τ ′ − A) sinh τ ′ (cosh τ ′ − A)2 + b2 . (8) Figuro 3: Doppleraj faktoroj. 3.a) D− kaj D+ kiel funkcio (11) de loko x de lum-eligo el la moviĝanta fonto. Sagoj montras la pozitivan fluon de tempo. Radiado el x−, −a, kaj x+ ne kolor-ŝanĝos, D=1. La valoroj de x− kaj x+ estas en (12). Radiadoj el preterpasoj x=0 havos D=1/A, do ili estos ruĝ-delokigataj. 3.b) D kiel funkcio (13) de tempo t de referenc-sistemo S. Vidu ke D = 1 kiam t− = sinh τ t=0, kaj t+=sinh τ , estante τ ′ kaj τ ′ prezentataj en (9). Vidu ankaŭ la asimptoton D=−t−A. Figuro (2.a) montras ke ne estas Dopplera efiko (D = 1) trifoje: := −2 sinh−1 b2 + 8a+ b) , τ ′ = 0 , τ ′ := 2 sinh−1 b2 + 8a− b) . (9) Nun ni kalkulasD kiel funkcio de la propratempo τ de enigo de lumo al observanto. Uzante (3) kaj (6) ni havigas sinh τ ′ = 2(τ 2 −A2) τ(τ 2 − C2) + A (τ 2 − C2)2 + 4(τ 2 − A2) , C2 := A2 + b2 + 1 ; (10) substituante tiun kaj cosh τ ′ = 1 + sinh2 τ ′ en ekvacio (8) ni havigas la deziratan Doppleran faktoron D(τ). Figuro (2.b) montras tiun funkcion. Nun ni kalkulu la Doppleran faktoron kiel funkcio de la loko x de signal-eligo. Uzante (2), (3) kaj (4) en (8), vidu ke Dǫ(x) = x+ A+ (x+ A)2 − 1 x2 + b2 , ǫ := |t|/t. (11) Figuro (3.a) montras la du funkciojn D−(x) kaj D+(x). Observu la tri lokojn el kiuj la lumo eligita ne kolor-ŝanĝos: x− := b2 + 8a+ b) , x = −a , x+ := − b2 + 8a− b) . (12) Fine ni montras D kiel funkcio de la inercia tempo t de eligo: D(t) = 1 + t2 + x2 + b2 , x(t) = 1 + t2 − A . (13) Figuro (3.b) montras tiun funkcion. Observu la asimptoton D= −t−A kiam t → −∞. Figuro 4: Doppleraj faktoroj kun b = 0. 4.a) D kiel eksponencialaj funkcioj (14) de propratempo τ ′ de moviĝanta fonto. Estas nekontinueco de D en momentoj ∓τ ′ , prezentataj en (5). 4.b) D kiel funkcio (15) de propratempo τ de observanto. Estas nekontinueco de D en momentoj ∓t0 de trapaso. Kiam τ =a ne estas kolor-ŝanĝo. Vidu la asimptoton τ =−A; do la observanto ricevas la unuajn signalojn en ĉi tiu momento, nefinie viol-delokigataj. La valoro de t0 kaj de τ estas en (5). 3 Trapaso Nun ni studas la interesan okazon ĉe b=0. Anstataŭ preterpasi, la fonto pasas tra la observanto. Tio generas nekontinuecon ∆D je la Dopplera faktoro. Por havigi D kiel funkcio de τ ′, τ , x kaj t, ni faras b = 0 en ekvacioj (8), (10), (11), (13), respektive. Pri D(τ ′) ni havigas D(τ ′) = exp(−ǫxτ ′) , (14) kie ǫx estas la signumo de x. Estas ǫx =+1 kiam τ ′ <−τ ′ kaj τ ′ > τ ′ , kaj estas ǫx =−1 kiam < τ ′<τ ′ . Vidu figuron (4.a). Pri D(τ), figuro (4.b) montras la funkcion D(τ) = (A+ τ)−1 , −A < τ < −t0 , τ > t0 , (A− τ)−1 , −t0 < τ < t0 . Pri D(x) ni havigas Dǫ(x) = x+ A− ǫǫx (x+ A)2 − 1 . (16) Vidu figuron (5.a), observante la tempan sinsekvon i, ii, iii, iv, v, vi. Fine, pri D(t), Dx(t) = 1 + t2 − tǫx , (17) kie ǫx estas la signumo de x. Estas ǫx = +1 kiam t < −t0 kaj t > t0, kaj estas ǫx = −1 kiam −t0<t<t0. Vidu figuron (5.b). La nekontinueco ∆D de Dopplera faktoro D estas facile kalkulata ekde la antaŭaj rezultoj: 1 + v0 1− v0 1− v0 1 + v0 1− v20 A2 − 1 = 2 sinh τ ′ = 2 t0 . (18) Figuro 5: Doppleraj faktoroj kun b = 0. 5.a) D kiel funkcio (16) de la loko x de lum-eligo. Sagoj montras la pozitivan fluon de la tempo. Vidu la asimptoton D=2(x+ A). Estas nekontinueco de D en x=0. La valoro de τ ′ estas en (5). 5.b) D kiel funkcio (17) de t. Estos nekontinueco de D en lum-eligoj kiam ∓t0, (5). Vidu asimptoton D=−2t. La valoro de τ ′ estas en (5). 4 Komentoj Ni konsideris restantan observanton ĉe inercia sistemo S kaj moviĝantan fonton, kiel figuro (1.a). Ni montris, vidu figuron (3.a), ke lumo eligita el x=0 (minimuma distanco al observanto, ĉe S) estas enigota ruĝ-delokigate. Tamen, en la antaŭa artikolo [1] (vidu ĝian figuron 1), kie la fonto restas ĉe inercia sistemo Σ kaj la observanto moviĝas, lumo enigita kiam la observanto estas en x = 0 estas vidata kun viol-delokigo. Tio malsamo estas natura en special-relativeco, kaj ni plieksplikos tion en estonta artikolo de revizio; vidu ankaŭ [3]. Ĉi tie ni vidis en figuroj (2.b) kaj (4.b), ke la observanto ricevas signalojn el la fonto nur poste sia finia propratempo τ=−A. Ĉi tiu kontrastas kun ekvacioj (19) kaj (17) de [1], kiuj indikas ke la observanto ricevas signalojn je ĉiuj momentoj −∞<τ <∞. La kialo de tiu malsameco estas, ke ĉe [1] la tempo τ estas propratempo de moviĝanta observanto, kvankam en ĉi tiu artikolo la tempo τ estas propratempo de restanta observanto. La fluoj de la du tempoj estas tre malsamaj. En estonta artikolo ni prezentos sistemojn kies fonto kaj observanto ambaŭ estas akcelataj. Kelkaj el ĝiaj rezultoj estas mirindaj. Citâoj [1] F.M. Paiva kaj A.F.F. Teixeira, Relativeca Dopplera efekto ĉe unuforme akcelata movo – I, http://arxiv.org/abs/physics/0701092 [2] B. Rothenstein et al., Relativistic Doppler effect free of “plane wave” and “very high fre- quency” assumptions, Apeiron 12 (2005) 122-135 [3] F. M. Paiva kaj A. F. F. Teixeira, La relativeca tempo - I, http://arxiv.org/abs/physics /0603053 Enkonduko Dopplera efiko Trapaso Komentoj

0704.1131

Finding (or not) New Gamma-ray Pulsars with GLAST

Finding (or not) New Gamma-ray Pulsars with GLAST Scott M. Ransom National Radio Astronomy Observatory, 520 Edgemont Rd., Charlottesville, VA, 22901, USA Abstract. Young energetic pulsars will likely be the largest class of Galactic sources observed by GLAST, with many hundreds detected. Many will be unknown as radio pulsars, making pulsation detection dependent on radio and/or x-ray observations or on blind periodicity searches of the gamma-rays. Estimates for the number of pulsars GLAST will detect in blind searches have ranged from tens to many hundreds. I argue that the number will be near the low end of this range, partly due to observations being made in a scanning as opposed to a pointing mode. This paper briefly reviews how blind pulsar searches will be conducted using GLAST, what limits these searches, and how the computations and statistics scale with various parameters. Keywords: pulsars, gamma rays, data analysis PACS: 95.75.Wx, 95.85.Pw, 97.60.Gb INTRODUCTION Pulsar emission mechanism(s) from radio to gamma-rays are poorly understood after 40 yrs of work. Yet despite this fact, pulsars are high-precision tools that probe a variety of topics in both fundamental physics and astrophysics. EGRET detected pulsed emission from at least 6, and probably 7−9, young pulsars at energies >∼ 100 MeV [1]. In addition, several tens of the (primarily Galactic) unidentified EGRET sources are likely pulsars [e.g. 2, 3, 4]. With its much larger effective area and improved angular resolution, GLAST will almost certainly detect hundreds of pulsars. The majority of those pulsars will remain unidentified though (and useless as physics tools), unless pulsations are detected through 1) “folding” of the events using timing ephemerides for known radio pulsars; 2) searches of associated x-ray sources to find radio-quiet Geminga-like pulsars [5], or very faint radio pulsars [6, 7]; or 3) “blind” searches of the gamma-ray events [8]. This paper discusses the latter, and perhaps most difficult, option. The known gamma-ray pulsars have similar characteristics [for a review, see 1]. They are mostly non-variable sources with fairly flat energy spectra in the gamma-ray regime, but with spectral cutoffs around or just above 1 GeV. The flat energy spectra imply that most of the photons detected by the Large Area Telescope (LAT) will be in the ∼100-300 MeV range, where the point spread function has a width >∼ 1 ◦. The gamma-ray pulse shapes are complex with two relatively sharp pulses which provide higher harmonic content in searches and make their detection easier. Most of the GLAST mission will be spent in sky survey mode, where the whole sky is scanned every ∼3 hours1. On average, a point in the sky will be within the LAT field of view ∼ 1/6 of the time. Pointed observations would increase this fraction to ∼ 1/2 (with Earth occultations preventing higher on-source efficiency). This loss in efficiency due to scanning (more specifically, the resulting decrease in the number of source events Ns during a particular viewing period Tview) will make coherent pulsation searches considerably more difficult and less sensitive. BLIND SEARCHES FOR GAMMA-RAY PULSARS Blind searches of low count rate event data are typically conducted using either Fourier techniques (i.e. binning events into a time series, computing one or more FFTs, and analyzing the resulting amplitude and possibly phase spectra; [e.g. 9, 8]) or via brute-force epoch folding (i.e. assembling a pulse profile by determining the pulse phase of each event from a trial ephemeris and then computing a probability of non-uniformity for the profile; [e.g. 10, 11]). Optimal sensitivity to pulsations comes from searches that treat all of the data in a “coherent” fashion, meaning that the pulsar’s rotational phase is accurately tracked over the full observation or analysis duration Tview, from first event to last. 1 For details, see the GLAST mission website: http://glast.gsfc.nasa.gov http://arxiv.org/abs/0704.1131v1 http://glast.gsfc.nasa.gov 5−sig 10−sigm > 15−sigma detection non−detection Pointing Scanning FIGURE 1. Left) The number of independent Fourier “bins” (i.e. 1/Tview in Hz) a signal with frequency derivative ḟ (bottom) and frequency 2nd derivative f̈ (top) would drift in an observation of duration Tview. To preserve sharp features in pulse profiles, search codes need to account for signal drift to a small fraction (1−10%) of a Fourier bin. Right) The 95% confidence-level detectability of a blind event-folding search with Tview=20 days for a pulsar with a Gaussian pulse profile with FWHM=15% in pulse phase. The “sigmas” represent Gaussian significance in standard deviations after accounting for the number of trials searched. The arrow (representing a factor of ∼3 increase in accumulated events during Tview and which can be translated anywhere on the plot) shows how the significance improves for pointed observations (which are still affected by Earth occultations) as opposed to scanning observations. In this case, a 20-day pointing results in an ∼8-σ detection, whereas 20-days of scanning gives a non-detection. The real problem with gamma-ray pulsation searches comes from the fact that the sources have very low count rates of 10−7−10−8 photons(>100 MeV) cm−2 s−1, which for the LAT corresponds to roughly 0.2−2 events per hour. Therefore, very long observations lasting from weeks to years are required to make significant detections. And unfortunately, young pulsars are notoriously badly behaved over such timescales. Ideally, we would only need to search over the unknown spin frequency f of a pulsar. However, young pulsars rapidly spin-down (i.e. have large frequency derivatives ḟ ), they exhibit timing noise which manifests itself in searches as significant frequency 2nd derivatives f̈ (and for the noisiest pulsars, many higher order derivatives as well), and they occasionally, on month or year timescales, “glitch” and instantaneously change their observed f and ḟ . Figure 1 shows the number of independent frequency bins of width 1/Tview in Hz, corresponding to an accumulated phase error of one rotation, over which a pulsar signal will “drift” during a coherent search for a variety of realistic values of f and f̈ . Pulsar spin-down becomes important for energetic young pulsars in a day or two, and so all gamma-ray searches must account for an unknown ḟ . Timing noise can become important after a few weeks or months. Since gamma-ray pulse profiles contain sharp features, and therefore their pulsed signals contain many harmonics, search codes must significantly oversample the ∼ 1/Tview or ∼ 1/T view spacings in f and ḟ to maintain optimal sensitivity. A potentially bigger problem, as shown by Chandler et al. [8], are positional errors or uncertainties ε , which cause apparent changes in the observed spin frequency δ f ∼ 10−6εmrad f10 sinθ Hz, frequency derivative δ ḟ ∼ 2× 10−13εmrad f10 cosθ Hz s−1, and frequency 2nd derivative δ f̈ ∼ 4 × 10−20εmrad f10 sinθ Hz s−2, where εmrad is the position error in milliradians, f10 is the spin frequency in tens of Hz, and θ is the time varying angle between the Earth’s velocity vector and the line of sight to the pulsar. For candidate GLAST pulsar sources, ε will be milliradians (arcminutes) in size, making the blind detection of even stable Geminga-like pulsars much more difficult. The intrinsic ḟ , timing noise, glitch frequency, and position error effects place upper limits on the useful Tview for coherent pulsation searches to durations as short as ∼10 days for the most energetic and active pulsars, or a few months for older, slower, and more stable pulsars. Because of the scanning nature of the GLAST mission, there will be fewer counts accumulated for a particular source over the specified Tview by a factor of ∼3 as compared to a pointed observation. This decrease in accumulated counts per unit time greatly impacts coherent search sensitivity and is the reason why scanning is not the optimal observing mode for pulsar studies. Pulsation Detection and Computational Considerations The power P in a periodic gamma-ray signal is P∼ 1+αN2s /Nt , where α is a pulse-shape dependent factor (0.4−0.9 for the known gamma-ray pulsars)[e.g. 12]. The total number of events Nt is the sum of the source (pulsed) events Ns and background (unpulsed) events Nb. The probability that noise fluctuations produce a certain search power Po is exponentially related to the power: Prob(P ≥ Po) ∝ exp(−Po)Ntrials, where Ntrials is the number of independent trials searched. Ntrials is proportional to the range of f and ḟ searched, but more importantly, to T view. In general then, to optimize search sensitivity, we want large Ns to maximize P, yet a relatively short Tview (appropriate for the limits discussed in the last section) to reduce Ntrials and the computational complexity. This rule exposes another problem for scanning-mode observations of pulsars: if Tview is increased by a factor of ∼3 to recover the events “lost” compared to a pointed observation (assuming the pulsations remain well-behaved over the longer Tview), a detected signal will have the same power as from the shorter pointed observation, yet the probability that the signal is significant will decrease by a factor of ∼ 33 = 27 due to the larger Ntrials searched. Computationally, the complexity of epoch-folding searches scales roughly as T 4view (i.e. Nt ×Ntrials), while FFT- based techniques fare slightly better and scale as T 3view logTview. Moderate sized computer clusters can currently handle epoch-folding searches of hundreds to thousands of events for viewing periods of up to several weeks (see Figure 1). For longer Tview or more limited computing resources, one can use evolutionary [14] and/or heirarchical [13] tech- niques to greatly reduce the computational burden of coherent searches without greatly impacting the overall sensi- tivity, if the average pulsation flux is time invariant. Coherent searches with GLAST with Tview <∼ 1−2 months should be sensitive to pulsars with photon fluxes (>100 MeV) of a few times 10−8 to a few times 10−7 photons cm−2 s−1 for scanning-mode observations, or as low as ∼2×10−8 photons cm−2 s−1 for pointed observations. Finally, incoherent searches are possible where the absolute phase of pulsations is not tracked over the full Tview, but only over much shorter “windows” Twin. Incoherent searches are less sensitive than coherent ones for the same Tview, but they use many fewer computing resources and enable searches with much longer Tview [9, 15]. The basic idea limits the duration of the coherently processed intervals Twin such that a signal with any reasonable ḟ or f̈ stays within a single independent frequency bin (of width 1/Twin Hz) or fraction thereof. These windowed analyses are then combined without phase information, but with corrections for ḟ and/or f̈ effects occurring between the intervals, and analyzed. Initial simulations show that one might detect relatively stable pulsars with photon fluxes (>100 MeV) as low as 1−2×10−8 photons cm−2 s−1 with Tview ∼ 1 yr and Twin ∼ 1 day. Without phase information, however, the folding of all events over such long durations to accumulate high-quality pulse profiles may be difficult or even impossible. In summary, instabilities of young pulsars, source positional uncertainties, and the default survey/scanning mode for the mission will make blind pulsation searches with GLAST much more difficult than early estimates had predicted. It is therefore likely that GLAST will blindly discover only tens of new pulsars rather than hundreds. Acknowledgements: Thanks go to Paul Ray, Julie McEnery, Mallory Roberts, Maura McLaughlin, and Kent Wood for useful discussions. REFERENCES 1. D. J. Thompson, “Gamma ray pulsars,” in ASSL Vol. 304: Cosmic Gamma-Ray Sources, edited by K. S. Cheng, and G. E. Romero, 2004, p. 149. 2. M. A. McLaughlin and J. M. Cordes, ApJ 538, 818 (2000). 3. M. A. McLaughlin and J. M. Cordes, ArXiv Astrophysics e-prints (2003), astro-ph/0310748. 4. M. Kramer et al., MNRAS 342, 1299 (2003). 5. J. P. Halpern and S. S. Holt, Nature 357, 222 (1992). 6. J. P. Halpern et al., ApJ 552, L125 (2001). 7. M. S. E. Roberts et al., ApJ 577, L19 (2002). 8. A. M. Chandler et al., ApJ 556, 59 (2001). 9. M. van der Klis, “Fourier Techniques in X-ray Timing,” in Timing Neutron Stars, (NATO ASI Series), edited by H. Ögelman, and E. P. J. van den Heuvel, Kluwer, Dordrecht, 1989, p. 27. 10. O. C. de Jager, B. C. Raubenheimer, and J. W. H. Swanepoel, Astron. Astrophys. 221, 180 (1989). 11. P. C. Gregory and T. J. Loredo, ApJ 398, 146 (1992). 12. R. Buccheri, M. E. Ozel, and B. Sacco, Astron. Astrophys. 175, 353 (1987). 13. S. M. Ransom, “Fast Search Techniques for High Energy Pulsars,” in ASP Conf. Ser. 271: Neutron Stars in Supernova Remnants, edited by P. O. Slane, and B. M. Gaensler, 2002, p. 361. 14. K. T. S. Brazier and G. Kanbach, Astron. Astrophys. 116, 187 (1996). 15. W. B. Atwood, M. Ziegler, R. P. Johnson, and B. M. Baughman, ApJ 652, L49 (2006). astro-ph/0310748 Introduction Blind Searches for Gamma-ray Pulsars Pulsation Detection and Computational Considerations

0704.1132

Did time begin? Will time end?

arXiv:0704.1132v2 [astro-ph] 14 May 2007 DID TIME BEGIN? WILL TIME END? Paul H. Frampton University of North Carolina at Chapel Hill. http://arxiv.org/abs/0704.1132v2 Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter 1: Why do many other scientists believe time began at a Big Bang? . . . . . . . . 7 Chapter 2: Smoothness of the Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 Chapter 3: Structure in the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 4: Dark Matter and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Chapter 5: Composition of the Universe’s Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 Chapter 6: Possible Futures of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Chapter 7: Advantages of Cyclic Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Chapter 8: Summary of Answers to the Questions: Did Time Begin? Will Time End? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Preface Did time begin at a Big Bang? Will the present expansion of the universe last for a finite or infinite time? These questions sound philosophical but are becoming, now in the twenty-first century, central to the scientific study of cosmology. The answers, which should become clarified in the next decade or two, could have profound implications for how we see our own role in the universe. Since the original publication of Stephen Hawking’s A Brief History of Time in 1988, the answers to these questions have progressed as a result of research by the community of active theoretical physicists including myself. To present the underlying ideas requires discussion of a wide range of topics in cosmology, especially the make up of the energy content of the universe. A brief summary of my conclusions, that of three different possibilities concerning the history and future of time, the least likely is the conventional wisdom (time began and will never end) and most likely is a cyclic model (time never begins or ends), is in the short final Chapter which could be read first. To understand the reasoning leading to my conclusions could encourage reading of my entire book. My hope in writing this, my first popular book, is that it will engender reflection about time. Many a non-scientist may already hold a philosophical opinion about whether time begins and ends. This book’s aim is to present some recently discovered scientific facts which can focus the reader’s consideration of the two brief questions in my title. Paul H. Frampton Chapel Hill Chapter 1 WHY DO MANY OTHER SCIENTISTS BELIEVE TIME BEGAN AT A BIG BANG? Our everyday perception of the universe comes from looking up at the sky to see the Sun in the daytime and, more particularly, to see thousands of stars in the night-sky. Surely some of the oldest questions since the beginnings of human thought are: How large is the universe? Did it ever begin? What are the principal constituents of the present universe? Will time ever end? Cosmology is the name for the scientific study of the universe. The present time is an unprecedented age for cosmology because it is fair to say that in the last five years we have learned more in cosmology than in all previous human history. Despite this enormous and exciting growth of our knowledge as a result of many impressive observations, the universe has become more enigmatic in many ways. The more we learn, the more the extent of our ignorance becomes manifest. Cosmology has recently answered some of the old questions and in this Chapter we shall give answers to the first two: How large is the universe? How long ago did it, or at least the present expansion era, begin? We can all agree that the expansion stage we are presently in began a finite time ago but, as I shall explain later in ths Chapter, it is not obvious that time itself began then, if ever. We do know how large the visible universe is, meaning how far away are the most distant galaxies whose light can have reached us on the Earth. It is theoretical possible, and even favored in some theoretical scenarios, that our universe is actually very much larger, than the visible universe. In some very speculative scenarios the universe is spatially finite with non-trivial topology. This is at present not readily testable so we shall be content to try to convey just how gigantic is the visible part. The observational means by which we know accurately the size of the universe will not concern us here but sufficient to say that present studies using the Hubble Space Telescope combined with the largest (up to 10 meters in diameter) ground-based optical telescopes tell us the size of the visible galaxy to an accuracy of a few per cent. This sort of accuracy has been achieved only since the turn of the 21st century. Cosmological distances are so much bigger than any distance with which we may be familiar that it is not easy to grasp or comprehend them even in our imagination. So let us begin with the largest distance which is easily comprehensible from the viewpoint of our experience. A very long airplane ride may take 15 hours and go 9000 miles, a significant part of half way around the Earth. People who travel a lot like some physicists may take such a flight a few times each year. One knows that the plane has a ground speed of about 600 miles per hour and the discomfort of sitting, especially in economy class, for such a long time gives a strong impression of just how far that distance is. Of course, people a hundred years ago would never travel that far in a day but now we do and it gives us a feel for the size of the planet so that is a length distance from which we can begin. The next larger distance to think about is the distance between the Earth and the Moon. This is about thirty times the distance of the plane ride and so would take some three weeks at the same airplane speed, a few days in a NASA spacecraft. The distance to the Moon is thus imaginable: if you walked at four miles an hour non-stop without sleep it would take about eight years to arrive and another eight years to return. Nevertheless, the arrival on the Moon of astronauts Armstrong and Aldrin in July 1969 was one of the most memorable events of the last century. Only partly was it due to the distance to the Moon, it was equally the concept of humans walking for the first time on an astronomical object other than the Earth. The Moon is visible in the night sky, and equally often in the daytime thanks to its reflection of light from the Sun. The Sun is by far our nearest star and its radiated energy is crucial to the possibility of life on Earth. How far away is the Sun? It is about ten thousand times the length of the airplane ride and would take about twenty years to reach at the speed of the airplane. Not that any sane person would want to go there with a surface temperature well above that of molten iron. The Sun is about four hundred times further away than the Moon, and is already at such a large distance that it far exceeds anything with which we are familiar. This sets the scale of the Solar System with the Earth, rotating on its axis once a day, orbiting once a year around the Sun at a distance of some ninety-three million miles. Other planets, Mercury and Venus, circulate inside the Earth’s orbit while six others including Mars, Jupiter and Saturn orbit outside the Earth. It is almost inconceivable that any human being will travel outside of the Solar System in our lifetimes just because of its enormous size. Yet on the scale of the visible universe the Solar System is, in contrast, unimaginably tiny and insignificant. So if there were no life other than on the Earth the universe would seem to be an almost absurdly large object if life were its primary goal. In addition to the Moon and some planets, we can see thousands of stars with the naked eye. Most of these stars are similar to our Sun but appear much dimmer because of their distance. How distant are even the nearest stars? The answer is some two million times the distance to the Sun. So whereas we can reach the Sun in twenty years at the speed of an airplane to reach the nearest star in twenty years would require a two million times faster speed. A quick calculation shows this takes six hundred miles per hour into thirty-five thousand miles per second. To put such a speed into perspective, the speed of light is about one hundred and eighty thousand miles per second. This means our imaginary airplane, suitably coverted as a spacecraft, must travel at one fifth of the speed of light just to reach the nearest star in twenty years. Here we see the limitations to any travel possibilities not only in our lifetime but what would seem to be for ever. According to the theory of relativity, which there is no reason to doubt, nothing can travel faster than the speed of light. So even if the human lifetime is extended by medical advance to two hundred years or even a thousand years it is impossible to travel during one lifetime more than a few hundred times the distance to the nearest star. But the galaxy to which our Solar System belongs extends about ten thousand times the distance to the nearest star. So it would seem impossible ever to leave our particular galaxy which is known as the Milky Way from its appearance spreading across the night sky. There are a couple of holes in this argument. First, according to relativity time slows down as one travels at approaching the speed of light. Second, it is conceivable that some cryogenic method might be devised to slow down the speed of life and greatly enhance the effective human lifetime. Even so, to travel outside our galaxy does seem forever impossible and cosmology may remain a spectator sport. One hundred years ago it was generally believed that the the universe was comprised of only the Milky Way. The size of the our galaxy is only ten thousand times the distance to the nearest star which is itself two hundred thousand times the distance to the Sun. That is, the galaxy size is two billion times (one billion is a thousand million) the Earth-Sun distance. The size of the galaxy seems to be relatively independent of time and so in ignorance of a universe very much bigger than a single galaxy it was believed, before the 1920s, that the universe was itself static, neither expanding nor contracting. When the general theory of relativity was proposed in 1915 this state of the observational knowledge stymied what could have been predicted, namely the overall expansion of the universe. This expansion, which is a key feature of the universe and will lead us to the conclusion that it had a definite beginning, became an option only by observations somewhat later in the 1920s. Now we arrive at the final leap in the distance scale. The visible universe turns out to be about four hundred thousand times the size of the Milky Way, very much larger than previously imagined. That is, not only is the Solar System of neglible size with respect to the universe but so is even the entire Milky Way. In fact, in theoretical cosmology galaxies are treated as point particles! And the human race may be confined forever to be inside one of these points! We have seen that the size of the galaxy is tremendously bigger, by a factor of billions, than the distance to the Sun. Yet the visible universe is so much larger again than a galaxy that to study it each galaxy may be regarded as just a single dot within it. This should communicate an idea in words of just how big the visible universe is. Now we show how we know the present expansion (and possibly time itself) had a beginning some fourteen billion years ago. As already discussed, the size of the Milky Way has not expanded or contracted signifi- cantly since it was formed some ten billion years ago. Within the Milky Way the Sun and the Solar System appeared about five billion years ago. The Earth is a little younger, about four and a half billion years. The point is the general arrangement of the Sun and planets in the Solar System has not markedly changed in the last few billion years. During that time we may regard the galaxy and its contents as of a constant size. A truly astonishing revelation comes when we study the same question for the entire universe including hundreds of thousands of the the billions of galaxies outside of the Milky Way. The issue is what is their typical motion relative to our galaxy? Here it is important to understand a phenomenon well-known in physics called the Doppler effect. It is a more familiar phenomenon for sound waves than for light. When a train blows its whistle and passes a listener the pitch of the whistle falls from a higher note to a lower note. In fact not only the whistle but the entire train noise exhibits the same Doppler effect. Why does this happen? It is because the motion of the train towards the listener compresses the sound waves to a shorter wavelength and, because the veloc- ity of sound is unaltered, to a higher frequency. Similarly, when the train is moving away the sound waves are stretched and the frequency lowers. The pitch for a stationary train would lie between the two pitches while approaching and receding. The shift in frequency is calculable simply in terms of the ratio of the speed of the train and the speed of sound. Exactly the same Doppler effect occurs for light. If a galaxy is approaching our galaxy, its light appears with a higher frequency. For the visible spectrum the highest frequency is for blue light so we may say that the light is blue-shifted. On the other hand, if the galaxy is receding from ours its light appears shifted to a lower frequency and is red-shifted toward the red or lowest frequency end of the visible spectrum. In fact what are observed are the spectral lines of light emitted from known atoms and whose frequencies are accurately known here on Earth. If all the lines are systematically shifted towards the blue then the galaxy hosting these atoms is approaching the Milky Way: if toward the red then it is receding from us. This can be made precise by a mathematical expression for the frequency shift which gives the approach or recession speed as a fraction of the speed of light. When this is studied for a large number of galaxies it might be expected that roughly half would be blue-shifted and half red-shifted if the overall universe were static and the galaxies were moving randomly. What was observed, however, to the astonishment of Hubble and Einstein in 1929 is that almost all galaxies are red-shifted. Apart from a few galaxies in our immediate neighborhood like the nearby Andromeda galaxy all the hundreds of thousands of more distant galaxies measured are receding from us. This means that the entire universe is expanding and the galaxies are moving away from us and as we shall see later, from each other. This phenomenon is called the Hubble expansion. The next important question is how does the recession velocity depend on the distance of the galaxy from the Milky Way? The galaxies can be classified into types such as spiral, elliptical, irregular and the total light emitted may be assumed to show regularity within each type. But the apparent brightness on Earth depends on the distance and falls off as an inverse square law. So the apparent brightness can be converted into a distance. When the recession velocity is compared to the distance a very important regularity appears. This is the most significant discovery in cosmology called Hubble’s law which states that the recession velocity is proportional to the distance. The ratio of the velocity and the distance is thus a constant, the Hubble parameter. Its value is notoriously difficult to measure but now we do know it to be close to seventy, within ten percent, in certain units. The units which are not crucial to the general discussion are kilometers per second per megaparsec where a megaparsec is the distance light travels in about three million years. How does this tell us when the present expansion phase began? This requires the use of the equations of the general theory of relativity together with two assumptions. The first assumption is that the universe at the large scale is the same, on average, in each of the three directions of space. This is called isotropy and is supported by observations which indicate no preferred direction exists in the universe. The second assumption is that, on average, all positions in the universe are equivalent. This means that in all galaxies (hypothetical!) observers would see the recession of other galaxies according to Hubble’s law. This assumption is called homogeneity. The combination of these two assumptions, isotropy and homogeneity, is known technically as the cosmological principle. Combining the cosmological principle with the general theory of relativity gives rise to mathematical equation known after its inventor as the Friedmann equation which character- izes the expansion of the universe in terms of a scale factor which is a function of time and specifies the typical distance between galaxies. Inserting the known Hubble parameter and the present known composition of the universe then enables us to calculate the scale for all past time. Note, in passing, that we cannot do this for future times with confidence because we do not know with certainty how the composition of the universe will evolve in the future. For the past we have good confidence and we find a striking conclusion: run in reverse, the contracting universe is seen mathematically to shrink to a point at a well-defined past time. At that time the universe may have begun in some unimaginably powerful explosion called the Big Bang. Our present expansion seems to have a beginning and we know when. It was some 13.7 billion years ago, give or take a two hundred million years. The age of the universe is now established to an accuracy of about two percent. There is one serious problem with extrapolating the Friedmann’equation back in time, namely that about 13.7 billion years ago the equation becomes singular. The density and temperature become infinite and the classical theory of general relativity breaks down. Thus the Big Bang, only much more recently named, was the initial singularity to the early workers of the 1920s and 1930s. Within the theory therefore we know that the Big Bang must be avoided. A common response is to invoke quantum mechanics. From the fundamental constants, the speed of light, Newton’s gravitational constant and Planck’s constant one can construct a time known as the Planck time which is a tiny fraction (10−44) of a second. One may say that at that short time after the would-be Big Bang the classical theory of general relativity must break down. Quantum gravity must play a role and no completely satisfactory theory exists. So one may argue that quantum mechanics rescues the day. Indeed an entire filed known as quantum cosmology has been built up around such a ruse but without really new insights. In this book we shall not appeal to quantum mechanics in this way but examine whether the Big Bang can be avoided in a purely classical context. Before proceeding there is one amusing anecdote about the origin of the graphic term Big Bang which seems so apt to describe the beginning of the universe. Before the scenario we have just described was firmly established a competing theory was the steady-state theory which postulated that despite the Hubble expansion there was a steady state and no beginning because of the continuous spontaneous creation of new galaxies. As a derogatory term for the competing theory, Big Bang was coined by a leading exponent of the steady-state theory. Unfortunately for him, the Big Bang theory and not his viewpoint was confirmed by subsequent measurements. There is one alternative view (we shall discuss another view in Chapter 7) of the Big Bang where the lifetime of the (generalized) universe is infinite. The process of the Big Bang is in that view something which has occurred repeatedly, indeed infinitely, resulting in an infinite number of different universes of which we are aware of just one. This is technically called eternal inflation and the resultant universe becomes a multiverse for obvious reasons. It is difficult, if not impossible, to test eternal inflation because the other universes would seem to be forever hidden from our view. The best chance may be to make a probabilistic treatment of the multiverse to estimate the probability of the universe we observe having the properties it has in terms of its fundamental consituents or building blocks. Some research is indeed being pursued along this line. In this book we shall assume the beginning of the present expansion era to have taken place approximately fourteen billion years ago followed, as will be discussed, by an infla- tionary era of rapid expansion. The latter explains two different kinds of extraordinary smoothness observed. We know that there was temperature uniformity to one part in one hundred thousand in the universe when it was only four thousand years old. Then there is the proximity of the observed density of the universe to a special value known as the critical density which would, without inflation, require preternatural fine-tuning in the early universe. Inflation appears now to be ubiquitous in almost all theoretical cosmology, in one form or another. As we shall discuss, it can account for the otherwise-puzzling smoothness properties of the universe. On the other hand, it is exceedingly difficult to make direct measurements which are sensitive to such an early era, the inflationary era which occurred even earlier than a billionth of a second after the would-be Big Bang. Normal observations involving electromagnetic radiation go back only to a few hundred thousand years after the would-be Big Bang, far too recent to study inflation. Studies of abundances of light elements like helium and hydrogen probe indirectly back to a cosmic time of one second after the would-be Big Bang. Potential neutrino astronomy measurements could directly probe a similar era. The only chance of direct observation of the inflationary era would appear to be by gravitational radiation - waves created by string gravitational fields in the early universe. The observability of such radiation depends on how early inflation took place, the earlier being the easier to detect. For later inflation it looks presently impossible to detect such gravity waves. The word ”presently” is essential since how technology will evolve, and what consequent scientific apparatus will be enabled, by the end of the 21st century is impossible to predict. It is a lesson from the history of physics that to decree anything impossible is a dangerous prediction. One may ask what happened before the Big Bang, if it did occur? This is beyond scientific investigation and it is easier to assume that time began then. Very ambitious and speculative theories discuss prior times using ideas such as T-duality in string theory or eternal inflation with its resultant multiverse. If such theories become testable and shed light on the physics of our universe then they must be taken very seriously in a more general domain of applicability. At present, such ideas remain pure speculation. Another question which we shall address at length in this book is what will happen to the universe in the future? This is understood less than the past, and depends critically on the properties of the newly-discovered Dark Energy which comprises almost three-quarters of the total energy density of the universe. Concerning space one will ask whether it too, like past time, can be finite in extent. Is it possible that by proceeding in a straight line one will return, after a finite time and distance, to the starting point due to a non-trivial topology of space? There is no compelling evidence for this possibility though certain data on the cosmic microwave background radiation can be interpreted as supporting such an assumption. Alternative explanations for the data come from arguments about cosmic variance or from small distortions in the hypothetical inflaton potential, so the case for non-trivial spatial topology is not at all strong. If there were non-trivial topology, it could be of one of three types. Positive curvature corresponds to a closed universe, negative curvature to an open universe and flat, without curvature, is the geometry predicted by inflation. The local properties in such a universe satisfy the same general relativity equations as for the case of infinite space with trivial topology; only the global toplogical properties differ so it is not obvious from, say, study of our galaxy alone which option Nature chooses. The notion of non-trivial topology of space necessarily introduces at least one fundamental length which, to be consistent with observational data, must be comparable to the radius of the visible horizon of a few gigaparsecs. A common question by an educated non-physicist is into what does the universe expand or, equivalently, what is ”outside” the universe? So let us try to give a clear answer. The answer is not obvious only because of the limitations to the human imagination. All of us can easily imagine three spatial dimensions but four is enormously more difficult. Unfortunately the spacetime manifold of the universe is itself four dimensional and this is both why the question naturally arises and why the answer is slightly elusive. If we scale down by one dimension there is an analogous situation which is, by contrast, very easy to grasp. Take a balloon with spots on the surface to represent galaxies. As time passes we inflate the balloon and the spots get further apart as for the expansion of the universe. Now a two-dimensional being on the balloon surface may ask: into what is this two dimensional space expanding? The answer is that there is nothing ”outside” the two-dimensional surface as obviously it is a closed surface without boundary. Similarly the three-dimensional space of our universe has no boundary and no ”outside”. Among so many interesting yet unanswered questions, the one about the beginning of the present expansion 13.7 billion years ago seems settled. The expansion itself was universally accepted only in 1965 as a result of the discovery of the remnant background microwave radiation. The uncertainty of the future scenario for the universe is under much study as a result of the discovery of dark energy dating from 1998. Thus we are at a very exciting time in the subject. The establishment of a finite length of the present expansion of about fourteen billion years is clearly of fundamental importance which could be equated by establishment of a finite spatial extent to the universe. There is absolutely no compelling evidence for such an idea although some data from the WMAP analysis of the cosmic background radiation, particularly the unexpectedly small values of the low multipoles, has been interpreted as suggestive of finite size and non-trivial topology. Certainly these cosmological considerations change our picture of our own history. Finally, in our discussion of the universe’s longevity, it is important that we use a linear time, rather than logarithmic time, in the above discussion. The two are dramatically different. Firstly in logarithmic time the age becomes infinite. But the difference can be better seen in a concrete analogy. Suppose we condense the entire cosmic history of fourteen billion years into one day of twenty-four hours starting and ending at midnight. First we use linear time. The nucleosyn- thesis takes place just a trillionth of a second after midnight; recombination and the surface of last scatter are three seconds later; galaxy formation starts around 1.40am; the Sun is created about 4pm in the afternoon and Julius Caesar invades Gaul about a hundredth of a second before midnight. But if we map the same history using logarithmic time starting at the Planck time (since we must now start at a finite time in the past) then the occurrence of major events looks completely different. Nucleosynthesis waits until 5pm in the afternoon; recombination and the surface of last scatter are at 10pm in the evening; galaxy formation begins at 11.25pm; the Sun is created at 11.48pm and Julius Caesar appears now only a trillionth of a second before midnight. This illustrates how the use of linear time in cosmology effects the relative spacing of subsequent events. From the viewpoint of fundamental physics more happened in the first second of the Big Bang than has happened in the subsequent fourteen billion years, more in parallel with a logarithmic picture of time. But it is in linear time, with which we are familiar in measuring all everyday events, that the time since the would-be Big Bang does have the finite value of 13.7 billion years. The answer to the question in this Chapter’s title has already been alluded to, that many other scientists (if not this author) explain away the initial singularity of the Friedmann equation by an appeal to quantum mechanics and quantum gravity. The absence of a fully satisfactory theory of quantum gravity can act as a further security for such scientists as no one can definitively refute the argument. However, the tentative attempts at quantum gravity has problems. The concept of the wave function of the universe, as employed in quantum cosmology, is problematic for several reasons, not least of which is that the observer is inside the system. The Planck time is much shorter than the time expected to pass between the would-be Big Bang and the onset of inflation. Explaining away the initial singularity by quantum mechanical arguments was useful only when it was faute de mieux. This is no longer the case. I find it more satisfactory to make a cosmological model where the density and temper- ature are never infinite. This precludes a Big Bang and replaces it with a different picture of time where time never begins and never ends. This is in contrast with the standard cosmological model where time begins at the Big Bang and never ends during an infinite future expansion. While I cannot prove rigorously that the conventional wisdom is wrong, it does entail the singularities and concomitant breakdown of general relativity that we have mentioned. The existence of plausible alternatives now, however, makes the Big Bang idea less plausible. As we shall show later in the book, there is an alternative version where time begins at a Big Bang and ends in a ”Big Rip” at a finite time in the future. I regard this as prefrbale aesthetically to the conventional picture. But best of three possibilities about time is the ”infinite in both directions”, past and future” as exemplified by a cyclic model My student and I constructed only in the twenty-first century based on the dark energy component. Obseravations of dark energy and its properties especially its equation of state will confirm or refute such more satisfactory ideas about time which insist there was never a Big Bang. Chapter 2 SMOOTHNESS OF THE UNIVERSE When we look further at the universe in the large, with the galaxies treated as point particles, there are even more surprises in store beyond those already discussed which include the gigantic size, the cosmic expansion and the fact that the present expansion phase had a beginning. What we have discussed already seems consistent however not obvious to the naked eye. From looking at the night sky we might most naturally suppose, as people did for many hundreds of years before the 20th century, that the stars had been there forever and would remain so. With the naked eye no further progress could ever have been made. Powerful telescopes see objects, galaxies, far more distant and completely outside the Milky Way. They are moving away at high speed, higher as their distance increases. Ironically the only other galaxy visible to the naked eye outside the Milky Way, the Andromeda galaxy, happens to be moving towards us! But almost all the other galaxies are receding from us. To discuss the unexpectedly high smoothness of the universe, technically called the hori- zon problem, we need to introduce the concept of temperature. The universe is filled with radiation, electromagnetic radiation, which is currently extremely cold. It is at a temper- ature of about three degrees above absolute zero. The value of this temperature varies inversely as the scale factor which characterizes the size of the universe. Consequently as the universe expands the temperature is falling even lower. Turning that around, if we ask instead about the past the temperature was higher. In the history of the universe, at least as far as the electromagnetic radiation is concerned, a most important thing happened about three hundred thousand years after the would-be Big Bang at which time the visible universe was about one thousand times smaller than it is now. At that time the temperature was therefore one thousand times higher than now. This means it was very hot, at three thousand degrees. It turns out this is the maximum temperature below which hydrogen atoms can survive as bound states. A hydrogen atom consists of a proton which forms the hydrogen nucleus around which one electron orbits. At temperatures above three thousand degrees the atom has a high propensity to ionize into a separate proton and electron. Indeed at all times before this special ”recombination” time the protons and electrons existed separately in what is called an ionized plasma. After the special time the hydrogen atoms existed as bound states. Technically this special occurrence is illogically but for ever called recombination. The name is illogical, it should be just combination, because the protons and electrons had never previously been combined! Electromagnetic radiation is composed of massless elementary particles called photons travelling at the speed of light. Photons are scattered by charged particles but not by neutral ones. This is why the recombination occurrence is so important for the electromagnetic radiation in the universe. Before recombination there was an ionised plasma of electrons and protons. Such charged particles can scatter the photons. This means that the universe was opaque until recombination. Photons could not travel in straight lines at the speed of light because of scattering by charged particles in the plasma. After recombination, in contrast, the charged electrons and protons became bound into neutral hydrogen atoms. The universe became transparent as it allowed photons to propagate freely through it in straight lines since neutral atoms have no interactions with photons. This means that the photons detected from the universe as a whole, technically called the cosmic microwave background radiation, have been able to travel along straight lines at the speed of light for the full fourteen billion years since the recombination occurred. This provides us with a unique and extremely valuable means of studying the state of the universe just three hundred thousand years after the beginning of the present expansion phase, the would-be Big Bang. Much of the recent progress in cosmology is due to the vastly improved experimental information since 1992 on the details of this background radiation. At a temperature of three degrees above absolute zero the wavelengh of these relic photons is about ten centimeters or four inches. This is in the part of the electromagnetic spectrum called microwave and is of similar wavelength to the radiation used in a domestic microwave oven. When they started out at recombination, from what is technically known as the surface of last scatter, their wavelength was one thousand times smaller or a tenth of a millimeter. This is because the wavelength of a photon goes inversely as its energy. Energy is proportional to temperature and the temperature was at that time one thousand times higher than now. Another equivalent viewpoint is that the wavelength of the photon has been stretched by a factor of a thousand, like everything else, during its propagation to Earth from the surface of last scatter. The background microwaves are invisible to the eye which can detect or see only a narrow part of the electromagnetic spectrum in the visible region where the wavelength is about a hundred thousand times smaller. But sensitive detectors of the microwave radiation have been flown on satellites and in the upper atmosphere suspended from large helium balloons. These detectors are looking out far beyond all the galaxies to the surface of last scatter which represents the most distant surface ever visible by electromagnetic radiation. To see further will require either neutrinos or gravitational radiation but such more exotic observations are reckoned to be extremely difficult. To explain why there is something extremely remarkable about the observed smoothness of the universe at the surface of last scatter requires that we take a close look at relativity theory. One prediction of relativity theory is that nothing can move at a speed greater than that of light. This applies not only to particles but also to any type of information. In particular, a causative phenomenon may not create any effect faster than the speed of light. We are therefore influenced only by events in the earlier universe that are such as can transmit information to us. Technically in special relativity theory we say that an event can be affected only by earlier events that are in its backward light cone. This requirement is called causality and is extremely restrictive on possible physical theories. Relativity theory has passed many tests and there is every reason to believe in it and in this consequent requirement of causality. When the microwave radiation coming from the surface of last scatter is analysed it is found that its temperature distribution is astonishingly smooth. The temperature is exactly the same in all directions to an accuracy of one in a hundred thousand. At that level there are exceedingly tiny fluctuations which are themselves extremely important in the formation of structure. But the important observation for the present discussion is that the distribution of temperature over this most distant surface is very, very smooth. Why is that so surprising? It is surprising because in the simplest Big Bang picture it would violate the sacred principle of causality. Why is that? Because the diameter of the present visible universe is roughly thirty billion light years and hence the diameter at the time these photons were emitted was one thousand times smaller or about thirty million light years. A light year is the distance light travels in one year. The age of the universe at recombination was just three hundred thousand years so information could have travelled only three hundred thousand light years. The radius of the surface of last scatter is a hundred times bigger. So on the two-dimensional surface there are the square of one hundred, namely ten thousand, regions of the surface which have never been in causal contact in the simplest Big Bang scenario. So therefore it is amazingly enigmatic that the temperature distribution is uniform within all these ten thousand causally-disconnected regions to an accuracy of one part in a hundred thousand! This is the first example of smoothness. It is very important to understand the previous paragraph because it suggests that the Big Bang theory has to be modified. Before describing the most popular modification of the Big Bang which accommodates this feature, we shall describe a second example of smoothness which will also be accommodated in the same modification, called inflation, to be described at the end of this chapter. The second example of smoothness observed for the present universe is different but has a similarity with the first that it is a property of the present universe which, when we extrapolate back in time, seems utterly perplexing in the context of Big Bang theory. As already discussed in Chapter 1 the mathematical underpinning of the expanding universe is provided by the general theory of relativity combined with the assumptions of isotropy and homogeneity which make up the cosmological principle. This leads to a simple differential equation for the time dependence of the scale factor. This equation tells us that the ultimate fate of the universe would appear to depend on the overall density of energy and matter in the universe. Of course we are talking only about the averaged density and not a local density where there is structure such as galaxies and stars. There is a critical value for this density which is such that the universe would expand forever but more and more slowly such that it will slow to a stop after an infinitely long time. With this critical density another viewpoint is that the positive kinetic energy of the motions of the galaxies and other matter in the universe exactly balances the gravitational energy associated with the gravitational attraction. Energy is conserved so the natural endpoint is where the galaxies come to rest at infinite separations and therefore no residual gravitational attraction. If the density is less than this critical density the universe’s geometry has negative curva- ture and it will expand for ever without coming to a halt. This negative curvature universe is technically termed open. The final possible case where the density is larger than the critical density has positive curvature. In it the expansion will eventually halt followed by a recon- traction to a Big Crunch. In other words, for such a closed universe the gravitational energy more than overcomes the kinetic energy. These simple arguments ignore the complications introduced by dark energy. We shall return to this issue. The special case where the density equals the critical density is at the borderline between open and closed and is called flat. The three different geometries can be characterized by whether the angles of a triangle add to less or more than, or equal to the canonical one hundred and eighty degrees. This is the total of the three angles in flat geometry, also called Euclidean geometry the simplest form of geometry taught at school and very well known since the times of the Greek civilization. The other two cases are examples of non-Euclidean geometries, discovered by mathematicians in the 19th century. An example of positive curvature is provided by the surface of a sphere such as the Earth. Drawing a triangle with one vertex at the North pole and the other two vertices on the equator a quarter of a circle from each other gives a triangle that has all three angles equal to ninety degrees and a total of two hundred and seventy degrees. The larger total characterizes a positive curvature as in a closed universe. In a negative curvature as in an open universe the three angles of a triangle sum to less than one hundred and eighty degrees. The present total density of mass and energy is equal to the critical density with an accuracy of a few percent. In other words, the angles in any triangle really add up to one hundred and eighty degrees to good accuracy. This fact is very surprising in the Big Bang theory and suggests that a component is missing. Why is that? In the equations which describe the evolution of the scale factor the flat geometry is an unstable solution. This means that if the universe is flat now it most probably was always exactly flat. Deviations from flatness either in the open or closed direction become rapidly magnified as the universe expands. To be so close to flat now the density one second after the Big Bang must have equalled the critical value to one part in a hundred trillion. We have described two different forms of extreme smoothness exhibited by our present universe in the large: the uniformity of the temperature over the surface of last scatter to an accuracy of one part in a hundred thousand despite the surface being comprised of ten thousand causally disconnected parts; and the fact that the geometry is flat after such a long evolution of 13.7 billion years. A priori both of these circumstances are exceptionally unlikely. This latter unlikeliness can be compared to balancing a pencil on its sharpened point and finding it still so balanced a very long time later: any physicist or non-physicist would demand an explanation. The appropriate modification of the Big Bang model is suggested by study of theories for particle physics at high energies. Such theories involve dramatic transitions between different phases as the temperature falls in the early universe. As the universe cools it is possible for it to become temporarily trapped in a state with peculiar properties called the wrong vacuum. In the short time that it spends in this wrong vacuum or ground state it is normal that the universe undergoes a very rapid exponential expansion. To lead to adequate smoothness it is important that the rapid expension be by at least twenty-eight orders of magnitude in scale. This assumed period of super-fast expansion is called inflation. It offers an explanation of both of the smoothness properties, the horizon problem and the flatness problem, in one fell swoop. The horizon problem is solved because the entire present visible universe arises through inflation from just one tiny causally connected region. The flatness is explained because the inflation renders any pre-existing curvature negligible. This is like taking a balloon and inflating it (without bursting!) to an extraordinarily large size. The surface which at the beginning was strongly curved would become essentially flat at the end of the process. Indeed the only way known to accommodate the two shortcomings associated with the smoothness properties of the Big Bang scenario is to assume an inflationary era in the very early universe. There is no direct evidence for inflation but the idea is regarded as likely to survive in some form or other and provides an important new ingredient in the Big Bang theory which became established in the 1960s. One of the most intriguing aspects of inflation is that it provides a possible mechanism which disrupts the total smoothness initially by only exceptionally tiny perturbations. This is an effect of quantum mechanics during the inflationary era in the first billionth of a second after the would-be Big Bang. These quantum fluctuations exit from the comoving Hubble radius, characterizing the distance which is causally connected during the inflationary era, and can re-enter very much later after some hundreds of thousands of years or more. The question is whether these quantum fluctuations can lead to the later perturbations needed to seed the large scale structure of the universe. This requires our understanding, more clearly than at present, the connection between the quantum fluctuations which exit the horizon and the resultant classical perturbations seen in the study of the cosmic background microwave radiation from the surface of last scatter. Nevertheless, it is a seductive idea that gigantic structures such as clusters of galaxies can be regarded as enormous amplifications of once ultra-microscopic quantum effects. Without inflation the extreme homogeneity and flatness of the present universe requires an extraordinary amount of fine tuning of the initial conditions in the early universe. One may argue that since we are dealing with just one unique universe the starting point could be arbitrarily special. But physicists dislike fine tuning since it reduces the power of expla- nation. In this case let us illustrate just how special the initial conditions would need to have been. At the surface of last scatter corresponding to a redshift of about one thousand we have seen that the density perturbations are at a level of one in a hundred thousand, already very tiny but accommodated in inflationary scenarios. Without inflation these perturbations would originate from very much smaller perturbations earlier in cosmic time. Roughly we may expect perturbations to evolve linearly with the expansion of the universe. Thus if we extrapolate back in time from the last scatter surface the density perturba- tions become progressively smaller. Going back to one second after the would-be Big Bang they have already diminished by a factor of a million so that they are then only one part in a hundred billion. This is so close to perfect homogeneity that it is an extremely fine-tuned situation meaning that it is almost infinitely unlikely among all initial conditions. More generally, a theory is considered fine-tuned if there are extremely small (in this case the relative size of the perturbations) numbers which are not zero but whose size are not natu- rally explained. The technical term ”naturalness” was invented for such a case. Naturalness implies the absence of fine tuning of the parameters of the theory. If we go back even further in time, the situation becomes all the more unnatural. At the Planck time the red-shift has increased by another twenty-two orders of magnitude so the perturbations of the unadorned Big Bang model must then be at the level of a zero followed by thirty-two zeros and then a non-zero digit! Such an absurdly small number is taken as a sign that something is seriously inadequate about the theory. While it does not violate any physical law to assume such an initial condition it violates common-sense and is not what any self-respecting theorist will include as part of a theory. In other words, it will accommodate the observations but does not explain why there is such very high homegeneity in terms of any other physical principles. The situation concerning flatness is quite similar. As we have already discussed the proximity of the present total energy density to the critical energy density implies that if we extrapolate back in time the proximity becomes accurate to the extent that the ratio of the total density to the critical density becomes exactly one to an incredible accuracy. Going back to one second after the Big Bang, this ratio must equal to one with an accuracy of one in a hundred trillion, again an intolerable amount of fine-tuning from the theoretical physicist’s viewpoint. At earlier time it becomes even more intolerable because at the Planck time the proximity of the ratio to one must have been precise to better than one part in a trillion trillion trillion. This seems an inevitable part of the unadorned Big Bang picture. It does not refute that general picture, for which there is very much support including the nature of the cosmic expansion, the properties of the cosmic microwave background radiation and the successes of nucleosynthesis of the light elements. What is does mean, however, is that an augmentation of Big Bang theory is needed, particularly for the very early universe, to replace the assumption of such very extremely fine-tuned initial conditions. One can imagine alternatives to inflation which could explain such fine-tuned initial conditions. One attempt has been to invoke the collisions of three-dimensional objects, or ”branes”, colliding together in a higher-dimensional space. This can have the effect of inducing the required amount of homogeneity and flatness just because the collision effects the whole three-dimensional space equally. So far such an ”ekpyrotic” scenario has run into a wall of criticisms concerning the technical details although such an imaginative idea is worth exploring as a potentially viable alternative to inflation. Of course, the assumption of extra spatial dimensions is itself a big step to take! But only time will tell. Yet another speculative possibility is to assume that the Big Bang originates from some pre-existing phase which transitions into the present universe in such a way that the nec- essary initial conditions are automatic. Such a ”Pre-Big-Bang” picture could shed light on the deep conceptual problems of the extremely early universe. Once we move forward in time from the surface of last scatter the density perturbations which are already present grow further and at the same time new ones enter the horizon. Gravitational instability leads to stronger growth of perturbations and creation of large scale structures. Structure is characterized by densities substantially larger than the mean cosmological density. At present the mean density is well measured and can be expressed in a variety of units. The dimension of density is of energy per volume and it turns out that it can be simply expressed as about ten electron volts per cubic millimeter. An electron volt is the energy acquired by an electron in traversing a potential difference of one volt. This mean energy density is of course extremely small by any everyday standard. It is one billion trillion trillion times smaller than the density of water. As we look at non-trivial structure the over-density, which is the technical name for the mean density of the structure compared to the mean background cosmological density, increases. For a cluster of galaxies, the overdensity is about ten to one hundred while for a galaxy it is ten thousand. When we come down in size to the Solar System, the overdensity is closer to a hundred billion. To get an idea of how empty the universe is: if the Solar System were filled uniformly at the density of water it would have about the same mass as the whole universe! Bear in mind that the Solar System is infinitesimally small compared even to the Milky Way and completely negligible in size with respect to the entire visible Universe. If we consider compact objects like the Sun or the Earth, the overdensity becomes the same huge factor of a million trillion trillion since their density is comparable to that of water, within a factor of a few. The conclusion is that, although there are such very strong fluctuations in density at cosmologically small scales, if we consider scales large compared to clusters of galaxies, say more than ten Megaparsecs, then the universe becomes to a good approximation exception- ally smooth both with regards to its homogeneity and it isotropy, being the same in all directions. Such large-scale smoothness is all important in concocting an appropriate theory of time and of the universe because it implies that the simple Friedmann equation derived from the assumptions of smoothness and of general relativity is very likely to be the correct choice, and we shall continue to make this assumption throughout the rest of the book, Chapter 3 STRUCTURE IN THE UNIVERSE Having raised the enigma of the smoothness of the universe in the large, it is now appro- priate to address the opposite issue of why there exists structure in the form of very many galaxies each containing very many stars? One such galaxy is the Milky Way containing our Sun around which our Earth orbits once per year. Recent work provides a plausible explanation for the origin and evolution of such structures. The structures are only a small perturbation of the smooth universe and yet its most fascinating aspect. The explanation is based on the idea that the structures are ”seeded” in the extremely early universe from effects of quantum mechanics, a theory formulated to describe the smallest scales associated with atoms, and now relevant, because of the enormous expansion of the universe since the Big Bang, to gigantic structures such as clusters of galaxies. In the observations of the surface of last scatter there is a one in a hundred thousand deviation from complete smoothness. This tiny effect must lead to the structure formation. There are two aspects to explain: the primordial origin in the very early universe of these fluctuations and then their subsequent evolution into the observed galaxies and stars. We first discuss the origin of fluctuations and this requires a familiarity with quantum mechanics. The precursor of quantum mechanics is classical mechanics which was set out in the 1680s in the Principia, the book written by Isaac Newton. Based on simple laws such as force equals mass times acceleration, classical mechanics successfully describes the motions of objects resulting from pushes and pulls. This includes not only objects of everyday sizes but, when augmented by the law of classical gravity enunciated in the same book, objects like the Moon circulating around the Earth and the Earth orbiting the Sun. Classical mechanics was so successful that Newton’s Principia dominated physics for the next two hundred years. It is impossible now to imagine any single publication dominating the physics community for anything like so long. In 1864, classical mechanics was extended further by Maxwell’s theory of classical elec- trodynamics which successfully accounted for the motions of charged particles in electric and magnetic fields. The combination of classical mechanics and classical electrodynamics formed the foundations of theoretical physics as it existed at the end of the 19th century. This great edifice of knowledge was such a source of pride to physicists that at least one great physicist announced around the beginning of the twentieth century that physics was essentially over because everything was understood. Such hubris was short-lived. In the early part of the 20th century, two serious limitations to the applicability of the foundations of theoretical physics were found, both of which led to revolutionary advances. The first limitation was associated with the concept of the aether. Aether was invented as a hypothetical medium through which electromagnetic waves could propagate. It seemed at the time inconceivable that such a wave could travel through vacuum and would need aether as an analog of the air through which sound waves propagate. But if there is such an aether, the Earth is moving through it on its orbit around the Sun at a speed of about one ten-thousandth of the speed of light. Therefore, it was argued, the speed of light should be different whether the light travels parallel or anti-parallel or perpendicular to the Earth’s motion. A precise interferometric experiment was completed in 1887 intended to measure such differences. The surprising result was that the speed of light did not depend on direction. The conclusion was that the concept of some medium such as an aether at rest in the cosmos was erroneous. This aether conundrum and the negative result from the interferometer experiment were in the air, so to speak, at the beginning of the 20th century and a crucial ingredient in the invention of special relativity theory. Relativity leads to the prediction, already mentioned, that information cannot be transmitted faster than the speed of light as well as the prediction that the speed of light is always the same in vacuum for any direction, and for any source and observer. The rise and fall of the aether idea may have a current analog. Many theoretical physicists believe that the understanding of dark energy, to be discussed in later chapters, could require another revolution equally as profound as relativity or quantum mechanics. The dark energy could play a role in theoretical physics of the 21st century of sparking a new revolutionary idea just as the aether cunundrum did in the 20th century. The second limitation of classical physics is the one which led to the invention of quantum mechanics in the 1920s. It was first noticed in the properties of heat radiated from a very hot body, and was heightened by a lack of understanding of the stability of atoms. For example, the hydrogen atom is comprised of a positively charged proton which acts as the hydrogen nucleus and a negatively charged electron orbiting around it. Regarding the much heavier proton as at rest and the electron circulating around it then, according to classical theory of electrodynamics, the electron must radiate continuously and lose energy. After a tiny fraction of a second it will spiral into the nucleus and the atom will collapse. The same is true for all atoms of the periodic table and so classical theory predicts that every atom in the universe can live for much less than one second. Needless to say, this total disaster for the classical theory was not foreseen by the great man earlier announcing the end of physics. This led after some false steps to the invention of quantum mechanics to replace classical mechanics especially for describing the inner workings of the atom. In the limit of large everyday sizes the quantum theory reverts to the classical theory so that all the previous successes are maintained. But at the atomic size, one important prediction of quantum mechanics is that energy is not continuous but exists only in discrete amounts called quanta. Thus the atomic electron does not radiate energy continuously but in discrete amounts. Once the electron is in its lowest energy state no radiation is possible according to quantum mechanics and the stability of atoms follows. Quantum mechanics succeeds where the 19th century classical theory fails completely. Quantum mechanics has many other successes but one particular feature merits our present attention. According to this theory there is an uncertainty in the value of all quantities which would classically be precisely knowable. For example, the position of the electron in the hydrogen atom cannot be specified like that of the Moon going around the Earth or of the Earth going around the Sun. Instead we know only a probability distribution, or wave function, which tells us what the probability of finding the electron in some region of space is. The total probability of finding the electron somewhere is necessarily one just as in the classical case. It is just that on the atomic scales the position and speed of the electron have a quantum fuzziness which is a subtle aspect of the theory. A related phenomenon of quantum mechanics is that the vacuum is no longer empty but is alive with spontaneous creation and annihilation of particle-antiparticle pairs which may live a very short time immediately to be replaced by others. To describe this situation needs a marriage of quantum mechanics and special (not general!) relativity called quantum field theory. The essential point is that the vacuum value of a quantum field is continuously fluctuating rather than having a definite value as it would in a classical description. This uncertainty is very important for understanding the structure formation in the early universe. This concept has no couterpart in everyday life. For an intuitive picture in the imagination think of a very sharply focused photograph as the classical theory and then put it very slightly out of focus. To capture the nature of quantum mechanics the out-of-focus scale needs to be of atomic size since everything is fluctuating by about that much and not more. At this point, there seems to be no possible connection between quantum mechanical uncertainty and the stars and galaxies. But that would overlook the enormous amount by which the universe has expanded. In the inflationary scenario, as mentioned already, the vacuum is assumed to roll in a potential between a higher and a lower energy value as part of a phase transition between an unstable vacuum and a stable one. As the quantum field, technically called a scalar or more specifically an inflaton field, rolls down the potential the universe undergoes a very rapid expansion. From a purely classical viewpoint, at the end of inflation the consequent surface of last scatter would have perfect smoothness properties with no variations at all to seed structure formation. In quantum field theory, however, the field must fluctuate while rolling down the potential and these quantum fluctuations can lead to the tiny ripples on the surface of last scatter. There are some intermediate steps in realising this idea but it may work out. To be honest, the exact size of the ripples at one part in one hundred thousand cannot be predicted from inflation theory. That is the amount necessary to evolve into the structure observed much later as well as being the size of the fluctuations observed by COBE (1992) and WMAP (2003,2006) measurements of the cosmic microwave background radiation at the surface of last scatter. It is perfectly natural the such ripples at such a level originate from the quantum mechanical fuzziness during inflation. The inflationary period is where the scale factor defining the size of the universe increases by at least twenty-eight orders of magnitude. If the expansion is exponential in time it implies that the Hubble parameter is constant during inflation. The Hubble parameter is the ratio of the time derivative of the scale factor to the scale factor itself. A property of an exponential is that its derivative equals to itself. The comoving visible horizon decreases as the inverse of the scale factor which means that the fluctuations go outside of the horizon during inflation. Another way of looking at it is the fluctuations are expanded exponentially but the Hubble size remains constant. At a much later time, very long after inflation, the fluctuations re-enter the horizon. While outside the horizon the fluctuations are essentially frozen and re-enter as classical fluctuations ready to grow linearly with expansion and seed the structure of galaxies and the stars. On a clear night when one looks at the thousands of stars, and the Milky Way meandering across the sky, there is (only for the last twenty years) the new insight that they can originate from quantum fluctuations which happened 13.7 billion years ago. These fluctuations from quantum uncertainty during inflation have three very specific and desirable properties. Firstly, they are adiabatic and seed density fluctuations systematically in the various components like photons, neutrinos, baryons and dark matter which are present in the universe. Adiabatic is a term referrring to the fact that no heat is transferred into or out of a comoving volume. Secondly, the perturbations are gaussian which means that the different wave numbers are all uncorrelated and each has a probabilistic distribution of a particularly simple and expected type. Third and finally, provided the potential governing the inflaton field during inflation satisfies conditions technically called slow-roll requirements, the spectral index which describes the relative power in different wave numbers is predicted to be close to one. All of these three properties are required to make a successful simulation of observed structure formation. This is how inflation not only solves the smoothness properties, known as the horizon and flatness problems, but even more impressivly can give a connection between quantum mechanics and the large scale structure observed in the universe. It is an appealing idea that the extremes of the very large and the very small can be so intimately related. Now we address the second question raised above concerning structure formation: how do the ripples on the surface of last scatter grow into the interesting and beautiful galaxies? The evolution of structure from the fluctuations as they enter the horizon involves nothing beyond Newton’s law of gravitation as enunciated in the 1680s. This gravitational attraction acts principally on the nonbaryonic dark matter but also on the baryonic component some six times smaller. From the time of recombination when the visible universe was one thousand times smaller than today until when it was merely twenty times smaller no significant structure was formed. This era is picturesquely known to specialists as the dark ages. After recombination the photon and neutrino components remain negligible and evolve respectively into the current cosmic microwave background which is very well studied and a relic neutrino background which has yet to be detected. During the dark ages the fluctuations in all components begin by growing linearly with the expansion of the universe. Eventually gravitational attraction causes local perturbations of nonbaryonic dark matter to grow nonlinearly and become more singular until, according to one scenario, the first baryonic stars can form with a mass of perhaps one hundred times the solar mass. Newton’s law is adequate for this. Such massive stars collapse under their own gravitational attraction. As nonlinear effects dominate, the system must be analyzed semi-analytically or by computer simulation. Such simulations confirm that starting from the type of perturbations already discussed, the end results for the present universe using the known quantities of nonbaryonic dark matter and baryons can lead to a structure similar to that observed in galaxy surveys. The dark matter first clumps and then the baryons follow suit to yield clusters and superclusters of galaxies as well as the large voids which are observed. There are problems of detail at small scales; for example, simulations typically give too many small satellite galaxies for each large galaxy, and too much dark matter accumulates at galactic cores. But the three-dimensional distribution of structure on all larger scales gives rise to pictures which look to the eye, and more importantly to detailed statistical analysis, indistinguishable from the real universe. To accommodate structure it is necessary to use both general relativity and quantum mechanics, though not together at any particular scale, with the assumptions of the cosmo- logical principle and inflation. General relativity with the cosmological principle gives rise to the underlying geometry of the expanding universe. Quantum uncertainties in the inflaton field provide a possible and plausible primordial origin of fluctuations which exit the horizon in the very early expanding universe during inflation and re-enter the horizon much more recently. The evolution of these fluctuations into galaxies and stars requires only Newton’s law of gravity which is not a separate assumption but a component of general relativity. The theories of quantum mechanics and general relativity are applied separately to dif- ferent regimes in this picture of theoretical cosmology. This is fortunate because there is no fully established marriage of general relativity and quantum mechanics into a consistent theory. The leading candidate is string theory and one very active area of research is the application of string theory to theoretical cosmology in a subject called string cosmology, a field so new it is still evolving towards a more precise definition. Earlier attempts to connect string theory with particle phenomenology, which describes the interactions of quarks and leptons deduced at high-energy colliders, have so far been relatively unsuccessful. Whether support for string theory will come first from cosmology or from phenomenology is an inter- esting open question. Only time will tell. The standard model of particle phenomenology was first invented in the 1960s and 1970s. First a unification of electrodynamics and weak interactions was proposed by Shel- don Glashow in 1960 later completed in 1967 by Abdus Salam and Steven Weinberg in a form that was conjectured by them to be a consistent quantum field theory, formally called renormalizable. The fact that the theory really was renormalizable was proven in the early 1970s by Gerard ’t Hooft and Martinus Veltman. It was shown by them that the theory was equally as consistent and as amenable to precise unambiguous calculation as is quan- tum electrodynamics, the marriage of quantum mechanics with the classical electrodynamics theory of the 1860s. In the 1970s a parallel development was the evolution of a similar field theory of the strong interactions called quantum chromodynamics or QCD. The combination of QCD with electroweak theory comprises the standard model. Its detailed predictions have held up remarkably well. Even fourty years after the original proposal all experimental data, with one exception, agree with the predictions, up to an impressive one in a thousand accuracy. The exception is the non-zero neutrino mass first established in 1998 and which leads to the necessity of some modification, still under intensive study, of the standard model. The standard model led to the successful prediction of new elementary particles including the charmed quark and, following the bottom quark discovery, of the top quark. There are six known flavors of strongly-interacting quark which fall into the three doublets: (up, down), (charm, strange), (top, bottom). There are also three corresponding doublets of non-strongly-interacting elementary particles called leptons: electron, muon and tau with a partner neutrino for each type. These all group into three quark-lepton ”families”. The standard model also led to the prediction of new types of weak interactions called neutral currents, discovered experimentally at CERN in 1973 and the weak intermediate bosons W and Z discovered, also at CERN, in 1983. All in all, this theory has been spec- tacularly successful and forms the basis of a theory for at least all the luminous matter seen in stars and galaxies and presumably all the baryonic matter. At the level of microscopic, subatomic scales it can be said without hesitation that the standard model is the greatest achievement of theory in the second half of the twentieth century. It is therefore manda- tory to ask how extending this successful model can accommodate features of observational cosmology particularly the nonbaryonic dark matter. It has been a major industry for at least the last thirty years to extend the standard model in various directions. One idea is to unify strong with electroweak interactions into a grand-unified theory. Studies of this type of theory at high temperatures actually led to the original idea for cosmological inflation. Just as with the uncertainty of how early, or equivalently at how high a temperature, inflation took place, there is an uncertainty in the energy scale at which grand unification happens. In the earliest and simplest such theories the unification scale was very high suggesting an extremely early inflation merely one trillion- trillion-trillionth of a second after the Big Bang. There are more recent unification schemes where unification as well as inflation could take place at much lower energies but, on whether such an alternative is correct, the jury is still out. One other well-studied extension of the standard model is based on a serious technical issue within it. There is a scalar particle, the Higgs boson, necessary for accommodating the symmetry breaking between the electromagnetic and weak interactions, which gives rise to violent infinities called quadratic divergences. These seem to render the theory inconsistent and lead to the consideration of an extension called supersymmetry. In such an extension each standard model particle has a new partner: a quark has a squark, a lepton has a slepton, and so on. It is not economical but does resolve the ”naturalness” problem associated with quadratic divergences. It also leads to a candidate particle, known as the neutralino, which could constitute the nonbaryonic dark matter which is established as making up almost one quarter of the cosmological energy density. The above outline gives just a rudimentary idea of the very strong interrelationship between particle theory on the one hand and theoretical cosmology on the other. At an earlier time, say around 1980, particle theorists treated cosmology with some condescension because the cosmological data were so inaccurate compared to the reproducible precision data from high-energy accelerators. This has by now completely changed as the data on the cosmological microwave background radiation in 2003 has achieved high-enough accuracy to be characterized as ”precision cosmology”. The two studies of the very small (particles) and the very large (cosmology) have become inextricably intertwined. Most university physics departments now combine these two disciplines in a single group of researchers. The subject of string theory falls neatly into the rubric of ”Particles, Strings and Cosmology”, the title of one well-known series of international conferences. In academia, strings unify not only the aspects of particle phenomenology and theoretical cosmology but also stimulate common threads of research in physics and mathematics departments. Chapter 4 DARK MATTER AND DARK ENERGY In this and the next chapter we discuss the present make-up of the universe. Baryons are the stuff of which everyday objects are made of so we start with a brief disussion of that component. There are two methods to estimate the energy density due to baryons in the universe. One which was first used in the 1960s is by calculating the formation of helium and other light elements in the early universe about one minute after the would-be Big Bang. The other which has been possible only since 2000 is by analysis of the relative heights of the odd and even acoustic peaks in the anisotropy of the cosmic microwave background at a time three hundred thousand years after the would-be Big Bang. These two methods which analyze therefore quite different cosmological epochs agree very well with each other. The result is that the baryons make up about four percent of the total critical energy density. The visible luminous baryons corresponding to the stars that shine add up to only about one per cent so the other three percent is invisible and labelled baryonic dark matter. This four percent of the total energy density is the only part of which we have a clear understanding. The present understanding of the ”dark” compinents should make theoretical cosmologists very humble but what an opportunity for young people entering the field that we have such limited understanding of ninety-six per cent of our universe. Nonbaryonic dark matter comprises about twenty-three percent of the critical density or some six times the baryonic density. Nonbaryonic dark matter is much more mysterious than baryonic matter because it is not something familiar in everyday life. Its presence has been strongly suspected already since the observations of Zwicky in 1933 but despite seventy years of study its nature is still a subject only of speculation. Nevertheless, two good candidates for nonbaryonic dark matter are undiscovered particles hypothesised by considering extensions of the standard model of particle physics. This standard model is a gauge field theory which describes the interactions of quarks and leptons. It has two pieces: one describing the strong interactions of quarks and known as quantum chromodynamics (QCD), the other describing a unification of electromagnetic and weak interactions known as electroweak theory. The two pieces are so far unconnected. Nevertheless, it is seductive to attempt to unify the strong and electroweak pieces into a single theory technically called a grand unified theory. In such a theory all the three couplings for the strong, electromagnetic and weak forces unify at a very high energy far above familiar collider energies. Such a grand unified theory has many attractive features and leads to remarkable pre- dictions. The most striking prediction is the instability of the proton which implies that all objects are eventually unstable. Of course, the proton lifetime must be exceedingly long just because material is very stable and even stronger limits are placed by the non-observation of proton decay in dedicated experiments. Nevertheless there are certain technical problems with the theory. For example, the electroweak theory contains a scalar field known as the Higgs boson which plays a crucial role in breaking the symmetry between electromagnetic and weak interactions. This scalar field must be included in the grand unified theory but then quantum corrections, technically called quadratic divergences, naturally force the Higgs field to have an extremely heavy mass near to the grand unified scale. But that destroys its capability to play the appropriate role in the symmetry breaking phenomenon. Another issue is that phenomenologically the unification of the three couplings does not occur very precisely within the grand unified theory. One way of ameliorating both of these two problems is to extend the standard model to what is called the supersymmetric standard model and to unify it in a supersymmetric grand unification. The additional assumption of supersymmetry is a mathematically elegant symmetry though for which there is no experimental evidence. It has the role of predicting a superpartner for each particle in the standard model: for example, for each quark there is a superpartner called a squark. The quark has spin one half while the squark has spin zero. Each superpartner has a spin differing by one half unit from its progenitor particle. In the limit of exact supersymmetry the particle and superparticle must have precisely the same mass. But this is experimentally excluded because, for example, a selectron at the same mass as the electron would be easy to detect and is clearly ruled out by experiment at that mass. Therefore supersymmetry is broken and the superpartners, if they exist, must in general be considerably heavier, in general, than their normal counterpart. The assumption of supersymmetry has two advantages. First, when the Higgs boson is incorporated into a supersymmetric grand unified theory the Higgs mass is not effected by quadratic divergences. This is true even with supersymmetry breaking provided such breaking is of a special kind known technically as ”soft” and at a scale close to the scale characterizing electroweak breaking. Secondly, the unification of the three couplings be- comes phenomenologically more precise when the superpartners are included than in the case without supersymmetry. While admitting that the supersymmetry idea has still no support from experiment, there is no shortage of theoretical work along this line and the design of detectors for the next generation of colliders is arranged at least partially to find superpartners if they exist. What has this got to do with nonbaryonic dark matter in the universe? This needs a little more explanation to understand how the supersymmetric standard model seems to provide a good candidate particle to play the role of nonbaryonic dark matter. In the supersymmetric standard model there is a natural discrete symmetry called technically R symmetry which means that the number of superparticles is conserved in every process. Such an R symmetry is necessary in the supersymmetric grand unification to avoid a proton decay lifetime too short for agreement with experiment. One consequence of R symmetry is that there must exist a lightest supersymmetric particle (LSP) which is absolutely stable because there is no lighter superpartner into which it can decay while conserving the crucial R parity. This LSP is typically a linear combination of superpartners of the Higgs (Higgsino), the photon (photino), and the B gauge boson (bino). Such an LSP is generically called the neutralino. The neutralino should have a mass of about one hundred times the proton mass and in- teract with strength characteristic of the weak interaction. As such, it provides an example of a class of particles invented for the purpose of comprising the nonbaryonic dark matter called Weakly Interacting Massive Particles (WIMPs). When one calculates how many such neutralinos survive annihilation in the early universe one is gratified to find that it fits per- fectly the amount of nonbaryonic dark matter which is observed. So the neutralino which follows from the assumption of supersymmetry is a good candidate to be all of the nonbary- onic dark matter. It works so well that some theorists regard this as a main motivation for belief in supersymmetry because the resultant neutralino gives such a natural candidate for a cosmological WIMP. There is a second candidate to play the role of the nonbaryonic dark matter arising from a different extension of the standard model of particle phenomenology from a quite disparate line of argument. It is called the axion and its motivation is somewhat less than that for the neutralino but it is worth to describe as yet another example of how there is such a strong interrelationship between cosmology and particles. In the theory of strong interactions, QCD, already mentioned there is an issue concerning why the strong interactions respect certain dicrete symmetries. These symmetries are parity (P) or mirror reflection and CP which is the product of parity with charge conjugation which interchanges matter with antimatter. Both of these symmetries, P and CP, are violated by the weak interaction but appear to be well respected in strong interactions. Nevertheless there is a particular term which is allowable in the QCD theory which violates both P and CP and of which the coefficient must be of magnitude less than one ten-billionth to avoid conflict with experiments on bound states, particularly on the electric dipole moments of mercury atoms and neutrons. One way to solve this problem is to impose and subsequently break an additional sym- metry. This procedure gives rise to an additional particle called the axion. It can act as the nonbaryonic dark matter. The axion theory has certain technical problems such as that the theory appears inconsistent when combined with gravitational interactions. The axion mass lies around a trillionth of the proton mass and so is a hundred trillion time lighter than the neutralino. These two candidates underline how little we understand the nonbaryonic dark matter: the range of possible mass for this missing ingredient actually lies somewhere between the axion mass and a million solar masses, a stunning range in mass of some sixty-nine orders of magnitude. Thus, although the neutralino and axion are plausible candidates, the true solution chosen by Nature for the nonbaryonic dark matter remains enigmatic. Given that almost one quarter of the cosmological energy density is nonbaryonic dark matter, roughly six times as much as the baryonic matter, the question is how to detect it. The arguments involving the study of galactic rotation curves, of the virial theorem in clusters, and of the general cosmological observations of the cosmic microwave background radiation, the large-scale structure and the type-1A supernovae are all indirect methods. They imply the fact that there must exist such nonbaryonic dark matter. But of what is it comprised? In particular is it made up of weakly interacting massive elementary particles (WIMPs) with mass of order a hundred times the proton mass, or the much lighter axion, or is it made of far larger entities with the mass of the Sun or larger? It is a measure of our present ignorance that such extreme possibilities are still viable. Next, given that this nonbaryonic dark matter has a specific form, what are its interactions with the baryonic matter? This issue of the interactions is especially relevant to the possibility of detection. The key question is how strong are the interactions with ordinary matter. We know that there is a gravitational attraction from the method of its indirect detection. If the WIMP were the neutralino state of supersymmetry then it must also have a weak interaction with ordinary matter. In that case one method of detection is to use bolometric detectors, typically large crytals cooled to very low temperature. If a WIMP particle strikes the crystal, and interacts, it may deposit energy in the form of vibrations, or phonons, in the crystal. Such phonons could then be detected. Such searches have been made and even claims of a positive signal have occurred, but such claims have not yet been reproducible and are therefore generally disregarded. The detection of the nonbaryonic dark matter will be an extraordinary claim. Extraordinary claims need backing by extraordinary evidence and that, so far, does not exist. Another possibility for detection is from the process of annihilation of dark matter with dark antimatter. If we assume this produces normal photons and electron-positron pairs then the subsequent positron annihilations could be detected by their gamma ray emission. Indeed there is a suspiciously large number of positrons near to the center of the Milky Way and it has been suggested that these may originate from annilihilation of dark matter. Unfortunately there are other explanations for the occurrence of such positrons and so the evidence for direct detection of dark matter is not yet compelling from that source either. The direct detection of nonbaryonic dark matter could be rendered extremely difficult (one must never say impossible!) if its interactions are only gravitational just because of the weakness of the gravitational interaction. For example, in such a case even if dark matter does annihilate against its antimatter, it would produce only more dark matter in the form of ”dark photons” or whatever dark particles are relevant. It is only an assumption that the nonbaryonic dark matter posseses significant non-gravitational interactions with ordinary matter. In such a case we might have to be content with indirect detection for some time but that would be frustrating in the sense that the identification of the constituent particles of nanbaryonic dark matter would remain elusive and would not act to motivate the building of models extending the standard model of particle phenomenology. A more optimistic scenario is that there are significant non-gravitational forces and that direct detection will occur in the near future. This would provide an important guide to model building especially when the nature of the interactions between the dark and ordinary matter are further explored experimentally. Enthusiasts for supersymmetry are naturally encouraged by the fact that identification of the WIMP with a neutralino very naturally gives the correct dark matter density after the WIMP-annihilations have taken place. Indeed this is taken by some as a strong suggestion that supersymmetry is correct. Another such evidence is the improved unification of the couplings in a supersymmetric grand unified theory. On the other hand, the main motivation for supersymmetry is to provide naturalness for the Higgs boson in the sense of cancelling quadratic divergences in the underlying field theory. If such naturalness can be obtained without supersymmetry then that motivation would be removed, although supersymmetry still plays an important role in the construc- tion of superstrings as a potential theory for quantum gravity. In that case, however, the supersymmetry can be broken at a very high energy scale, perhaps as high as the Planck scale, and there would then be no reason to expect to see sparticles at masses accessible to the next generation of colliders. It can be seen from such arguments that the direct detection of dark matter in non- accelerator experiments could help clarify precisely what particles to expect to be produced at high-energy colliders. It is in such a sense that the interrelationship between cosmology and particle theory becomes even more important. We have so far discussed the baryonic component of four percent and the nonbaryonic dark matter component which is twenty-three percent. Yet the total energy is consistent with the critical density. This means that some seventy-three percent of the total mass-energy density has not been listed. This additional dominating component has been discovered only since 1998 and is called dark energy. What is the difference of dark energy from dark matter? To understand the distinction it is necessary to introduce the ideas of pressure and equation of state. Baryonic matter and dark matter exert zero pressure, the corresponding particles are essentially at rest and the equation of state which is defined as the pressure divided by the density is equal to zero. Photons which necessarily travel at the speed of light do exert a positive pressure and their equation of state is equal to plus one third. These two values of zero and plus one third are the most familiar equations of state applicable to dust and radiation respectively. Dark energy is strangely different and, in particular, exerts a negative pressure. In its simplest manifestation dark energy corresponds to a cosmological constant with equation of state equal to minus one, or pressure equal to minus the density. Negative pressure has no example in everyday life. Normally, with positive pressure, exerting a force on the piston confining a cylinder of gas will compress the gas and increase the positive pressure. For a cylinder of dark energy, however, the force would increase the volume, completely contrary to the physical intuition we develop from everyday experience. We shall discuss much more about the equation of state for dark energy in the final two chapters of the book. Here we recall the early history of the cosmological constant, the precursor of modern dark energy. After general relativity was first published by Einstein in 1915, he followed up with its application to cosmology in his very important paper of 1917 which may justifiably be regarded as the start of what we now regard as theoretical cosmology. At that time, it was generally believed that the Milky Way galaxy was identical to the entire visible universe and that this universe was a static situation without overall expansion or contraction. We now know that the universe is really four hundred thousand times bigger and that the expansion can be seen only in the much larger theater in which the Milky Way and all other galaxies are treated as mere points. Ironically, an observational astronomer named Slipher had published a few years earlier, in 1912, his result of observing far more red shifts than blue shifts. It seems likely that Slipher was seeing the expansion of the universe but he naturally thought he was looking at stars within the Milky Way rather than at other galaxies outside the Milky Way. In reality, Einstein in 1917 missed a golden opportunity to predict the cosmological expansion which follow quite straightforwardly from his general relativity theory combined with the cosmological principle. Instead, he added a negative cosmological term which made the theory less elegant but allowed a static universe. Only much later in 1929 did Einstein realize from Hubble’s publication that the universe is really expanding. He must have, at least figuratively, kicked himself for not sticking to the most elegant version of general relativity. A moral of this story is that a theorist, including Einstein, may not take his own theory seriously enough. A sequel is that since 1998 the effect known as dark energy which resembles a cosmological constant, though now with a positive sign, has appeared and looks more than likely to remain as a robust part of cosmological theory. Should one regard this new development as something which vindicates the mistake of Einstein in not leaving the term out from his cosmological application of general relativity? This new discovery of an effect looking like a cosmological constant should not be attributed to the far-sightedness of Einstein. It arises not from the expansion of the universe but from the even more surprising fact that the rate of expansion is accelerating. Noone had any notion of this even ten years ago, let alone in 1917, and it has no connection to Einstein’s motivation for adding such a term in his theory. Rather it shows that there is only a small finite number of ways to modify the theory of general relativity while preserving all of its symmetry principles. It is surprising that the most economical version fails but future understanding of the additional piece associated with dark energy may shed light on how to discuss gravity correctly and relate it to the other interactions. One ambitious attempt at a consistent theory of quantum gravity is string theory and one place where one would have hoped for insight from string theory is certainly in the issue of the cosmological term. This has not been forthcoming and string theory actually seems slightly to favor a negative rather than the observed positive sign for the cosmological constant. This is still probably not the end of string theory input, mainly because the theory is still plagued by the present inability to select between an astronomical number of candidate lowest-energy states. It is unclear whether this hurdle to progress will be jumped over in the near or distant future in order to confront string theory with the real world. In any case, even if string theory turns out not to be the correct theory of quantum gravity, it has already, by a remarkable technical argument going by the name of “correspondence” between string and gauge theories, provided very interesting possibilities for gauge theories that could address some of the shortcomings of the standard model of particle phenomenology. The tiny cosmological constant and the cosmic coincidence of the dark energy density being now comparable to the matter density may be explicable in field theory models or alternatively by exotic ”phantom menace” models with an unexpected equation of state as well as bizarre consequences for the distant future fate of the universe, as we shall discuss in further detail especially in the final chapter. Chapter 5 COMPOSITION OF THE UNIVERSE’S ENERGY The newcomer to the cosmological mass-energy menu, and the dominant component, is the dark energy. Since dark energy is considerably more obscure even than dark matter, and because dark energy could play for the 21st century the role played by the aether for the 20th, the way in which dark energy was discovered and quickly established deserves discussion as it is a fascinating story. In 1929 Hubble discovered that the recession velocity of a galaxy is proportional to its distance. The constant of proportionality, the Hubble parameter, is now known within a few per cent error to be seventy in the units mentioned earlier: kilometers per second per megaparsec. What had been assumed until 1998 was that the rate of expansion was decelerating. With no cosmological term, whether the universe were open, flat or closed the natural expectation from general relativity is that the rate of expansion is slowing. One of the major problems with the evaluation of the Hubble parameter is the reliable estimation of distance to other galaxies. Until relatively recently this required a sequence of steps called the cosmological distance ladder, each step introducing its own error. Finally the resultant distance and the corresponding Hubble parameter could be uncertain by a factor two. This distance ladder can be avoided if there is a so-called standard candle with known luminosity at a very great distance, ideally from when the visible universe was, say, half its present size. Such an object is provided by a supernova of a particular type, type-1A. A supernova is a gigantic thermonuclear explosion which happens when a star collapses under its own gravitational attraction. The most straightforward case is called core-collapse and does not involve interaction with any other object. A particularly consistent type involves instead one member of a binary pair of stars when one of them becomes a supernova. This is called technically a supernova of type-1A, because it has no hydrogen in it spectrum (hence not Type 2) and because it accretes mass from its binary companion (hence not Type 1B or 1C). By the study of nearby type-1A supernovae it is found that the peak luminosity is simply related to the rise and fall time in luminosity. This regularity means that we can estimate the absolute luminosity and hence distance, by the inverse square law, to such type-1A supernovae. The red-shift and hence the recession velocity can be measured also and so very distant type-1A supernovae provide points on the Hubble plot which are exceptionally distant and because they act as standard candles they by-pass the error-inducing process of the cosmological distance ladder. The key point is that the supernova type-1A can be discovered as its light output is increasing then observed as it reaches a maximum then declines gradually in intensity. The rise and fall in intensity typically takes from a few weeks to a few months. Since it is not known where and when a supernova will occur, it is important to make the discovery to have a wide-angle telescope which may be quite small typically four meters in diameter. Once the discovery is made a very large telescope up to ten meters in diameter or, in space, the Hubble Space telescope, can follow the subsequent evolution with greater accuracy. What has been established phenomenologically for the nearby supernovae type-1A is that the duration of the light curve characterizing the increase and decrease of the light intensity is closely correlated with the absolute value of the peak intensity: the longer the time period, the brighter the supernova and vice versa. Thus, observation of the time dependence of the light intensity translates into the measurement of the peak intensity and hence the distance. The theory of supernovae type-1A is not sufficiently developed to derive this relationship between light curve and peak intensity from a purely theoretical perspective but empirical data strongly support the correlation. During the 1990s such type-1A supernovae were pursued by two groups of observers and the data analyzed to give a consensus on the unexpected result. Instead of slowing down as everyone expected the expansion rate of the universe is accelerating. Further analysis suggested that this acceleration has been happening for only a fraction of the time since the Big Bang and that the transition from decelerated to accelerated expansion happened at some time more recent than when the visible universe was half it present size. These data suggest that there is a significant component of the total energy density which exerts a negative pressure tending to blow the universe apart. One example is a positive cosmological constant. Fortunately there are independent checks for such a dramatic and unexpected result. If it were only the supernovae, it could be that they appear dimmer than expected merely be- cause of obscuration by intermediate dust clouds. Alternatively there could be a systematic evolution effect between the very distant and nearby supernovae. Both of these possibilities already seem unlikely. For example, the obscuration would be expected to effect different colors differently and there is no sign of that. Also the similarity of spectra between the distant and nearby type-1A supernovae would argue against any significant evolutionary effect. The observations of these type-1A supernovae was undertaken beginning around 1992 especially by a group based in Berkeley. Presumably, it was not anticipated at that time the extent to which such study would revolutionize theoretical cosmology. In 1995 a second group based in Harvard was converted to join the chase as an independent group to check on the nature of the output results. Despite some false starts the two groups eventually agreed on the result that the expansion of the universe is accelerating. This result was completely unexpected and so it was very important to have the two independent groups making the observations and analysis. A second independent approach is to analyze the anisotropy of the cosmic background radiation. By decomposition into multipoles one can plot bumps, technically called acoustic peaks, which reflect the acoustic oscillations of the baryon-photon plasma immediately before recombination. The detailed positions and heights of these acoustic peaks are sensitive to the values of the energy components and so can give independent evaluations for the amount of dark energy as well as of baryons and the dark matter. This is because the photon trajectories from the surface of last scatter to Earth depend on the geometry of the intervening spacetime, whether it is Euclidean or non-Euclidean and, if the latter, whether the curvature is positive or negative. It is possible to find analytical expressions for the acoustic peak positions in this way. In addition, there are public software codes which will calculate the detailed shapes of the peaks for any input cosmological parameters. Physically we may say that the causally connected horizon represents only a small frac- tion, on the order of a fourty-thousandth of the full visible horizon. The angle subtended at Earth by that causally connected region depends on whether the photons traverse straight lines, as in a flat geometry, or if their trajectories are bent by a curvature of space. This will effect which multipole is enhanced in the observations of the anisotropy. For a flat geometry the prediction which depends on the time of recombination is that the multipole equal to about two hundred will be the location of the first acoustic peak. Further harmonics will appear at multiples of this value in first approximation. If the curvature is positive the peak will appear at a lower multipole while if curvature is negative it will appear at a higher multipole so this measurement give a direct estimate of the nature of the geometry back to the surface of last scatter. The baryon content effects the relative heights of the odd and even acoustic peaks in a calculable way. The most detailed such survey was published in 2003 from the WMAP satellite. It confirms that seventy-three percent of the total energy is in the form of a dark energy, while four percent is in baryons and the remaining twenty-three percent in dark matter. The errors in these percentage estimates are quite small. The WMAP data does not pin down the equation of state for the dark energy. Although it is consistent with the value minus one as for a cosmological constant, it allows values both more positive than this and especially considerably more negative, down to -4/3. Finally, there is a third independent means of analysis. From the large scale galaxy surveys such as the so-called 2dF Galaxy Redshift Survey and the Sloan Digital Sky Survey the total amount of dark matter plus baryons can be deduced. The result is some thirty percent of the critical density with an error of some three percent. This is in agreement with the other two estimates. We note that each of these two galaxy surveys included about 500,000 galaxies. To understand better how the dark matter component is estimated, it is necessary to examine the gravitational attraction and behavior within a cluster of galaxies. There is a general theorem, technically known as the virial theorem, applying to any such gravitation- ally bound system, which relates the kinetic energy to the potential energy. This is used to estimate the amount of gravitational mass in a cluster. By comparing this with the esti- mated mass of the luminous matter one can check for any excess and identify such as dark matter. At the length scales of cluster and superclusters, just as for individual galaxies, it is found that dark matter is present at a level some thirty times the density of the luminous matter, or very roughly thirty percent of the critical density with errors of three per cent. If we input the observation that the geometry is flat from the CMB analysis then there is clearly a missing component, other than the luminous and dark matter, to be identified with the dark energy. The important point is that the study of galaxy redshifts and particularly the virial theorem for clusters of galaxies points to the necessary presence of dark matter. At larger distance scales it becomes more important but the consensus is that it does not approach the critical density even for scales approaching the size of the visible universe. Instead the convergence is on an amount of dark plus luminous matter approximately thirty percent of the critical density. The precise WMAP data confirms a number twenty-seven per cent for this quantity. Taken as a whole there is a cosmic concordance which is difficult to dispute. All three approaches lead inevitably to the conclusion that there is almost 3/4 of the universe (seventy- three percent) in dark matter, twenty-three percent nonbaryonic dark matter and four per- cent baryons. Only the four percent is something we really understand! The remaining dark side of the universe is truly enigmatic. Even the baryonic component has a majority which is not dirctly detectable. At most one per cent of the critical density can be accounted for by the luminous matter in stars and galaxies. The remainder of the baryonic component must be in the form of objects too cold or light to sustain thermonuclear power. These could be Jupiter-sized objects, brown dwarfs, etc. Nonbaryonic dark matter could be composed of objects from the LSP particle, a kind of WIMP, with a mass of about a hundred times the proton mass all the way up to objects with a million times the solar mass. This range of masses which stretches over sixty orders of magnitude reveals our ignorance on this major component. Even this level of ignorance pales by the side of the excitement created by the discovery of dark energy. This could be merely a reappearance of Einstein’s cosmological constant, albeit with the sign reversed, and with a totally different motivation from Einstein’s. If so, there is still a fundamental issue of fine-tuning which remains unresolved. If the dark energy is something more exotic then its understanding may well require a revolution in theory. That is why the dark energy provides so much excitement. The fact that it is so badly understood provides motivation for pushing again the limits of human intelligence and creativity. The end result of the cosmic concordance is that the make-up of the menu of the total cosmic energy density is therefore 4% baryonic matter, 23% nonbaryonic dark matter and 73% dark energy. The errors on these partitionings are quite small and its now seems well established from the cosmic concordance that this picture is accurate and unlikely to change significantly in the future, The surprising and even slightly embarrassing aspect of this overall composition, and one may add enigmatic, is that only the 4% contribution of baryons is well understood. The remaining 96% of the dark side of the universe is amazingly little understood. It stands as a wonderful opportunity for the upcoming generation of physicists to seize upon and make their reputation on understanding dark matter and energy. The dark matter may be already implicit in certain extensions of the standard model of particle phenomenology, for example the lightest supersymmetric partner in the supersym- metric extension of the standard model. For the dark energy, on the other hand, it may require some further understanding of the fundamental theory of quantum gravity to make real progress in its description. From observations in the near future it will be possible to pin down the equation of state for dark energy. This means the ratio of the pressure to the density. The density is always positive but for dark energy the pressure is significantly negative and the question is: how negative? It is known that the equation of state must be less than negative one third to accommodate an accelerating cosmic expansion. It is important to know whether the equation of state is greater than, equal to, or more negative than negative one. The future of the universe depends strongly on this equation of state of the dark energy. Its evaluation will require more accurate data on the cosmic microwave background radiation and on Type-1A supernovae. We shall discuss the future fate of the dark energy and thence of the universe in the final chapter but let us briefly preview here one of the more fascinating aspects for the case where the equation of state lies below minus one. In this case there is the peculiar feature that the universe ends at a finite future time when the scale factor diverges to infinity. This has been called in the physics literature the Big Rip. The dark energy density increases without bound as a positive power of the scale factor, becoming also infinite at the time of the Big In a philosophical sense, this picture is quite attractive because it introduces a kind of symmetry between the past and the future in the sense that both are finite in linear time. The physics of the final billion years before the Big Rip is quite interesting even if it is extremely unlikely that, even if it will occur, anyone will be around to observe it. It has been computed how the dark energy density increases to such a level that, assuming that there are non-negligible fluctuations in it, it will gradually rip apart the existing structure. Some billion years before the Big Rip, clusters of galaxies are the first to be erased. Galaxies are similarly dismembered some sixty million years before the Big Rip. Three months or so before the end, the Solar System becomes unbound. Just a half hour before the Rip the Earth will explode. Finally, a ten-million-trillionth of a second before the final gasp, all atoms will be torn apart and dissociated. If this scenario is correct, we are living in a cosmological era some time between the formation of intricate structure from an exquisitely-smooth universe and a future time when all the structure will be torn apart just before a Big Rip. So it is no surprise that we are living when there exists complex structure. That is concerning the dark energy properties and we discussed dark matter in the previous chapter. On both dark aspects out knowledge is inadequate so perhaps we should look again at the small four per cent of baryonic matter which we do understand considerably better, and there is therefore much more that we can say. The baryonic matter, like everything else, experiences the gravitational interaction which is classically well described by the general theory of relativity. The non-gravitational inter- actions are well-described by the standard model of particle phenomenology. But even this standard model has plenty of open questions, not the least of which is the large number of free parameters which must be matched to experiment. With massless neutrinos there were already nineteen and now, with the massive neutrino sector, that number of parameters has increased to twenty-eight. Let us go through these parameters which some all-enveloping theory would be expected to explain. The masses of the quarks and leptons are twelve quantities which remain mysterious. Apart from the apparent ordering into three quark-lepton families, they might as well be twelve random numbers. Maybe they are, but physicists would much prefer an underlying theory to postdict them. That would be a very major step in understanding. There are mixing angles and phases: three angles and a phase for the quarks; three angles and three phases for the leptons. These ten bring the total to twenty-two. The remaining parameters include three gauge couplings: one for QCD and two for the electroweak theory. Finally there are the Higgs boson mass, the weak scale (or the mass of the W boson) and, the most peculiar of all, a parameter called theta bar which controls an unwanted symmetry violation in the QCD of strong interactions. Even if all these twenty-eight parameters could be somehow derived the standard model still remains inadequately unified because the quite similar theories of QCD and electroweak forces are disjoint and it is tempting to put them together in a so-called grand unified theory. One striking test of this idea is the prediction that the proton, and hence all matter, is unstable. The lifetime is predicted to be at least a billion trillion times the age of the universe. An experiment cannot wait that long, of course, but instead one can observe the astronomical number of protons in thousands of tons of purified water for a few years with similar sensitivity. Such water is usually surrounded by large photomultiplier tubes and placed deep underground. If a proton decays, for example into a pion and a positron, the decay particles will travel faster than the speed of light in water and consequently produce radiated light to be detected by the tubes. Such experiments have so far failed to detect proton decay, although the extraordinarily sensitive water detectors have made revolutionary discoveries concerning neutrinos. For example, the first extrasolar neutrinos were so detected in 1987 from a supernova in the Large Magellanic Cloud lying outside the Milky Way. Similarly, it was first demonstrated compellingly in 1998 that neutrinos have non-zero mass by studying the angular distribution of events produced by cosmic rays in such a giant underground water detector in Japan originally designed to look for proton decay. The next step in developing the standard model is likely to be prompted by the data generated in the Large Hadron Collider at CERN, scheduled to start producing data in the year 2008. It is expected that entirely new particles will be discovered including the Higgs boson and hopefully others. The others could be those predicted by supersymmetry or large extra dimensions. Perhaps more likely they will be surprising additional states such as gauge bosons, fermions and scalars somehow arranged to cure the unnatural quadratic divergences in the scalar sector of the standard model. Chapter 6 POSSIBLE FUTURES OF THE UNIVERSE It might be hoped that with such a detailed understanding of the history of the universe we are in a position to predict confidently its future. That this is not at all the case, we shall discuss in this Chapter. Such a prediction would be possible were the dark energy component absent so let us first discuss that hypothetical case. The Friedmann equations that arise from general relativity under the assumption of the cosmological principle assign a special role to the critical density. If the total density of baryons and nonbaryonic dark matter are less than this then the universe is open with negative curvature and will expand for ever without coming to rest. The rate of expansion will decelerate. If the total baryons plus nonbaryonic dark matter density is greater than the critical density then the universe has positive curvature and will eventually stop expanding after a finite time then recontract to a Big Crunch. Finally, if the baryonic plus dark nonbaryonic densities add to exactly the critical density then the universe is flat, the kinetic eenrgy equals the gravitational potential energy and so it will expand forever coming to rest at an asymptotic infinite time. Notice that the division between baryons and nonbaryonic dark matter is irrelevant in all three scenarios since both have the identical equation of state, namely zero pressure. In all cases, the rate of expansion will decelerate. The presence of dark energy spoils such simple prognostications mainly because the equation of state for dark energy is unknown. Not only is it unknown but it may depend on time giving even more possibilities for the future. The simplest possibility is that the dark energy equation of state is constant and equal to minus one. This is precisely the case for a cosmological constant. For this possibility, while the dark matter density falls like the inverse cube of the scale factor, the dark energy density remains constant. Clearly therefore the dark energy becomes more and more important as time progresses. At present, by a coincidence, the dark en- ergy and dark matter have approximately equal roles. In the distant past the dark matter dominated while in the distant future dark energy will take over. In this case, the scale factor will grow exponentially with a constant Hubble parameter and no recontraction can be foreseen. The universe becomes more and more dilute in its energy density. Thus, although the dark energy played little role in the formation of structure in the past, it certainly plays the major role in deciding the future fate of the universe. This is the main change in our understanding of the future fate of the universe in the last five years. It was thought until about five years ago that the matter content, including just baryons and nonbaryonic dark matter, was the whole story. Dark energy was unknown. Let us take a more exotic and theoretically-disfavored case where the dark energy equa- tion of state is more negative than minus one and constant. This is consistent, possibly even favored by the status of the CMB observations, particularly those by the WMAP collaboration. This means that the dark energy has an energy density which actually grows with time, rather than being a constant as in the previous case. It is not surprising therefore that for this scenario the dark energy takes over even more rapidly and what happens is that the scale factor gauging the size of the universe becomes infinite after only a finite time. This blow up in the scale factor needs some elaboration. Firstly, this will not happen any time soon. The earliest could be in about ten billion years. The Earth will have been swallowed as the Sun enters its red giant phase after another five billion years so any life surviving on Earth will be have been annihilated long before this so-called Big Rip. But if there are observational astronomers safely away from the Sun but still in the Milky Way, they will cease to see galaxies outside of our Local Group because they will have expanded away to infinity. It will be as though the Local Group were the entire universe. Ironically, this is similar to the way the universe was perceived in 1917 when the first real theoretical cosmology paper was published. At this time, locally-bound systems held together by gravitational attraction can continue with their local time but the notion of cosmological time will end. One could melodramat- ically call the phenomenon the end of time. But it is important to distinguish two kinds of time: local and cosmological. Only the cosmological version ends and people locally can still wear watches as usual, though only temporarily as we shall see. This ending of cos- mology is possible only in the case of the equation of state below minus one. Why is this theoretically-disfavored? One assumption typically made in general relativity is that the energy density is not negative in any inertial frame related by special relativity. This is called the weak energy condition and is based on intuition of what is plausible. With such an assumption, an equation of state below minus one is impossible. Nevertheless, it is a question which must be settled by observation not by theory. At first the observers of the type-1A supernovae presented their data assuming the weak energy condition obtained. But they learned better not to listen to theorists and to keep an open mind. Plotting the data with an unconstrinaed equation of state is much more useful to the theorist. The first data release by the WMAP collaboration studying the CMB made similar assumptions about the weak energy condition but they have now re-analyzed their data without such an unnecessary assumption. If the dark energy is going to require revolutionary new theory for its understanding then the less prejudiced the input the better. Clearly at risk, especially in this exotic scenario, is classical general relativity itself and what could be more interesting? Of course, such a radical departure is rightly resisted and regarded as a last resort by conservative theorists. But let us pursue this radical possibility even further. The presence of a negative energy density is suggestive of instability and that the dark energy may be able to decay into a lower energy state, for example one where there is no dark energy. If this is what is technically called a first-order phase transition, like water boiling into steam then the metastable dark energy is analogous to superheated water. Such superheated water is water carefully heated in a very clean vessel to above its boiling point. It may remain in this state for a considerable time but eventually a bubble of steam will exceed a critical size and expand to precipitate boiling of all the water. The idea of a critical radius appears also for the dark energy but here the critical radius is truly astronomical, at least of galactic size. This means that there is a huge barrier to the decay and in this way one can reconcile the metastability of dark energy with the fact that it has existed already for billions of years. As already mentioned, one further aspect of the super-negative equation of state is that it leads to an unusual future for the universe. The scale factor characterizing the expansion of the universe becomes infinite after a finite future time, maybe just a few tens of billions of years. This may not bother planet Earth too much because after another five billion years or so our Sun will swell up into a red giant, as its store of thermonuclear power depletes, and engulf the Earth and any life surviving here. Nevertheless this Big Rip expansion would effect small as well as large objects. Above we mentioned that people could wear watches as usual. This was not completely true because, apart from there possibly being no people to wear watches, the dark energy density grows so rapidly towards an infinite value that even stars like the Sun, even if already a red giant, and smaller things like watches and individual atoms may get torn apart by this overwhelmingly repulsive gravitational force generated by dark energy. Whether or not this happens depends on the causal structure of the resultant spacetime which in turn depends on the fluctuation spectrum of the dark energy. Present data are consistent with perfect smoothness of the dark energy but it seems more likely that there exist fluctuations for it just as there are for dark matter. If the equation of state, on the other hand, is more positive than minus one then the future growth of the dark energy can be less rapid than for a cosmological constant. Especially in a situation where the equation of state is time dependent such that it actually becomes positive at some future time. After that time the universe will become dominated by the baryons and nonbaryonic dark matter and so will evolve just as in the situation without dark energy. An equation of state more positive than minus one occurs naturally in a theory with a scalar field which plays a role for dark energy similar to that played for inflation by the scalar inflaton. Such a theory is generically called quintessence. In some versions it may be possible that the same scalar field plays the role of the inflaton and of the quintessence field. This is called quintessential inflation. There is plenty of freedom in inventing a quintessence model and it is never completely clear whether quintessence is any more than a parametrization of ignorance. Nevertheless, it is a lively topic of research and can give some theoretical frameworks for comparison with the observational data. So the fate of the universe is as unknown as the equation of state of the dark energy. The most conservative possibility is that the latter is constant and equals minus one. It is then the cosmological constant introduced by Einstein in 1917 for a completely different reason and with the opposite sign. This still leaves the severe issue of fine tuning in that the value of the constant is over one hundred and twenty orders of magnitude smaller than would naturally appear in a gravitational theory. More exciting surely, and no less likely, is a radical equation of state more negative than minus one. Such an outcome of observations would lead to a crisis in theoretical physics as severe as the one created by the aether issue over a hundred years ago. It could then, by the dictum that necessity is the mother of invention, lead to dramatic and revolutionary progress in our understanding of the underlying theory. It used to be thought, before the discovery of the dark energy component, that the future fate of the universe depended simply on whether the matter content ΩM satisfied ΩM > 1 which gives a positive curvature closed universe which will stop expansion after a finite time and recontract to a Big Crunch. Or if ΩM < 1 there is a negative curvature open universe which expands forever. Finally if ΩM = 1 there is a flat universe which also expands forever, coming to rest at asymptotically infinite positive time. The present knowledge is that Ωtotal = 1 (at least very nearly) and so the universe is flat. This resembles most the ΩM = 1 flat universe without dark energy. But the dark energy component introduces a major uncertainty, especially with regard to its equation of state. It is possible that this equation of state will never be known sufficiently accurately from observation that one can extrapolate into the future with absolute confidence. To that extent the future fate of the universe may never be known with certainty. One other possible future scenario is that the theory itself will become more certain. For example, a successful theory of quantum gravity that explained the past history of the universe could be trusted to predict the future. This could be from string theory which is the most promising candidate for a consistent theory of quantum gravity at this time. On the other hand there is such a variety of candidate vacua in string theory, possibly 10500 or more, that it may require a better theory to make a definite prediction. Such a successful theory of quantum gravity would be expected also to shed light on the origin of the universe. What happened before inflation? Was inflation inevitable? Was there an eternal repetition of inflationary eras leading to an infinite number of universes, called a multiverse? Some of these questions edge towards the limits of scientific enquiry and may never (a dangerous word!) be answered with certainty. Some workers use the principle that our universe must be such as to permit the evolution of intelligent life. This so-called anthropic principle can be invoked to account for the values of the dimensionless physical parameters. The majority of people find this device unscientific because it replaces physical explanation by a principle which involves biology. Still any theory which is to be trusted to predict the future fate of our universe must surely be able to provide information pertaining to its origins. Both questions are valid lines of scientific enquiry. It can and has been argued that the future fate of the universe is not within physics because the prediction cannot be tested. This seems to be a semantic question because in principle it can be tested if only one has the patience to wait for a few billion years. At present we can be sure of our extrapolation back to a temperature of the equivalent of one hundred proton masses which obtained some ten-billionth of a second after the Big Bang. As high-energy colliders push back the energy frontier this will recede by another order of magnitude or two towards a trillionth of a second after the Big Bang. This is still too late for inflation and evidence for inflation will necessarily be more indirect. Most compelling would be observation of the gravity waves emanating from the inflationary era but for these to be of detectable strength requires that the charateristic energy scale of inflation be very high, not too far below the Planck energy. It remains to be seen whether Nature chooses such a high inflation energy scale. As we have seen, direct evidence for inflation is particularly elusive just because, if it occurred, it did so only in the first billionth of a second, possibly even in the first trillion- trillion-trillionth of a second, after the Big Bang. Electromagnetic radiation measurements such as optical or radio telescopes are sensitive only back to the surface of last scatter some hundred thousand years later. Abundances of helium and hydrogen and other light isotopes can tell us indirectly about one minute after inflation, still a very long time by early universe standards. Neutrinos may tell us directly about the same period. The only possible direct evidence for inflation would seem to be possible by detecting weak gravity waves emanating from that era especially if the inflation takes place extremely early on. This is an exciting possibility for which we will have to wait for at least a few more years. The discussions of the fate of the universe can be criticized as untestable because it will be many billions of years before the actual event. It can even be said that the discussion of such matters is not even physics, or science. This seems unnecessarily semantic since if a theory is successful in accommodating what is known about the past it is very natural to ask what it predicts for the very distant future. The time scales involved in cosmology are typically gigantic compared to the human lifetime. But in physics, especially cosmology, there is nothing significant about the biological scale of a human lifetime. It is equally valid to ask what happens to the universe in the next hundred billion years as to enquire about the decay of an unstable particle in a tiny fraction of a second. There is no difference in principle. The separation of physics from biology is called into question also in the use of the “Anthropic Principle”: that the universe must be such as to allow intelligent life which can observe it. The use of this principle is quite controversial since it is a way of avoiding a physical explanation for certain aspects and parameters describing our universe. While it is true that without intelligent observers the science of cosmology would be impossible, it seems to be a “cop out” on scientific explanation to appeal to such an idea to say, for example, that the lifetime of the universe must be above some minimal value or that the couplings and parameters in the underlying fundamental theory must be within small ranges around their observed values or else the cosmic evolution would be so different as to disallow the creation of life. So it is the preferable path to study physics without the need to input facts from biology with the idea that the occurrence of intelligent life is not of central importance to the physics rather than constraining the physics such that life is possible. This attitude is more likely to lead to explanations which are satisfying. As we have seen, in the minimsl Big Bang scenario, the previous history of the universe is finite with an age of 13.7 billion years. In some future scenarios, particularly where the dark energy has an equation of state less than minus one, the cosmos ends at a finite time in the future when the scale factor diverges to infinity. This provides a philosophically pleasing symmetry between the future and the past. One possibility, for which there exists no substantial evidence but which has some theo- retical appeal, is that the universe is finite also spatially. This can occur if the universe has non-trivial topology in space. If this were the case, it would provide even a more symmetric view between space and time. The recent growth in knowledge about the universe has been astonishing but, as usually happens, answering several questions gives rise to others. At present there are great enigmas about the universe. Perhaps the most striking mysteries are: what is the dark matter? what is the dark energy? The existence of such enigmas is very healthy for research in the field because they signal the distinct possibility that revolutionary advances in the theory, equally as dramatic as relativity and quantum mechanics were in the previous century, will be necessary. When old ideas are tried and fail it is all the more likely that intelligence and creativity of theorists is the only avenue to advance knowledge and understanding. Chapter 7 ADVANTAGES OF CYCLIC COSMOLOGY We have, by now, discussed how the past and the future lifetimes of the universe may be finite by virtue of the Big Bang and the Big Rip respectively. The Big Bang was first discussed in the 1920’s and some physicists found it disturbing that the density and temper- ature of the universe apparently become infinite at a finite time (now known as 13.6 billion years) in the past. This concern led several of the leading scientists to explore an alternative where the universe cycles between expansion to a turnaround then contracting to a bounce, such that the temperature and density always remain finite. This happens an infinite number of times thus eliminating any start or end of time. However, the theorists in the 1920s and 1930s could not construct a model with this cyclic property. The principal obstruction was entropy and the second law of thermodynamics. During each cycle the entropy necessarily increases so that subsequent cycles become longer and expand to a larger size. Extrapolating back in time the cycles become shorter and smaller until eventually one arrives a Big Bang again with the same issues of initial conditions as in the non-cyclic universe. This property removes the motivation to study the cyclic scenario. As we have discussed at length in the two previous chapters, the accelerated expansion of the universe discovered only in 1998 requires a large fraction, over seventy percent, of the Universe to be in the form of dark energy. This is something we now know which the theory giants of old (Friedmann, LeMaitre, De Sitter, Einstein, Tolman) did not know. Therefore, we may consider whether dark energy can help overcome the obstruction which entropy provided to oppose construction of a consistent cyclic cosmology which could preclude a start and an end. To understand how this problem is addresses, we need to explain the concept of entropy. Consider a room full of air molecules. The number of molecules is exter- mely large, typically of order 1030 or one million trillion trillion. Written out this is 1, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000 molecules! These molecules are moving with high velocity, typically about 300 meters per second and constantly colliding with one an- other. One might think that describing such a system is hopeless, which it is in detail such as the motion of any individual molecule.But the very large numbers lead to simple laws for the entire system. This is the physics of thermodynamics and staistical mechanics of which a principal architect was Boltzmann. Let us think of the following question. The air molecules are about twenty per cent oxygen and eighty per cent nitrogen. The oxygen is crucial to us to breathe and survive: without any we will pass out within a few minutes, and then die. How do we know that the volume surrounding our nose and mouth will not be depleted of all oxygen molecules for a few minutes? Why are there not ambulances waiting for just such an emergency? Anwering this question will introduce the concepts of entropy and the second law of thermodynamcs which play such a central role in cyclic cosmology. The molecules is a room can be in an astronomical number of configurations with regard to the positions and velocities of all the individual molecules. Statistical mecahnics is the study of such configurations and its basic assumption is that every configuration is equally likely. The system will quicky evolve into a configuration which corresponds to the maimum probability. Such configurations are those of highest entropy which is a measure of he disorder of the system. The highest entropy states are favored by an enormous factor. For such an equilibrium state of highest entropy one can calculate the velocity distri- bution of the molecules and the temperature to extreme precision because of the very high statistics involved. Coming back to our question the configuration with no oxygen molecules in the vicinity of a person’s nose and mouth for a few minutes is a highly ordered state of lower entropy and of extremely low probability. Boltzmann’s second law of thermodynamics (the first and the third are not relevant here) states that the entropy of an isolated sys- tem always remains constant or increases; entropy never decreases. Thus a room full of air molecules will never lower the entropy to this unlikely configuration. We can calculate the probability of it happening using statistical mechanics. The result is the following, as I have sometimes asked to a class: if I write the probability as 0.0000000...... on the blackboard with each zero occupying one inch where will be the first non-zero digit? Will it be at the end of the blackboard? At the other side of campus? a hundred miles away? The amazing answer is that it will not be the visible universe so no ambulances are necessary. Transition to such a configuration would violate the second law of thermodynamics be- cause the entropy would decrease. It is important, however, to emphasize that such an unlikely configuration would not violate any fundamental law of physics in the microscopic regime. The second law is only a statistical law, yet because of the huge number of molecules the statistical law might as well be an exact one for practical purposes. Historically it was this subtle distinction bewtween an exact law which is never violated and a statistical law which is practically never violated for a system of a million trillion trillion molecules that led to only slow acceptance of Boltzmann’s profound result, especially by mathematicians, possibly contributing to his eventual suicide. Another central aspect of statistical mechanics is provided by phase transitions such as from ice to water or water to steam. The thermodynamics of phase transitions was first systematized in a general treatment of staistical mechanics by the earliest american professor of theoretical physics, Gibbs. The ideas of entropy and the second law can be applied to the universe as a whole. The entropy of the present universe can be estimated as ∼ 1088 which is a one followed by eight-eight zeros. This entropy lies mainly in the radiation, also in the matter both dark and luminous. As the universe expands the entropy necessarily increases according to the second law. The entropy of the radiation component remains constant and we say the it expands adiabatically; the entropy associated with matter graudally increases as a result of irreversible processes. In the very early universe, entropy increased during inflation by a factor ∼ 1084, so comparison with the present entropy reveals that the entropy at the beginning of inflation was extremely low, ∼ 104. This last number is essentially zero on the scale of the later entropies. In our discussion we will come across two magnitudes of the entropy: ≥ 1088 will be called large entropy, ≤ 104 will be called small or essentially zero entropy. As can be seen, a cyclic universe with periodic entropy must involve a dramtic decrease in entropy to compensate for the huge increase in entropy at inflation, but how can any decrease be consistent with the second law of thermodynamics which demands increasing entropy? This is the question that stymied progress toward an oscillating universe in the 1920s and 1930s. But why did the cyclic universe seem an attractive alternative to the non-cyclic scenario? Once it was realized by Friedman in 1922 and independently by LeMaitre in 1927 that the equations of general relativity led naturally to an expanding universe, not a quai-static one as originally proposed by Einstein in 1917, it was also seen that there was one undesirable feature. Namely, as one extrapolates into the past at a finite time (now known as 13.7 billion years) the temperature and density become infinite, and the scale factor reduces to zero. This leads to the idea of the explosion of a primordial ”atom” or a Big Bang at the initial time t = 0. But the problem is that the classical equations cease to be applicable at this singularity, and general relativity cannot hold. One reponse is that at a sufficiently early time, the Planck time, ∼ 10−44 s, after the Big Bang, the effects of quantum mechanics must enter so that, in any case, the classical Friedman equations fail and singular behavior at t = 0 with infinite density and temperature and vanishing scale factor could be avoided if we knew a satisfactory theory of quantim gravity. Such a complete theory is unknown but in attempts at such quantum cosmology various attempts have been to formulate satisfactory initial conditions for the quantum version of the Big Bang. Without entering into techicalities, the most studied such ideas are due to Vilenkin, and to Hartle and Hawking. It is fair to say that the jury is still out on these valiant attempts to circumvent the Big Bang singularity. At the classical level the Friedman equations make a hypothesis known as the Cosmolog- ical Principle. This has two components: (i) the universe is assumed perfectly homogeneous, and (ii) it is assumed to be perfectly isotropic or rotationally symmetric. Now it is clear that neither component of the Cosmological Principle is exactly valid at smaller scales such as the size of galaxies and clusters of galaxies but at extremely large scales, an order of magnitude above cluster sizes, it does seem to be a good approximation. Nevertheless, a legitimate question for classical general relativity is whether relaxing either homogeneity or isotropy or both could avoid the Big Bang singularity?. An answer to this queation was offered in the 1960s by Hawking and Penrose who showed under general conditions that a past singularity was inevitable and did not depend on the Cosmological Principle. However, the Hawking and Penrose no-go theorem necessarily made assumptions, one of which will be seen to be important. They assumed the physically plausible requirement that the energy density never be negative, since such an energy density seemed to have no physical interpretation. As we shall see, it is this assumption which must be avoided in making a workable cyclic cosmology. Going back to the 1920s, it seemed desirable to avoid the Big Bang singularity even classically. One attractive possibility was a cyclic theory in which the density and temper- ature remain always finite and the scale factor is never zero. We have already discussed the independent attempts by Friedman and LeMaitre respectively in 1922 and 1927. It is significant that when he heard a seminar at California Institute of Technology in 1931 by LeMaitre on his research Einstein was very enthusiastic in his reception of the cyclic ides. De Sitter and Einstein published jointly on the topic in 1932. A particularly clear and endearingly modest discussion of the role of entropy in cyclic cosmology is in the book Relativity, Thermodynamics and Cosmology by Richard Chase Tolman. To add a personal anecdote, when I was studying for the Final Honour School in Oxford in 1965 and had immediate access to several hundreds of physics books, my personal fa- vorite book was always Tolman’s. I do recall spending hours then intrigued by the apparent contradiction between the attractive idea of cyclic cosmology and the second law of thermo- dynamics; the contents of Tolman’s book, however, did not appear on my examination! One important fact about the universe, discovered only in 1998, and obviously unknown in the 1920s and 1930s is that the expansion is actually accelerating. As we have previ- ously discussed, this led to the identification of over seventy per cent of the energy as dark energy. This leads to out question of whether dark energy can intercede in the apparent contradiction? Before explaining a positive response to this question, let us make some general consid- erations of cyclicity and entropy in an attempt to make the solution seem inevitable. A cyclic universe goes through four stages: Bounce −→ Expansion −→ Turnaround −→ Contraction −→ Bounce, repeated an infinite number of times. During expansion, entropy initially increases by an enormous factor ∼ 1084 during infla- tion then more gradually increases due to irreversible processes in the matter component. We recall that the radiation expands with constant entropy and the dark energy has zero entropy. Before discussing the turnaround, consider the contraction phase which presnts its own peculiar issues. One is that matter and dust will form structure more readily than during expansion. Black holes, if present, will expand and more will form, further impeding any smooth contraction. There is an even more serious problem with matter content during contraction, namely that several phase transitions must take place in reverse to proceed successfully back to a bounce in the early universe. For example the phenomenon of recombination would require, in reverse, that neutral atoms dissociate into protons and electrons as the temperature increase. This would decrease entropy and be statistically impossible as a violation of the second law of thermodynamics. There are other phase transitions such as the so-called quantum chromodynamics transtion at a few hundred MeV where quarks and gluons become confined into hadrons. Also, the weak transition at about one hundred thousand MeV at which electroweak forces separate into weak and electromagnetic forces. How can a contracting universe pass in reverse through such phase transitions without violating the second law of thermodynamics? One possibility which is seductive but which we shall quickly discard is that during contraction the ”arrow of time” reverses. This is an assumption made to argue that entropy may decrease during the contracting phase. What does it mean? The arrow of time refers to the second law of thermodynamics of a staistical mechanical system for which increase of entropy defines a ”forward” direction of time. At a biological level, we remember the past but not the future, and we grow older so for us there is surely a well-defined arrow of time. For statitistical systems in physics the arrow of time is equally well defined by the behavior of the entropy. The entropy problem could be solved very simply if we were allowed to adopt a reversal of the arrow of time during contraction and entropy could then decrease. If we consider this further, however, it is only a semantic device corresponding to unacceptable physics although it has been used by some authors (including, for a few days, by this one!) as a last resort to address the thorny cyclicity problem. In a universe with a reversed arrow of time, statistical mechanics would become nonsensical because equilibrium states of, say, an ideal gas would proceed to diseqilibriate into very unlikely configurations, evidently nonsensical. Reversal of the arrow of time is therefore an absurdity better to avoid if at all possible. Fortunately we shall find a solution to cyclicity in which the arrow of time keeps moving forward in the conventional sense. To proceed toward an acceptable solution, we need to introduce a couple of new concepts: branes and causal patches. Branes were first emphasize in the 1990s on the basis of sting theory, particularly by Polchinski. String theory was believed in the 1980s to be a theory of one-dimensional extended objects, namely strings, interacting with one another. As such, it was already a remarkably consistent geenralization of quantum field theory which included within it both gauge field theory and general relativity. Moreover, it provided a finite theory of quantum gravity for the first time. Around 1990, however, it was shown that a theory of only one-dimensional strings is not internally consistent and must include higher dimensional ”branes”, a shortened version of the word ”membranes”. In this language, strings are 1-branes. In a ten dimensional supersting theory branes appear in addition as p-branes for all p between 2 and 9. Which values of p appear depends on the particular superstring considered; for example, a 9-brane in 10 spacetimes dimensions is a space-filling brane. Of special interest to cosmology is the 3-brane. This notion has by now been abstracted from string theory leading to the idea of a brane world in which non-gravitational physics is restricted to the 3-brane on which we are assumed to live. Strong and electroweak inter- actions are restricted to this ”TeV” brane, while gravity alone spreads also into the extra spatial dimension or dimensions in which the TeV 3-brane happens to be embedded. In the theory of brane worlds one remains agnostic about the number of extra space dimensions which can vary from one up to the six predicted by superstrings or seven as allowed in M theory. In our discussion we shall for simplicity assume only one extra space dimension, or a total of five spacetime dimensions. One popular version of the brane world involves two parallel 3-branes where one is the TeV 3-brane and the second is a ”Planck” 3-brane from which gravity originates. This theory offers an attractive expalnation of why gravitation is so weak in our observed world. It is remarkable, for example, that the electric force between two protons separated by, say, an atomic size (or any other separation) is some fourty orders of magnitude larger than the gravitational attraction; that is a one followed by fourty zeros. Another reflection of the weakness of gravity is that a small magnet will hold an object to a refrigerator door against the gravitational pull of the entire Earth. For cosmology, our universe will be the TeV brane so in such a brane world one can derive a Friedman equation for this TeV brane which has a crucial modification. Without going into technical details, a new term appears which involves a parameter with the dimension of a density and this critical density signals when the expansion of the universe will stop and turnaround inot a contractin mode. This same critical density determines what will be the bounce temperature at which the contracting universe will stop and bounce into a new expanding mode. The critical density is related to the brane tension and the new brane term in the Fried- man equation plays a central role in our solution for the cyclic universe. A central role is played equally by the concept of causal patch. To introduce the causal patch we must revisit the Big Rip where the present expansion of the universe ends at a finite time in the future when the universe rips itself apart and the dark energy density, as well as the scale factor diverges to infinity. As one approaches the Big Rip, the structures in the universe successively disintegrate beginning with the largest scales. An approximate rule is that once the dark energy density reaches the average density of a gravitationally bound system that system will become unbound. This implies that first the clusters of galaxies disintegrate followed by the galaxies then the solar system. Nothing is immune to this disintegration process and eventually even atoms, atomic nuclei and nucleons - protons and neutrons - will become unbound. These are just the smallest systems we currently know and if there exist yet smaller higher density bound sytems they too would be subject unbinding process. As another example which is important for the sequel, black holes are themselves torn apart on the approach to the Big Rip. At a time somewhat later than the unbinding of a system, the bound components become causally disconnected, meaning that they cannot communicate even at the speed of light before the universe ends. Eventually we may regard the universe itself as disintegrating into a huge number > 1084 of causal patches which are disjoint and separate. The idea now is to delay the brane induced turnaround until a trillion trillionth of a second or less before the would be rip. At this time, one causal patch contains no quarks or leptons and certainly no black holes. The causal patch at the turnaround contains only dark energy and a small number of low energy photons. Its entropy is small in the sense discussed earlier, meaning it may have entropy equal to ten, or some such number. Before the unbinding process, on the other hand, the entropy was large by the same discussion, meaning it was at least a one followed by eight-eight zeros. This is essentially the reverse of the vast increase of entropy experienced during inflation and so may naturally be called deflation. We have now assembled all the pieces of our proposed cyclic universe. It will involve ending the present expansion at a finite time in the future, typically of the order of a trillion years if the equation of state of the dark energy is just below minus one say, −1.01. At the turnaround, only one causal patch will be retained as our universe and it contracts with constant small entropy: the radiation or photons contracts with no increase of entropy and the dark energy has zero entropy. What is equally important is that the contracting universe is empty of all matter whether dust or black holes and thus can contract without confronting all the difficulties discussed earlier such as enhanced structure formation, and growth of black holes as well as formation of new black holes. The contracting universe, much smaller than its expanding progenotor, continues until the dark energy again reaches the critical value . Then the contraction ceases at a certain bounce temperature and the universe cycles into inflationary expansion again with the en- tropy increasing from a small to a large value. The temperature at which this happens is related to the critical density which is a free parameter in the model. This provides a possible model for a cyclic universe which respects the second law of thermodynamics. It has two ingredients which could not have been foreseen in the 1920s and 1930s by leading theorists who attempted such a model. The first is dark energy which was deduced from the observed accelerated expansion rate first in 1998. The second is the idea of a higher dimensional brane world which has emerged in theoretical physics equally recently. Another hurdle of more recent vintage was the singularity theorems of Hawking and Penrose from the 1960s. These assumed an everywhere positive energy density while a dark energy with equation of state less than minus one violates such an assumption. Only time will tell whether the cyclic model we have discussed will survive closer scrutiny. A general observation, however, is that a cyclic universe which avoids classically any singular or infinite behavior in density or temperature, and which avoids any beginning or end of time, seems more aesthetically acceptable than a cosmology which originates from a singularity. This new viewpoint on the future of the universe is possible only given the amazing developments in both theory and observations since the very end of the twentieth century. The conclusion is that conventional cosmology with time starting at the Big Bang and continuing for ever in an infinite expansion now has plausible alternatives, especially this cyclic model where time never begins or ends. Chapter 8 SUMMARY OF ANSWERS TO THE QUESTIONS: DID TIME BEGIN? WILL TIME END? There are three different futures for the universe which we have discussed. One key question is whether the present expansion phase will continue for an infinite time, as in the conventional wisdom, or will it stop after a finite time of order a trillion years? If it stops, a second question is whether our universe will then contract and bounce cyclically? Although we cannot say with certainty which future is correct, progress in addressing this question has been so rapid that it is possible to order these three futures, according only to aesthetics, in decreasing probability with conventional wisdom third. The scenarios in 1. and 2. assume the equation of state of dark energy ω is slightly below −1, say ω = −1.01, while scenario 3. adopts ω = −1 exactly. 1. Most likely. The present expansion will end after a finite time, the universe will contract, bounce and repeat the cycle. A cyclic universe. Time had no beginning and will have no end. This presumes that the entropy problem has been resolved as discussed in this book. 2. Next most likely. The present expansion will end after a finite time in a Big Rip. Time began in the Big Bang some 13.7 billion years ago and will end some trillion years in the future. 3. Least likely. The present expansion will continue for an infinite time as for a cosmolog- ical constant. Time began 13.7 billion years ago and will never end. This is the prevailing conventional wisdom which is here being challenged. GLOSSARY Aether Hypothesized, but later discredited, medium through which electromagnetic radia- tion propagates. In physically interpreting Maxwell’s equations it was thought that for light to propagate through vacuum, as from the Sun to the Earth, it was necessary for there to be “something”, the aether, through which the waves were transmitted. One consequence was that light should travel at different speeds according to whether the direction was parallel, anti-parallel, or orthogonal to the motion of the Earth through the putative aether. A land- mark experiment in Cleveland, at what is now Case University, by Michelson and Morley in 1887 showed that the speed of light did not so vary with direction, thus strongly disfavoring the aether hypothesis. Baryonic matter Matter, including the stuff we and all everyday things are made of, comprised by mass almost entirely of baryons: protons and neutrons. This forms about 4% of the energy of the universe, and is the only major component which is well understood. The microscopic physics of this component is well described, at least up to energies of hundred times the proton mass, by the standard model of particle phenomenology. Big Bang By using the Friedmann equation and the known values for the contributions of matter and radiation on the right-hand-side thereof, the expansion history of the universe can be reliably traced back to one ten billionth of a second after the would-be Big Bang when a primordial fireball is often regarded as exploding. In cyclic models this Big Bang is avoided but it is Widely regarded, though not in this book, as the beginning of time some 13.7 billion years ago. Bounce The transition from contraction to expansion phases in a cyclic cosmology. Branes Multi-dimensional objects which occur in string theory. These objects were identified in string theory in the 1990s and have led to ideas about the weakness of the force of gravity especially that gravity alone propagates in extra dimension(s) while the other forces are confined to our four-dimensional brane which bounds the higher-dimensional “bulk” space. In particular, attempts have been made to replace inflation, couched in the language of quantum field theory in four dimensional spacetime, with a higher-dimensional brane-world set-up where the Big Bang is initiated by the collision of two branes in an extra space dimension. European laboratory for high-energy physics near Geneva, Switzerland. Under construc- tion there is the largest particle collider in history in which protons will collide with total energy of 14,000 times the proton mass. It is expected that the elusive Higgs boson will be discovered, together with (unknown) other particles which will hopefully shed light on the mysteries of the standard model. As well as its large complex of particle accelerators, CERN has a theory group which, if one counts both CERN employees and visitors, is the largest group of particle theorists anywhere in Europe, probably the world. COsmic Background Explorer. This was a satellite experiment to measure the cosmic microwave background (CMB) and was the first to detect the small anistropy at a level of about one in one hundred thousand. These small perturbations seed the formation of large scale structure in the universe including galaxies and stars. Cosmic microwave background Gas of photons which fills the universe. The study of this radiation is one of the most fruitful methods to investigate the early universe from the time of some 400,000 years after the Big Bang. At this recombination era the universe became transparent to photons and they propagated unimpeded. The gas is thoroughly eqilibriated and has a spectrum corre- sponding now to a temperature of 2.725 degrees above absolute zero. The CMB was origi- nally discovered by accident in 1965 by Penzias and Wilson who were looking for anisotropic microwaves correlated with the plane of the Milky Way. Cosmological principle The assumption that the universe is Homogeneous and Isotropic. On the largest scales, these properties are well approximated by the observed universe and using this cosmo- logical principle simplifies the differential equation, the Friedmann equation, characterizing the expansion of the universe. Cosmology The scientific study of the universe. This includes the finite time back to the Big Bang and the finite or infinite time future discussed in this book. The beginning of modern theoretical cosmology can be taken as the paper, despite its flaws, by Einstein in 1917 where he first applied general relativity to the whole universe. One cannot help wishing he had anticipated the work by Friedmann, Lemâıtre and Hubble in the 1920’s: what a paper that would have been! Critical density A special value of mean energy density of the universe. Since the WMAP data was released in 2003 we now believe the actual energy density is very close (within 2%), and possibly extremely close, to this value, meaning that the geometry of the universe is flat or Euclidean. If the density is larger (smaller) than the critical value the universe is closed (open) and has positive (negative) curvature. Such closed and open universes were widely considered until the recent observational discovery that the universe is flat. Cyclic Cosmology To avoid an infinite density and temperature at the Big Bang theorists since the 1920s have studied the possibility of cyclic cosmology where these quantities remain finite but the universe expands to a turnaround then contracts to a bounce and again expands. If this cycle can repeat an infinite number of times it avoids any beginning or end of time. For decades an apparently insuperable theoretical hurdle was the second law of thermodynamics but the discovery of dark energy has led to renewed interest in the idea. Dark matter About 27% of the total energy of the Universe. It is comprised of Baryonic matter and, about six times as much, Nonbaryonic dark matter. The nonbaryonic matter’s composition is unknown but can comprise particles with any mass between one billionth of the proton mass to one millionth of the solar mass, a range of 60 orders of magnitude. Popular candidates arising from extension of the standard model are the neutralino of super- symmetry and the axion arising from the strong CP problem. The neutralino is one example of a WIMP or Weakly Interacting Massive Particle. The dark matter sector may be richer than suggested by a single constituent and equally as complex as the baryonic matter sector but as a first hypothesis it is natural to assume a single dark matter constituent. Doppler effect The frequency of a wave which may be sound or light is effected by the relative motion of the source and observer. This effect was first discovered by Doppler in 1845. In a dramatic demonstration of his discovery, he arranged for trumpeters to play on an open wagon of a moving train and for musicians with perfect pitch to listen to the change of pitch as the train passed. This physics, applied to electromagnetic waves, played a central role 80 years later in the discovery of the expansion of the universe. Earth Third nearest planet to the Sun and the only known location of life. In the universe there must be either only one or many such locations: two would be much harder to comprehend than one. The reason is this. The Milky Way contains one hundred billion stars similar to our Sun; the visible universe contains one hundred billion galaxies similar to our Milky Way so there are some ten billion trillion stars. Over one hundred planets have been discovered near relatively nearby stars so there are presumably some hundred billion trillion planets. The probability of life originating on a given planet is more or less than one in one hundred billion trillion. If it is much less, we are likely alone; if the probability is much larger there should be many life bearing planets. The probalility of life formation on a given planet is unlikely to be close to the reciprocal of the total number of planets in the visible universe. Electromagnetic radiation Oscillating and propagating electric and magnetic fields. Examples are X rays, radio waves, visible light. The cosmic microwave background is an important example, originating from the surface of last scatter. At the present temperature of 2.725 degrees the peak frequency is about one millimeter and therefore in the microwave region. Friedmann equation Mathematical equation governing the time dependence of the Scale factor derived by Friedmann in 1922. It was independently derived by Lemâıtre in 1927 and is in some books called the Friedmann-Lemâıtre equation. In this book for simplicity it is called the Friedmann equation. General Relativity Theory relating gravitation to the geometry of space introduced by Einstein in 1915 after concentrating for ten years on how to generalize special relativity to the case of accelerating coordinate systems. An important intuition was that a falling person does not experience the force of gravity, implying the equality of gravitational and inertial mass, as expressed by the equivalence principle. The theory explained the discrepancy in the perihelion of Mercury when calculated from Newton’s law and made two predictions: the bending of light confirmed in the solar eclipse expedition of 1919 and the red-shift of “falling” light confirmed by Pound and Rebka in 1960. Helium The second lightest element after hydrogen. Also, by far the second most common atom after hydrogen. The fact that the abundances are about 23% by mass of helium and 77% by mass of hydrogen strongly support Big Bang nucleosynthesis; the other evidences for the Big Bang are the discovery of the cosmic expansion and of the cosmic microwave backgound. Homogeneous Uniformly distributed or the same at every position. At large distances, the universe does appear to become more and more homogeneous and so this, together with isotropy, are the traditional assumptions made about the universe in theoretical cosmology. Hubble Law Recession velocity is proportional to distance. This was established by Edwin Hubble in 1929 and is the most important discovery made in cosmology before the accelerated rate of expansion was discovered in 1998.. Hubble’s original data was so inaccurate that it’s even more impressive that he drew the correct conclusion. This expansion of the universe convinced Einstein, as he expressed it to Hubble on a visit to California in 1931, that his 1917 paper on cosmology had contained the “blunder” of introducing a cosmological constant. Hubble parameter Ratio of recession speed to distance. Its units are kilometers per second per megaparsec. In these units the value is now known to be within a few percent of 70. Historically, its value was controversial because of uncertainties in distance measurement. For several decades values as different as 50 and 100, both with quite small errors, were forcefully defended by various groups. Hubble’s original value was even worse, close to 500, back in 1929. Hubble Space Telescope An orbiting instrument with a 2.4 meter primary mirror deployed by NASA astronauts in 1990. The HST has been one of the most successful scientific instruments, having made a series of important discoveries. Hydrogen The lightest element whose atomic nucleus contains one proton. The atome contains also one electron. This is the simplest atomic system and so formed the first target of quantum mechanics which explaines its detailed spectrum. About 75% of the atoms in the universe are hydrogen, while 25“metallic” atoms is negligible. Inflation Period of very rapid expansion during the first trillionth of a second after the Big Bang. Appended to the Big Bang, inflation solves two severe fine-tuning issues: the horizon and flatness problems, as discussed in Chapter Two of this book. Inflaton Hypothesized particle underlying inflation. Inflation was motivated by the scalar po- tentials and phase transitions occurring in grand unified theories of elementary particles. The inflationary era is expected to originate from the variation in the value of a scalar field named the inflaton. Different assumptions about the inflaton potential leads to alternative versions of inflation theory which observations may discriminate. Isotropic The same in all directions, also known as rotational invariance. The CMB, in particular, is isotropic to an accuracy of one part in one hundred thousand and this represents the horizon problem for the unadorned Big Bang theory. The surface of last scatter from which the CMB photons originate contains about fourty thousand regions never causally connected so why should their temperature be the same to this accuracy? The isotropy, by itself, strongly suggests that something like inflation occurred in the early universe. Large Hadron Collider at CERN. It will start test running in 2007 and collide protons in 2008 at a center-of-mass energy of 14 TeV, seven times the previous highest energy. In this new energy regime, it is expected to discover the Higgs boson and hopefully other new particles and forces which will shed light on some of the mysteries of the standard model. Mercury Nearest planet to the Sun. The known discrepancy of the orbit of Mercury from the Newtonian prediction was crucial in the development of general relativity. The perihelion is the closest approach of a planet to the Sun. It was known already before Einstein’s work that there was an anomalous precession in the perihelion of Mercury of 43 seconds of arc per century. General relativity was able to explain this discrepancy accurately and to make two other predictions as classic tests: the bending of light by the Sun, confirmed in 1919 an expedition to view the Hyades cluster during a six-minute-long total solar eclipse from northern Brazil and Principe, a small island West of equatorial Africa, in 1919 and the red-shift of “falling” photons, first confirmed by Pound and Rebka at Harvard in 1960. When the eclipse results were announced in November 1919 the exceptional media reaction rendered Einstein a famous personality for the rest of his life. When the expedition’s leader, Eddington, was asked at that time by a journalist if it was true only three people understood general relativity, his self-revealing reply was: ”Who is the third?” Milky Way Name of the galaxy containing the Earth. It is just one of at least one hundred billion galaxies in the expanding universe. The Milky Way contains some hundred billion stars comparable to our Sun. The Sun is thirty thousand light-years from the core of the Milky Way whose luminous matter stretches out to one hundred thousand light-years in spiral arms. From outside, the Milky Way would presumably look like one of the many other observed spiral galaxies although such a picture is not likely anytime soon. The Milky Way contains about ten times as much mass in dark matter stretching a few hundred thousand light-years from the core, far enough to mingle with the corresponding dark matter halo of neighbouring Andromeda. Naturalness Criterion that a theory should not contain unexplained large ratios of numbers. If there exists an unexplained large dimensionless ratio, it is often called a hierarchy. As much has been written about naturalness and hierarchy as any other topic in particle theory. It explains the popularity of low-scale supersymmetry despite no experimental evidence. Its importance was stressed strongly in 1971 by the eminent theorist K.G. Wilson who, however, much later in 2004 called his assertion a “blunder” when he emphasized instead that very large ratios can and do occur naturally in Nature. Neutrino Elementary particle which experiences only the weak interaction. Until 1998, the neu- trino mass could have been zero but since then we know that flavors (e, µ, τ) of neutrino oscillate into each other which requires that there are non-zero mass differences. All the three masses must be small, less than one millionth of the electron mass. The mixing angles, on the other hand, are large, near maximal for two angles out of three; this is different from quarks where all three mixing angles are small. The non-zero mass of neutrinos increases the number of free parameters in the standard model from 19 to 28. Nonbaryonic Dark Matter 23% of the energy of the universe. It clumps like baryonic matter but non-luminous. The consitutents of dark matter are completely unknown and can have mass ranging anywhere from a billionth of a proton mass to a millionth of the solar mass, a range of 60 orders of magnitude. Popular candidates arising from motivated extension of the standard model are the neutralino particle of supersymmetry and the axion associated with solution of the string CP problem. The neutralino is expected with a mass of about 100 times the proton mass and exemplifies a class of dark matter particles called WIMPs (Weakly Interacting Massive Particles). The axion is expected to be near the light end, between one billionth and one millionth of the proton mass. The dark matter could be much more complicated and contain a wide variety of constituents, just as does the luminous baryonic matter. Nontrivial topology An idea that space has special properties at the largest distance scales. There is no evi- dence for this but the accurate data on the CMB will allow detailed searches for unexpected properties such as “circles in the sky” where circles in opposite directions exhibit correlations which reflect a periodicity of space. So far, no such correlations have been found but if they were, it would change significantly our notion of space dimensions. Photon Massless elementary particle which is the smallest unit of electromagnetic radiation. The photon was used in his explantion of the photoelectric effect by Einstein in 1905 but it was not universally accepted until 1923 with the demonstration of the Compton effect in which light must be treated as particulate. Two prominent skeptics of the interim period were Planck himself in 1913 who, in nominating Einstein to the Prussian Academy asked the committee to overlook the work on photons as an example where Einstein “may have gone overboard in his specualtions”; and Bohr, in his 1922 Nobel lecture, said that photons are “not able to shed light on the nature of radiation”. Principia Exceptional book by Isaac Newton published in 1687, rightly regarded as the beginning of modern theoretical physics. For the next two hundred years this book, now of only historical interest, had huge influence. It contains in particular the three laws of motion and the universal law of gravity. Although the author invented calculus, he remarkably avoided using it in the book, although this made many of his proofs more complicated. Over three hundred years later, non-calculus courses in introductory physics are still offered by many universities. Quantum Mechanics Theory invented in 1925 which explains spectra and stability of atoms. According to classical theory an electron circulating around a nucleus would rapidly lose energy by radia- tion. In quantum mechanics, a basic assumption is that the energy is quantized and cannot be lower than that of a ground state which is the stable situation of an atom. Quantum mechanics explained the spectra of atomic radiation as well as many other hitherto profound mysteries such as the distinction between conductors and insulators of electricity. It intro- duces a fundamental constant of Nature which was discovered phenomenologically in 1900 by Planck. Recombination Cosmological era when charged particles bind to form neutral atoms and universe be- comes transparent to electromagnetic radiation. This creates the surface of last scat- ter from which the CMB radiates. “Recombination” is a misnomer because the electrons and protons were never combined. Scale factor Measures distance between galaxies and depends on time according to the Friedmann equation. A useful analogy is the surface of a balloon with spots painted on its surface. As the balloon inflated the spots separate from each other. The galaxies separate similarly as the universe expands. Scientific Method Systematic making of experimental observations, framing hypotheses and theories which make predictions followed by more experimental tests. The ancient Greeks proceeded in a different fashion by assuming that human thought alone could discover the laws of physics. It required the intellects of people like Galileo and Newton, after a remarkable two thousand years of dark ages, to identify experiments and observations as the driving force of physics with human thought secondary. They realized that all humans are less smart than Nature. Solar System Our Sun and its eight planets including Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. General relativity has been tested with good precision on the length scales which characterize the solar system. The visible universe has a size 14 orders of magnitude bigger and general relativity is not tested with comparable precision at such length scales. Special Relativity Theory which accommodates motion comparable to the speed of light, proposed by Ein- stein in 1905 on the basis of a symmetry of Maxwell’s equations which he generalized also to classical mechanics, and all of theoretical physics. It placed time on a similar footong to the three space coordinates and led to the notion of four-dimensional spacetime. The predic- tions of special relativity for motion at ultra-high speed have been confirmed to impressive accuracy. Steady-state theory A once popular, now discredited, alternative to the Big Bang, in which the universe is quasi-static. It postulated that the dilution of matter caused by the universe’s expansion was compensated by the continuous creation of new matter. The discovery of the CMB in 1965 laid the steady-state theory to rest. String theory An elegant and consistent mathematical framework proposed as a marriage of quantum mechanics and general relativity, based on the premise that the fundamental entities are not point particles but extended objects, the simplest being one-dimensional strings. Ex- perimental effects of quantum gravity are probably so extremely small that they will not be detected in the foreseeable future so it will be impracticable to determine whether string theory is the correct theory of quantum gravity. String theory has a dual description by gauge field theories and hence provides a source of interesting ideas of how to extend the established standard model of particle phenomenology. Independently of whether string theory is the correct theory of gravity (a question unlikely to be answered anytime soon), it is the only known consistent extension of quantum field theory from point particles to extended objects As such it provides an important insipration for model-building in the simpler, and more realistically testable, gauge field theories. By far the nearest and most understood star. Its present age is about five billion years and it is expected to run out of hydrogen fuel in another similar time. The next nearest star is remarkably a million times further away and much harder to study. Our understanding of the Sun’s evolution over its five-billion year history is remarkably good and enshrined in a theory called the Standard Solar Model. Surface of last scatter An opaque wall corresponding to recombination where photons of cosmic microwave background originate. The photons we observe in the CMB have travelled uninterrupted through the universe for over 13 billion years from this surface which makes an effectively- opaque wall behind which (or earlier than which) electromagnetic radiation cannot pene- trate. T-duality Mathematical symmetry of string theory which relates very small and very large distance scales. In application of string theory to cosmology, T-duality plays a central role and leads to the suggestion that there is a maximum temperature in the past and that before that there might exist a T-dual universe in which distance scales were interchanged between the very large and very small. This is difficult to visualize but follows simply from the string theory equations. Turnaround The transition from expansion and contraction phases in a cyclic cosmology. Venus Second nearest planet, after Mercury, to the Sun. The two inner planets provide some of the best checks of general relativity theory. Weakly Interacting Massive Particle, a candidate for the cold dark matter particle. One example is the neutralino of supersymmetric theories but the WIMP is a more general concept of a particle which might be discovered at the LHC. Wilkinson Microwave Anisotropy Probe, whose first data was released in February 2003 and initiated “Precision Cosmology” by virtue of analysis of its data leading to values of the cosmic parameters with unprecedentedly small errors.

0704.1133

FUSE Observations of the Dwarf Novae UU Aql, BV Cen, and CH UMa in Quiescence

FUSE Observations of the Dwarf Novae UU Aql, BV Cen, and CH UMa in Quiescence1 Edward M. Sion, Patrick Godon2, Fuhua Cheng Astronomy and Astrophysics, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA [email protected], [email protected], [email protected] Paula Szkody Department of Astronomy, University of Washington, Seattle, WA 98195, USA [email protected] ABSTRACT We report on FUSE spectra of three U Gem-type, long period, dwarf novae, UU Aql, BV Cen and CH UMa taken during their quiescence intervals. We discuss the line identifications in their spectra and attempt to characterize the source(s) of their FUV flux distribution. Archival IUE spectrum of CH UMa and BV Cen in quiescence were identified as having a matching flux level with the FUSE spectra and these were combined with each FUSE spectrum to broaden the wavelength coverage and further constrain model fits. Multi-component synthetic spectral fits from our model grids, consisting of single temperature white dwarfs, two-temperature white dwarfs, accretion disks and white dwarfs plus accretion disks, were applied to the FUSE spectra alone and to the combined FUSE + IUE spectra. We present the results of our model analyses and their implications. Subject headings: accretion, accretion disks - novae, cataclysmic variables - stars: dwarf novae - stars: individual (UU Aql, BV Cen, CH UMa) - white dwarfs 1Based on observations made with the NASA-CNES-CSA Far Ultraviolet Spectroscopic Explorer. FUSE is operated for NASA by the Johns Hopkins University under NASA contract NAS5-32985 2Visiting at the Space Telescope Science Institute, Baltimore, MD 21218, USA; [email protected] http://arxiv.org/abs/0704.1133v1 – 2 – 1. Introduction Dwarf novae (DNe) are close interacting binaries in which a Roche-lobe filling main sequence-like dwarf transfers matter with angular momentum through a disk onto a white dwarf (WD). The rapid disk accretion during outburst, due to a thermal instability that causes cyclic changes of the accretion rate, releases gravitational potential energy identified as the DN outburst. The high accretion rate (∼ 10−8 to 10−9 M⊙/yr) outburst phase (which lasts a few days to weeks) is preceded and followed by a low accretion rate (∼ 10−11M⊙/yr) quiescence stage. This DN behavior is believed to be punctuated every few thousand years or more by episodes of catastrophically unstable thermonuclear burning, the classical nova explosion. Perhaps the least understood topic in CV/DN research (along with what drives the wind outflow in outburst) is the state and structure of the boundary layer and accretion disk during quiescence and the physics of how long term accretion of mass, angular momentum and energy affects the WD. Our studies with archival IUE, and HST/STIS have found that ∼50% of the DNe in quiescence are dominated (i.e., > 60% of UV flux) by a component of FUV flux other than the WD called the ”accretion disk”; ∼25% are dominated by the WD and ∼25% have nearly equal contribution of WD and accretion disk (40-60% each) (Urban & Sion 2006). A number of studies (Araujo-Betancor et al. 2003; Sion 1991, 1999; Urban & Sion 2006) have shown that CV WDs above the gap are typically on-average 10,000K hot- ter than CV WDs below the period gap (almost certainly a consequence of higher time- averaged accretion rates of systems above the gap but possibly with system total age also being a factor). Since the white dwarf surface temperature is crucial for understanding CV evolution and whether CVs evolve across the period gap, the use of cooling ages and long term evolutionary model sequences with accretion (including the effects of nova explosions, Townsley & Bildsten (2003)) must rely on the empirical WD temperature of the photo- sphere in equilibrium with long term compressional heating from accretion. The work of Townsley & Bildsten (2002, 2003) allows measured Teff ’s of CV WDs to be converted to the accretion rate per unit WD surface area averaged over the thermal time of the WD envelope. Unfortunately, there are far fewer systems with reliably known WD properties above the period gap compared with below the gap, thus impeding detailed comparisons between the two groups. For example, among CVs below the gap, there are now roughly 20 systems with reliable WD temperatures but only 5 systems above the gap with reliable WD temper- atures. The primary reason for this disparity is that in long period CVs with higher mass transfer rates, the disks may remain optically thick even during quiescence, making the disk contribution to the total flux typically larger in systems above the gap. Hence, it is more – 3 – difficult to disentangle the white dwarf flux contribution from that of the accretion disk. As part of our effort to increase the sample of CV degenerates with known properties above the gap, we have used FUSE and IUE archival spectra to analyze three long period dwarf novae, UU Aql, BV Cen, and CH UMa. For UU Aql, system properties were adopted from Ritter & Kolb (2003) and from Szkody (1987). For BV Cen parameters were adopted from Ritter & Kolb (2003). For CH UMa, we adopted values from Friend et al. (1990). For all three systems, the distances were the same as those in Urban & Sion (2006) where the Warner (1995) and Harrison et al. (2004) Mv(max) versus Porb relations, calibrated with trigonometric parallaxes, were used. The reddening values were the same as those quoted in Urban & Sion (2006) which were from Verbunt (1987), laDous (1991) and Bruch & Engel (1994). The dwarf nova systems analyzed in this work are UU Aql, BV Cen, and CH UMa. In Table 1, the observed properties of these dwarf novae are summarized by column as follows: (1) system name; (2) dwarf nova subclass with UG denoting a U Gem-type system; (3) orbital period in days; (4) the recurrence time of dwarf nova outbursts in days; (6) the apparent magnitude at minimum (quiescence); (7) the apparent magnitude in outburst; (8) secondary spectral type; (9) orbital inclination in degrees; (10) white dwarf mass in solar masses; (11) secondary star mass in solar masses; (12) adopted reddening value and ; (13) distance in parsecs. 2. Observations and Data Reduction The instrumental setup and exposure details of the FUSE spectra of BV Cen, CH UMa and UU Aql in quiescence are provided in Table 2. The LWRS was used in all cases since it is least prone to slit losses due to the misalignment of the four FUSE telescopes. All the spectra were obtained in time tag (TTAG) mode, and each one of them consists of 7 individual exposures (corresponding to 7 FUSE orbits). It is clear that the relatively poor FUSE spectral quality of the spectra speaks to the requirement for more observing time. Nevertheless, we deemed that there was sufficient S/N to warrant a first attempt multi- component FUV analysis of each system All the data were reduced using CalFUSE version 3.0.7. In this version of CalFUSE the data are maintained as a photon list: the intermediate data file - IDF. Bad photons are flagged but not discarded, so the user can examine and combined data without re-running CalFUSE. For each target, we combined the individual exposures (using the IDF files) and channels to create a time-averaged spectrum weighted in the flux in each output datum by – 4 – the exposure time and sensitivity of the input exposure and channel of origin. The details are given here. The spectral regions covered by the spectral channels overlap, and these overlap regions are then used to renormalize the spectra in the SiC1, LiF2, and SiC2 channels to the flux in the LiF1 channel. We then produce a final spectrum that covers almost the full FUSE wavelength range 905 − 1182Å. The low sensitivity portions of each channel are discarded. In most channels there exists a narrow dark stripe of decreased flux in the spectra running in the dispersion direction. This stripe has been affectionately known as the ”worm” and it can attenuate as much as 50% of the incident light in the affected portions of the spectrum. The worm has been observed to move as much as 2000 pixels during a single orbit in which the target was stationary. The ”worm” appears to be present in every exposure and, at this time, there is no explanation for it. Because of the temporal changes in the strength and position of the worm, CalFUSE cannot correct target fluxes for its presence. Here we take particular care to discard the portion of the spectrum where the so-called worm ’crawls’, which deteriorates LiF1 longward of 1125Å . Because of this the 1182−1187Å region is lost. We then rescale and combine the spectra. When we combine, we weight according to the area and exposure time for that channel and then rebin onto a common wavelength scale with a 0.1Å, 0.2Å, and 0.5Å resolution. In the observing log given in Table 2, the entries are by column: (1) gives the target, (2) FUSE spectral data ID, (3) the aperture used, (4) the date and time of observation, (5) the (good) exposure time in seconds, (6) central wavelength, and (7) S/N. The FUSE spectra for the three systems, UU Aql, BV Cen and CH Uma are displayed in Figures 1, 2, and 3 respectively. A quantitative sense of the relative data quality is provided by the signal to noise for the three FUSE spectra. We binned the data by 0.1Å for which the S/N of UU Aql, BV Cen, and CH UMa is 5.15, 5.9, and 3.6, respectively. For UU Aql, Fig. 1 reveals a rich line spectrum with numerous lines of molecular hydrogen, interstellar species and possible accretion disk or photospheric absorption features. The spectrum reveals a downturn in the continuum shortward of 1000Å. The spectrum does not exhibit any evidence of emission lines from the source, the only emission lines are from air glow and heliocoronal (e.g. sharp emissions lines from C iii (around 977Å) and the Ovi doublet). In figure 2, the FUSE spectrum of BV Cen has a variety of interstellar and stellar features but has a continuum shape distinctly different from UU Aql. The broad C iii absorption feature around 1175Å is definitely from the source, and the C iii (around 977Å) and the Ovi doublet broad emission features could also possibly be associated with one of the FUV components in BV Cen. All the other sharp emission features are either heliocoronal or geocoronal in origin. – 5 – In figure 3, CH UMa reveals numerous absorption lines due to highly ionized and singly ionized metals. It has a continuum energy distribution similar to UU Aql. There are def- initely some broad emission lines from the source itself. The most prominent one is the Ovi doublet (the right component being strongly attenuated by molecular hydrogen sharp absorption lines), C iii (both around 977Å and 1175Å) which also seems to be in emission and a tentative identification of N iv emission in the short wavelengths. The source is also contaminated with sharp emission lines due to air glow. Here too the N i & N ii are geo- coronal in origin, and the sharp peaks on top of the broad C iii (977Å) and Ovi emissions are heliocoronal in origin. We note that ISM molecular hydrogen absorption is affecting the continuum. Since the wavelength range covered by FUSE overlaps with HST/STIS or IUE in the region of C iii between 1170Å and 1180Å, a much broader FUV wavelength coverage is afforded by combining the spectra when the flux levels of the two spectra in the wavelength overlap region match closely enough. We found archival IUE spectra matching the FUSE spectra of two of the three systems (CH UMa and BV Cen) but unfortunately no HST spectra exist for the three systems. The FUSE + IUE combination of spectra rests on the assumptions that (1) differences between the two spectra in orbital phase and (2) in time after the last outburst can be ignored. Given the long exposure times of the FUSE and IUE spectra, the questionable reliability of the orbital ephemerides, including UU Aql’s and (3) the limited S/N of the quiescent spectra, the influence of phase-dependent variations is not considered. However, the time since the last outburst as well as the brightness state of the system at the times of the FUSE and IUE observations for the three systems is considered in detail using AAVSO archival light curve data. For UU Aql, the FUSE spectrum was obtained approximately 50 days after its last major dwarf nova outburst, however, the IUE spectrum was obtained during the transition to a brightening that appeared not to be a major outburst. Therefore, we have excluded the IUE spectrum from use in combination with the FUSE data for UU Aql. For BV Cen both the FUSE and IUE spectra were obtained during quiescence but the FUSE spectrum was acquired approximately 159 days after the last outburst while the IUE spectrum was taken roughly 50 days after the last outburst. Thus, the FUSE spectrum probably recorded a greater degree of white dwarf cooling than the IUE spectrum obtained closer to the last outburst. Likewise for CH UMa, both the FUSE and IUE spectra were obtained during quiescence but the FUSE spectrum appears to have been obtained roughly 125 days after the last outburst while the IUE spectra were acquired about 83 days after the last outburst. Since the e-folding times for white dwarf cooling in both of these systems following the outburst heating episode is typically shorter than the above two post-outburst intervals, it is probably acceptable to combine the FUSE and IUE spectra for BV Cen and – 6 – CH UMa. In Table 3 we present the observing log of IUE observations which matched the FUSE flux level in the wavelength overlap region of these two systems. The entries by column are (1) the target name, (2) the observation ID, (3) aperture, (4) dispersion mode, (5) date of the observation, (6) time of mid-exposure, and (6) the exposure time in seconds. Thus, our analysis was carried out first for the FUSE spectra of the three systems and then separately for the combined FUSE plus IUE data of BV Cen and CH UMa. 3. Multi-Component Synthetic Spectral Fitting Our data analysis and modeling involves the full suite of multi-component (accretion disk, white dwarf photosphere, accretion belt) synthetic spectral codes, which we have uti- lized in our spectral fitting of FUSE and IUE data. Based upon our expectation that the accreting white dwarf is an important source of FUV flux in these systems during quies- cence, we carried out a high gravity photosphere synthetic spectral analysis first. The model atmosphere (TLUSTY200; Hubeny (1988)), and spectrum synthesis codes (SYNSPEC48 and ROTIN4 Hubeny & Lanz (1995)) and details of our χ2ν (χ 2 per degree of freedom) minimization fitting procedures are discussed in detail in Sion et al. (1995) and will not be repeated here. To estimate physical parameters, we generally took the white dwarf photo- spheric temperature Teff , log g, and rotational velocity vrot and chemical abundances as free parameters. We normalize our fits to 1 solar radius and 1 kiloparsec such that the distance of a source is computed from d = 1000(pc) ∗ (Rwd/R⊙)/ S, or equivalently the scale factor , is the factor by which the theoretical flux (integrated over the entire wavelength range) has to be multiplied to equal the observed (integrated) flux. The grid of WD models extended over the following range of parameters: log g = 7.0, 7.5, 8.0, 8.5, 9.0; Teff/1000 (K) = 22, 23, ..., 75; Si = 0.1, 0.2, 0.5, 1.0, 2.0, 5.0; C = 0.1, 0.2, 0.5, 1.0, 2.0, 5.0; and vrot sin i (km s −1) = 100, 200, 400, 600, 800. For the synthetic accretion disk models, we used the latest accretion disk models from the optically thick disk model grid of Wade & Hubeny (1998). The range of disk model parameters varies as follows: WD mass (in solar units) value of 0.35, 0.55, 0.80, 1.03, and 1.21; orbital inclination (in degrees) of 18, 41, 60, 75 and 81. The accretion rate ranges from 10−10.5M⊙yr −1 to 10−8.0M⊙yr −1 by increments of 0.5 in logṀ . For each dwarf nova, we adopted the following procedure. First, we masked out all of – 7 – the obvious emission features and artifacts in both the FUSE and IUE spectra of each object. Second, we carried out synthetic spectral fits using and/or combining model components in this order: a white dwarf model alone, accretion disk model alone, combination white dwarf plus accretion disk model, and two-temperature white dwarf model (the latter to simulate a hotter equatorial region as well as a cooler photosphere at higher latitudes). For accretion disk fits, we ”fine-tuned” the derived accretion rate of the best-fitting disk model by changing the accretion rate in increments of 0.1 over the range 0.1 to 10, on the assumption that the disk fluxes scale linearly over that range. In Table 4, we indicate where we masked any strong emission features, artifacts, or negative fluxes in the FUSE and IUE spectra of each object. 4. Synthetic Spectral Fitting Results The noise level of the FUSE spectra precludes the opportunity to extract reliable pa- rameters for the accreting white dwarfs in these three systems. This is especially true for deriving rotational velocities which rely on well-resolved, strong absorption lines arising in the photosphere. The rotational velocity is also affected by underlying emission filling of absorption features and by the chemical abundances one uses. With these caveats in mind, we proceeded to apply our grid of WD photosphere models (keeping the chemical abundance fixed at solar) and accretion disk synthetic spectra. For any dwarf nova in quiescence, a single temperature white dwarf model should be a reasonable first approximation as the source of the FUV flux. For UU Aql, we adopted two possible distances, 150 pc and 350 pc and carried out detailed fits for both values. For a distance of 350 pc, the best-fit WD model to the FUSE spectrum gave Teff = 27, 000K, Rwd/R⊙ = 1.13× 10−2 and a χ2 = 0.963. This best-fitting model is shown in figure 4. The continuum of the model gives a fair representation of the observed continuum down to about 1060Å but there is a shortfall of model flux relative to the data at wavelengths shorter than 1030Å. For the same distance, an accretion disk alone yielded a best fit with χ2 = 1.14, an accretion rate of 5 × 10−11M⊙/yr and inclination i = 41 degrees, Mwd = 0.8M⊙. This disk fit is shown in figure 5. The model disk continuum, unlike the WD, fails to match the flux level of the data between about 1090Å and 1180Å and the solar abundance accretion disk model fails to provide a sharp absorption features. A combination white dwarf plus accretion disk yielded a modest improvement with χ2 = 0.71, Ṁ = 1.6× 10−11M⊙/yr, i = 41 degrees and Mwd = 0.8M⊙. Finally we tried two temperature (WD + belt) fits. The best fit two temperature WD yields χ2 = 0.74 with the – 8 – cooler white dwarf portion ( Teff = 24, 000K) giving 57% of the flux and the hotter belt (Tbelt = 33, 000K) providing 43% (figure 6). In general, for a distance d = 150 pc, the model fits to UU Aql’s FUSE spectrum are worse than for 350 pc. Applying single temperature white dwarf fits, we obtained Teff = 17, 000K, Rwd/R⊙ = 1.06 × 10−2 and a χ2 = 1.13. An accretion disk alone yielded a best fit with χ2 = 1.23, an accretion rate of 1.3 × 10−11M⊙/yr and inclination i = 41 degrees and Mwd = 0.8M⊙. A combination white dwarf plus accretion disk yielded a modest improvement with χ2 = 1.13, Ṁ = 1.6× 10−11M⊙/yr, i = 41 degrees and Mwd = 0.8M⊙. A two temperature white dwarf (WD + belt) gives a best-fit with a χ2 = 1.02, with a 17,000K white dwarf providing 77% of the FUV flux and the belt giving 23% of the flux. In view of the much better agreement of the models with the observations of UU Aql for our assumed distance of 350 pc than for a distance of 150 pc, the closer distance can be ruled out. For CH UMa, the FUSE spectrum is very noisy and underexposed. We took into account our adopted distance of 300 pc in the model fitting, we fixed the WD mass at Mwd = 1.2M⊙ (Log g = 9.0, with a radius of ≈ 4, 000km), and fixed the disk inclination at the published value of 18 degrees. A single temperature white dwarf fit to CH UMa had χ2 = 0.227, and yielded a best fit Teff = 29, 000K, and a distance of 310pc. This model however did not fit very well in the shorter wavelengths (<1020Å). A lowest χ2 fit for this same WD mass was obtained for T=40,000K, with χ2 = 0.208, but it yielded a distance of 600pc, twice the adopted estimate of the distance. Since the Ritter catalog give a mass of 1.95M⊙ (well above the Chandraskhar mass limit for a WD), we decided to try a larger mass with a correspondingly smaller radius which lead to a smaller emitting surface area and therefore a shorter distance. We assumed M = 1.38M⊙ (logg = 9.5, with a radius of ≈ 2, 000km) and found that the best fit was for T = 40, 000K. This model yielded a distance of 307pc with χ2 = 0.199. Since the absorption features around 1120Å - 1150Å are not pronounced in the observed spectrum, we decided to increase the rotational velocity to improve the fit of this model. However, a better result was obtained by simply reducing the abundances of Si and C to 0.01 their solar value. This low Si and C model had a χ2 = 0.184 and a distance of 314pc. This model is shown in Figure 7. Because of the low S/N of the spectrum the assessment of the error on the temperature estimate is of the order of 5,000K, namely Twd = 40, 000± 5, 000K. Though the WD fit yielded the lowest value of χ2, we tried accretion disk fits alone to CH UMa’s FUSE data. The best fit gave a χ2 = 0.213, an accretion rate of 5× 10−12M⊙/yr for Mwd = 1.2M⊙ and i = 18 degrees. Various attempts to fit the combined FUSE + IUE spectra of CH UMa met with limited success as summarized in Table 5. The best combination fit was for an accretion disk plus – 9 – WD with the disk contributing 79% of the FUV flux. However, these fits were less satisfactory than the fits to the FUSE spectrum alone. Because the FUSE spectrum of CH UMa is very noisy, and because of the broad emission lines and the absence of strong absorption lines, it is difficult to assess which model (disk or WD) is the best solution. However, the absence of strong absorption lines would favor the accretion disk model because of Keplerian broadening. Also because of the poor quality of the spectrum, composite models (WD+accretion disk, two-temperature WD) did not improve the fit. In view of all of the above, we cannot be confident that we have determined the temperature of the white dwarf in CH UMa. Next we analyzed the combined FUSE + IUE spectra of CH UMa. The best white dwarf-only fit yielded Teff = 31, 000K, log g = 9 a χ 2 = 3.02 and Rwd/R⊙ = 6.02 × 10−3. For accretion disk models only, the best fit occurred with a χ2 = 2.02, corresponding to Ṁ = 3× 10−9M⊙/yr. This disk fit is a modest improvement of the fit to CH UMa since the χ2 value was lowered to 2.02. The best-fitting white dwarf + accretion disk model resulted in only a modest, statistically insignificant improvement. The χ2 value was lowered to 1.87, the accretion rate was 6.4× 10−11M⊙/yr with the WD Teff = 22, 000K and the scale factor yielding a white dwarf radius of 5.84 × 10−3R⊙. In this composite disk plus white dwarf fit, the WD contributes 21% of the FUV flux while the accretion disk contributes 79% of the FUV flux. We also tried a two-temperature WD solutions with the best-fitting model consisting of a 26,000K WD providing 77% of the flux and a hot accretion belt/ring with Tbelt = 50, 000K giving 23% of the UV flux. However, this two-temperature fit was no better than the WD + accretion disk fit. For BV Cen we dereddened the spectrum assuming E(B-V)=0.10, and first tried single temperature white dwarf fits with a white dwarf mass = 0.83M⊙, and used solar chemical abundances. We found the best-fitting white dwarf model to have Teff = 40, 000± 1000K, log g = 8.3, V sin i = 500 km/s ±100 km/s. This fit yielded χ2 = 0.2701 and a distance d=435pc (see figure 8). We note here that the WD solution fits the following absorption features quite accurately: C ii (1066Å), S iv (1073Å), N ii (1084Å), S iiv (1122.5Å, 1128.3Å), and S iiii (1140Å- 1146Å). For an optically thick, steady state accretion disk alone, we chose Mwd = 0.80M⊙, i = 60 degrees. The best fit we obtained has a mass accretion rate of 10−8.5M⊙yr −1, too large for dwarf nova quiescence. Moreover, the longer wavelength part of the spectrum is rather flat and unable to fit the absorption features around 1120Å and 1130Å. This fit, with i=60deg, leads to a distance of 1255pc. In order to fit these absorption features, one needs to assume an inclination of 18 degrees, inconsistent with the known inclination of the system (62 degrees). Such a model leads to a distance of more than 2000pc, again inconsistent – 10 – with all estimates of the system distance (500pc). Therefore, based on the parameters of the system, the disk solution is completely inconsistent. We tried composite model fits, but they also led to very poor results. We also combined BV Cen’s FUSE + IUE spectra but they led to very poor fits, much worse than the fits to the FUSE spectrum alone. Since the WD solution is consistent with the parameters of the system and fits the absorption features of the spectra, it is clear that the favored solution of the FUSE spectrum of BV Cen is Twd = 40, 000± 2, 000K. 5. Discussion Our principal objective of determining the surface temperatures of the white dwarfs in these three dwarf novae during quiescence has met with mixed results. From synthetic spectral fits to the FUSE spectra of the three long period dwarf novae, UU Aql, CH UMa and BV Cen, and the FUSE + IUE archival SWP spectra of CH UMa we have presented preliminary evidence that during quiescence, their accreting white dwarfs all have surface temperatures hotter than 20,000K. Unfortunately, all three temperatures have considerable uncertainty due to the low S/N of the FUSE spectra and IUE spectra as well as the difficulty of disentangling the flux contribution of the second component of FUV flux or ”accretion disk” during quiescence. Of the three systems in this study, we regard our estimate of the Teff of the WD in UU Aql to be the most reliable since that system appears to be dominated in the FUV by the white dwarf flux. For UU Aql, we used both assumed distances, 150 pc and 350 pc, and tried the composite fits (WD + disk) to UU Aql’s FUSE spectrum. However, the quality of the fits for WD-only, disk-only, WD + disk, and WD + ”belt” were all roughly comparable and thus it is difficult to distinguish the best-fit case. For both distances, statistically insignificant improvements in the fits result when a white dwarf and accretion disk are combined or a two-temperature WD (WD + belt) is applied to the FUSE data. The results in Table 5 illustrate the difficulty. Qualitatively, the composite fits involving combinations of white dwarf plus accretion disk or accretion belt models look marginally more reasonable than the fits that involve single component (WD or Disk). It appears that for both distances, the white dwarf component is the dominant source of FUV flux and that the Teff of the WD is probably between 17,000K and 27,000K (say 22,000±5,000K). For BV Cen, the WD model fit to the FUSE spectrum gave the best result, both in fitting actual features of the observed spectrum and in leading to consistent values of the system parameters. This was not the case for the disk model and the composite models. From the best WD model fit we obtained that the WD of BV Cen must have a temperature – 11 – of about 40,000K. For CH UMa, the broad and dominant emission lines together with the poor S/N (and possible detector noise at very short wavelengths) of the FUSE and IUE spectra precluded the opportunity to obtain conclusive results. However, the results for the FUSE spectrum were of a much higher quality than for the FUSE + IUE combined spectrum. Therefore we adopt the FUSE results for CH UMa with the possibility that the WD temperature could be as high as 40,000K. In Table 6, we list the dwarf novae above the period gap whose white dwarfs have surface temperature determinations. In Column (1) we give the system name, column (2) the orbital period; column (3) the surface temperature and column (4) the temperature reference. As seen in Table 6, there is now a sample of eight long period dwarf novae of which roughly seven have relatively secure white dwarf temperatures obtained during quiescence. In the case of BV Cen, the inclination is expected to be high. Thus, it is plausible to expect that the quiescent accretion disk may be blocking the direct radiation from the accreting white dwarf. The relatively poor FUSE spectral quality of CH UMa underscores the need for re- observation with longer exposure times. However, analysis will remain hindered until more reliable information on white dwarf masses and distances becomes available. Until then, the conclusions in this work must be regarded as preliminary. 6. Acknowledgements We thank an anonymous referee for helpful comments and corrections. PG wishes to thank the Space Telescope Science Institute for its kind hospitality. This work was supported by NSF grant AST05-07514 and NASA grants NAG5-12067 and NNG04GE78 to Villanova University. This research was based on observations made with the NASA-CNES-CSA Far Ultraviolet Spectroscopic Explorer. FUSE is operated for NASA by the Johns Hopkins University under NASA contract NAS5-32985. – 12 – REFERENCES Araujo-Betancor, S., et al. 2003, ApJ, 583, 437 Bruch, A., & Engel, A.1994, A&AS, 68, 41 Friend, M.T., Martin, J.S., Connon-Smith, R., & Jones, D.H.P. 1990, MNRAS, 246, 654 Gilliland, R. 1982, ApJ, 263, 302 Godon, P., Seward, L., Sion, E.M., & Szkody, P. 2006, AJ, 131, 2634 Harrison, T.E., Johnson, J.J., McArthur, B.E., Benedict, G.F., Szkody, P., Howell, S.B., Gelino, D.M. 2004, AJ, 127, 460 Hartley, L.E., Long, K.S., Froning, C.S., & Drew, J.E. 2005, ApJ, 623, 425 Hubeny, I. 1988, Comput. Phys. Comm., 52, 103 Hubeny, I., & Lanz, T. 1995, ApJ, 439, 875 laDous, C. 1991, A&A, 252, 100 Ritter, H., & Kolb, U. 2003, A&A, 404, 301 Sion, E.M. 1991, AJ, 102, 295 Sion, E.M. 1999, PASP, 111, 532 Sion, E.M., Cheng, F., Godon, P., Urban, J.A., & Szkody, P. 2004, AJ, 128, 1834 Sion, E.M., Cheng, F., Long, K.S., Szkody, P., Huang, M., Gilliland, R., Hubeny, I. 1995, ApJ, 439, 957 Sion, E.M., Cheng, F., Szkody, P., Sparks, W.M., Gänsicke, B.T., Huang, M., & Mattei, J. 1998, ApJ, 496, 449 Sion, E.M., Szkody, P., Gänsicke, B.T., Cheng, F., LaDous, C., Hassall, B. 2001, ApJ, 555, Szkody, P. 1987, ApJS, 63, 685 Townsley, D., & Bildsten, L. 2002, ApJ, 565, 35 Townsley, D., & Bildsten, L. 2003, ApJ, 596, L227 – 13 – Urban, J., & Sion, E.M. 2006, ApJ, 642, 1029 Verbunt, F. 1987, A&A, 71, 339 Wade, R.A., & Hubeny, I. 1998, ApJ, 509, 350 Warner, B. 1995, Cataclysmic Variable Stars (Cambridge University Press) This preprint was prepared with the AAS LATEX macros v5.2. – 14 – Table 1: Dwarf Nova Parameters System Porb trec Vq Vo Sec. i Mwd EB−V Distance Name (days) (days) (deg) (M⊙) (pc) BV Cen 0.610108 150 12.6 10.5 G5-8V 62± 5 0.83± 0.1 0.10 500 CH UMa 0.343 204 15.3 10.7 K4-M0V 21± 4 300 UU Aql 0.14049 71 16.0 11.0 M2-4V 150-350 Table 2. FUSE Observations Target Data ID Aperture Time of Observation texp λcentered S/N(0.1Å) UU Aql C1100301000 LWRS 30x30 2004-05-16 13h:48m:00s 16,121s 1035Å 5.15 BV Cen D1450301000 LWRS 30x30 2003-04-13 20h:26m:00s 26,545s 1035Å 5.9 CH UMa D1450201000 LWRS 30x30 2003-04-02 22h:00m:00s 17,311s 1035Å 3.6 Table 3. IUE Observations Target SWP No. Aperture Dispersion Date Time 0f Mid-Exposure texp(s) BV Cen 26623 Lg Low 08/16/85 17:17:00 14,400 CH UMa 56270 Lg Low 12/06/95 12:15:49 13,800 – 15 – Table 4. Masked Spectral Regions Target Emission Features UU Aql for λ < 1050Å if Fλ > 6× 10−15ergs s−1cm−2Å−1 for 1050 Å < λ < 1185Å if Fλ > 1.8× 10−14ergs s−1cm−2Å−1 975-985Å CH UMa for λ < 957Å if Fλ > 1× 10−14ergs s−1cm−2Å−1 970-980, 987-994, 1023-1042, 1071-1090, 1107-1136, 1167-1180, 1200-1260, 1285-1315, 1380-420, 1532-1560, 1630-1650, 1846-1940Å BV Cen for λ < 1185Å if Fλ > 2.5× 10−14ergs s−1cm−2Å−1 1190-1220, 1320-1340, 1375-1415, 1530-1570, 1625-1650, 1835-1880Å Table 5. UU Aql, BV Cen, and CH UMa Fitting Results System Spectrum d χ2 Twd Tbelt Ṁ % Flux Fig (pc) (103K) (103K) (M⊙/yr) (Contribution) UU Aql FUSE 350 0.96 27 - - 100(WD) 4 FUSE 150 1.13 17 - - 100(WD) - FUSE 350 1.13 - - 5× 10−11 100(Disk) 5 FUSE 150 1.24 - - 1× 10−11 100(Disk) - FUSE 350 0.714 24 - 2× 10−11 73(WD)/27(Disk) - FUSE 150 1.13 17 - 3× 10−11 60(WD)/40(Disk) - FUSE 350 0.738 24 33 - 57(WD)/43(Belt) 6 FUSE 150 1.021 16 29 - 77(WD)/23(Belt) - CH UMa FUSE 314 0.184 40 - - 100(WD) 7 FUSE 313 0.213 - - 5× 10−12 100(Disk) - FUSE+IUE 300 3.02 31 - - 100(WD) - FUSE+IUE 300 2.02 - - 3× 10−9 100(Disk) - FUSE+IUE 300 1.87 22 - 6.4× 10−11 21(WD)/79(Disk) - FUSE+IUE 300 2.00 26 50 - 77(WD)/23(Belt) - BV Cen FUSE 435 0.27 40 - - 100(WD) 8 FUSE 1255 0.27 - - 3× 10−9 100(Disk) - – 16 – Table 6: Surface Temperatures of White Dwarfs in Dwarf Novae above the Period Gap System Name Period Teff References (min) (Kelvin) BV Cen 878.6 40,000: This paper RU Peg 539.4 49,000 Sion et al (2004) Z Cam 417.4 57,000 Hartley et al. (2005) RX And 302.2 34,000 Sion et al. (2001) SS Aur 263.2 31,000 Sion et al (2004) U Gem 254.7 31,000 Sion et al. (1998) WW Ceti 253.1 26,000 Godon et al. (2006) UU Aql 202.3 27,000: This paper – 17 – Figure Captions Fig. 1.— The flux Fλ (ergs/cm 2/s/Å) versus wavelength (Å) FUSE spectrum of the U Gem- type dwarf nova UU Aql during quiescence. The identifications of the strongest neutral and ionized line features as well as rotational-vibrational transitions of molecular hydrogen are marked with vertical tick marks. The molecular hydrogen absorption are modestly affecting the continuum, slicing it at almost regular wavelength intervals. We identify the most prominent molecular hydrogen absorption lines by their band (Werner or Lyman), upper vibrational level (1-16), and rotational transition (R,P, or Q) with lower rotational state (J=1,2,3). Air glow line features are indicated with the earth symbol. The C iii (977Å) and Ovi doublet sharp emissions are due to heliocoronal emission often present in some of the FUSE channels when the target (i.e. UU Aql) is faint. Fig. 2.— The flux Fλ (ergs/cm 2/s/Å) versus wavelength (Å) FUSE spectrum of the U Gem- type dwarf nova BV Cen during quiescence. The identifications of the strongest neutral and ionized line features as well as rotational-vibrational transitions of molecular hydrogen are marked with vertical tick marks. Air glow line features are indicated with the earth symbol. The O i emissions are also due to air glow. The broad C iii (977Å) and Ovi (doublet) features might be real broad emissions from the source, as these correspond to the left wing of the Lyman γ & β (respectively) and not much flux is expected there. The N i and N ii sharp emission features are geocoronal. Fig. 3.— The flux Fλ (ergs/cm2/s/Å) versus wavelength (Å) FUSE spectrum of the U Gem- type dwarf nova CH UMa during quiescence. The identifications of the strongest neutral and ionized line features as well as rotational-vibrational transitions of molecular hydrogen are marked with vertical tick marks. Several terrestrial line features are indicated with the earth symbol. There are some broad emission features associated with the source: we identify here the Ovi doublet, C iii (at 977Å and 1175Å). There could be some N iv emission in the short wavelengths (between 920Å and 925Å), but the increase of flux in the very short wavelengths suggests that most the flux at λ < 930Å could be higher orders of the hydrogen Lyman series associated with air glow contamination and possibly some detector noise. Fig. 4.— A single temperature WD fit to the FUSE spectrum of UU Aql for a distance of 350 pc. The best-fit yielded Teff = 27, 000K ±3000K, log g = 9. The white dwarf contributes 100% of the FUV flux. Fig. 5.— An accretion disk-only fit to the FUSE spectrum of UU Aql for a distance of 350 pc. The best-fit corresponded to Mwd = 0.8M⊙, i = 41 degrees, Ṁ = 5× 10−11M⊙/yr. The accretion disk contributes 100 % of the FUV flux. – 18 – Fig. 6.— A WD + belt fit to the FUSE spectrum of UU Aql for a distance of 350 pc. The dotted line is the white dwarf flux component, the dashed line is the belt flux component and the solid line is the combined two-temperature fit. This best fit two temperature WD yields χ2 = 0.74 with the cooler white dwarf portion ( Teff = 24, 000K) giving 57% of the flux and the hotter belt (Tbelt = 33, 000K) providing 43%. Fig. 7.— A single temperature WD fit to the FUSE spectrum of CH UMa. The best- fit yielded Teff = 40, 000 ± 1, 000K, log g = 9.5 and a projected rotational velocity of Vrot sin i = 200km/s. The portions of the spectrum that have been masked are shown in blue and include emission lines (either intrinsic to the source or due to air glow) and ISM molecular hydrogen absorption lines (which have been matched using a simple ISM model to identify the exact location of the ISM lines). Therefore the fit is between the model and the red portions of the observed FUSE spectrum. The absence of strong absorption features in the lower panel were modeled assuming low Si and C abundances and led to a lower χ2 than the same model with solar C and Si abundances but with a high projected rotational velocity (≈ 1, 000km/s). Fig. 8.— A single temperature WD fit to the FUSE spectrum of BV Cen. The best-fit yielded Teff = 40, 000K±1000K, a projected rotational velocity of 500km/s, and d=435pc, assuming M = 0.83M⊙. the FUSE plus IUE spectrum of UU – 19 – – 20 – – 21 – – 22 – – 23 – – 24 – – 25 – – 26 – Introduction Observations and Data Reduction Multi-Component Synthetic Spectral Fitting Synthetic Spectral Fitting Results Discussion Acknowledgements

0704.1134

Experimental and theoretical study of light scattering by individual mature red blood cells by use of scanning flow cytometry and discrete dipole approximation

Microsoft Word - RBC_preprint.doc Experimental and theoretical study of light scattering by individual mature red blood cells by use of scanning flow cytometry and discrete dipole approximation Maxim A. Yurkin, Konstantin A. Semyanov, Peter A. Tarasov, Andrei V. Chernyshev, Alfons G. Hoekstra, and Valeri P. Maltsev M. A. Yurkin, K. A. Semyanov, P. A. Tarasov, A. V. Chernyshev, and V. P. Maltsev are with the Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of Sciences, Institutskaya 3, Novosibirsk 630090 Russia A. G. Hoekstra is with the Faculty of Science, Section Computational Science, of the University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands M. A. Yurkin is also with the Faculty of Science, Section Computational Science, of the University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands A. V. Chernyshev and V. P. Maltsev are also with Novosibirsk State University, Pirogova 2, Novosibirsk 630090 Russia V. P. Maltsev’s e-mail is [email protected] Abstract Elastic light scattering by mature red blood cells (RBCs) was theoretically and experimentally analyzed with the discrete dipole approximation (DDA) and the scanning flow cytometry (SFC), respectively. SFC permits measurement of angular dependence of light-scattering intensity (indicatrix) of single particles. A mature RBC is modeled as a biconcave disk in DDA simulations of light scattering. We have studied the effect of RBC orientation related to the direction of the incident light upon the indicatrix. Numerical calculations of indicatrices for several aspect ratios and volumes of RBC have been carried out. Comparison of the simulated indicatrices and indicatrices measured by SFC showed good agreement, validating the biconcave disk model for a mature RBC. We simulated the light-scattering output signals from the SFC with the DDA for RBCs modeled as a disk-sphere and as an oblate spheroid. The biconcave disk, the disk-sphere, and the oblate spheroid models have been compared for two orientations, i.e. face-on and rim-on incidence. Only the oblate spheroid model for rim-on incidence gives results similar to the rigorous biconcave disk model. Keywords: red blood cell, discrete dipole approximation, non-spherical particle light scattering, scanning flow cytometry OCIS code: 170.0170, 170.1530, 170.1470, 290.5850, 120.5820 Introduction The scattering of electromagnetic waves by dielectric particles is a problem of great significance for a variety of applications ranging from particle sizing and remote sensing to radar meteorology and biological sciences.1,2 In the area of medical diagnostics, understanding how a laser beam interacts with blood suspensions or a whole-blood medium is of paramount importance in quantifying the inspection process in many commercial devices and experimental setups that are used widely for in vivo or in vitro blood measurements.3 These measurements are focused mainly on electromagnetic scattering properties of red blood cells (RBCs), the type of cell that are most numerous in the blood. Usually, RBCs measure from 6.6 to 7.5 µm in diameter, however, cells with a diameter greater than 9 µm (macrocytes) or less than 6 µm (microcytes) have been observed. These cells are nonnucleated (among vertebrates, only the red cells of mammalians lack nuclei) biconcave discs that are surrounded by thin, elastic membranes and filled with hemoglobin. They are soft, flexible, and elastic and therefore move easily through the narrow blood capillaries. The primary function of these cells is to carry oxygen from the lungs to the body cells.3 The light scattering properties of blood are based on a solution of the single- electromagnetic-scattering problem for a RBC. Moreover RBCs can play an important role in verification of solutions of the direct light-scattering problem for nonspherical particles because of their simple internal structure and stable biconcave discoid shape. Most of the relevant work solves the single-electromagnetic-scattering problem by a RBC analytically either basically by use of Mie4, Fraunhofer5 and anomalous diffraction theories5,6 or numerically with the aid of the T-matrix approach7 by treating a real RBC as a volume equivalent spherical4, spheroidal,7 or ellipsoidal5,8 dielectric particle. In these cases the assumptions can be considered exact only because of the special experimental conditions when RBCs become spheroidal or ellipsoidal. In contrast, only a few papers have dealt with light scattering by a real nondeformed RBC.9,10 Shvalov et al.10 made use of the single- and double-wave Wentzel-Kramers-Brillouin approximations and were able to reproduce analytically the experimental results for forward scattering angles in the range of 15° to 35°. Mazeron and Muller,11 with the aid of a physical optics approximation, presented small-angle forward-scattering patterns by a RBC whose axisymmetric geometry was obtained through complete rotation of a Cassini curve. At present RBCs form the border of the size range where effective methods of light scattering simulation can be applied. Tsinopoulos and Polyzos12 studied scattering by nondeformed, average-sized RBCs illuminated by a He-Ne laser beam. They examined various orientations of the RBC with respect to the direction of the incident light and computed scattering patterns in the forward, sideways, and backward directions. They used a boundary-element method appropriately combined with fast-Fourier-transform (FFT) algorithms. Another method to simulate light scattering by arbitrary shaped particles is the discrete dipole approximation (DDA).13 The latest improvements in the DDA14 and its implementation on parallel supercomputers15 permit simulation of light scattering by particles with the sizes of RBCs. The next generation of flow cytometers, the scanning flow cytometer (SFC),16 permits measurement of the angular dependence of the light-scattering intensity of single particles at a speed of O(102) particles/s. In the research reported in this manuscript, light scattering by mature RBCs was theoretically and experimentally analyzed with the DDA and a SFC, respectively. In the numerical simulations we modeled a mature RBC as a biconcave disk.17 The effect of RBC orientation relative to the direction of the incident light was studied. We carried out numerical calculations of indicatrices for several axis ratios and volumes of RBC. Comparison of the simulated indicatrices with those measured with the SFC showed a good agreement, validating the biconcave disk model for a mature RBC. We simulated light- scattering output signals from a SFC with DDA for a RBC modeled as a disk-sphere (a symmetric discoid layer cut out of a sphere) and an oblate spheroid and compared those results with results of biconcave disk model for two orientations, face-on and rim-on incidence. Theory Optical Model of the RBC A mature red blood cell (RBC) can be modeled as a biconcave discoid. A RBC is composed of hemoglobin (32%), water (65%), and membrane tissue (3%) and does not contain any nucleus.18 The shape of the RBC was described by Fung et al.17: ( )422 8579.05262.11583.0165.0)( xxxdxT −+−= , (1) where T is a thickness of RBC (along the axis of symmetry), x is a relative radial cylindrical coordinate x = 2ρ /d (−1 ≤ x ≤ 1), ρ is a radial cylindrical coordinate, and d is the diameter of the RBC. In order to vary the ratio of maximum thickness and diameter independently on the diameter of a RBC we have rewritten Eq. (1) in the following form: ( )422 8579.05262.11583.012)( xxxdxT −+−= ε , (2) where ε = Tmax/d is an aspect ratio of maximum thickness and diameter. The RBC shape is shown schematically in Fig. 1. Discrete Dipole Approximation The DDA permits simulation of light scattering by a particle modeled as a finite array of polarizable points. The points acquire dipole moments in response to the local electric field. The dipoles interact by means of their electric fields,19,20 and the DDA is also sometimes referred to as the coupled dipole approximation.21 The theoretical base for the DDA, including Fig. 1. Shape of a mature RBC and its orientation with respect to the incident radiation. xyz is a laboratory reference frame and x′y′z′ is a reference frame tied to the RBC (z′ is the axis of symmetry); inc and sca are propagation vectors for incident and scattering radiation respectively. radiative reaction corrections, was summarized by Draine and Flatau.13 The performance of the DDA was significantly improved by introducing complex-conjugate gradient methods for solving systems of linear equations and a FFT algorithm for matrix-vector multiplication (summarized by Draine and Flatau13). Hoekstra et al.15 parallelized the DDA and showed that the algorithm runs efficiently on distributed memory computers, provided that the number of dipoles per processor is large enough. They run DDA simulations with O(107) dipoles, limited only by the available amount of memory.15 The size of a sub-volume that relates to the number of dipoles that form a particle must be small enough to ensure that response to an electromagnetic field is the response of an ideal induced dipole. The size should be in range λ/20 < d < λ/10.13,22 In some cases we can even use d ≥ λ/10. Comparison of light-scattering calculation performed with the DDA approximation and with exact Mie theory for a homogeneous sphere is a traditional way to determine the accuracy of the DDA. Using the parallel DDA code produced by Hoekstra et al.,15 we compared DDA simulations with Mie calculations for a sphere with a diameter of 4.56 μm and a relative refractive index of 1.10; the wavelength was 0.6328 μm. The relative errors in DDA simulations as a function of the scattering angle are presented in Fig. 2 for three different discretization sizes: λ/5, λ/10, and λ/20. The relative error for discretization size of λ/5 is ∼10% for scattering angles smaller then 50°. This degree of precision is good enough for comparison with experimental curves for which the scattered intensity was measured in a range of scattering angles from 10° to 50°. In the sequel of the paper, the size of the dipoles was varied in the range λ/11–λ/8 for different sizes of RBCs because of requirements for lattice regularity in the current implementation of the parallel FFT algorithm.15 Experimental Equipment and Procedures The experimental part of this study was carried out with a SFC that permits measurement of the angular dependency of light-scattering intensity (indicatrix) in a region ranging from 5° to 100°. The design and basic principles of the SFC were described in detail elsewhere.16 The laser beam of the SFC is directed coaxially with the hydrodynamically focused flow which 0 30 60 90 120 150 180 Scattering angle θ, deg a = λ/5 a = λ/10 a = λ/20 Fig. 2. Relative errors of DDA simulations for several discretization sizes. The DDA simulations were compared with the exact Mie solution for a sphere with diameter d = 4.56 μm and relative refractive index m = 1.10. carries analyzed cells. Most disklike-shaped RBCs attain the testing zone of the SFC in rim- on orientation relative to the direction of the incident laser beam. The current set-up of the SFC provides a measurement of the following combination of Mueller matrix elements23: ϕθϕθϕθ 1411s )],(),([d)( SSI , (3) where Is(θ ) is the output signal of the SFC and θ and ϕ are the polar and azimuthal angles, respectively. After integration over azimuthal angle ϕ the second term in Eq. (3) vanishes, because of the axisymmetry of RBCs (see Eq. (A5) of Appendix A for details; we also obtained the same result numerically for all simulations). Therefore, the SFC output signal will be proportional to S11 integrated over the azimuthal angle. To compare the experimental and theoretical light scattering from RBCs we used the DDA for calculation of the indicatrices. Our SFC setup allowed reliable measurements in the angular range 10°–50° (because of the operational range of the analog-digital converter). A sample containing approximately 106 cells/ml was prepared from fresh blood with buffered saline used for dilution. We used the SFC to continuously measure 3000 indicatrices of RBCs. Each of them was compared with each of the calculated theoretical indicatrices (for details see next section) by a χ 2 test with a weighting function: 10 15 20 25 30 35 40 45 50 Scattering angle θ, deg β = 90° β = 70° β = 50° β = 30° β = 0° Fig. 3. (a) Initial and (b) modified indicatrices calculated by DDA for five orientations of the single biconcave disk relative to the direction of incident light beam. theorexp )()]()([ θθθ where Iexp(θi) is the experimental indicatrix measured at N angular values θi (θ1 = 10°, θN = 50°), Itheor(θi) is the theoretical indicatrix calculated at the same angles, w(θi) is a weighting function, defined as: sin)( 2 iiw πθ , (5) which is intended to suppress experimental errors (a detailed discussion is presented in Ref. 24). An experimentally measured RBC is said to have characteristics of the closest (by χ 2) theoretical indicatrix, if their χ 2 distance is less than a threshold. The threshold was set empirically to 40. The optical model of the RBC was used in calculations with the profile described by Eq. (2). The RBC diameter was varied from 6 to 9 µm. We assume that the imaginary part of the refractive index is negligible at the wavelength λ of 0.6328 µm. The refractive index of 10 15 20 25 30 35 40 45 50 V = 100 μm3 β = 90° d = 8.28 μm, ε = 0.294 d = 7.60 μm, ε = 0.380 d = 6.75 μm, ε = 0.542 Scattering angle θ, deg Fig. 4. Modified indicatrices of biconcave disks with different diameters and fixed volume. 10 15 20 25 30 35 40 45 50 V = 110 μm3, ε = 0.595 V = 100 μm3, ε = 0.541 V = 92 μm3, ε = 0.498 d = 6.75 μm β = 90° Scattering angle θ, deg Fig. 5. Modified indicatrices of biconcave disks with different volumes and fixed diameter. the surrounding medium (saline) was 1.333. The RBC refractive index was fixed at 1.40 that falls into the region of typical variation of a RBC refractive index (1.39 – 1.42)25 and results in a relative refractive index of 1.05. Results and discussions Using the DDA, we studied the effect of the biconcave disk orientation on the indicatrices. Angular orientation β is the angle between the axis of the biconcave disk symmetry and the direction of the incident beam (Fig. 1). We used the profile defined by Eq. (2) with the following typical RBC characteristics17: diameter of 8.28 µm, aspect ratio of 0.323 (resulting in a volume of 110 μm3), and relative refractive index of 1.05. The indicatrices for five orientations are shown in Fig. 3(a). To provide an effective comparison of experimental and theoretical light-scattering data we modified the indicatrices by multiplication by weighting function w(θi) [Eq. (5)]. The multiplication corresponds to the standard Hanning window procedure that strongly reduces the effects of the discontinuities at the beginning and the end of the sampling period of the SFC.24 Moreover, this function resembles the SFC instrument function16 and improves visual inspection of indicatrices, as the logarithmic scale can be replaced by a linear one. The modified indicatrices for the five orientations are shown in Fig. 3(b). The orientation of the biconcave disk modifies the indicatrix substantially. For instance, the mean scattering intensity, in the angular range from 15° to 50°, has a reduced intensity for symmetric orientation (β = 0°), whereas the oscillating structure vanishes for increasing orientation angle. Therefore the mean light-scattering intensity and indicatrix structure are good indicators of a mature RBC orientation in scanning flow cytometry. Practically, it is interesting to study the sensitivity of indicatrix structure to variations in the volume and diameter of the biconcave disk when the orientation corresponds to orthogonal directions of the biconcave disk symmetry axis and incident beam (i.e. for β = 90°). The last requirement is caused by the orientating effect of the hydrofocussing head of the SFC in which nonspherical particles are oriented with a long axis along the flow lines.9 We calculated the indicatrices of RBC with a typical volume of 100 μm3 while we varied the biconcave disk diameter. The results of the DDA simulation are shown in Fig. 4. Variation of the diameter does not change the intensity substantially, whereas the visibility of the oscillating structure, which characterizes the relative difference between maximum and minimum values,16 is reduced with increasing biconcave disk diameter. Additionally, we varied the thickness of the biconcave disk, which resulted in variation of volume with a constant diameter. The RBC volume is the most important hematological index measured with modern automatic hematology analyzers.3 Flow cytometry25 and Coulter’s cell3 are the instrumental solutions utilized for measurement of RBC volume distribution. The modified indicatrices of biconcave discs calculated for three volumes with a typical diameter of 6.75 µm are shown in Fig. 5. The light-scattering intensity does not depend on variation of the biconcave disk volume or thickness. There is the same tendency in the indicatrix structure as for Fig. 4: the visibility of the oscillating structure increases for increasing aspect ratio (of thickness and diameter). Our simulation of light scattering by mature RBCs allows us to conclude that the orthogonal orientation of a biconcave disk relative to the direction of incident beam does not provide sufficient sensitivity of the indicatrix to the RBC characteristics. Coincidence of biconcave disk symmetry axis and direction of incident beam (i.e. β = 0°) gives an advantage in the solution of the inverse light-scattering problem because in that case the oscillating structure of the indicatrix is much more sensitive to variation of RBC characteristics. Unfortunately, as it is shown below, such orientation is improbable in SFC. Table 1. The parameters of RBCs for preliminary calculations.a V, μm3 d, μm 86 92 100 105 110 6.08 0.638 – – – – 6.33 0.565 0.605 – – – 6.51 – 0.556 0.604 – 0.665 6.75 0.466 0.499 0.542 0.569 0.596 6.84 – – 0.521 – 0.573 7.01 – – 0.484 – 0.532 7.60 0.327 0.349 0.380 0.399 0.418 8.28 – – 0.294 0.418 0.323 a Values are aspect ratios for selected diameter and volume. Dashes indicate that indicatrices were not calculated. 10 15 20 25 30 35 40 45 15 20 25 30 35 40 45 50 V = 100 μm3 d = 6.84 μm β = 80° 2 = 30 theory experiment V = 100 μm3 d = 8.28 μm β = 90° 2 = 30 V = 110 μm3 d = 6.75 μm β = 80° 2 = 36 V = 92 μm3 d = 6.75 μm β = 90° 2 = 33 V = 92 μm3 d = 6.33 μm β = 90° 2 = 31 Scattering angle θ, deg Scattering angle θ, deg V = 110 μm3 d = 7.6 μm β = 90° 2=28 Fig. 6. Experimental and theoretical modified indicatrices of individual mature RBCs. Values of χ 2 differences are shown. Practically, the DDA cannot be used to fit experimental indicatrices because the DDA calculations require approximately 10 h for a fixed biconcave disk orientation (the computations where carried out on 8 nodes of a Beowulf computer with nodes running at 750 MHz). To solve this problem we made calculations of indicatrices for several biconcave disks with different diameters and volumes, presented in Table 1, to fill a small database to be used in the inverse problem. Values in the cells as listed in the table are aspect ratios for selected diameter and volume, whereas dashes indicate that indicatrices of RBC with these characteristics were not calculated. For each set of diameter and volume, four indicatrices were calculated for orientation angles β (= 60°, 70°, 80°, 90°). Calculated theoretical indicatrices were used for determination of characteristics of experimentally measured RBCs by the χ 2 test as explained in the previous section. A few representative results of our comparison by the χ 2 test are presented in Fig. 6. One can see that theoretical indicatrices fit experimental curves well. The mean scattering intensity (intensity integrated over the angular interval) of the measured indicatrices is in a good agreement with the mean intensity of theoretical indicatrices calculated for orientation angles β of 70° and 90° [Fig. 3(b)], and exceeds the mean intensity for smaller orientation angles. This fact allows us to conclude that orientation of RBCs in the capillary of a SFC is close to orthogonal (β = 90°), which agrees with our previous results.9 This result justifies our choice of orientation angles for preliminary calculations of RBCs’ indicatrices. We used all the experimental indicatrices that passed the χ 2 test to plot a distribution of mature RBCs over the orientation angle, as presented in Fig. 7. This distribution proves once again that orthogonal orientation is much preferable for RBCs in the capillary of a SFC and gives an estimation of deviation from orthogonal orientation. Contrary to the DDA, the T-matrix method allows one to reduce a time of light- scattering calculation substantially.2 However the biconcave disk shape complicates the light- scattering simulation in the T-matrix method. This method can be effectively applied to a particle with disk-sphere or oblate spheroid shapes that can be used as models of RBCs. Therefore, we have compared the indicatrices of particles shaped according to Eq. (2) and of particles with the following shape geometries: disk-sphere and oblate spheroid. The comparison was performed for the diameter-volume-equal particles. The parameters of the biconcave disk were as follows: d = 7.60 μm, ε = 0.380 (V = 100 μm3), and m = 1.05. The light scattering for two orientations of the particles relative to the direction of the incident beam, rim-on and face-on incidence, was computed. The indicatrices of the biconcave disk and disk-sphere are shown in Fig. 8. This figure allows us to conclude that RBC can be modeled by a disk-sphere only for simulation of light scattering in the angular interval ranging from 10° to 15°. An oblate spheroid is the most popular model in simulation of light scattering of individual RBCs. This model was used in simulation of light scattering by the T-matrix method.7 We have calculated the indicatrices of diameter-volume-equal spheroids in rim-on and face-on orientations of the oblate spheroid and RBC. The results of these calculations, shown in Fig. 9, allow us to conclude that RBC can be modeled by an oblate spheroid over a wide angular interval only for rim-on incidence. This conclusion is in agreement with the boundary-element methodology12 and discrete sources method26 applied to study of light scattering of red blood cell. However the validity of such substitution for calculation of RBC indicatrices should be further studied for different RBC sizes with respect to the certain problem, where these indicatrices are to be used. 60 70 80 90 Orientation angle β, deg Fig. 7. Distribution of mature RBCs over an orientation angle obtained using χ 2 test. 10 15 20 25 30 35 40 45 50 disk Scattering angle θ, deg Fig. 8. The modified indicatrices of the biconcave disk and the diameter-volume-equivalent disk- sphere: (a) rim-on and (b) face-on incidence. Conclusion Our simulation of light scattering of a mature RBC has shown that the indicatrix is sensitive to the RBC shape and the DDA (or some other method for which no simplifications of the RBC shape are assumed) should be used in a study of formation of the indicatrix for RBCs with various characteristics. However, light scattering of RBCs can be simulated with the T-matrix method for rim-on incidence by use of the oblate spheroid model. Fortunately, the hydrodynamic system of the SFC delivers mature RBCs into the testing zone in this specific orientation. This performance of the SFC gives a chance of solving the inverse light- scattering problem for mature RBCs, e.g. by parameterization or use of a neural network because T-matrix simulation requires substantially less computing time than the DDA algorithm. However, the precision of such algorithms when developed should be tested using realistic indicatrices obtained e.g. by DDA simulations. Therefore, need for improvement of current DDA code arises. Such an improvement can not only provide enough testing indicatrices for the problem described above but also make feasible such tasks as solving an inverse light-scattering problem for any orientation of RBCs relative to the incident beam. 10 15 20 25 30 35 40 45 50 Scattering angle θ, deg oblate sheroid Fig. 9. The modified indicatrices of the biconcave disk and the diameter-volume-equivalent oblate spheroid: (a) rim-on and (b) face-on incidence. This research was supported by Russian Foundation for Basic Research through the grant 02-02-08120-inno and 03-04-48852-a, by Siberian Branch of the Russian Academy of Sciences through the grant 115-2003-03-06, and by the NATO Science for Peace program through grant SfP 977976. Appendix A Let incident radiation propagate along the z-axis, and assume that a particle has a symmetry plane containing this axis. We will investigate then the properties of the Mueller matrix integrated over complete azimuthal angle ϕ at a fixed polar angle θ. Not restricting generality (as we consider the integral over the complete azimuthal angle), we can consider that the x-axis resides in the symmetry plane of the particle. We divide the integral into two parts and then group them: ∫∫∫∫ ),−+)=)+)=), ϕθϕθϕϕθϕϕθϕϕθϕ ](,([d,(d,(d(d SSSSS . (A1) Let us consider two scattering problems for angles (θ,ϕ) and (θ,−ϕ). We rotate the laboratory reference frame about the z-axis by angles ϕ and −ϕ for the first and the second problems respectively (it is the same as rotating everything else – the particle, the incident and scattering direction, and the electric field vectors – in the backward direction). These operations do not change the scattering matrices, therefore ),=), − 0(( θϕθ ϕSS and ),=),− 0(( θϕθ ϕSS , (A2) where Sϕ, S−ϕ are scattering matrices for particles (pϕ and p−ϕ) rotated about the z-axis by angle ϕ and −ϕ respectively, relative to its initial orientation (p0). Let us denote operator of plane zx reflection as Pzx and operator of rotation around the axis z by angle ϕ as Rϕ. Then ( ) ( ) ϕϕϕϕϕϕϕ ppRpPRpRPRpP zxzxzx ==== −− 00ooo , (A3) where the first equation is an identity for any operand (which can be easily verified since operators involved do not affect z coordinates) and the third one exploits the assumption that particle has a zx symmetry plane in its initial orientation. Equations (A2) and (A3) imply that S(θ,ϕ) + S(θ,−ϕ) can be considered as a sum of scattering matrices for the same scattering geometry but for particles which are mirror images of each other with respect to the scattering plane. It was shown6 that such a sum gives a scattering matrix of the form: . (A4) Then Eq. (A1) obviously implies that the Mueller matrix integrated over the complete azimuthal angle will be of the same form. Therefore we have proved that, if particle has a symmetry plane containing the z-axis (coincident with the propagation vector of the incident radiation), then 0=∫ )(d ϕϕ ijS , for 2,1=i and 4,3=j or vice versa. (A5) It is clear that, for a body of rotation, any plane containing symmetry axis is as well a symmetry plane. Therefore a plane containing both the symmetry axis and the axis z is symmetric for the body of rotation regardless of its orientation. Hence Eq. (A5) is automatically proved for axisymmetric particles. References 1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983). 2. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. Light Scattering by Nonspherical Particles, Theory, Measurements, and Applications (Academic Press, San Diego, 2000). 3. S. M. Lewis, B. J. Bain, I. Bates, and M. I. Levene, Dacie & Lewis Practical Haematology (W B Saunders, Churchill Livingstone, London, 2001). 4. J. M. Steinke and A. P. Shepherd, “Comparison of Mie theory and the light scattering of red blood cells,” Appl. Opt. 21, 4335–4338 (1982). 5. G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof and R. M. Heethar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. 32, 2266–2272 (1993). 6. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981). 7. A. M. K. Nilsson, P. Alsholm, A. Karlsson, and S. Andersson-Engels, “T-matrix computations of light scattering by red blood cells,” Appl. Opt. 37, 2735–2748 (1998). 8. P. Mazeron and S. Muller, “Light Scattering by ellipsoids in physical optics approximation,” Appl. Opt. 35, 3726–3735 (1996). 9. J. He, A. Karlsson, J. Swartling, and S. Andersson-Engels, “Light scattering by multiple red blood cells,” J. Opt. Soc. Am. A, 21, 1953-1961 (2004). 10. A. N. Shvalov, J. T. Soini, A. V. Chernyshev, P. A. Tarasov, E. Soini, and V. P. Maltsev, “Light-scattering properties of individual erythrocytes,” Appl. Opt. 38, 230-235 (1999). 11. P. Mazeron and S. Muller, “Dielectric or absorbing particles: EM surface fields and scattering,” J. Opt. 29, 68-77 (1998). 12. S. V. Tsinopoulos and D. Polyzos, “Scattering of He-Ne-Laser Light by an Average-Sized Red-Blood- Cell,” Applied Optics 38, 5499-5510 (1999). 13. B. T. Draine and P. J. Flatau, “Discrete-Dipole Approximation for Scattering Calculations,” J. Opt. Soc. Am. A. 11, 1491-1499 (1994). 14. B. T. Draine, “The discrete dipole approximation for light-scattering by irregular targets,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic Press, San Diego, 2000), pp. 131-145. 15. A. G. Hoekstra, M. D. Grimminck, and P. M. A. Sloot, “Large scale simulation of elastic light scattering by a fast discrete dipole approximation,” Int. J. Mod. Phys. C 9, 87-102 (1998). 16. V. P. Maltsev, “Scanning flow cytometry for individual particle analysis,” Rev. Sci. Instruments 71, 243- 255 (2000). 17. Y. C.Fung, W. C. Tsang, and P. Patitucci, “High-resolution data on the geometry of red blood cells,” Biorheology 18, 369-385 (1981). 18. P. Mazeron, S. Muller, and H. El. Azouzi, “Deformation of erythrocytes under shear: a small-angle light scattering study,” Biorheology 34, 99-110 (1997). 19. E. M. Purcel, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973). 20. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848-872 (1988). 21. S. B.Singham, G. C. Salzman, “Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation,” J. Chem. Phys. 84, 2658-2667 (1986). 22. A. G. Hoekstra and P. M. A. Sloot, “Dipolar unit size in coupled-dipole calculations of the scattering matrix elements,” Opt. Lett. 18, 1211-1213 (1993). 23. V. P. Maltsev and K. A. Semyanov, Characterisation of Bio-Particles from Light Scattering (Inverse and Ill-Posed Problems Series, VSP, Utrecht, 2004) 24. K. A. Semyanov, P. A. Tarasov, A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. 43, 5110-5115 (2004). 25. K.A. Semyanov, P.A. Tarasov, J.T. Soini, A.K. Petrov, and V.P. Maltsev, “Calibration free method to determine the size and hemoglobin concentration of individual red blood cells from light scattering,” Appl. Opt. 39, 5884-5889 (2000). 26. E. Eremina, Y. Eremin, and T. Wriedt, “Analysis of light scattering by erythrocyte based on discrete sources method,” Opt. Comm. 244, 15–23 (2005).

0704.1135

Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy

September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy Eckehard W. Mielke Departamento de F́ısica, Universidad Autónoma Metropolitana–Iztapalapa, Apartado Postal 55-534, C.P. 09340, México, D.F., MEXICO [email protected] Fjodor V. Kusmartsev Department of Physics, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom [email protected] Franz E. Schunck Institut für Theoretische Physik, Universität zu Köln, 50923 Köln, Germany [email protected] A possible equivalence of scalar dark matter, the inflaton, and modified gravity is analyzed. After a conformal mapping, the dependence of the effective Lagrangian on the curvature is not only singular but also bifurcates into several almost Ein- steinian spaces, distinguished only by a different effective gravitational strength and cosmological constant. A swallow tail catastrophe in the bifurcation set indicates the possibility for the coexistence of different Einsteinian domains in our Universe. This ‘triple unification’ may shed new light on the nature and large scale distribution not only of dark matter but also on ‘dark energy’, regarded as an effective cosmological constant, and inflation. 1. Introduction The origin of the principal constituents of the Universe in the form of dark matter (DM) and dark energy (DE) (or rather dark tension due to its negative pressure), remains a major puzzle of modern cosmology and particle physics. Ad hoc proposals like modifying Newton’s law of gravity as in Milgrom’s MOND (Modified Newtonian Dynamics) are difficult to reconcile with relativity, or need a phase coupling to a complex scalar.3 The particle physics’ view is to leave gravity untouched but postulate WIMPs (Weakly Interacting Massive Particles) like axions, dilatons or neutralinos as dark matter candidates.66 The dominant non-visible “dark” fraction of the total energy of the Universe is http://arxiv.org/abs/0704.1135v1 September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 2 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck known to exist from its gravitational effects. Since the dark matter part is distributed over rather large distances, its interaction, including a possible self-interaction,60 must be weak. The main candidates for such weakly interacting particles are the (universal) axion or the lightest supersymmetric particle, as the neutralino in most models, cf. Ref.66 Recently, an observed excess of diffuse gamma rays has been attributed8 to the annihilation of DM in our Galaxy. The flux of the presumed neutralino annihilation allows a reconstruction of the distribution of DM in our Galaxy. Most probable is a pseudo-isothermal profile but with a substructure of two doughnut-shape rings in the galactic plane. It is believed that these transient substructures have their origin in the hierarchical clustering of DM into galaxies. However, there are reservations concerning the internal consistency in the interpretation of the observations: Given the same normalization for the cross section of the DM particle, it appears unlikely1 that DM annihilation is the main source of the extragalactic gamma-ray background (EGB). Moreover, then the large accompanied antiproton flux needs to meet several constraints.5 On the other hand, heterotic string theory provides a very light universal axion which may avoid12 the strong CP problem in quantum chromodynamics (QCD).50 Given the existence of such almost massless (pseudo-)scalars, it has been speculated that a coherent non-topological soliton (NTS) type solution of a nonlinear Klein- Gordon equation can account for the observed halo structure, simulating a Bose- Einstein condensate of astronomical size;16,17,34 cf. Ref.53 In particular, a Φ6 toy model43 yields exact Emden type solutions in flat spacetime, including a flattening37 of halos with ellipticity e < 1 as observed via microlensing.14,54 Both views are to some extend physically equivalent, i.e., scalar dark matter minimally coupled to Einstein’s general relativity (GR) is equivalent4,27,49,55,59 to a modified gravity in the relativistic framework of higher-order curvature Lagrangians. Such effective Lagrangians may also arise from the low-energy limit of (super-) strings. The model proposed by Carroll et al.6 also uses such an equivalence of a nonlinear higher-order curvature Lagrangian, in order to explain the present cos- mic acceleration, but takes resort to an 1/R type curvature Lagrangian, which is unbounded for weak gravitational fields. Quite generally, the dependence of the effective Lagrangian L on the scalar curvature R is singular and cannot always be represented analytically in the (R,L) plane. More stunning is our recent finding55 that, although nonlinear, a L = L(R) type Lagrangian bifurcates in several branches of almost linear Lagrangians 2κeff (R− 2Λeff) (1) distinguished only by the effective gravitational constant κeff and a cosmological constant Λeff . This indicates a unifying picture of dark matter, ‘dark energy’53 with an equation of state parameter wDE = p/ρ ≃ −1, and modified gravity which may account, on different scales, for inflation,37,40 dark matter halos36,43 of galaxies or even dark September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 3 matter condensations, the so-called boson stars23,24,41,58 as candidates of MACHOs (Massive Compact Halo Objects). The ‘landscape’ view25 of (super-)strings suggests also such a triple unification based on coherent, possibly oscillating scalar field with a mass larger than the Hubble scale at the present epoch, i.e. m > H∗ = 10 −23 eV. 2. Gravitationally coupled scalar fields Conventionally, dark matter and inflation can be modeled by a real (or complex) scalar field φ with self-interaction U(|φ|2) minimally coupled to gravity with the Lagrangian density LDM = | g | gµν(∂µφ ∗)(∂νφ)− 2U(|φ|2) , (2) where κ = 8πG is the gravitational constant in natural units, g the determinant of the metric gµν , and R the scalar curvature of Riemannian spacetime with Tolman’s sign conventions.63 A constant potential U0 = Λ/κ would simulate the cosmological constant Λ. In a wide range of inflationary models, the underlying dynamics is simply that of a single scalar field — the inflaton — rolling in some underlying potential. This scenario invented by Linde26 is generically referred to as chaotic inflation due to its choice of initial conditions. Many superficially more complicated models can also be rewritten in this framework. 3. General metric of a flat inflationary universe A spatially flat (k = 0) Friedman-Robertson-Walker (FRW) Universe with metric ds2 = dt2 − a2(t) dr2 + r2 dθ2 + sin2 θdϕ2 is nowadays favored by observations.61 Its temporal evolution of the generic model (2) is determined by the two autonomous first order equations Ḣ = −3H2(1 + wDE) = κU(φ)− 3H2 =: V (H,φ) , (4) φ̇ = ± 3H2 − κU(φ) = ± V (H,φ) , (5) where H = ȧ(t)/a(t) is the Hubble expansion rate. The function V (H,φ) will turn to be the “height function” in Morse theory.46 Observe that V ≤ 0 in order to avoid scalar ghosts. For the FRW metric, the Lagrangian density (2) reduces to L = − 3 ȧ2a+ φ̇2 − U(φ) a3 . (6) Since the shift function is normalized to one for the FRW metric, the canonical momenta are given by P = ∂L/∂ȧ = −6Ha2/κ and π = ∂L/∂φ̇ = a3φ̇. The September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 4 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck Hamiltonian or “energy function” is given by φ̇2 + V (H,φ) a3 ∼= 0 (7) and, due to (5), vanishes for all solutions, which is a familiar constraint in GR. Using the Hubble expansion rate H := ȧ(t)/a(t) as the new “inverse time” coordinate t = t(H) = κŨ − 3H2 , (8) we were able to find the general solution:56 a = a(H) = a0 exp κŨ − 3H2 . (9) ds2 = κŨ − 3H2 )2 − a0 2 exp κŨ − 3H2 dr2 + r2 dθ2 + sin2 θdϕ2 , (10) φ = φ(H) = ∓ 3H2 − κŨ , (11) where Ũ = Ũ(H) := U(φ(t(H))) is the reparametrized inflationary potential. The singular case Ũ = 3H2/κ with wDE = −1 leads in (4) to the de Sitter inflation with exponential expansion a(t) = a0 exp( Λ/3 t), which for a later time becomes physically unrealistic, since the inflationary expansion eventually needs to merge into the usual Friedman cosmos. Therefore, in explicit models we use the ansatz Ũ(H) = , (12) for the potential, where g(H) = V (H,φ(t(h))) is a nonzero function which should provide the graceful exit from the inflationary phase to the Friedman cosmos. In order to have a positive acceleration during the inflationary phase, but on the other hand a real scalar field in (11), its allowed range is −H2 < g < 0. This H–formalism facilitates considerably the reconstruction of the inflaton potential.36,40,56 In second order this is based on an Abel equation13 for the primordial density ǫ = −g/H2. 3.1. Classification by catastrophe theory A general classification of all allowed inflationary potentials and scenarios has al- ready been achieved by Kusmartsev et al.22 via the application of catastrophe theory to the Hamilton–Jacobi type equations (4) and (5) regarded as an autonomous non- linear system. In phase space, the equilibrium states are given by the constraint {Ḣ, φ̇} = 0. The critical or equilibrium points, respectively, of this system are determined by September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 5 V (Hc, φc) = 0. This constraint is globally fulfilled by κU(φ) = 3H Λ, where the Hubble expansion rate is constant, i.e. HΛ =: Λ/3 corresponding to wDE = −1. For φ̇ = 0 and Λ 6= 0, we recover the de Sitter inflation. In general, the Jacobi matrix −6H κU ′ ±6H(−2κV )−1/2 ∓κU ′(−2κV )−1/2 , (13) of the system (4) and (5) depends crucially on U ′ = dU/dφ. Since its determinant vanishes, i.e. det J = 0, the system is always degenerate. For the analysis of stability, it suffices therefore to consider only (4) and to reconstruct φ later. Since H and φ are independent variables, we can introduce the non–Morse superpotentialW (H,φ), defined via V := −∂W/∂H , and analyze the system with the aid of catastrophe theory. From (4) we obtain W (H,φ) = H3 − κU(φ)H + C(φ) , (14) where C is an arbitrary function of φ. The function W is already in canonical form in H–space, and belongs to a Whitney surface2,21 or to the Arnold singularity class A2. The corresponding Whitney surface has only one control parameter, the potential U . Thus, an evolution of critical points is determined via the values of the potential U . Let us analyze the types of critical points at different fixed values of the control parameter U . If Uc := U(φc) < 0, the equation has no stable critical point due to the shape of the Whitney surface. However if Uc > 0, there are two critical points: a stable one at Hc = Uc/3 and an unstable one at Hc = − Uc/3. The value Uc = 0 is the bifurcation point. Provided this is also an extremal of V , we necessarily have ∂V/∂H |c= −6Hc = 0, ∂V/∂φ |c= κU ′c = 0, and φ̇c = 0. Thus, also the critical points of the Klein–Gordon equation for the scalar field are involved. Hence, the Hubble parameter has to vanish and φc is a double zero of the potential U . The Hessian of (4) takes the form Hess(V ) = 0 κU ′′ . (15) The sub–determinants of the Hessian are ∆0 = ∂ 2V/∂H2 = −6 < 0 and ∆1 = detHess(V ) = (∂2V/∂H2) (∂2V/∂φ2)−(∂2V/(∂H∂φ))2 = −6κU ′′. For a maximum of the potential U , i.e. U ′ = 0 and U ′′ < 0, the function V possesses a maximum; a minimum of the potential U , however, corresponds to a saddle point of V . 3.2. Reheating Since U(φc) can be associated with the latent heat of the Universe in this epoch, we could demonstrate:22 The critical points of the non–Morse potential W (H,φ) determine the evolution in the inflationary phase. Along the minima and maxima Hc = ± U(φc)/3, the in- flaton moves from the slow–roll to the hot regime. The saddle points of W , i.e. more precisely, the minima of V , determine the onset of reheating. September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 6 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck 3.3. Scale-invariant spectrum as a limiting point It is now possible to relate the “slow–roll” condition, for the velocity of the infla- tionary phase, to the critical points resulting from catastrophe theory. For inflation (with ä > 0) the two “slow–roll parameters” are, in first order approximation, ǫ = −g/H2 and η = −dg/d(H2) , (16) where g = V (H,φ) is the “graceful exit function”. In this reduced dynamics, they are effectively determined by the first and second derivatives of the re- duced non–Morse function W (H), i.e., more precisely by ǫ = (1/H2)(∂W/∂H) and η = (1/2H)∂2W/(∂H)2. They will also determine the density fluctuations of the early Universe. The deviation ∆ := (n− 1)/2 from the scale-invariant Harrison-Zeldovich spec- trum with n = 1 leads in the first order slow-roll approximation to the differential equation − ǫ = ∆ (17) ǫ = −∆+AH2 , V = ∆H2 −AH4 (18) as solutions for the density and the graceful exit function, respectively. Then the non-Morse potential turns out to be W (H,φ) = − V dH = −∆ H5 + C(φ) (19) which together with (14) provides us with the reparametrized potential κŨ(φ) = H2 − A H4 , (20) which for ∆ ≃ 0 is positive in the leading order. In the next order slow-roll approximation arises a nonlinear Abel equation13 with a continuous spectrum for n < 1 and a discrete ‘blue‘ one for n > 1, see Ref.57 for more details. This division of the spectrum persists in the next to second order slow-roll approximation.36 More important is our finding that in both higher order approximations the Harrison-Zeldovich spectrum with n = 1 stays a limiting point, which agrees quite well with recent constraints from the observations of WMAP.20 4. Higher–order curvature Lagrangians via field redefinition Nonlinear modifications of the Einstein–Hilbert action are of interest, among oth- ers, for the following reasons: First, some quadratic models can be renormalized when quantized, cf. Ref.38 Second, specific nonlinear Lagrangians have the prop- erty that the field equations for the metric remain second order as in GR; these are the so–called Lovelock actions which arise from dimensional reduction of the Eu- ler characteristics, cf. Ref.47 In Yang’s theory of gravity, this topological invariant September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 7 induces an effective cosmological constant for instanton solutions residing in Ein- stein spaces.33,38 At times,64 this renormalizable model is referred to as Yang-Mielke theory of gravity. As an instructive example of higher-derivative theories of gravity,52 we consider the Lagrangian density L = L(R) | g | = | g |, (21) where β is a dimensionless coupling constant. Through the conformal change30 gαβ → g̃αβ = Ωgαβ with Ω = 2κ , (22) of the metric, this Lagrangian can be mapped to the usual Hilbert–Einstein La- grangian with a particular self–interacting scalar field. Indeed, Starobinsky62 con- sidered earlier such models for inflation. The scalar field, the inflaton, will arise via lnΩ (23) from the nonlinear parts of a higher–order Lagrangian L = L(R) in the scalar cur- vature R. Recently, such modified gravity models are considered as alternatives7,48 for dark matter or even dark energy. 4.1. Reparametrized Lagrangian Instead of studying the resulting complicated nonlinear field equations of higher– order curvature Lagrangians, we follow the equivalence proof of Ref.27 for the con- formal frame (22). Then, our inflaton Lagrangian density (2) acquires the form L = 1 | g | RΩ− 2κΩ2Û(Ω) , (24) where Û(Ω) := U(φ(Ω)) = U 3/2κ lnΩ is the reparametrized potential. Thus in our approach, the inflaton or dark matter scalar will not be regarded as an independent field, but is induced via (23) by the non–Einsteinian pieces of the general Lagrangian L = L(R). Solving for the potential via the method of Helmholz, we obtain Û(Ω) = H(R)/Ω2 = /Ω2 . (25) If we identify the conformal factor with the field momentum via Ω = 2κdL/dR, the bracket in (25) can be regarded35 as a Legendre transformation L → H(R) = RdL/dR − L from the original Lagrangian (2) to the general nonlinear curvature scalar Lagrangian L = L(R). Then, the parametric reconstruction R = 2κ exp 2κ/3φ 2U(φ) + , (26) September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 8 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck Fig. 1. The qualitative form of a Whitney surface with local extrema in the case of a nonlinear curvature Lagrangian L(R) depending on φ as control parameter. L = exp 2κ/3φ U(φ) + of the higher–order effective Lagrangian L(R) from the self–interacting inflaton potential U(φ) arises.4 Here the scalar field plays merely the role of a control pa- rameter. The form of the Whitney surface with its local valleys and mountains is qualitatively shown in Fig. 1. 5. Bifurcations with effective ‘dark energy’ In order to model self-interacting dark matter, let us consider the potential U = m2|φ|2 1− χ|φ|4 , (28) where m is the mass of an ultra-light scalar and χ the coupling constant of the nonlinear self-interaction. It provides us with a solvable non-topological soliton type model of dark matter halos43,44 even with toroidal substructures45 and a reasonable approximation of the rotation curves of dark matter dominated galaxies. Moreover, the predicted scaling relation11 fits almost ideally astronomical observations. In Ref.54 we predicted the effects of such scalar field halos for microlensing. September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 9 5.1. Free field In the string landscape,25 a triple unification of inflation, DM and DE suggests itself based on a simple quadratic potential (28) with χ = 0 corresponding to a free massive field. Then, our scalar field toy model likewise incorporates ‘dark energy’ in a rather novel way: The exact parametric solution of the equivalent nonlinear Lagrangian L(R) for the ‘free’ field reads R = 6m2xex(1 + x), (29) xe2x(2 + x), (30) where x := lnΩ under the reality condition Ω > 0. -1 -0.5 0 0.5 1 1.5 2 Fig. 2. The swallow tail behavior of the Lagrangian L(R) for a massive ‘free’ field (χ = 0 and κ = m = 1). The dependence of L(R) given in Fig. 2 is rather surprising and represents the bifurcation set of the swallow tail catastrophe2,21,23 associated with some higher dimensional grand manifolds. The scalar field φ, its mass m as well as χ play the role of control parameters. According to the theory of singularities (more widely known as catastrophe theory, cf. Arnol’d2), this bifurcation set indicates that the Lagrangian manifolds are associated with two local “minima” and one “maximum” (and saddle points at the meeting points of the grand manifold). Each of the “min- ima” merges with the “maximum” at the cuspoidal points A and B and then dis- appears. The minima (the semi-infinite segments A and B) are characterized by a positive second derivative of the Hamiltonian H(Ω) with respect to the momentum Ω, i.e. d2H/dΩ2 > 0. They correspond to an effective Lagrangian with vanishing cosmological constant. For the “maximum”, i.e. the segment AB, this derivative is negative and the ef- fective Lagrangian has a modified gravitational constant and a positive cosmological constant, κeff = eκ , Λeff = 3m 2/(2e) , (31) September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 10 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck respectively. Fig. 3. Whitney surface for the nonlinear curvature Lagrangian L(R) corresponding to the model (28) with χ = 0.3 and m = κ = 1 depending on φ as control parameter. (φ in units of 3/(2κ) 5.2. Self-interacting scalar In the nonlinear case, the exact parametric solution L(R) reads R = 6m2xex 1 + x− , (32) 2 + x− 27χ x4 − 9χ . (33) The resulting extremal curve of the Whitney surface is drawn in Fig. 3, and its projections L = L(R), R = R(φ), and L = L(φ) in Fig. 4. For χ 6= 0, the bifurcation diagram in Fig. 5 is of higher rank than for the free field. Near the center, we find the butterfly part of the catastrophe. But, for each non-vanishing χ, there exists a further cusp far away from the center, for a large negative R value and L close to zero. From this cusp, the curve finally returns to the origin, for L values very close to zero so that it cannot be seen in Fig. 5 or 6. In an enlargement of the additional cusp in Fig. 7, one can see that L(R) actually becomes negative and forms a cusp. The effective strength of gravity and the value of cosmological constant κeff = κe −x , Λeff = 3m , (34) September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 11 Fig. 4. Projections of the Whitney surface in Fig. 3 for the nonlinear curvature Lagrangian L(R) corresponding to a nonlinear scalar (28) with χ = 0.3 and m = κ = 1. 0 5 10 15 20 25 30 Fig. 5. The wigwam singularity for the nonlinear curvature Lagrangian L(R) for a NTS model with χ = 0, 0.1 (long dashes), 0.2 (short dashes), 0.3 (dashed) and κ = m = 1; here only the butterfly part near the center is shown. respectively, depend now on both, the mass m and, via x as a solution of R = 0, on the coupling constant χ of the scalar field, and need to be constrained by cosmological data. In quintessence models,65 e.g., the crossover scale of the scalar field is |φc| = exp(−1/α)M2 /m, where α = 1/138 is about Sommerfeld’s fine structure constant and MPl = 1/ κ the reduced Planck mass. Then the tiny observed cosmological constant of the present epoch ΛDE = m 2φ2c = exp(−2/α)M4Pl ≃ (10−3eV)4 (35) is roughly reproduced for small χ. It is important to stress that within the range limited by cuspoidal points, all September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 12 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck -20 -10 0 10 20 30 Fig. 6. For the NTS model with χ = 0.3 and κ = m = 1, the wigwam catastrophe of the Lagrangian L(R) can be decomposed into a butterfly plus additional elementary catastrophe. Here, the butterfly catastrophe arises near the origin; at about (R,L) = (−250,−0.015) there is a further cusp, cf. Fig. 7; the curve finally returns to the origin which cannot be seen due to our scale. -240 -220 -200 -180 -160 -0.04 -0.02 Fig. 7. Continuation of Fig. 6. Enlarging the additional cusp for the nontopological soliton field with χ = 0.3 and κ = m = 1. three states may coexist with each other. Thus our bifurcation set, cf. Ref.55 for a generalization with χ 6= 0, indicates that each local patch of the Universe may have a different strength and cosmological constant (‘dark energy’) controlled by the mass m of the scalar field, but the effective Lagrangian has approximately the same Einsteinian form (1). In “maximal” domains, inflation may be still going on, thus realizing prospective ideas of Linde.26 5.3. Natural inflation from the axion? In QCD, after integrating out the fermion fields, its generating functional including a topological Pontrjagin term for the Yang-Mills gauge fields induces an effective September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 13 axion potential U = Λ4QCD [1− cos(φ/fφ)] . (36) This potential displays a periodicity with a period of 2πfφ, has a minimum at φ = 0, as required, and leads to the induced axion mass of mφ = Λ QCD/fφ. A quintaxion may also be induced by spacetime torsion.39 Within natural inflation,51 such a potential has been proposed for an axion coupling constant close to the Planck scale. For simplicity let us assume in the following that fφ ≃ 3/(2κ). Then, following the general prescription (26)-(27), we find the reparametrized solution R = 2κ exp (x) Λ4QCD [2− 2 cos (x) + sin (x)] , (37) L = exp (2x) Λ4QCD [1− cos (x) + sin (x)] . (38) Fig. 8 exhibits a swallow tail behavior similar as for a NTS potential with χ = 0. -1 -0.5 0 0.5 1 1.5 2 -15000 -10000 -5000 Fig. 8. The axion gives rise again to a swallow tail catastrophe in the Lagrangian L(R), with the choice of ΛQCD = 0.1, κ = 1, and fφ = 1. 6. Domain structure of dark matter and induced dark energy Baryonic particles (such as protons and neutrons) account for visible matter in the Universe, that is only a small fraction of the observed total matter. The major part of it is a mysterious DM, which only interacts gravitationally. On the other hand according to our model, the Universe splits into almost Einsteinian domains with – depending on the physical scale and eras – a different gravitational strength and cosmological constant, the latter being most likely a representation of DE. A natural question is how these domains arise? To answer this, we have to look into the initial inflationary stage of the Universe, long before decoupling of radiation and matter, even before grand unified phase transitions where only an initial pre-field existed. At the initial stage of the quantum era, the fundamental pre-field may have existed September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 14 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck in a form of a scalar or vector gauge fields. Without lack of generality we can limit ourselves to a scalar field φ. During this quantum era, the scalar field was strongly fluctuating. Due to the rapid inflation, the size of the Universe as well as the size of each individual fluc- tuation have increased enormously while the amplitude remains almost the same. During this fast spacelike expansion, just after the stretching, these space fluctu- ations of φ were frozen while loosing their quantum nature due to the increasing size. The decoupling of radiation and visible matter has imprinted fluctuations on the electromagnetic spectrum which we observe now in the anisotropy microwave background radiation. However, we cannot see such ripples of the visible matter after the decoupling. Due to the gravitational interaction the process of large scale formation has started just after the decoupling. And the state of nearly homoge- neously distributed visible matter existing after the decoupling has eventually been transformed into clusters of galaxies and star structures which we see nowadays. Ob- viously, the decoupling of the dark part of matter from radiation and visible matter may have happen even earlier, when the Universe was in the form of a quark-gluon plasma. However, after decoupling the dark part of matter must be subjected to the same gravitational forces and to a similar large scale formation. Since its total mass is significantly larger than the visible mass, the formation of large structures of DM may have a weak influence on the visible matter. Dark matter will have an opposite effect by dictating where and how the visible matter should be distributed, probably localized around clusters of DM. In the case of scalar field DM, its peculiarities such as self-interaction may also have contributed to the DE distribution. DM cannot be seen directly by traditional observations but can be inferred from more sophisticated gravitational lensing,28 revealing that DM exists in a form of a loose network of filaments, growing over time, which intersect in massive structures at the locations of clusters of galaxies. This finding is consistent with the conventional theory of large scale structure formation, where a smooth distribution of DM collapses first into filaments and then into clusters, forming a cosmic scaffold,28 then accumulating visible matter and later newly born stars. The primary candidate for DM is a scalar field. If a scalar field exists, then it may have different amplitudes or different mass densities in different space regions. Such a distribution of φ has been created after the inflation of the Universe. It is associated with different amplitudes of the frozen fluctuations. The process of large scale structure formation is rather similar in each of these regions. Our re- sults indicate that in each of these different regions the gravitational constant may vary,55 depending on the amplitude of the scalar field of the given frozen space fluctuation. In other words, in each of these regions defined by the original frozen fluctuations there will arise different Newtonian potentials. This is consistent with recent studies9 showing that, during expansion, a vector field having two different initial amplitudes will give rise to different Newtonian potentials. This difference in turn drives self-consistently enhanced growth of the density perturbations which enables tiny perturbations to grow into the large structures we see today. September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 15 6.1. Cosmic domains Our studies indicate that these different Newtonian potentials correspond to differ- ent effective gravitational constants in various regions of the Universe. If this is true, the data of recent observations28 should be reinvestigated. In regions with larger gravitational constants one may see that the gravitational lensing will be stronger and vice versa. We regard areas with different gravitational potentials as cosmic do- mains characterized by their own gravitational and cosmological constants. There occurs a coexistence of these domains (which primordially may be of topological origin18,19). On the boundaries of the domains, changes in the gravitational and cosmological constants are expected to be abrupt. In spite of the random or fluctu- ational origin of these domains, the values of these gravitational and cosmological constants take only a few universal numbers, corresponding to only a finite number of types of domains. For small curvature, each of these states can be described approximately by the same effective Lagrangian (1) with different gravitational constant κeff and cosmo- logical constant Λeff and emerge as a fixed point of the conformal transformation from one side and universal classes of the smooth differential mappings from the other side.21 These spaces are approximately Einstein spaces, but of different grav- itational strength as well as with different cosmological constants. The distribution of such cosmological constants in the Universe corresponds to DE induced by the scalar field distribution. On a very large scale, φ is homogeneously distributed and therefore we expect that DE will be distributed homogeneously, depending on the details of inflationary cosmology. 6.2. Domains arising from free fields If DM and DE were associated with a massive free scalar field gravitationally coupled or/and associated with axions (see, Sec. 5.3) there will arise three types of domains: 1) The first type of domains, I, is described by Einstein‘s theory with conven- tional gravitational constant and vanishing cosmological constant. 2) The second type of domains, II, is described by an Einsteinian model having a very large gravitational constant and a vanishing cosmological constant. Such type of domains may be gravitationally unstable, leading eventually to their contraction and ultimately to a gravitational collapse and phenomena resembling supernova ex- plosions. For very large masses, we may expect that these explosion will be stronger than standard supernovae. Our approach may explain the fact why some supernovae progenitors seem to have exceeded15 the Chandrasekhar limit. 3) The third type of domains, III, are described by an Einsteinian model with large gravitational and positive cosmological constant ΛIIIeff given by (34). Then we expect that such domains are still expanding according to effective Friedman equa- tions and probably their sizes are also increasing. Such expansion may be ended abruptly and such domains will be transformed into domains of type I. One may speculate that the largest part of the Universe is occupied by domains September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 16 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck of type I, corresponding to spaces with zero cosmological constant. The next largest proportion of the Universe is expected to be occupied by domains of type III. Those parts of the Universe occupied by domains of type II are decreasing with time. Each disappearance of such domain would be accompanied by supernova type explosions. On the average, the cosmological constant inferred from recent observations is ΛDE = VIIIΛ VI + VII + VIII , (39) where the spacelike volumes VI, VII and VIII are associated with domains of type I, II, and III, respectively. In summary, our bifurcation set (see Fig. 1 and Fig. 2) indicates that the Universe is locally described by Einstein’s GR and effectively split into domains. The splitting originated from primordial quantum fluctuations which were frozen during inflation. Each domain or local patch of the Universe may have different gravitational strength and may have zero or negative pressure associated with DE. In domains of type II associated with a positive cosmological constant the inflation may be still going on. It will be stopped exactly at the bifurcation point A and this domain will be transformed into a domain of type I, as in Fig. 2. For a massive free scalar field and for axions the bifurcation set is the swallow tail catastrophe. There are two cusps at the points A and B which are associated with the highest singularities of the differential mappings. They are also related to the domain boundaries. In each of these cuspoidal points A and B of the bifurcation diagram, the minimum of some grand manifold merges with the maximum.22 Each local minimum (the segment A and the semi-infinite segment B) is associated with two types of domains, I and II, respectively. Each local maximum (saddle) of the grand manifold is associated with an expanding domain of type III (the segment AB). The effective Einsteinian gravity for this domain is stronger than that in the domain of type I, depending on the mass of the scalar field. Domains with a positive effective cosmological constant are still in an expanding inflationary phase. The ‘strong’ gravity state, the domain II (the semi-infinite segment B) with a negative or vanishing cosmological constant corresponds to deflation. Since the Uni- verse is split into dynamical regions associated with different gravity, their bound- aries change with time and present some kind of cosmological strings or membranes. Changes of the boundary are associated with some sort of local phase transition re- minding us of phase separations like those arising in first order phase transitions. At some instant, all these domains are in quasi-equilibrium with each other although their boundaries changes with time as the membranes are moving. At very long time scales (much larger than the Planck time) when inflation in the domains of type III will eventually be ceased, only stable phases, like the domains of type I and some small concentration of domains of type II will remain. These domains would produce explosions resembling supernovae with masses well beyond of the Chandrasekhar limit, similar to those observed.15 September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 17 6.3. Domains arising from self-interacting fields In case of self-interacting scalar fields with χ 6= 0, the classification will be richer (see Figures 4, 5 and 6), with five types of domains. The bifurcation diagram as- sociated with such a wigwam catastrophe or A5 singularity according to Arnold’s classification2 consists of four cusps or four A2 singularities, with a dramatic change in the gravitational and cosmological constants. There occurs a general decoupling of the wigwam catastrophe A5 into a butterfly catastrophe A4, and the elementary cusp A2, depending on the values of R: If the butterfly catastrophe arises near the origin or at small values of R, the additional cuspoidal point is associated with a very large value of R ≃ −250. For some fixed values of the coupling constant χ = 0.1, 0.2, 0.3, the butterfly part of bifurcation diagram is presented in Figs. 5. In comparison with the free massive field, the in- clusion of the self-interaction associated with the nonzero parameter χ gives rise to two extra cusps leading to the appearance of two extra branches in the bifurcation diagram, again associated with effective Einsteinian spaces. Although these almost linear branches do exist at positive R, their major part is related to negative val- ues of R. For one of these branches the cosmological constant vanishes while for the other one the cosmological constant decreases when the value of χ increases, and the part of the branch associated with R > 0 decreases. In general, the self- interaction induced by the field φ increases the strength of gravity and decreases the cosmological constant. Let us describe in more detail the transitions in the bifurcations: Starting from the first branch, denoted as I, through the first cuspoidal point we arrive at the second branch II which is very similar to the case of a free massive scalar field, although its length decreases significantly when χ increases and the cosmological constant vanishes at any value of χ. This branch corresponds to the limit φ → 0 of small scalar fields, corresponding to the conformal factor Ω = 1. Therefore, on this branch we recover the linear Hilbert-Einstein Lagrangian LHE = R/2κ, as ex- pected. With decreasing Ω, we will come to the next cuspoidal point, where the new branch III has a positive cosmological constant and gravity becomes stronger. With increasing value of χ the length of this branch increases. Thus, there will be more domains having positive cosmological constant and a gravitational strength higher than the conventional. By decreasing Ω until reaching the next cuspoidal point of the bifurcation diagram, we arrive at the branch IV. The spacetime associated with this branch has a very strong gravity and negative cosmological constant. Similar to the free massive case, this branch probably is gravitationally unstable. Finally, for very small Ω, we will arrive at the branch V with R < 0, where gravity is the strongest but without cosmological constant. Similar to the previous branch IV, the domain V is unstable, as well. Thus in a Universe filled with self-interacting scalar fields, we would expect that a rather strong phase separation into further domains will arise. September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 18 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck 7. Discussion For dilute DM the Lagrangian has the standard Hilbert-Einstein form LHE = R/2κ. For dense DM, i.e. large values of the DM scalar, the resulting effective Lagrangian is LHE = R/2κeff, only the slope is less steep, i.e. the effective gravitational coupling κeff > κ is, by “renormalization”, 3,65 larger than Newton’s. This resembles some aspects of MOND, but with the advantage that we still work in the standard general relativistic framework. Depending on its sign, the cosmological constant Λeff in a bifurcation approximated for small R by (1) can also model DE or an accelerating phase of the present epoch of the Universe (like ‘anti-gravity’). There also may arise “gravitational screening” for κeff smaller than Newton’s. MACHOs can now be described by some specific nonlinear modifications of the Hilbert-Einstein action. In the light of our finding we have to comment on recent observations,29 which constrain the density of MACHO type objects in the Universe by measuring the brightness of high redshift type Ia supernovae relative to low redshift samples. These data favor DM made of microscopic particles (such as WIMPs) or scalar and vector gauge fields over MACHOs with masses between 10−2 and 1010 solar masses. This provides another evidence that our approach correctly describes the frozen space fluctuations which lead to a formation of cosmic domains with a different density of DM and DE in the Universe. Acknowledgments We would like to thank Burkhard Fuchs, Humberto Peralta, and Konstantin Zioutas for helpful discussions. One of us (EWM) thanks Noelia, Markus Gérard Erik, and Miryam Sophie Naomi for encouragement. References 1. S. Ando, “Can dark matter annihilation dominate the extragalactic gamma-ray back- ground?”, Phys. Rev. Lett. 94, 171303 (2005). 2. V.I. Arnol’d: Catastrophe Theory, Third, revised and expanded edition (Springer-Verlag, Berlin 1992). 3. J.D. Bekenstein, “Relativistic gravitation theory for the MOND paradigm”, Phys. Rev. D 70, 083509 (2004). 4. J. Beńıtez, A. Maćıas, E.W. Mielke, O. Obregón, and V.M. Villanueva, “From inflationary COBE potentials to higher–order curvature scalar Lagrangians”, Int. J. Mod. Phys. A 12, 2835 (1997). 5. L. Bergström, J. Edsjö, M. Gustafsson, and P. Salati, “Is the dark matter interpretation of the EGRET gamma excess compatible with antiproton measurements?”, JCAP 0605, 6 (2006). 6. S.M. Carroll, V. Duvvuri, M. Trodden, and M.S. Turner, “Is cosmic speed-up due to new gravitational physics?”, Phys. Rev. D 70, 043528 (2004). 7. S.M. Carroll, I. Sawicki, A. Silvestri, and M. Trodden, “Modified-source gravity and cosmo- logical structure formation”, New Journal of Phys. 8, 323 (2006). 8. W. de Boer, “Evidence for dark matter annihilation from galactic gamma rays?”, New Astron. Rev. 49, 213 (2005). 9. S. Dodelson and M. Liguori, “Can cosmic structure form without dark matter?”, Phys. Rev. Lett. 97, 231301 (2006). 10. B. Fuchs, “Constraints on the decomposition of the rotation curves of spiral galaxies”, Proc. September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 19 of the 4th International Workshop on the Identification of Dark Matter (IDM 2002), York, England, 2-6 Sep 2002, e-Print: astro-ph/0212485. 11. B. Fuchs and E.W. Mielke, “Scaling behaviour of a scalar field model of dark matter haloes”, Mon. Not. R. Astron. Soc. 350, 707–709 (2004). 12. M.K. Gaillard and B. Kain, “Is the universal string axion the QCD axion?”, Nucl. Phys. B 734, 116 (2006). 13. A. Garćıa, A. Maćıas, and E.W. Mielke, “Stewart–Lyth second–order approach as an Abel equation for reconstructing inflationary dynamics”, Phys. Lett. A 229, 32 (1997). 14. H. Hoekstra, H.K.C. Yee, and M.D. Gladders, “Properties of galaxy dark matter halos from weak lensing”, Astrophys. J. 606, 67 (2004). 15. D.A. Howell et al., “The type Ia supernova SNLS-03D3bb from a super-Chandrasekhar-mass white dwarf star”, Nature 443, 308-311 (2006). 16. W. Hu, R. Barkana, and A. Gruzinov, “Fuzzy cold dark matter: The wave properties of ultralight particles”, Phys. Rev. Lett. 85, 1158 (2000). 17. K.R.W. Jones and D. Bernstein, “The self-gravitating Bose-Einstein condensate”, Class. Quantum Grav. 18, 1513 (2001). 18. T.W.B. Kibble, “Some implications of a cosmological phase transition”, Phys. Rept. 67, 183 (1980). 19. T.W.B. Kibble, “Symmetry breaking and defects”, arXiv:cond-mat/0211110. 20. W.H. Kinney, E.W. Kolb, A. Melchiorri, and A. Riotto, “Inflation model constraints from the Wilkinson microwave anisotropy probe three-year data”, Phys. Rev. D 74, 023502 (2006). 21. F.V. Kusmartsev, “Application of catastrophe theory to molecules and solitons”, Phys. Rept. 183, 1 (1989). 22. F.V. Kusmartsev, E.W. Mielke, Yu.N. Obukhov, and F.E. Schunck, “Classification of infla- tionary Einstein–scalar–field–models via catastrophe theory”, Phys. Rev. D 51, 924 (1995). 23. F.V. Kusmartsev, E.W. Mielke, and F.E. Schunck, “Gravitational stability of boson stars”, Phys. Rev. D 43, 3895 (1991). 24. F.V. Kusmartsev, E.W. Mielke, and F.E. Schunck, “Stability of neutron and boson stars: a new approach based on catastrophe theory”, Phys. Lett. A 157, 465 (1991). 25. A.R. Liddle and L.A. Urena-Lopez, “Inflation, dark matter and dark energy in the string landscape”, Phys. Rev. Lett. 97, 161301 (2006). 26. A. Linde: Inflation and Quantum Cosmology (Academic Press, San Diego, 1990). 27. G. Magnano and L.M. Sokolowski, “Physical equivalence between nonlinear gravity theories and a general–relativistic self–gravitating scalar field”, Phys. Rev. D 50, 5039 (1994). 28. R. Massey et al., “Dark matter maps reveal cosmic scaffolding”, Nature 445, 286-290 (2007). 29. R.B. Metcalf and J. Silk, “New constraints on macroscopic compact objects as dark matter candidates from grav. lensing of type Ia supernovae”, Phys.Rev.Lett. 98, 071302 (2007). 30. E.W. Mielke, “Conformal changes of metrics and the initial–value problem of general rela- tivity”, Gen. Rel. Grav. 8, 321 (1977). 31. E.W. Mielke, “Note on localized solutions of a nonlinear Klein–Gordon equation related to Riemannian geometry”, Phys. Rev. D 18, 4525–4528 (1978). 32. E.W. Mielke, “Mass formular for solitons with quantized charge”, Lett. Nuovo Cim. 25, 424 (1979). 33. E.W. Mielke, “On pseudoparticle solutions in Yang’s theory of gravity”, Gen. Rel. Grav. 13, 175–187 (1981). 34. E.W. Mielke, B. Fuchs, and F.E. Schunck: “Dark matter halos as Bose-Einstein condensates”, Proc. of the Tenth Marcel Grossman Meeting on General Relativity, Rio de Janeiro, 2003, M. Novello, S. Perez-Bergliaffa, and R. Ruffini, eds. (World Scient., Sing. 2006), p. 39–58. 35. E.W. Mielke and F.W. Hehl, “Comment on: General relativity without the metric”, Phys. Rev. Lett. 67, 1370 (1991). 36. E.W. Mielke and H.H. Peralta, “Eigenvalues of the third order slow-roll approximation of inflation”, Phys. Rev. D 66, 123505 (2002). 37. E.W. Mielke and H.H. Peralta, “Flattened halos in a nontopological soliton model of dark matter”, Phys. Rev. D 70, 123509 (2004). 38. E.W. Mielke and A.A. Rincón Maggiolo, “Duality in Yang’s theory of gravity”, Gen. Rel. http://arxiv.org/abs/astro-ph/0212485 http://arxiv.org/abs/cond-mat/0211110 September 20, 2021 4:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in MG11chair10 20 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck Grav. 37, 997–1007 (2005). 39. E.W. Mielke and E.S. Romero (2006), “Cosmological evolution of a torsion-induced quin- taxion”, Phys. Rev. D 73, 043521 (2006). 40. E.W. Mielke and F.E. Schunck, “Reconstructing the inflaton potential for an almost flat COBE spectrum”, Phys. Rev. D 52, 672 (1995). 41. E.W. Mielke and F.E. Schunck, “Boson stars: Alternatives to primordial black holes?”, Nucl. Phys. B 564, 185 (2000). 42. E.W. Mielke and F.E. Schunck, “Are axidilaton stars massive compact halo objects?”, Gen. Rel. Grav. 33, 805–813 (2001). 43. E.W. Mielke and F.E. Schunck, “Non-topological scalar soliton as dark matter halo”, Phys. Rev. D 66, 023503 (2002). 44. E.W. Mielke, F.E. Schunck, and H.H. Peralta, “Scalar soliton in Newtonian gravity modelling dark matter halos”, Gen. Rel. Grav. 34, 1919–1930 (2002). 45. E.W. Mielke and J.A. Vélez Pérez, “Toroidal halos in a nontopological soliton model of dark matter”, Phys. Rev. D 75, 043504 (2007). 46. J. Milnor: Lectures on Morse Theory, Ann. Math. Studies No. 51 (Princeton University Press, Princeton, New York, 1963). 47. F. Mueller-Hoissen, “From Chern-Simons to Gauss-Bonnet”, Nucl. Phys. B 346, 235 (1990). 48. S. Nojiri and S.D. Odintsov, “Modified f(R) gravity consistent with realistic cosmology: From matter dominated epoch to dark energy universe”, Phys. Rev. D 74, 086005 (2006). 49. O. Obregon, L.A. Urena-Lopez, and F.E. Schunck, “Oscillatons formed by nonlinear gravity,” Phys. Rev. D 72, 024004 (2005). 50. R.D. Peccei, “Reflections on the strong CP problem”, Nucl. Phys. Proc. Suppl. 72, 3 (1999). 51. C. Savage, K. Freese, and W.H. Kinney, “Natural inflation: Status after WMAP 3-year data”, Phys. Rev. D 74, 123511 (2006). 52. R. Schimming and H.J. Schmidt, “On the history of fourth order metric theories of gravita- tion”, NTM Schriftenr. Gesch. Naturw. Tech. Med. 27, 41 (1990). 53. F.E. Schunck, “A scalar field matter model for dark halos of galaxies and gravitational redshift”, astro-ph/9802258. 54. F.E. Schunck, B. Fuchs, and E.W. Mielke, “Scalar field haloes as gravitational lenses”, Mon. Not. R. Astron. Soc. 369, 485–491 (2006). 55. F.E. Schunck, F.V. Kusmartsev, and E.W. Mielke, “Dark matter problem and effective curvature Lagrangians”, Gen. Rel. Grav. 37, 1427–1433 (2005). 56. F.E. Schunck and E.W. Mielke, “A new method of generating exact inflationary solutions”, Phys. Rev. D 50, 4794 (1994). 57. F.E. Schunck and E.W. Mielke, “Eigenvalues of the Stewart–Lyth equation for inflation with a blue spectrum”, Phys. Lett. B 485, 231–238 (2000). 58. F.E. Schunck and E.W. Mielke, “TOPICAL REVIEW: General relativistic boson stars”, Class. Quantum Grav. 20, R301 (2003). 59. F.E. Schunck and O. Obregón, “Self-gravitating complex scalar fields conformally trans- formed into higher-order gravity: Inflation and boson stars”, preprint (1997). 60. D.N. Spergel and P.J. Steinhardt, “Observational evidence for self-interacting cold dark matter”, Phys. Rev. Lett. 84, 3760 (2000). 61. D.N. Spergel et al., “First-year Wilkinson microwave anisotropy probe (WMAP) Observa- tions: Determination of cosmological parameters”, Astrophys. J. Suppl. 148, 175 (2003); “Wilkinson microwave anisotropy probe (WMAP) three year results: Implications for cos- mology”, arXiv:astro-ph/0603449. 62. A.A. Starobinsky, “A new type of isotropic cosmological models without singularity”, Phys. Lett. 91B, 99 (1980). 63. R.C. Tolman: Relativity, Thermodynamics, and Cosmology (Clarendon Press, Oxford 1934). 64. D. Vassiliev, “Pseudoinstantons in metric affine field theory”, Gen.Rel.Grav. 34, 1239 (2002). 65. C. Wetterich, “Conformal fixed point, cosmological constant, and quintessence”, Phys. Rev. Lett. 90, 231302 (2003). 66. K. Zioutas, D.H.H. Hoffman, K. Dennerl, and Th. Papaevangelou, “What is dark matter made of?”, Science 306, 1485 (2004). http://arxiv.org/abs/astro-ph/9802258 http://arxiv.org/abs/astro-ph/0603449

0704.1136

Stability of Chaplygin gas thin-shell wormholes

Stability of Chaplygin gas thin–shell wormholes Ernesto F. Eiroa1,∗, Claudio Simeone2,† 1 Instituto de Astronomı́a y F́ısica del Espacio, C.C. 67, Suc. 28, 1428, Buenos Aires, Argentina 2 Departamento de F́ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pab. I, 1428, Buenos Aires, Argentina November 6, 2018 Abstract In this paper we construct spherical thin–shell wormholes supported by a Chaplygin gas. For a rather general class of geometries we introduce a new approach for the stability analy- sis of static solutions under perturbations preserving the symmetry. We apply this to worm- holes constructed from Schwarzschild, Schwarzschild–de Sitter, Schwarzschild–anti de Sitter and Reissner–Nordström metrics. In the last two cases, we find that there are values of the param- eters for which stable static solutions exist. PACS number(s): 04.20.Gz, 04.40.Nr, 98.80.Jk Keywords: Lorentzian wormholes; exotic matter; Chaplygin gas 1 Introduction Traversable Lorentzian wormholes [1] are solutions of the equations of gravitation associated to a nontrivial topology of the spacetime: their basic feature is that they connect two regions (of the same universe or two separate universes [1, 2]) by a throat. For the case of static wormholes the throat is defined as a minimal area surface satisfying a flare-out condition [3]. To fulfill this, wormholes must be threaded by exotic matter that violates the null energy condition [1–4]; it was shown by Visser et al [5], however, that the amount of exotic matter needed around the throat can be made as small as desired by means of an appropriate choice of the geometry of the wormhole. A well studied class of wormholes is that of thin–shell ones, which are constructed by cutting and pasting two manifolds [2,6] to form a geodesically complete new one with a shell placed in the joining surface. This makes such wormholes of particular interest because the exotic matter needed for the existence of the configuration is located only at the shell. Stability analysis of thin–shell wormholes under perturbations preserving the original symmetries has been widely developed. A linearized analysis of a thin–shell wormhole made by joining two Schwarzschild geometries was performed by Poisson and Visser in Ref. [7]. Later, the same method was applied to wormholes constructed using branes with negative tensions in Ref. [8], and the case of transparent spherically symmetric thin–shells and wormholes was studied in Ref. [9]. The linearized stability analysis was extended to Reissner–Nordström thin–shell geometries in Ref. [10], and to wormholes with a cosmological constant in Ref. [11]. The case of dynamical thin–shell wormholes was considered in e-mail: [email protected] e-mail: [email protected] http://arxiv.org/abs/0704.1136v2 Ref. [12]. The stability and energy conditions for five dimensional thin–shell wormholes in Einstein– Maxwell theory with a Gauss–Bonnet term were studied in Ref. [13], while thin–shell wormholes associated with cosmic strings have been treated in Refs. [14]. Other related works can be found in Refs. [15]. The requirement of matter violating energy conditions relates the study of wormholes to mod- ern cosmology: Current day observational data seem to point towards an accelerated expansion of the universe [16]. If General Relativity is assumed as the right gravity theory describing the large scale behavior of the universe, this implies that its energy density ρ and pressure p should violate the strong energy condition. Several models for the matter leading to such situation have been proposed [17]. One of them is the Chaplygin gas [18], a perfect fluid fulfilling the equation of state pρ = −A, where A is a positive constant. A remarkable property of the Chaplygin gas is that the squared sound velocity v2s = A/ρ 2 is always positive even in the case of exotic matter. Though introduced for purely phenomenological reasons (in fact, not related with cosmology [19]), such an equation of state has the interesting feature of being derivable from string theory; more precisely, it can be obtained from the Nambu–Goto action for d−branes moving in (d+2)−dimensional space- time if one works in the light-cone parametrization [20]. Besides, an analogous equation of state, but with A a negative constant, was introduced for describing cosmic strings with small structure (“wiggly” strings) [21]. Models of exotic matter of interest in cosmology have already been considered in wormhole con- struction. Wormholes supported by “phantom energy” (with equation of state p = ωρ, ω < −1) have been studied in detail [22]. A generalized Chaplygin gas, with equation of state pρα = −A (0 < α ≤ 1), has been proposed by Lobo in Ref. [23] as the exotic matter supporting a wormhole of the Morris–Thorne type [1]; there, as a possible way to keep the exotic matter within a finite region of space, matching the wormhole metric to an exterior vacuum metric was proposed. If, instead, a thin–shell wormhole is constructed, exotic matter can be restricted from the beginning to the shell located at the joining surface. In the present paper we study spherically symmetric thin–shell wormholes with matter in the form of the Chaplygin gas (the generalized Claplygin gas introduces a new constant and non trivial complications in the equations, which possibly cannot be solved in an analytical way). We introduce a new approach for the study of the stability under radial perturbations. The more complex case of the stability analysis of thin–shell wormholes under perturbations that do not preserve the symmetry has not been addressed in previous works, even for simpler metrics and equations of state, so we consider it beyond the scope of this article. In Section 2 we apply the Darmois–Israel formalism to the cut and paste construction of a generic wormhole with the Chaplygin equation of state imposed on the matter of the shell. In Section 3 we perform a detailed analysis of the stability under spherically symmetric perturbations. In Section 4 we analyze the specific cases of the Schwarzschild and the Reissner–Nordström geometries, and we also consider the inclusion of a cosmological constant of arbitrary sign. In Section 5 the results are discussed. We adopt units such that c = G = 1. 2 Wormhole construction Let us consider a spherically symmetric metric of the form ds2 = −f(r)dt2 + f(r)−1dr2 + r2(dθ2 + sin2 θdϕ2), (1) where r > 0 is the radial coordinate, 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π are the angular coordinates, and f(r) is a positive function from a given radius. For the construction of the thin–shell wormholes, we choose a radius a, take two identical copies of the region with r ≥ a: M± = {Xα = (t, r, θ, ϕ)/r ≥ a}, (2) and paste them at the hypersurface Σ ≡ Σ± = {X/F (r) = r − a = 0}, (3) to create a new manifold M = M+ ∪ M−. If the metric (1) has an event horizon with radius rh, the value of a should be greater than rh, to avoid the presence of horizons and singularities. This construction produces a geodesically complete manifold, which has two regions connected by a throat with radius a, where the surface of minimal area is located and the condition of flare- out is satisfied. On this manifold we can define a new radial coordinate l = ± 1/f(r)dr representing the proper radial distance to the throat, which is situated at l = 0; the plus and minus signs correspond, respectively, to M+ and M−. We follow the standard Darmois-Israel formalism [24,25] for its study and we let the throat radius a be a function of time. The wormhole throat Σ is a synchronous timelike hypersurface, where we define coordinates ξi = (τ, θ, ϕ), with τ the proper time on the shell. The second fundamental forms (extrinsic curvature) associated with the two sides of the shell are: K±ij = −n ∂ξi∂ξj , (4) where n±γ are the unit normals (n γnγ = 1) to Σ in M: n±γ = ± . (5) Working in the orthonormal basis {eτ̂ , eθ̂, eϕ̂} (eτ̂ = eτ , eθ̂ = a −1eθ, eϕ̂ = (a sin θ) −1eϕ), for the metric (1) we have that = K±ϕ̂ϕ̂ = ± f(a) + ȧ2, (6) τ̂ τ̂ = ∓ f ′(a) + 2ä f(a) + ȧ2 , (7) where a prime and the dot stand for the derivatives with respect to r and τ , respectively. Defining ] ≡ K+ , K = tr[Kı̂̂] = [K ı̂] and introducing the surface stress-energy tensor Sı̂̂ = diag(σ, p , pϕ̂) we obtain the Einstein equations on the shell (the Lanczos equations): − [Kı̂̂] +Kgı̂̂ = 8πSı̂̂, (8) which in our case correspond to a shell of radius a with energy density σ and transverse pressure p = p = pϕ̂ given by σ = − 1 f(a) + ȧ2, (9) 2aä+ 2ȧ2 + 2f(a) + af ′(a) f(a) + ȧ2 . (10) The equation of state for a Chaplygin gas has the form , (11) where A is a positive constant. Replacing Eqs. (9) and (10) in Eq. (11), we obtain 2aä+ 2ȧ2 − 16π2Aa2 + 2f(a) + af ′(a) = 0. (12) This is the differential equation that should be satisfied by the throat radius of thin–shell wormholes threaded by exotic matter with the equation of state of a Chaplygin gas. 3 Stability of static solutions From Eq. (12), the static solutions, if they exist, have a throat radius a0 that should fulfill the equation − 16π2Aa20 + 2f(a0) + a0f ′(a0) = 0, (13) with the condition a0 > rh if the original metric has an event horizon. The surface energy density and pressure are given in the static case by σ = − f(a0) , (14) 2πAa0 f(a0) . (15) The existence of static solutions depends on the explicit form of the function f . To study the stability of the static solutions under perturbations preserving the symmetry it is convenient to rewrite a(τ) in the form a(τ) = a0[1 + ǫ(τ)], (16) with ǫ(τ) ≪ 1 a small perturbation. Replacing Eq. (16) in Eq. (12) and using Eq. (13), we obtain (1 + ǫ)ǫ̈+ ǫ̇2 − 8π2A(2 + ǫ)ǫ+ g(a0, ǫ) = 0, (17) where the function g is defined by g(a0, ǫ) = 2f(a0 + a0ǫ) + a0(1 + ǫ)f ′(a0 + a0ǫ) 2f(a0) + a0f ′(a0) . (18) Defining ν(τ) = ǫ̇(τ), Eq. (17) can be written as a set of first order differential equations ǫ̇ = ν 8π2A(2 + ǫ)ǫ− g(a0, ǫ) 1 + ǫ 1 + ǫ . (19) Taylor expanding to first order in ǫ and ν we have ǫ̇ = ν ν̇ = ∆ǫ, (20) where ∆ = 16π2A− (a0, 0) = 16π 3f ′(a0) + a0f ′′(a0) , (21) which, by defining and M = , (22) can be put in the matrix form ξ̇ = Mξ. (23) If ∆ > 0 the matrix M has two real eigenvalues: λ1 = − ∆ < 0 and λ2 = ∆ > 0. The presence of an eigenvalue with positive real part makes this case unstable. As the imaginary parts of the eigenvalues are zero, the instability is of saddle type. When ∆ = 0, we have λ1 = λ2 = 0, and to first order in ǫ and ν we obtain ν = constant = ν0 and ǫ = ǫ0 + ν0(τ − τ0), so the static solution is unstable. If ∆ < 0 there are two imaginary eigenvalues λ1 = −i |∆| and λ2 = i |∆|; in this case the linear system does not determine the stability and the set of nonlinear differential equations should be taken into account. For the analysis of the ∆ < 0 case, we can rewrite Eq. (19) in polar coordinates (ρ, γ), with ǫ = ρ cos γ and ν = ρ sin γ, and make a first order Taylor expansion in ρ, which gives ρ̇ = sin γ cos γ(1 + ∆)ρ γ̇ = ∆cos2 γ − sin2 γ + h(γ)ρ, (24) where h(γ) is a bounded periodic function of γ. For small values of ρ, i.e. close to the equilibrium point, the time derivative of the angle γ is negative (the leading term ∆cos2 γ− sin2 γ is negative), then γ is a monotonous decreasing function of time, so the solution curves rotate clockwise around the equilibrium point. To see that these solution curves are closed orbits for small ρ, we take a time τ1 so that (ǫ(τ1), ν(τ1)) = (ǫ1, 0) with ǫ1 > 0. As the solution curve passing through (ǫ1, 0) rotates clockwise around (0, 0), there will be a time τ2 > τ1 such that the curve will cross the ǫ axis again, in the point (ǫ(τ2), ν(τ2)) = (ǫ2, 0), with ǫ2 < 0. As the Eq. (19) is invariant under the transformation composed of a time inversion τ → −τ and the inversion ν → −ν, the counterclockwise curve beginning in (ǫ1, 0) should cross the ǫ axis also in (ǫ2, 0). Therefore, for ∆ < 0 the solution curves of Eq. (19) should be closed orbits near the equilibrium point (0, 0), which is a stable center. The only stable static solutions with throat radius a0 are then those which have ∆ < 0, and they are not asymptotically stable, i.e. when perturbed the throat radius oscillates periodically around the equilibrium radius, without settling down again. 4 Application to different geometries In this section we analyze wormholes constructed with different metrics with the form of Eq. (1). 4.1 Schwarzschild case For the Schwarzschild metric we have that f(r) = 1− 2M , (25) where M is the mass. This geometry has an event horizon situated at rh = 2M , so the radius of the wormhole throat should be taken greater than 2M . The surface energy density and pressure 0 0.5 1 1.5 2 AM2H´10-3L Figure 1: Solutions of Eq. (28) as functions of AM2. For AM2 < (54π2)−1 there are three real roots, two of them positive au (full black line) and as (full grey line), and one negative a (dashed line); when AM2 = (54π2)−1 the two positive roots merge into a double one; and if AM2 > (54π2)−1 there is only one negative real root. for static solutions are given by σ = − a0 − 2M , (26) a0 − 2M , (27) with the throat radius a0 that should satisfy the cubic equation: 8π2Aa30 − a0 +M = 0. (28) Using that A > 0 and M > 0, it is not difficult to see that this equation has one negative real root and two non real roots if AM2 > (54π2)−1, one negative real root and one double positive real root if AM2 = (54π2)−1, and one negative real root and two positive real roots if AM2 < (54π2)−1. The solutions of cubic equation (28), plotted in Fig. 1, are given by1 −1− i 6AM + i 1− 54π2AM2 6AM + i 1− 54π2AM2 , (29) as0 = −1 + i 1 + i 6AM + i 1− 54π2AM2 6AM + i 1− 54π2AM2 , (30) the notation for the roots will be clear below au0 = 6AM + i 1− 54π2AM2 6AM + i 1− 54π2AM2 , (31) where the powers of complex numbers give the principal value. The negative root a has no physical meaning, so if AM2 > (54π2)−1 there are no static solutions. When AM2 = (54π2)−1 the positive double real solution of Eq. (28) is a0 = 3M/2 < rh, then no static solutions are present. For AM2 < (54π2)−1, the positive roots of Eq. (28) are M < as ≤ 3M/2 and 3M/2 ≤ au ; the first one is always smaller than rh, thus it has to be discarded, and the second one is greater than rh if AM2 < α0, where α0 ≈ 1.583× 10−3 < (54π2)−1 (obtained numerically). To study the stability of , we calculate ∆ for this case: ∆ = 16π2A− , (32) which, with the help of Eq. (28), can be simplified to give: (2a0 − 3M). (33) Then, using that au > 3M/2, it is easy to see that ∆ is always a positive number so, following Sec. 3, the static solution is unstable (saddle equilibrium point). Briefly, for the Schwarzschild metric if AM2 ≥ α0 no static solutions are present, and if AM2 < α0 there is only one unstable static solution with throat radius au 4.2 Schwarzschild–de Sitter case For the Schwarzschild–de Sitter metric the function f has the form f(r) = 1− r2, (34) where Λ > 0 is the cosmological constant. If ΛM2 > 1/9 we have that f(r) is always negative, so we take 0 < ΛM2 ≤ 1/9. In this case the geometry has two horizons, the event and the cosmological ones, which are placed, respectively, at −1 + i 3− (1 + i ΛM + i 1− 9ΛM2 ΛM + i 1− 9ΛM2 , (35) ΛM + i 1− 9ΛM2 ΛM + i 1− 9ΛM2 . (36) The event horizon radius rh is a continuous and increasing function of Λ, with rh(Λ → 0+) = 2M and rh(ΛM 2 = 1/9) = 3M , and the cosmological horizon radius rc is a continuous and decreasing function of Λ, with rc(Λ → 0+) → +∞ and rc(ΛM2 = 1/9) = 3M . If 0 < ΛM2 < 1/9 the wormhole throat radius should be taken in the range rh < a0 < rc, and if ΛM 2 = 1/9 the construction of the wormhole is not possible, because rh = rc = 3M . Using Eqs. (14) and (15), we obtain that the energy density and the pressure at the throat are given by σ = − + 3a0 − 6M , (37) -2 -1 0 1 2 M2H´10-3L Figure 2: Solutions of Eq. (39) as functions of ÃM2, where à = A + Λ/(12π2). We have that à is always positive or it has any sign, depending on Λ > 0 or Λ < 0, respectively. When à ≤ 0, there is only one real (and positive) root as (full grey line); for 0 < ÃM2 < (54π2)−1 there are three real roots, two of them positive au (full black line) and as (full grey line), and one negative (dashed line); when ÃM2 = (54π2)−1 the two positive roots merge into a double one; and if ÃM2 > (54π2)−1 there is only one negative real root. + 3a0 − 6M . (38) The throat radius a0 should satisfy in this case the cubic equation a30 − a0 +M = 0. (39) If we define à = A + Λ/(12π2), which in this case is a positive number, it is easy to see that Eq. (39) has the same form as Eq. (28), so the solutions of Eq. (39) are given again by Eqs. (29-31), with A replaced by Ã. These solutions are shown in Fig. 2 (the part of the plot with à > 0). With the same arguments of Sec. 4.1, if ÃM2 ≥ (54π2)−1 we have no static solutions. Using that 2M < rh < 3M , when ÃM 2 < (54π2)−1, as is always smaller than rh, so it has to be discarded, and au can be greater, equal or smaller than rh, depending on the values of the parameters. To study the stability of the static solution with throat radius au (if present) we obtain for this metric that ∆ is again given by Eq. (33). Using that au > 3M/2, it is straightforward to see that ∆ is a positive number, therefore, following Sec. 3, the static solution is unstable (saddle equilibrium point). 4.3 Schwarzschild–anti de Sitter case In the Schwarzschild–anti de Sitter metric the function f(r) is given again by Eq. (34), but now with a negative Λ. The event horizon for this metric is placed at |Λ|M + 1 + 9|Λ|M2 |Λ|M + 1 + 9|Λ|M2 . (40) -2 -1 0 1 2 M2H´10-3L Figure 3: Schwarzschild–anti de Sitter case: number of static solutions for given parameters M , Λ < 0 and à = A+Λ/(12π2). Region I: no static solutions; region II: one static unstable solution; region III: two static solutions, one stable and the other unstable; region IV: one static stable solution. The horizon radius rh is a continuous and increasing function of Λ, with values in the interval 0 < rh < 2M , with rh(Λ → −∞) = 0 and rh(Λ → 0−) = 2M . The wormhole throat radius a0 should be greater than rh. The energy density and the pressure at the throat are given by Eqs. (37) and (38), with the throat radius a0 satisfying Eq. (39). Using à = A+ Λ/(12π 2), which now can be positive, zero or negative, we have that Eq. (39) takes again the form of Eq. (28) with A replaced by Ã. Its solutions are shown in Fig. 2. For the stability analysis, it is not difficult to see that ∆ is again given by Eq. (33). Then we have five possible situations, depending on the different values of Ã: 1. When ÃM2 > (54π2)−1 there are no positive real solutions of Eq. (39), so we have no static solutions. 2. When ÃM2 = (54π2)−1, Eq. (39) has one positive double real root au = 3M/2 for which ∆ = 0, then following Sec. 3 it is unstable. For the existence of the unstable static solution a large enough value of |Λ| is needed, so that au > rh. 3. When 0 < ÃM2 < (54π2)−1, the solutions of Eq. (39) are given by Eqs. (29-31), with A replaced by à (see Fig. 2). Using that au > 3M/2 and M < as < 3M/2, it follows from Sec. 3 that the static solution with throat radius au is unstable (saddle equilibrium point) and the solution with radius as is stable (center). This stable static solution exists if |Λ| is large enough so that as > rh. 4. When ÃM2 = 0, Eq. (39) has only one real solution given by as = M . The associated wormhole solution exists if |Λ| is large enough so that M > rh and in this case it is stable (center) because ∆ is negative. 5. When ÃM2 < 0, we have that Eq. (39) has only one real solution given by as0 = 6|Ã|M + 1 + 54π2|Ã|M2 6|Ã| 6|Ã|M + 1 + 54π2|Ã|M2 , (41) which is an increasing function of à and lies in the range 0 < as < M (see Fig. 2). Then we have that ∆ < 0 and the solution is stable. Again, |Λ| should be large enough to have > rh. The number of static solutions for the different values of the parameters is shown in Fig. 3. When ÃM2 < (54π2)−1 there are static stable solutions if |Λ| is large enough so the condition of > rh is satisfied, which corresponds to the regions III and IV of Fig. 3. 4.4 Reissner–Nordström case The Reissner–Norsdtröm metric represents a charged object with spherical symmetry which has f(r) = 1− , (42) where Q is the charge. For |Q| < M this geometry has an inner and an outer (event) horizon given r± = M ± M2 −Q2, (43) if |Q| = M the two horizons merge into one, and when |Q| > M there are no horizons and the metric represents a naked singularity. When |Q| ≤ M the throat radius a0 should be taken greater than rh = r + so that no horizons are present in M. If |Q| > M the condition a0 > 0 assures that the naked singularity is removed. Replacing Eq. (42) in Eqs. (14) and (15), we obtain the energy density and pressure at the throat: σ = − − 2Ma0 +Q2 , (44) 2πAa2 − 2Ma0 +Q2 . (45) Replacing the metric (42) in Eq. (13), it is not difficult to see that the charge cancels out and the throat radius should satisfy the cubic equation (28) again, with its solutions given by Eqs. (29-31) and plotted in Fig. 1. As pointed out in Sec. 4.1 the number of roots of the cubic (28) are zero, one or two depending on if AM2 is, respectively, greater, equal or smaller than (54π2)−1. The solutions should satisfy that a0 > rh, then the number of static solutions will also depend on the value of the charge. As rh is a decreasing function of |Q|, for large values of charge there will be two static solutions with radius as and au . Following Sec. 3, the relevant quantity for the analysis of the stability is ∆, which is again given by Eq. (33). Using that when AM2 < (54π2)−1 we have < 3M/2 and au > 3M/2, it is easy to check that ∆ < 0 for as and ∆ > 0 for au , therefore is stable (center) and au is unstable (saddle). The special case where AM2 = (54π2)−1 and = 3M/2 is also unstable because ∆ = 0. The number of static solutions depends on the 0 0.5 1 1.5 2 2.5 AM2H´10-3L Figure 4: Reissner–Nordström case: number of static solutions for given parameters A, M and Q. Region I: no static solutions; region II: one static unstable solution; region III: two static solutions, one stable and the other unstable. values of the parameters A, M and Q. Using that au and as should be greater than rh, given by Eq. (43), and defining the functions 6AM + i 1− 54π2AM2 6AM + i 1− 54π2AM2 , (46) −1 + i 1 + i 6AM + i 1− 54π2AM2 6AM + i 1− 54π2AM2 , (47) it is easy to see that the three possible cases are: 1. No static solutions: when AM2 > (54π2)−1, or if α0 < AM 2 ≤ (54π2)−1 and |Q|/M ≤ α(2− α), where α0 ≈ 1.583 × 10−3. 2. One unstable static solution: when 0 ≤ AM2 < α0 and |Q|/M < β(2− β), or if α0 ≤ AM2 ≤ (54π2)−1 and α(2 − α) < |Q|/M ≤ β(2− β). 3. Two static solutions, one stable and the other unstable: when 0 ≤ AM2 < (54π2)−1 and |Q|/M > β(2− β). The three regions are plotted in Fig. 4. For |Q| > M and AM2 < (54π2)−1 there is always one stable static solution (center). Also one stable static solution can be obtained with values of |Q| smaller than M when AM2 is slightly smaller than (54π2)−1. These stable configurations correspond to values of the parameters within the region III of Fig. 4. A phase portrait of curves surrounding a static stable solution is shown in Fig. 5. -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 -0.015 -0.01 -0.005 0.005 0.015 Figure 5: Reissner–Nordström case: phase portrait around a stable static solution. The values of the parameters used in the plot are |Q|/M = 0.95 and AM2 = 1.85 × 10−3 and the radius of the throat is as /M = 1.406. The perturbation ǫ(τ) is defined by ǫ(τ) = a(τ)/as − 1 and ν(τ) = ǫ̇(τ) is its time derivative. The curves have different initial conditions and rotate clockwise. 5 Conclusions We have constructed spherical thin–shell wormholes supported by exotic matter fulfilling the Chaplygin gas equation of state. Such kind of exotic matter has been recently considered of particu- lar interest in cosmology as it provides a possible explanation for the observed accelerated expansion of the Universe. For the wormhole construction we have applied the usual cut and paste procedure at a radius greater than the event horizon (if it exists) of each metric. Then, by considering the throat radius as a function of time we have obtained a general equation of motion for the Chaplygin gas shell. We have addressed the issue of stability of static configurations under perturbations pre- serving the symmetry. The procedure developed has been applied to wormholes constructed from Schwarzschild, Schwarzschild–de Sitter, Schwarzschild–anti de Sitter and Reissner–Nordström ge- ometries. In the pure Schwarzschild case we have found that no stable static configurations exist. A similar result has been obtained for the case of Schwarzschild–de Sitter metric (positive cosmological constant). For the Schwarzschild–anti de Sitter geometry, we have found that the existence of static solutions requires that the mass M , the negative cosmological constant Λ and the positive constant A characterizing the Chaplygin fluid should satisfy ÃM2 ≤ (54π2)−1, with à = A+Λ/(12π2), and |Λ| great enough to yield a small horizon radius in the original manifold. If these conditions are verified, for each combination of the parameters, when 0 < ÃM2 < (54π2)−1 there is one stable solution with throat radius a0 in the range M < a0 < 3M/2, and for ÃM 2 ≤ 0 there is one stable configuration with a0 ≤ M . In the Reissner–Nordström case the existence of static solutions with charge Q requires AM2 ≤ (54π2)−1; we have found that if this condition is satisfied, then one stable configuration always exists if |Q|/M > 1. When |Q|/M is slightly smaller than one, there is also a stable static solution if AM2 is close to (54π2)−1. In this work, then, we have shown that if over-densities in the Chaplygin cosmological fluid had taken place, stable static configurations which represent traversable wormholes would be possible. Acknowledgments This work has been supported by Universidad de Buenos Aires and CONICET. Some calculations in this paper were done with the help of the package GRTensorII (which can be obtained freely at the address http://grtensor.org). References [1] M.S. Morris and K.S. Thorne, Am. J. Phys. 56, 395 (1988). [2] M. Visser, Lorentzian Wormholes (AIP Press, New York, 1996). [3] D. Hochberg and M. Visser, Phys. Rev. D 56, 4745 (1997). [4] D. Hochberg and M. Visser, Phys. Rev. Lett. 81, 746 (1998); D. Hochberg and M. Visser, Phys. Rev D 58, 044021 (1998). [5] M. Visser, S. Kar and N. Dadhich, Phys. Rev. Lett. 90, 201102 (2003). [6] M. Visser, Phys. Rev. D 39, 3182 (1989); M. Visser, Nucl. Phys. B328, 203 (1989). [7] E. Poisson and M. Visser, Phys. Rev. D 52, 7318 (1995). [8] C. Barceló and M. Visser, Nucl. Phys. B584, 415 (2000). [9] M. Ishak and K. Lake, Phys. Rev. D 65, 044011 (2002). [10] E.F. Eiroa and G.E. Romero, Gen. Relativ. Gravit. 36, 651 (2004). [11] F.S.N. Lobo and P. Crawford, Class. Quantum Grav. 21, 391 (2004). [12] F.S.N. Lobo and P. Crawford, Class. Quantum Grav. 22, 4869 (2005). [13] M. Thibeault, C. Simeone and E. F. Eiroa, Gen. Relativ. Gravit. 38, 1593 (2006). [14] E. F. Eiroa and C. Simeone, Phys. Rev. D 70, 044008 (2004); C. Bejarano, E. F. Eiroa and C. Simeone, Phys. Rev. D 75, 027501 (2007). [15] E. F. Eiroa and C. Simeone, Phys. Rev. D 71, 127501 (2005); F. Rahaman, M. Kalam and S. Chakraborty, Gen. Relativ. Gravit. 38, 1687 (2006); F. Rahaman, M. Kalam, and S. Chakraborti, gr-qc/0611134; E. Gravanis and S. Willison, Phys. Rev. D 75, 084025 (2007); F. Rahaman, M. Kalam, K.A. Rahman and S. Chakraborti, gr-qc/0703143; G. Dotti, J. Oliva and R. Troncoso, Phys. Rev. D 75, 024002 (2007). [16] A. Riess et al., Astron. J. 116, 1009 (1998); S. J. Perlmutter et al., Astroph. J. 517, 565 (1999); N. A. Bahcall, J. P. Ostriker, S. J. Perlmutter and P. J. Steinhardt, Science 284, 1481 (1999). [17] V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. A9, 373 (2000); P. J. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003); T. Padmanabhan, Phys. Rep. 380, 235 (2003). [18] A. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. B511, 265 (2001); N. Bilić, G.B. Tupper and R.D. Viollier, Phys. Lett. B535, 17 (2002); M.C. Bento, O. Bertolami and A.A. Sen, Phys. Rev. D 66, 043507 (2002); for a review see V. Gorini, A. Kamenshchik, U. Moschella and V. Pasquier, gr-qc/0403062. http://grtensor.org http://arxiv.org/abs/gr-qc/0611134 http://arxiv.org/abs/gr-qc/0703143 http://arxiv.org/abs/gr-qc/0403062 [19] S. Chaplygin, Sci. Mem. Moscow Univ. Math. Phys. 21, 1 (1904); H.-S-Tien, J. Aeron. Sci. 6, 399 (1939); T. von Karman, J. Aeron. Sci. 8, 337 (1941). [20] J. M. Bordemann and J. Hoppe, Phys. Lett B317, 315 (1993). [21] T. Vachaspati and A. Vilenkin, Phys. Rev. Lett. 67, 1057 (1991); T. Vachaspati, Phys. Rev. D 45, 3487 (1992); A. Arazi and C. Simeone, Mod. Phys. Lett. A 15, 1369 (2000). [22] P. F. González-Dı́az, Phys. Rev. Lett. 93, 071301 (2004); S. V. Sushkov, Phys. Rev. D 71, 043520 (2005); V. Faraoni and W. Israel, Phys. Rev. D 71, 064017 (2005); F. S. N. Lobo, Phys. Rev. D 71, 084011 (2005); F. S. N. Lobo, Phys. Rev. D 71, 124022 (2005); F. Rahaman, M. Kalam, N. Sarker and K. Gayen, Phys. Lett. B633, 161 (2006); K. A. Bronnikov and J. C. Fabris, Phys. Rev. Lett. 96, 251101 (2006); K. A. Bronnikov and A. A. Starobinsky, JETP Lett. 85, 1 (2007); Pisma Zh. Eksp. Teor. Fiz. 85, 3 (2007). [23] F. S. N. Lobo, Phys. Rev. D 73, 064028 (2006). [24] N. Sen, Ann. Phys. (Leipzig) 73, 365 (1924); K. Lanczos, ibid. 74, 518 (1924); G. Darmois, Mémorial des Sciences Mathématiques, Fascicule XXV ch V (Gauthier-Villars, Paris, 1927) ; W. Israel, Nuovo Cimento 44B, 1 (1966); 48B, 463(E) (1967). [25] P. Musgrave and K. Lake, Class. Quantum Grav. 13, 1885 (1996). Introduction Wormhole construction Stability of static solutions Application to different geometries Schwarzschild case Schwarzschild–de Sitter case Schwarzschild–anti de Sitter case Reissner–Nordström case Conclusions

0704.1137

Lattice refining loop quantum cosmology, anisotropic models and stability

IGPG–07/4–2 Lattice refining loop quantum cosmology, anisotropic models and stability Martin Bojowald∗ Institute for Gravitational Physics and Geometry, The Pennsylvania State University, 104 Davey Lab, University Park, PA 16802, USA Daniel Cartin† Naval Academy Preparatory School, 197 Elliot Street, Newport, RI 02841 Gaurav Khanna‡ Physics Department, University of Massachusetts at Dartmouth, North Dartmouth, MA 02747 Abstract A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in in- homogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In general, this leads to a new fea- ture of dynamical difference equations which may not have constant step-size, posing new mathematical problems. It is discussed how such models can be evaluated and what lattice refinements imply for semiclassical behavior. Two detailed examples illustrate that stability conditions can put strong constraints on suitable refinement models, even in the absence of a fundamental Hamiltonian which defines changes of the underlying lattice. Thus, a large class of consistency tests of loop quantum grav- ity becomes available. In this context, it will also be seen that quantum corrections due to inverse powers of metric components in a constraint are much larger than they appeared recently in more special treatments of isotropic, free scalar models where they were artificially suppressed. 1 Introduction Loop quantum cosmology [1] was designed to test characteristic effects expected in the full framework of loop quantum gravity [2, 3, 4]. Implementing symmetries at the kinematical ∗e-mail address: [email protected] †e-mail address: [email protected] ‡e-mail address: [email protected] http://arxiv.org/abs/0704.1137v1 quantum level allows explicit treatments of the dynamical equations while preserving basic features such as the discreteness of spatial geometry [5]. (See also [6, 7, 8, 9, 10, 11] for recent work on symmetry reduction in quantum theories.) Indeed, several new, initially surprising results were derived in different applications in cosmology and black hole physics. By now many such models have been studied in detail. As the relation of dynamics to that of a possible full framework without symmetries is not fully worked out, detailed studies can be used to suggest improvements of the equa- tions for physically viable behavior. Comparing results with full candidates for quantum dynamics can then provide stringent self-consistency tests of the overall framework. It is to be seen if, and how, such alterations of quantization procedures naturally result from a full quantization. The first example of this type related to the stability behavior of solutions to the difference equations of isotropic loop quantum cosmology, which was studied in [12, 13] and was already restrictive for models with non-zero intrinsic curvature. Another limita- tion, realized early on [14], occurs in the presence of a positive cosmological constant Λ. In an exact isotropic model, the extrinsic curvature scale is given by k = ȧ = 8πGa2Λ/3 which, due to the factor of a2, can be large in a late universe although the local curva- ture scale Λ might be small. Extrinsic curvature plays an important role since in a flat isotropic model it appears in holonomies on which loop quantizations are based in such a way that only eiαk with α ∈ R can be represented as operators, but not k itself [15]. Large values of k would either require one to use extremely small α in the relevant operators, or imply unexpected deviations from classical behavior. In fact, holonomies as basic objects imply that the Hamiltonian constraint is quantized to a difference rather than differential equation [16] since k in the Hamiltonian constraint (as in the Friedmann equation) is not directly quantized but only exponentials eiαk. These are shift operators instead of differen- tial operators. For a large, semiclassical universe a Wheeler–DeWitt wave function should be a good approximation to the basic difference equation of loop quantum cosmology [17] which, in a representation as a function of the momentum p = a2 conjugate to k, would be oscillating on scales of the order (a Λ)−1. This scale becomes shorter and shorter in an expanding universe, eventually falling below the discreteness scale of the difference equa- tion of loop quantum cosmology. At such a point, discreteness of spatial geometry would become noticeable in the behavior of the wave function (independently of how physical observables are computed from it) although the universe should be classical. This does not pose a problem for the general formalism, because it only shows that the specific quantization of the exact isotropic model used reaches its limits. Physically, this can be understood as a consequence of a fixed spatial lattice being used throughout the whole universe evolution. Exponentials eiαk in isotropic models derive from holonomies he(A) = P exp( Aiaτiė adt) of the Ashtekar connection along spatial curves e. All the freedom contained in choosing edges to capture independent degrees of freedom of the full theory reduces, in isotropic models, to the single parameter α which suffices to separate isotropic connections through all functions eiαk. The parameter α, from the full perspective, is thus related to the edge length used in holonomies. Using a fixed and constant α is analogous to using only edges of a given coordinate length, as they occur, for instance, in a regular lattice. In the presence of a positive cosmological constant, for any α a value of k will then be reached such that eiαk differs strongly from iαk. From the lattice perspective, this means that the local curvature radius becomes comparable to or smaller than the fixed lattice scale corresponding to α. Such a fixed lattice ceases to be able to support all small-scale oscillations relevant for a semiclassical geometry. This is not problematic if it occurs in a quantum regime where dynamics is indeed expected to differ from the classical one, but it poses a problem in semiclassical regimes. A better treatment has to refer to changing lattices, which is not easy to implement in a straightforward quantization of purely homogeneous models. In a dynamical equation closer to what is expected from the full framework, lattice refinements would take place during the evolution since full Hamiltonian constraint operators generally create new ver- tices of a lattice state in addition to changing their edge labels [18, 19]. While k increases with increasing volume, the corresponding α decreases since the lattice is being refined all the time. For a suitable lattice refinement, the increase in k can be balanced by the decrease of α such that αk stays small and semiclassical behavior is realized for any macroscopic volume even with Λ > 0. This provides an interesting relation between the fundamental Hamiltonian, which is responsible for the lattice refinement, and semiclassical properties of models. Testing whether an appropriate balance between increasing k and lattice re- finements can be reached generically can thus provide stringent tests on the fundamental dynamics even without using a precise full Hamiltonian constraint operator. This feature of lattice refinements was not mimicked in the first formulations of loop quantum cosmology [20, 21, 14, 22, 15] since the main focus was to understand small-volume effects such as classical singularities [23, 24]. In this context, lattice refinements appear irrelevant because only a few action steps of the Hamiltonian, rather than long evolution, are sufficient to probe a singularity. By now, perturbative regimes around isotropic models have been formulated in loop quantum cosmology which are inhomogeneous and thus must take into account lattice states and, at least at an effective level, lattice refinements [9]. One special version, corresponding to lattices with a number of vertices growing linearly with volume in a specific way referring to the area operator, has been studied in detail in isotropic models with a free, massless scalar [25]. Although the complicated relation to a full, graph-changing Hamiltonian constraint is still not fully formulated, such models allow crucial tests of the local dynamics. While isotropic models can easily be understood in terms of wave functions on a 1- dimensional discrete minisuperspace in terms of oscillation lengths [26], anisotropic models with higher-dimensional minisuperspaces can be more subtle. In such models, limitations similar to that of a cosmological constant have been observed as possible instabilities of solutions in classical regions or the lack of a sufficient number of semiclassical states [27, 28, 29]. For the partial difference equations of anisotropic models in loop quantum cosmology, stability issues can be much more severe than in isotropic models and thus lead to further consistency tests which might help to restrict possible quantization freedom (see, e.g., [30]). In this paper we therefore introduce the general setting of anisotropic models taking into account lattice refinements of Hamiltonian constraint operators, fo- cusing mainly on the anisotropic model which corresponds to the Schwarzschild interior. As we will see, the type of difference equations in general changes since they can become non-equidistant. This leads to new mathematical problems which we address here briefly, leaving further analysis for future work. The examples presented here already show that one can distinguish different refinement models by their stability properties. The refine- ment model corresponding to [25] turns out to give unstable evolution of the Schwarzschild interior, while a new version, whose vertex number also grows linearly with volume, is stable. Compared to isotropic models which are sensitive only to how the vertex number of a state changes with volume, anisotropic models allow one to test much more detailed properties. An appendix discusses subtleties in how homogeneous models faithfully represent inho- mogeneous states, mainly regarding the magnitude of corrections arising from quantizations of inverse metric components which often plays a large role in cosmological applications. 2 Difference equation for the Schwarzschild interior with varying discreteness scale Basic variables of a loop quantization are holonomies along lattice links and fluxes over transversal surfaces. For the Schwarzschild interior [31], the connection used for holonomies and the densitized triad used for fluxes take the form Aiaτidx a = c̃τ3dx+ (ãτ1 + b̃τ2)dϑ+ (−b̃τ1 + ãτ2) sinϑdϕ + τ3 cosϑdϕ (1) Eai τ = p̃cτ3 sin ϑ + (p̃aτ1 + p̃bτ2) sinϑ + (−p̃bτ1 + p̃aτ2) . (2) Coordinates (x, ϑ, ϕ) are adapted to the symmetry, with polar angles ϑ and ϕ along orbits of the rotational symmetry subgroup, and τj = − i2σj in terms of Pauli matrices. Spatial geometry is determined by the spatial line element, which in terms of the densitized triad components is ds2 = p̃2a + p̃ |p̃c| dx2 + |p̃c|dΩ2 (3) obtained from qab = Eai E i /| detEcj |. We will also use the co-triad eia, i.e. the inverse of eai = E | detEbj |, eiaτidx a = ecτ3dx+ (eaτ1 + ebτ2)dϑ+ (−ebτ1 + eaτ2) sinϑdϕ (4) with components sgnp̃c p̃2a + p̃ |p̃c| , eb = |p̃c| p̃b p̃2a + p̃ and ea = |p̃c| p̃a p̃2a + p̃ . (5) The phase space is spanned by the spatial constants (ã, b̃, c̃, p̃a, p̃b, p̃c) ∈ R6 with non- vanishing Poisson brackets {ã, p̃a} = γG/L0 , {b̃, p̃b} = γG/L0 , {c̃, p̃c} = 2γG/L0 where G is the gravitational constant and γ the Barbero–Immirzi parameter [32, 33]. Moreover, L0 is the size of a coordinate box along x used in integrating out the fields in d3xȦiaE ˙̃cp̃c + bp̃b + ˙̃ap̃a to derive the symplectic structure. The SU(2)-gauge transformations rotating a general triad are partially fixed to U(1) by demanding the x-component of Eai to point in the internal τ3-direction in (2). The U(1)-gauge freedom allows one to set ã = 0 = p̃a, still leaving a discrete residual gauge freedom (b̃, p̃b) 7→ (−b̃,−p̃b). The remaining variables can be rescaled by (b, c) := (b̃, Loc̃) , (pb, pc) := (Lop̃b, p̃c) . (6) to make the canonical structure L0-independent: {b, pb} = γG , {c, pc} = 2γG . (7) This rescaling is suggested naturally by holonomies, as written below, and fluxes which are considered the basic objects in loop quantizations. To express the elementary variables through holonomies, which unlike connection com- ponents will be promoted to operators, it suffices to choose curves along the x-direction of coordinate length τL0 and along ϑ of coordinate length µ since this captures all information in the two connection components, h(τ)x (A) = exp ∫ τLo dxc̃τ3 = cos + 2τ3 sin ϑ (A) = exp dϑb̃τ2 = cos + 2τ2 sin . (9) The quantum Hilbert space is then based on cylindrical states depending on the connection through countably many holonomies, which can always be written as almost periodic func- tions f(b, c) = µ,τ fµ,τ exp (µb+ τc) of two variables. These form the set of functions on the double product of the Bohr compactification of the real line, which is a compact Abelian group. Its Haar measure defines the inner product of the (non-separable) Hilbert space, in which states 〈b, c|µ, τ〉 = e i2 (µb+τc) µ, τ ∈ R . (10) form an orthonormal basis. Holonomies simply act by multiplication on these states, while densitized triad components become derivative operators p̂b = −iγℓ2P , p̂c = −2iγℓ2P using the Planck length ℓP = G~. They act as p̂b|µ, τ〉 = 12 γℓ P µ|µ, τ〉, p̂c|µ, τ〉 = γℓ2P τ |µ, τ〉 , (12) immediately showing their eigenvalues. To formulate the dynamical equation, one has to quantize the Hamiltonian constraint d3x ǫijk(−F kab + γ2Ωkab) EaiEbj | detE| where Ωkabτkdx a ∧ dxb = − sinϑτ3dϑ ∧ dϕ is the intrinsic curvature of 2-spheres, while F kab is the curvature computed from Aia ignoring the spin connection term sin ϑτ3dϕ. Following standard procedures a Hamiltonian constraint operator can be expressed in the basic op- erators. First, one replaces the inverse determinant of Eai by a Poisson bracket, following [19], ǫijkτ ajEbk | detE| = − 1 K∈{x,ϑ,ϕ} ǫabcωKc h (δ)−1 K , V } (14) with edge lengths ℓx0 = δL0 and ℓ 0 = δ, and left-invariant 1-forms ω c on the symmetry group manifold. For curvature components F kab one uses a holonomy around a closed loop F iab(x)τi = A(IJ) IJ − 1) + O((b2 + c2)3/2 A) (15) IJ = h −1 (16) and AIJ being the coordinate area of the loop, using the corresponding combinations of ℓI0. In these expressions, a parameter δ has been chosen which specifies the length of edges with respect to the background geometry provided by the symmetry group. Putting all factors together and replacing Poisson brackets by commutators, one has Ĥ(δ) = 2i(γ3δ3ℓ2P) −1 tr ǫIJK ĥ (δ)−1 (δ)−1 K [ĥ (δ)−1 K , V̂ ] + 2γ 2δ2τ3ĥ x [ĥ (δ)−1 x , V̂ ] = 4i(γ3δ3ℓ2P) 8 sin V̂ cos − cos δb V̂ sin 4 sin2 + γ2δ2 V̂ cos − cos δc V̂ sin which acts as Ĥ(δ)|µ, τ〉 = (2γ3δ3ℓ2P)−1 [2(Vµ+δ,τ − Vµ−δ,τ ) (18) ×(|µ+ 2δ, τ + 2δ〉 − |µ+ 2δ, τ − 2δ〉 − |µ− 2δ, τ + 2δ〉+ |µ− 2δ, τ − 2δ〉) + (Vµ,τ+δ − Vµ,τ−δ)(|µ+ 4δ, τ〉 − 2(1 + 2γ2δ2)|µ, τ〉+ |µ− 4δ, τ〉) on basis states. This operator can be ordered symmetrically, defining Ĥ symm := (Ĥ(δ) + Ĥ(δ)†), whose action is1 Ĥ(δ)symm|µ, τ〉 = (2γ3δ3ℓ2P)−1 [(Vµ+δ,τ − Vµ−δ,τ + Vµ+3δ,τ+2δ − Vµ+δ,τ+2δ)|µ+ 2δ, τ + 2δ〉 −(Vµ+δ,τ − Vµ−δ,τ + Vµ+3δ,τ−2δ − Vµ+δ,τ−2δ)|µ+ 2δ, τ − 2δ〉 −(Vµ+δ,τ − Vµ−δ,τ + Vµ−δ,τ+2δ − Vµ−3δ,τ+2δ)|µ− 2δ, τ + 2δ〉 +(Vµ+δ,τ − Vµ−δ,τ + Vµ−δ,τ−2δ − Vµ−3δ,τ−2δ)|µ− 2δ, τ − 2δ〉 (Vµ,τ+δ − Vµ,τ−δ + Vµ+4δ,τ+δ − Vµ+4δ,τ−δ)|µ+ 4δ, τ〉 −2(1 + 2γ2δ2)(Vµ,τ+δ − Vµ,τ−δ)|µ, τ〉 (Vµ,τ+δ − Vµ,τ−δ + Vµ−4δ,τ+δ − Vµ−4δ,τ−δ)|µ− 4δ, τ〉 . (19) Transforming this operator to the triad representation obtained as coefficients of a wave function |ψ〉 = µ,τ ψµ,τ |µ, τ〉 in the triad eigenbasis and using the volume eigenvalues Vµ,τ = 4π |(p̂c)µ,τ |(p̂b)µ,τ = 2π(γℓ2P)3/2 |τ |µ , a difference equation γ3/2δ3 (Ĥ(δ)symm|ψ〉)µ,τ = 2δ( |τ + 2δ|+ |τ |) (ψµ+2δ,τ+2δ − ψµ−2δ,τ+2δ) |τ + δ| − |τ − δ|) (µ+ 2δ)ψµ+4δ,τ − 2(1 + 2γ2δ2)µψµ,τ + (µ− 2δ)ψµ−4δ,τ |τ − 2δ|+ |τ |) (ψµ−2δ,τ−2δ − ψµ+2δ,τ−2δ) = 0 (20) results for physical states. (For small µ the equation has to be specialized further due to the remaining gauge freedom; see [31]. This is not relevant for our purposes.) 2.1 Relation to fixed lattices Although there are no spatial lattices appearing in the exactly homogeneous context fol- lowed here, the construction of the Hamiltonian constraint mimics that of the full theory. States are then associated with spatial lattices, and holonomies refer to embedded edges and loops. The parameter δ is the remnant of the loop size (in coordinates) used to act with holonomies on a spatial lattice. As one can see, this parameter is important for the resulting difference equation, determining its step-size. The above construction, using a constant δ, can be seen as corresponding to a lattice chosen once and for all such that the loop size is not being adjusted even while the total volume increases. As described in the 1Note that the first factor of 2 in the next-to-last line was missing in [31] and analogous places in subsequent formulas. This turns out to be crucial for the stability analysis below. In particular, with the corrected coefficient the quantization of the Schwarzschild interior in [31] is unstable for all values of γ. Possible restrictions on γ, as suggested in [30] based on a difference equation with the wrong coefficient, then do not follow easily but could be obtained from a more detailed analysis. introduction, this ignores the possible creation of new lattice vertices and links, and can be too rigid in certain semiclassical regimes. To express this clearly, we now construct holonomies which are not simply along a single edge of a certain length δ, but which are understood as holonomies along lattice links. We keep our coordinate box of size L0 in the x-direction as well as the edge length ℓ0. If this is a link in a uniform lattice, there are Nx = L0/ℓ0 lattice links in this direction, and a link holonomy appears in the form hx = exp(ℓ0c̃τ3) = exp(ℓ0cτ3/L0) = exp(cτ3/Nx) (21) when computed along whole lattice edges. Thus, a constant coefficient 1/Nx in holonomies corresponds to a fixed lattice whose number of vertices does not change when the volume increases. Lattice refinements of an inhomogeneous lattice state, on the other hand, can be mimicked by a parameterNx which depends on the phase space variables, most importantly the triad components. If this is carried through, as we will see explicitly below, the step-size of the resulting difference equation is not constant in the triad variables anymore. 2.2 Lattice refinements Let us now assume that we have a lattice with N vertices in a form adapted to the symmetry, i.e. there are Nx vertices along the x-direction (whose triad component pc gives rise to the label τ) and N 2ϑ vertices in spherical orbits of the symmetry group (whose triad component pb gives rise to the label µ). Thus, N = NxN 2ϑ . Since holonomies in such a lattice setting are computed along single links, rather than through all of space (or the whole cell of size L0), basic ones are hx = exp(ℓ 0 c̃τ3) and hϑ = exp(ℓ 0 b̃τ2), denoting the edge lengths by ℓ 0 and keeping them independent of each other in this anisotropic setting. Edge lengths are related to the number of vertices in each direction by ℓx0 = L0/Nx and ℓϑ0 = 1/Nϑ. With the rescaled connection components c = L0c̃ and b = b̃ we have basic holonomies hx = exp(ℓ 0 cτ3) = exp(cτ3/Nx) , hϑ = exp(ℓϑ0bτ2) = exp(bτ2/Nϑ) . (22) Using this in the Hamiltonian constraint operator then gives a difference equation whose step-sizes are 1/NI . So far, we only reinterpreted δ in terms of vertex numbers. We now turn our attention to solutions to the Hamiltonian constraint which, in the full theory, usually changes the lattice by adding new edges and vertices while triad eigenvalues increase. For larger µ and τ , the Hamiltonian constraint thus acts on a finer lattice than for small values, and the parameter N for holonomies appearing in the constraint operator is not constant on phase space but triad dependent. Due to the irregular nature of lattices with newly created vertices such a refinement procedure is difficult to construct explicitly. But it is already insightful to use an effective implementation, using the derivation of the Hamiltonian constraint for a fixed lattice, but assuming the vertex number N (µ, τ) to be phase space dependent. Moreover, we include a parameter δ as before, which now takes a value 0 < δ < 1 and arises because a graph changing Hamiltonian does not use whole lattice edges but only a fraction, given by δ.2 Effectively assuming in this way that the lattice size is growing through the basic action of the Hamiltonian constraint, we will obtain a difference equation whose step-size δ/N is not constant in the original triad variables. For the Schwarzschild interior, we have step sizes δ/Nϑ for µ and δ/Nx for τ . Go- ing through the same procedure as before, we end up with an operator containing flux- dependent holonomies instead of basic ones, e.g., Nx(µ, τ)hx = Nx(µ, τ) exp(cτ3/Nx(µ, τ)) which reduces to an Nx-independent connection component c in regimes where curvature is small. Keeping track of all prefactors and holonomies in the commutator as well as the closed loop, one obtains the difference equation C+(µ, τ) ψµ+2δNϑ(µ,τ)−1,τ+2δNx(µ,τ)−1 − ψµ−2δNϑ(µ,τ)−1,τ+2δNx(µ,τ)−1 +C0(µ, τ) (µ+ 2δNϑ(µ, τ)−1)ψµ+4δNϑ(µ,τ)−1,τ − 2(1 + 2γ 2δ2Nϑ(µ, τ)−2)µψµ,τ + (µ− 2δNϑ(µ, τ)−1)ψµ−4δNϑ(µ,τ)−1,τ +C−(µ, τ) ψµ−2δNϑ(µ,τ)−1,τ−2δNx(µ,τ)−1 − ψµ+2δNϑ(µ,τ)−1,τ−2δNx(µ,τ)−1 = 0 . (23) C±(µ, τ) = 2δNϑ(µ, τ)−1( |τ ± 2δNx(µ, τ)−1|+ |τ |) (24) C0(µ, τ) = |τ + δNx(µ, τ)−1| − |τ − δNx(µ, τ)−1| . (25) (A total factor NxN 2ϑ for the number of vertices drops out because the right hand side is zero in vacuum, but would multiply the left hand side in the presence of a matter term.) 3 Specific refinement models For further analysis one has to make additional assumptions on how exactly the lattice spacing is changing with changing scales µ and τ . To fix this in general, one would have to use a full Hamiltonian constraint and determine how its action balances the creation of new vertices with increasing volume. Instead of doing this, we will focus here on two geometrically motivated cases. Technically simplest is a quantization where the number of vertices in a given direction is proportional to the geometrical area of a transversal surface. Moreover, the appearance of transversal surface areas is suggested by the action of the full Hamiltonian constraint which, when acting with an edge holonomy, creates a new vertex along this edge (changing NI for this direction) and changes the spin of the edge (changing 2A precise value can be determined only if a precise implementation of the symmetry for a fixed full constraint operator is developed. Currently, both the symmetry reduction for composite operators and a unique full constraint operator are lacking to complete this program and we have to work with δ as a free parameter. This parameter is sometimes related to the lowest non-zero eigenvalue of the full area operator [15, 25]. From the inhomogeneous perspective of lattice states used here, however, there is no indication for such a relation. the area of a transversal surface). It also agrees with [25, 34], although the motivation in those papers, proposing to use geometrical areas rather than coordinate areas AIJ in (16), is different. Geometrically more intuitive is the case where the number of vertices in a given direction is proportional to the geometrical extension of this direction.3 The resulting difference equation will be more difficult to deal with due to its non-constant step-size, but naturally gives rise to Misner-type variables. This case will also be seen to have improved stability properties compared to the first one using areas. In both cases, N ∝ V is assumed, i.e. the lattice size increases proportionally to volume. This is not necessary in general, and we choose these two cases mainly for illustrative purposes. In fact, constant N as in [15] and N ∝ V first used in [25] are two limiting cases from the full point of view, the first one without creating new vertices and the second one without changing spin labels along edges since local lattice volumes V/N remain constant. In general, both spin changes and the creation of new vertices happen when acting with a Hamiltonian constraint operator. Thus, one expects N ∝ V α with some 0 < α < 1 to be determined by a detailed analysis of the full constraint and its reduction to a homogeneous model. Even assuming a certain behavior of N (V ) without analyzing the relation to a full constraint leaves a large field to be explored, which can give valuable consistency checks. We will not do this systematically in this paper but rather discuss a mathematical issue that arises in any such case: initially, one has to deal with difference equations of non-constant step-size which can be treated either directly or by tranforming a non-equidistant difference equation to an equidistant one. We first illustrate this for ordinary difference equations since partial ones, as they arise in anisotropic models, can often be reduced to this case. 3.1 Ordinary difference equations of varying step-size Let us assume that we have an ordinary difference equation for a function ψµ, which appears in the equation with µ-dependent increments ψµ+δN1(µ)−1 . To transform this to a fixed step- size, we introduce a new variable µ̃(µ) such that µ̃(µ+δ/N1(µ)) = µ̃(µ)+δµ̃′/N1(µ)+O(δ2) has a constant linear term in δ. (For the isotropic equation, N1 is the vertex number only in one direction. The total number of vertices in a 3-dimensional lattice is given by N = N 31 .) This is obviously satisfied if we choose µ̃(µ) := ∫ µN1(ν)dν. We then have ψµ+δ/N1(µ) = ψ̃µ̃(µ+δ/N1(µ)) = ψ̃µ̃+δ+ δiN (i−1)1 /N i1 = ψ̃µ̃+δ + ψ̃′ +O(δ3) (27) 3This behavior is introduced independently in [35] where “effective” equations, obtained by replacing connection components in the classical constraint by sines and cosines of such components according to how they occur in the quantized constraint, are analyzed for the Schwarzschild interior. The results are complementary to and compatible with our stability analysis of the corresponding difference equations below. We thank Kevin Vandersloot for discussions on this issue. where N (i)1 denotes the i-th derivative of N1. Thus, up to terms of order at least δ2 the new equation will be of constant step-size for the function ψ̃µ̃ := ψµ(µ̃). (The derivative ψ̃ µ̃ may not be defined for any solution to the difference equation. We write it in this form since such terms will be discussed below in the context of a continuum or semiclassical limit where derivatives would exist.) It is easy to see that, for refining lattices, the additional terms containing derivatives of the wave function are of higher order in ~ and thus correspond to quantum corrections. For N1(µ) ∝ µq as a positive power of µ, which is the expected case from lattice refinements related to the increase in volume, we have µN1(µ) 4πγℓ2P relating µ to an isotropic triad component p = 4πγℓ2Pµ/3 as it occurs in isotropic loop quantum gravity [14]. Moreover, ψ̃′ = N1(µ) = − i N1(µ) in terms of a curvature operator ĉ = 8πiγG~/3d/dp = 2id/dµ which exists in a continuum limit [17]. Thus, ψ̃′ ∝ )1+2q With q positive (or just larger than −1/2) for a refining lattice, there is a positive power of ~, showing that additional terms arising in the transformation are quantum corrections. This has two important implications. First, it shows that the correct classical limit is obtained if lattices are indeed refined, rather than coarsened, since q is restricted for corrections to appear in positive powers of ~. In anisotropic models, as we will see, the behavior is more complicated due to the presence of several independent variables. An analysis of the semiclassical limit can then put strong restrictions on the behavior of lat- tices. Secondly, we can implicitly define a factor ordering of the original constraint giving rise to the non-equidistant difference equation by declaring that all quantum correction terms arising in the transformation above should cancel out with factor ordering terms. We then obtain a strictly equidistant equation in the new variable µ̃. For example, a function N1(µ) = |µ| gives µ̃ ∝ |µ|3/2 such that the transformed difference equation will be equidistant in volume rather than the densitized triad component. For this special case, factor orderings giving rise to a precisely equidistant difference equation have been constructed explicitly in [25, 34]. 3.2 Number of vertices proportional to transversal area A simple difference equation results if the number of vertices is proportional to the transver- sal area in any direction.4 In the x-direction we have transversal surfaces given by symmetry 4Since this refers to the area, it is the case which agrees with the motivation of [25, 34]. orbits of area pc, using the line element (3), and thus Nx ∝ τ . Transversal surfaces for an angular direction are spanned by the x- and one angular direction whose area is pb, giving Nϑ ∝ µ. Each minisuperspace direction has a step-size which is not constant but independent of the other dimension. Moreover, due to the simple form one can transform the equation to constant step-size by using independent variables τ 2 and µ2 instead of τ and µ. Illustrating the general procedure given before, a function ψ̃τ2,µ2 acquires constant shifts under the basic steps, ψ̃(τ+nδ/τ)2 ,(µ+mδ/µ)2 = ψ̃τ2+2nδ+n2δ2/τ2,µ2+2mδ+m2δ2/µ2 = ψ̃τ2+2nδ,µ2+2mδ +O(τ −2) +O(µ−2) up to terms which can be ignored for large τ and µ. This is sufficient for a straightforward analysis in asymptotic regimes. Moreover, higher order terms in the above equation come with higher derivatives of the wave function in the form γ2ℓ4P ψ̃′ = −i(γℓ since q = 1 compared to the discussion in Sec. 3.1. Due to the extra factors of ~ (or even higher powers in further terms in the Taylor expansion) any additional term adding to the constant shift of ψ̃τ2,µ2 can be attributed to quantum corrections in a semiclassical limit. Accordingly, such terms can be avoided altogether by a judicious choice of the initial factor ordering of operators. 3.3 Number of vertices proportional to extension Geometrically more intuitive, and as we will see below dynamically more stable, is the case in which the number of vertices in each direction is proportional to the extension of that direction measured with the triad itself. This gives Nϑ ∝ |τ | and Nx ∝ µ/ |τ |, using the classical co-triad (4). (One need not worry about the inverse τ since the effective treatment of lattice refinements pursued here is not valid close to a classical singularity where an already small lattice with a few vertices changes. Singularities in general can only be discussed by a direct analysis of the resulting difference operators. Since only a few recurrence steps are necessary to probe the scheme around a classical singularity, equidistant difference operators are not essential in this regime. They are more useful in semiclassical regimes where one aims to probe long evolution times as in the examples below. Similar remarks apply to the horizon at µ = 0 which, although a classical region for large mass parameters, presents a boundary to the homogeneous model used for the Schwarzschild interior.) The behavior is thus more complicated than in the first case since the step size of any of the two independent variables depends on the other variable, too. First, it is easy to see, as before with quadratic variables, that the volume label ω = µ changes (approximately) equidistantly with each iteration step which is not equidistant for the basic variables µ and τ . But it is impossible to find a second, independent quantity which does so, too. In fact, such a quantity f(µ, τ) would have to solve two partial differential equations in order to ensure that f(µ+nδNϑ(µ, τ)−1, τ+mδNx(µ, τ)−1) ∼ f(µ, τ)+nδNϑ(µ, τ)−1∂µf(µ, τ)+mδNx(µ, τ)−1∂τf(µ, τ) changes only by a constant independent of τ and µ. This implies ∂µf(µ, τ) ∝ |τ | and ∂τf(µ, τ) ∝ µ/ |τ | whose only solution is f(µ, τ) ∝ µ |τ | which is the volume ω. We thus have to deal with non-equidistant partial difference equations in this case which in general can be complicated. A possible procedure to avoid this is to split the iteration in two steps since an ordinary difference equation can always be made equidistant as above (cancelling quantum corrections by re-ordering). We first transform τ to the volume variable ω which gives, up to quantum corrections, constant iteration steps for this variable. With the second variable still present, a higher order difference equation C0(µ, ω 2/µ2)(1 + 2δ/ω)µψµ(1+4δ/ω),ω+4δ + C+(µ, ω 2/µ2)ψµ(1+2δ/ω),ω+3δ −C−(µ, ω2/µ2)ψµ(1+2δ/ω),ω+δ − 2C0(µ, ω2/µ2)(1 + 2γ2δ2µ2/ω2)µψµ,ω −C+(µ, ω2/µ2)ψµ(1−2δ/ω),ω−δ + C−(µ, ω2/µ2)ψµ(1−2δ/ω),ω−3δ +C0(µ, ω 2/µ2)(1− 2δ/ω)µψµ(1−4δ/ω),ω−4δ = 0 (28) results with C0(µ, ω 2/µ2) = C±(µ, ω 2/µ2) = 2δ 1± 2δ derived from the original coefficients (24). The structure of this difference equation is quite different from the original one: not only is it of higher order, but now only one value of the wave function appears at each level of ω, rather than combinations of values at different values of µ. Note also that only the coefficient of the unshifted ψµ,ω depends on µ. This form of the difference equation is, however, a consequence of the additional rotational symmetry and is not realized in this form for fully anisotropic Bianchi models as we will see below. Proceeding with this specific case, we have to look at wave functions evaluated at shifted positions µ(1 + mδ/ω) with integer m. At fixed ω = ω0, we are thus evaluating the wave function at values of µ multiplied with a constant, instead of being shifted by a constant as in an equidistant difference equation. This suggests to use the logarithm of µ instead of µ itself as an independent variable, which is indeed the result of the general procedure. After having transformed from τ to ω already, we have to use τ as a function of µ and ω in the vertex number Nϑ, which is τ(µ, ω) = (ω/µ)2 after using ω = µ Thus, Nϑ(µ, τ(µ, ω)) = τ(µ, ω) = ω/µ now is not a positive power of the independent variable µ and we will have to be more careful in the interpretation of correction terms after performing the transformation. (The lattice is coarsened with increasing anisotropy at constant volume.) Naively applying the results of Sec. 3.1 to q = −1 would suggest that corrections come with inverse powers of ~ which would certainly be damaging for the correct classical limit. However, the factors change due to the presence of the additional variable ω0 even though it is treated as a constant. We have N ′ϑ/N 2ϑ = −1/ω0 = −(γℓ2P/2)3/2/V0 in terms of the dimensionful volume V , while it would just be a constant −1 without the presence of ω. The additional factor of ~3/2 ensures that corrections come with positive powers of ~ for the correct classical limit to be realized. For any ω0, we thus transform ψ̃µ(1+mδ/ω0) to equidistant form by using ψµ̃ = ψ̃µ(µ̃) with µ̃(µ) = log µ. This transformation is possible since the second label ω0 is now treated as a constant, rather than an independent variable of a partial difference equation. (Recall that for the type of difference equation discussed here there is only one variable, the volume, which is equidistant under all of the original discrete steps.) Despite of negative powers of some variables in the vertex numbers, we have the correct classical limit in the presence of ω. As before, the transformation is exact up to higher order terms which are quantum and higher order curvature corrections. Defining the original constraint operator ordering implicitly by the requirement that all those terms are cancelled allows us to work with an equidistant difference equation. 3.4 Bianchi models As mentioned before, the transformed difference equation does not become higher order for fully anisotropic Bianchi models. In this case, we have three independent flux labels µI , I = 1, 2, 3, and vertex numbers NI . Using vertex numbers proportional to the spatial extensions for each direction gives N1 = µ2µ3/µ1, N2 = µ1µ3/µ2 and N3 = µ1µ2/µ3. As in the difference equation for the Schwarzschild interior, the difference equation for Bianchi models [22] uses values of the wave function of the form ψµ1+2δ/N1,µ2+2δ/N2,µ3 . One can again see easily that the volume ω = |µ1µ2µ3| behaves equidistantly under the increments, ω(µ1 + 2δ/N1, µ2 + 2δ/N2, µ3) = µ1 + 2δ µ2 + 2δ µ1µ2µ3 + 4δ µ1µ2µ3 + 4δ2 = ω + 2δ +O(δ The leading order term of the difference equation in ω results from a combination C1ψµ1,µ2+2δ/N2,µ3+2δ/N3 + C2ψµ1+2δ/N1,µ2,µ3+2δ/N3 + C3ψµ1+2δ/N1,µ2+2δ/N2,µ3 ≈ C1ψ̃µ1,µ2+2δ/N2,ω+2δ + C2ψ̃µ1+2δ/N1,µ2,ω+2δ + C3ψ̃µ1+2δ/N1,µ2+2δ/N2,ω+2δ = C1ψ̃µ1,µ2(1+2δ/ω),ω+2δ + C2ψ̃µ1(1+2δ/ω),µ2 ,ω+2δ + C3ψ̃µ1(1+2δ/ω),µ2(1+2δ/ω),ω+2δ =: Ĉ+ψ̃ω+2δ(µ1, µ2) where we used 1/N1 = µ1/µ2µ3 = µ1/ω and defined the operator Ĉ+ acting on the dependence of ψ on µ1 and µ2. Thus, unlike for the Schwarzschild interior the difference equation does not become higher order in ω, and the highest order term does have a difference operator coefficient in the remaining independent variables. The recurrence proceeds as follows: We have a partial difference equation of the form Ĉ+ψ̃ω+2δ(µ1, µ2) + Ĉ0ψ̃ω(µ1, µ2) + Ĉ−ψ̃ω−2δ(µ1, µ2) with difference operators Ĉ± and Ĉ0 acting on the dependence on µ1 and µ2. In terms of initial data at two slices of ω we can solve recursively for Ĉ0ψ̃ω(µ1, µ2)+Ĉ−ψ̃ω−2δ(µ1, µ2) =: φ(µ1, µ2) and then, in each ω-step, use boundary conditions to solve the ordinary difference equation Ĉ+ψ̃ω+2δ(µ1, µ2) = φ(µ1, µ2) . Although the operator Ĉ+ itself is not equidistant, this remaining ordinary difference equation can be transformed to an equidistant one by transforming µ1 and µ2 as in Sec. 3.1 (using that ω is constant and fixed for this equation at any recursion step). With µ3(µ1, µ2, ω) = ω 2/µ1µ2, we have lattice spacings N1(µ1, µ2, ω) = ω/µ1 and N2(µ1, µ2, ω) = ω/µ2 in terms of ω which are already independent of each other. The two remaining vari- ables µ1 and µ2 are thus transformed to equidistant ones by taking their logarithms as encountered before. Note the resemblance of the new variables, volume and two logarithms a metric compo- nents at constant volume, to Misner variables [36]. This observation may be of interest in comparisons with Wheeler–DeWitt quantizations where Misner variables have often been used, making the Wheeler–DeWitt equation hyperbolic. 4 Application: Stability of the Schwarzschild interior Now that we have several possibilities for the lattice spacings, we consider their effect on the solutions of the Hamiltonian constraint. In particular, these solutions may have un- desirable properties reminiscent of numerical instabilities, as it was indeed noticed for the original quantization of the Schwarzschild interior in [28]. Also problems in the presence of a positive cosmological constant, described in the introduction, are of this type. Re- call that when one wishes to solve an ordinary differential equation, for example, there are various discrete schemes that ensure errors do not propagate as the number of time steps increases. Here we are in the opposite situation – instead of having the freedom to pick the discrete version of a continuous equation, the discrete equation itself is what is fundamental. Thus, like a badly chosen numerical recipe, some choices of the functions Nτ and Nϑ in the constraint equation may quickly lead to solutions that are out of control, and increase without bound. To test for this, we will use a von Neumann stability analy- sis [28] on the possible recursion relations. The essential idea is to treat one of the relation parameters as an evolution parameter, and decompose the rest in terms of orthogonal functions, representing “spatial” modes of the solution. This will give rise to a matrix that defines the evolution of the solution; if the matrix eigenvalues are greater than unity for a particular mode, that mode is unstable. In particular, a relation k=−M an+kψn+k = 0 is equivalent to a vector equation of the form ~vn = Q(n)~vn−1, where the column vector ~vn = (ψn+M , ψn+M−1, · · · , ψn−M+1)T . The evolution of an eigenvector ~w of the matrix Q(n) is given by ~wn = λw ~wn−1. Thus, when the size of the corresponding eigenvalue |λw| > 1, the values in the sequence associated to ~w will grow as well. With this in mind, we consider the choices of Nx and Nϑ discussed previously, starting with the case Nx = τ and Nϑ = µ. In the large µ, τ limit for this choice, the coefficients of the Hamiltonian constraint become C±(µ, τ) ∼ , C0(µ, τ) ∼ τ 3/2 In the asymptotic limit, the coefficients of the ψµ±2δ/µ,τ and ψµ,τ terms go to C0(µ, τ)µ. As we saw in Section 3.2, we can choose a different set of variables in which the step sizes are constant (up to ordering of the operators). Plugging these asymptotic values into the Hamiltonian constraint, and changing variables to µ̃ = µ2/2 and τ̃ = τ 2/2 gives 4τ̃(ψµ̃+2δ,τ̃+2δ − ψµ̃−2δ,τ̃+2δ + ψµ̃−2δ,τ̃−2δ − ψµ̃+2δ,τ̃−2δ) + µ̃(ψµ̃+4δ,τ̃ − 2ψµ̃,τ̃ + ψµ̃−4δ,τ̃) = 0. Because all the step sizes now are constants depending on δ, we define new parameters m,n such that µ̃ = 2mδ and τ̃ = 2nδ. Using m as our evolution parameter and n as the “spatial” direction, we decompose the sequence as ψ2mδ,2nδ = um exp(inω). With this new function, the recursion relation is written as 2in(un+1 − un−1)− (m sin θ)un = 0. This is equivalent to the vector equation [ − im sin θ 1 = Q(m,n) . (31) The eigenvalues of the matrix Q are −im sin θ ± 16n2 −m2 sin2 θ When the discriminant 16n2 − m2 sin2 θ ≥ 0, then |λ| = 1, and the solution is stable; however, there are unstable modes when 16n2 − m2 sin2 θ < 0. The most unstable mode corresponds to the choice sin θ = 1, giving instabilities in terms of the original variables when µ > 2τ . In this regime, all solutions behave exponentially rather than oscillating. This region includes parts of the classical solutions for the Schwarzschild interior even for values of µ and τ for which one expects classical behavior to be valid. The presence of instabilities implies, irrespective of the physical inner product, that quantum solutions in those regions cannot be wave packets following the classical trajectory, and the correct classical limit is not guaranteed for this quantization, which is analogous to that introduced in [25, 34]. The situation is different when we consider the choices Nx = |τ | and Nϑ = µ/ |τ |, where we will find a lack of instability. There is no choice of variables that allows us to asymptotically approach a constant spacing recursion relation, because of the mixing of the µ and τ variables in the step size functions. Thus, we will make the assumption that in the large µ, τ limit, the solution does not change much under step sizes δN−1x and δN To see how this affects the resulting stability of the solutions, we will look at a simpler example first. If we start with the Fibonacci relation Rτ ≡ ψτ+1 − ψτ − ψτ−1 = 0, then the two independent solutions are of the form ψτ = κ τ , where κ is either the golden ratio φ = (1 + 5)/2 or else −φ−1. Only the latter solution meets the criterion for stability, since |φ| > 1. When we change this relation to R̃τ ≡ ψτ+1/τn − ψτ − ψτ−1/τn = 0, (32) with n 6= 1, the situation changes – only one of the two solutions outlined above will solve the relation asymptotically. In particular, when we examine the error R̃τ we get when we plug κτ into the altered relation (32), i.e. R̃τ = κ τ (κ1/τ n − 1− κ−1/τn), the error is proportional to ψτ itself. As τ → ∞, therefore, the error for the κ = φ solution grows without bound, while that of κ = −φ−1 goes to zero. Thus, we see in this situation a relation between the stability and the asymptotic behavior of a solution. Returning to the Schwarzschild relation, in the large µ, τ limit the coefficient functions of the recursion relation are to leading order C±(µ, τ) ∼ 4δ, C0(µ, τ) ∼ In turn, the relation itself becomes 4(ψµ+2δ/ τ ,τ+2δ τ/µ − ψµ−2δ/√τ ,τ+2δ√τ/µ − ψµ+2δ/√τ ,τ−2δ√τ/µ + ψµ−2δ/√τ ,τ−2δ√τ/µ) +(ψµ+4δ/ τ ,τ − 2ψµ,τ + ψµ−4δ/√τ ,τ ) = 0. From this point on, we assume that we have a solution to this relation which does not vary greatly when, for example, µ is changed by ±2δ/√µ, and similarly for τ . Both Nx and Nϑ are constant to first order in shifts µ± 2δN−1x and similarly for τ , in the asymptotic limit. Thus, we assume that α = 2δN−1x and β = 2δN ϑ are constants, and use the scalings µ = αm and τ = βn. When this is done, we get an equation similar to the case when Nx = τ and Nϑ = µ, but with constant coefficients; this is the crucial difference that allows stable solutions to the case here. Using the decomposition ψαm,βn = un exp(imθ), we arrive at the matrix equation [ − i sin θ 1 . (33) The matrix here has eigenvalues λ with |λ| = 1 for all m,n, so the solution is stable. Using arguments as in the Fibonacci example, the non-equidistant equation of the second scheme is shown to be stable. 5 Conclusions Following [9], we explicitly introduced loop quantum cosmological models which take into account the full lattice structure of inhomogeneous states. Such lattices are in general refined by adding new vertices when acting with the Hamiltonian constraint. Thus, also dynamical equations even in homogeneous models should respect this property. Several interesting features arose: One obtains non-equidistant difference equations which, when imposed for functions on the whole real line as in isotropic loop quantum cosmology, are more restrictive than equidistant ones due to the absence of superselected sectors. This leaves the singularity issue unchanged since for this one only needs to consider a few steps in the equation. But a stability analysis of solutions and the verification of the correct classical limit in all semiclassical regimes can be more challenging. We presented an example for such an analysis, but also introduced a procedure by which one can transform the resulting equations to equidistant ones up to quantum corrections, which is sufficient for a semiclassical analysis. Interestingly, properties of the transformation itself provide hints to the correct semiclassical behavior. As a side-result, we demonstrated that one particular version of lattice refinements naturally gives rise to Misner-type variables. It is our understanding that this general procedure of defining lattice refining models mostly agrees with the intuition used specifically in isotropic models in [25], and adapted to anisotropic ones in [34].5 However, there are some departures from what is assumed in [25]. First, we do not see indications to refer to the area operator while the area spectrum was not only used in [25] to fix the constant δ and the volume dependence of the step size but in fact provided the main motivation. Secondly, due to this motivation [25] presents a more narrow focus which from our viewpoint corresponds to only one single refinement model. It has a vertex number proportional to volume, which is a limiting case not realized by known full Hamiltonian constraints, and puts special emphasis on geometrical areas to determine the vertex number. Finally, commutators for inverse volume operators are to be treated differently from [25], taking into account a lattice refining model which would not be possible in a purely homogeneous formulation. As shown in the appendix, this enlarges expected quantum corrections to the classical functions. We have discussed similar cases for illustration here, but keep a more general view- point on the refinement as a function of volume. A preliminary stability analysis for the Schwarzschild interior, consistent with [35] indeed suggests that a behavior different from what is suggested in [25] is preferred, which indicates that models can provide tight conditions for the general analysis of quantum dynamics. We emphasize that stability arguments as used here are independent of physical inner product issues since they refer to properties of general solutions. A general analysis as started here allows detailed tests of the full dynamics in manageable settings, which can verify the self-consistency of the framework of loop quantum gravity — or possibly point to limitations which need to be better understood. 5We thank A. Ashtekar for discussions of this point. Acknowledgements We thank Kevin Vandersloot for discussions. This work was supported in part by NSF grant PHY0554771. GK is grateful for research support from the University of Mas- sachusetts and Glaser Trust of New York. A Inverse volume terms in homogeneous models and lattice refinement We have seen in this paper, following [9], that Hamiltonian constraint operators with triad- dependent parameters in holonomies allow one to model lattice refinements faithfully, with interesting results and some improvements over the original non-refining models. However, as always there are also some features of inhomogeneous states and operators which are not present and difficult to mimic in homogeneous models. Thus, even models generalized in this way by allowing for lattice refinement effects have to be interpreted with great care. While qualitative effects can be investigated fruitfully to test the full framework, there is no basis for drawing quantitative conclusions. The prime example is that of commutator terms which appear ubiquitously in composite operators of loop quantum gravity, such as the coefficients of difference equations or also matter Hamiltonians. In the main construction of this paper we used holonomies associated with links of a lat- tice, rather than edges of a fixed coordinate length. This allows us, effectively, to take into account lattice refinements which change the number of vertices. It applies to holonomies (16) along a closed loop used to quantize curvature components which determine the step size of difference equations, but also to the link holonomies used in commutators to quan- tize inverse triad components based on (14). What is not modeled in homogeneous models is the fact that a lattice operator makes use of the local volume V̂v at a given vertex v where the commutator is acting, rather than the total volume V̂ = v V̂v of the whole box in which the lattice is embedded. In a fully inhomogeneous setting the difference does not matter since volume contributions from vertices not touched by the edge used in a commutator drop out in the end, [he, V̂ ] = v∈e[he, V̂v]. But in homogeneous models there is a difference since volume contributions from different vertices, in an exactly ho- mogeneous setting, are all identical. Thus, the total volume V = NVv is the number of vertices multiplied with the local volume Vv. Then, [h, V̂v] rather than [h, V̂ ] is expected as the contribution to constraint operators from the inhomogeneous perspective. In homoge- neous models as in [25], on the other hand, [h, V̂ ] is more straightforward to use. We now show that without corrections this would imply crucial deviations from the inhomogeneous behavior. It is easy to see that commutators differ depending on whether the local or total volume is used. For simplicity of the argument, we proceed with an isotropic situation where V = |p|3/2 in terms of the basic isotropic densitized triad component p. A local lattice flux, for a surface S intersecting only a single link, would be ρ = d2yp̃ = ℓ20p/L 0 = p/N 2/3 for links of coordinate length ℓ0, such that Vv = |ρ|3/2 = |p|3/2/N is the local volume. (We again use a coordinate box of size L20 and introduce rescaled flux variables p = L 0p̃.) Isotropic states are spanned by eiµc/2 where µ ∈ R is related to the triad eigenvalues by pµ = 4πγℓ Pµ/3. Using a link holonomy h ∼ eiℓ0c̃/2 = eic/2N which as a multiplication operator increases µ by 1/N 1/3, a commutator with the total volume will have eigenvalues of the form h−1[h, V̂ ] ∼ V (µ+1/N 1/3)−V (µ−1/N 1/3) = |p+4πγℓ2P/3N 1/3|3/2−|p−4πγℓ2P/3N 1/3|3/2 . If the local volume is used, on the other hand, we have to refer to local edge labels µe = 3ρ/4πγℓ2P, rather than using the total p. Thus, h−1e [he, V̂v] ∼ Vv(µe + 1)− Vv(µe − 1) = |ρ+ 4πγℓ2P/3|3/2 − |ρ− 4πγℓ2P/3|3/2 = N−1(|p+ 4πγℓ2PN 2/3/3|3/2 − |p− 4πγℓ2PN 2/3/3|3/2) . (35) For large volume, p ≫ N , both expressions give the correct classical limit 3 N−1/3 expected from {eic/2N1/3 , |p|3/2}. However, quantum corrections, i.e. deviations from this classical limit for finite p, are much larger for the second version using the local volume as it would occur in an inhomogeneous quantization. The smooth classical function dV/dp in a Poisson bracket appears in discretized form by the large step-size N2/3 in (35) rather than the small one N−1/3 in (34). Perturbative corrections, derived by Taylor expanding the difference terms and keeping higher order corrections to the classical expression, are thus larger. (This can have cosmological implications [37, 38, 39].) Non-perturbative effects as observed for the inverse scale factor operator in isotropic loop quantum cosmology which has an upper bound at finite volume [40], start to arise for p ∼ N 2/3 when the local volume is used but only at the much smaller p ∼ N−1/3 for the total volume. Since only the local volume is relevant for inhomogeneous quantizations, quantum corrections from inverse volume operators can be large. Unfortunately, this effect is more difficult to mimic in exact homogeneous models unlike the behavior of holonomies under lattice refinements and has therefore been overlooked in [25]. The connection components appearing in holonomies can simply be divided by a function N of triad components to implement shrinking edges due to subdivision. This is not possible for the volume itself to use a local version in a homogeneous model since, if we would divide the total volume by the appropriate function of triad components, only a constant would remain for an N proportional to volume and the commutator would be zero. The only way to have this effect faithfully implemented in a homogeneous model is to use higher SU(2) representations for holonomies in commutators but not for holonomies used in the loop to quantize curvature components. (This is not possible if one writes the constraint as a single trace, tr(hαhe[h e , V̂ ]) but can easily be done using the equivalent form tr(τihα) tr(τihe[h e , V̂ ]). We emphasize that higher representations for commutators are advocated here only in exactly homogeneous models to mimic inhomogeneous effects. Fully inhomogeneous operators usually need not refer to higher representations.) In a rep- resentation of spin j, matrix elements of holonomies contain exponentials exp(imc/2N 1/3) with −j ≤ m ≤ j, which increases the shifts in volume labels resulting from commutators. Resulting expressions for commutators can be found in [41, 42]. If the representation label j is of the order N , effects as they result from lattice refinements and using the local vol- ume are correctly implemented. Accordingly, corrections from inverse triad components quantized through commutators are much larger than they would otherwise be.6 References [1] M. Bojowald, Loop Quantum Cosmology, Liv- ing Rev. Relativity 8 (2005) 11, [gr-qc/0601085], http://relativity.livingreviews.org/Articles/lrr-2005-11/ [2] C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge, UK, 2004 [3] A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A status report, Class. Quantum Grav. 21 (2004) R53–R152, [gr-qc/0404018] [4] T. Thiemann, Introduction to Modern Canonical Quantum General Relativity, [gr-qc/0110034] [5] M. Bojowald and H. A. Kastrup, Symmetry Reduction for Quantized Diffeomor- phism Invariant Theories of Connections, Class. Quantum Grav. 17 (2000) 3009–3043, [hep-th/9907042] [6] M. Bojowald, Spherically Symmetric Quantum Geometry: States and Basic Opera- tors, Class. Quantum Grav. 21 (2004) 3733–3753, [gr-qc/0407017] [7] M. Bojowald, H. H. Hernández, and H. A. Morales-Técotl, Perturbative degrees of freedom in loop quantum gravity: Anisotropies, Class. Quantum Grav. 23 (2006) 3491–3516, [gr-qc/0511058] [8] J. Engle, Quantum field theory and its symmetry reduction, Class. Quant. Grav. 23 (2006) 2861–2893, [gr-qc/0511107] [9] M. Bojowald, Loop quantum cosmology and inhomogeneities, Gen. Rel. Grav. 38 (2006) 1771–1795, [gr-qc/0609034] [10] T. Koslowski, Reduction of a Quantum Theory, [gr-qc/0612138] [11] J. Engle, On the physical interpretation of states in loop quantum cosmology, [gr-qc/0701132] 6This is the reason why inverse triad corrections did not play a large role in recent numerical studies of the so-called µ̄-quantization, as repeatedly emphasized in that context [43, 25, 44, 45]. This is only the case because this quantization is a hybrid version of lattice refinements which are taken into account for holonomies but not for the local volume used in commutators. Inverse triad corrections are thus artificially, though subtly, suppressed in those quantizations, departing from quantizations of inhomogeneous models. http://arxiv.org/abs/gr-qc/0601085 http://relativity.livingreviews.org/Articles/lrr-2005-11/ http://arxiv.org/abs/gr-qc/0404018 http://arxiv.org/abs/gr-qc/0110034 http://arxiv.org/abs/hep-th/9907042 http://arxiv.org/abs/gr-qc/0407017 http://arxiv.org/abs/gr-qc/0511058 http://arxiv.org/abs/gr-qc/0511107 http://arxiv.org/abs/gr-qc/0609034 http://arxiv.org/abs/gr-qc/0612138 http://arxiv.org/abs/gr-qc/0701132 [12] M. Bojowald and K. Vandersloot, Loop quantum cosmology, boundary proposals, and inflation, Phys. Rev. D 67 (2003) 124023, [gr-qc/0303072] [13] M. Bojowald and G. Date, Consistency conditions for fundamentally discrete theories, Class. Quantum Grav. 21 (2004) 121–143, [gr-qc/0307083] [14] M. Bojowald, Isotropic Loop Quantum Cosmology, Class. Quantum Grav. 19 (2002) 2717–2741, [gr-qc/0202077] [15] A. Ashtekar, M. Bojowald, and J. Lewandowski, Mathematical structure of loop quantum cosmology, Adv. Theor. Math. Phys. 7 (2003) 233–268, [gr-qc/0304074] [16] M. Bojowald, Loop Quantum Cosmology IV: Discrete Time Evolution, Class. Quan- tum Grav. 18 (2001) 1071–1088, [gr-qc/0008053] [17] M. Bojowald, The Semiclassical Limit of Loop Quantum Cosmology, Class. Quantum Grav. 18 (2001) L109–L116, [gr-qc/0105113] [18] C. Rovelli and L. Smolin, The physical Hamiltonian in nonperturbative quantum gravity, Phys. Rev. Lett. 72 (1994) 446–449, [gr-qc/9308002] [19] T. Thiemann, Quantum Spin Dynamics (QSD), Class. Quantum Grav. 15 (1998) 839–873, [gr-qc/9606089] [20] M. Bojowald, Loop Quantum Cosmology: I. Kinematics, Class. Quantum Grav. 17 (2000) 1489–1508, [gr-qc/9910103] [21] M. Bojowald, Loop Quantum Cosmology III: Wheeler-DeWitt Operators, Class. Quantum Grav. 18 (2001) 1055–1070, [gr-qc/0008052] [22] M. Bojowald, Homogeneous loop quantum cosmology, Class. Quantum Grav. 20 (2003) 2595–2615, [gr-qc/0303073] [23] M. Bojowald, Absence of a Singularity in Loop Quantum Cosmology, Phys. Rev. Lett. 86 (2001) 5227–5230, [gr-qc/0102069] [24] M. Bojowald, Singularities and Quantum Gravity, In Proceedings of the XIIth Brazil- ian School on Cosmology and Gravitation, [gr-qc/0702144] [25] A. Ashtekar, T. Pawlowski, and P. Singh, Quantum Nature of the Big Bang: Improved dynamics, Phys. Rev. D 74 (2006) 084003, [gr-qc/0607039] [26] M. Bojowald, Dynamical Initial Conditions in Quantum Cosmology, Phys. Rev. Lett. 87 (2001) 121301, [gr-qc/0104072] [27] D. Cartin and G. Khanna, Absence of pre-classical solutions in Bianchi I loop quantum cosmology, Phys. Rev. Lett. 94 (2005) 111302, [gr-qc/0501016] http://arxiv.org/abs/gr-qc/0303072 http://arxiv.org/abs/gr-qc/0307083 http://arxiv.org/abs/gr-qc/0202077 http://arxiv.org/abs/gr-qc/0304074 http://arxiv.org/abs/gr-qc/0008053 http://arxiv.org/abs/gr-qc/0105113 http://arxiv.org/abs/gr-qc/9308002 http://arxiv.org/abs/gr-qc/9606089 http://arxiv.org/abs/gr-qc/9910103 http://arxiv.org/abs/gr-qc/0008052 http://arxiv.org/abs/gr-qc/0303073 http://arxiv.org/abs/gr-qc/0102069 http://arxiv.org/abs/gr-qc/0702144 http://arxiv.org/abs/gr-qc/0607039 http://arxiv.org/abs/gr-qc/0104072 http://arxiv.org/abs/gr-qc/0501016 [28] J. Rosen, J.-H. Jung, and G. Khanna, Instabilities in numerical loop quantum cos- mology, Class. Quantum Grav. 23 (2006) 7075–7084, [gr-qc/0607044] [29] G. Date, Pre-classical solutions of the vacuum Bianchi I loop quantum cosmology, Phys. Rev. D 72 (2005) 067301, [gr-qc/0505030] [30] D. Cartin and G. Khanna, Wave functions for the Schwarschild black hole interior, Phys. Rev. D 73 (2006) 104009, [gr-qc/0602025] [31] A. Ashtekar and M. Bojowald, Quantum Geometry and the Schwarzschild Singularity, Class. Quantum Grav. 23 (2006) 391–411, [gr-qc/0509075] [32] J. F. Barbero G., Real Ashtekar Variables for Lorentzian Signature Space-Times, Phys. Rev. D 51 (1995) 5507–5510, [gr-qc/9410014] [33] G. Immirzi, Real and Complex Connections for Canonical Gravity, Class. Quantum Grav. 14 (1997) L177–L181 [34] D.-W. Chiou, Loop Quantum Cosmology in Bianchi Type I Models: Analytical In- vestigation, Phys. Rev. D 75 (2007) 024029, [gr-qc/0609029] [35] D.-W. Chiou and K. Vandersloot, in preparation [36] C. W. Misner, Mixmaster Universe, Phys. Rev. Lett. 22 (1969) 1071–1074 [37] M. Bojowald, H. Hernández, M. Kagan, P. Singh, and A. Skirzewski, Formation and evolution of structure in loop cosmology, Phys. Rev. Lett. 98 (2007) 031301, [astro-ph/0611685] [38] M. Bojowald, H. Hernández, M. Kagan, P. Singh, and A. Skirzewski, Hamiltonian cosmological perturbation theory with loop quantum gravity corrections, Phys. Rev. D 74 (2006) 123512, [gr-qc/0609057] [39] M. Bojowald, H. Hernández, M. Kagan, and A. Skirzewski, Effective constraints of loop quantum gravity, Phys. Rev. D 75 (2007) 064022, [gr-qc/0611112] [40] M. Bojowald, Inverse Scale Factor in Isotropic Quantum Geometry, Phys. Rev. D 64 (2001) 084018, [gr-qc/0105067] [41] M. Bojowald, Quantization ambiguities in isotropic quantum geometry, Class. Quan- tum Grav. 19 (2002) 5113–5130, [gr-qc/0206053] [42] M. Bojowald, Loop Quantum Cosmology: Recent Progress, In Proceedings of the International Conference on Gravitation and Cosmology (ICGC 2004), Cochin, India, Pramana 63 (2004) 765–776, [gr-qc/0402053] [43] A. Ashtekar, T. Pawlowski, and P. Singh, Quantum Nature of the Big Bang: An An- alytical and Numerical Investigation, Phys. Rev. D 73 (2006) 124038, [gr-qc/0604013] http://arxiv.org/abs/gr-qc/0607044 http://arxiv.org/abs/gr-qc/0505030 http://arxiv.org/abs/gr-qc/0602025 http://arxiv.org/abs/gr-qc/0509075 http://arxiv.org/abs/gr-qc/9410014 http://arxiv.org/abs/gr-qc/0609029 http://arxiv.org/abs/astro-ph/0611685 http://arxiv.org/abs/gr-qc/0609057 http://arxiv.org/abs/gr-qc/0611112 http://arxiv.org/abs/gr-qc/0105067 http://arxiv.org/abs/gr-qc/0206053 http://arxiv.org/abs/gr-qc/0402053 http://arxiv.org/abs/gr-qc/0604013 [44] K. Vandersloot, Loop quantum cosmology and the k = −1 RW model, Phys. Rev. D 75 (2007) 023523, [gr-qc/0612070] [45] H.-H. Xiong and J.-Y. Zhu, Effect of the Inverse Volume Modification in Loop Quan- tum Cosmology, [gr-qc/0702002] http://arxiv.org/abs/gr-qc/0612070 http://arxiv.org/abs/gr-qc/0702002 Introduction Difference equation for the Schwarzschild interior with varying discreteness scale Relation to fixed lattices Lattice refinements Specific refinement models Ordinary difference equations of varying step-size Number of vertices proportional to transversal area Number of vertices proportional to extension Bianchi models Application: Stability of the Schwarzschild interior Conclusions Inverse volume terms in homogeneous models and lattice refinement

0704.1138

Testing Disk Instability Models for Giant Planet Formation

Testing Disk Instability Models for Giant Planet Formation Alan P. Boss Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road, NW, Washington, DC 20015-1305 [email protected] ABSTRACT Disk instability is an attractive yet controversial means for the rapid formation of giant planets in our solar system and elsewhere. Recent concerns regarding the first adiabatic exponent of molecular hydrogen gas are addressed and shown not to lead to spurious clump formation in the author’s disk instability models. A number of disk instability models have been calculated in order to further test the robustness of the mechanism, exploring the effects of changing the pressure equation of state, the vertical temperature profile, and other parameters affecting the temperature distribution. Possible reasons for differences in results obtained by other workers are discussed. Disk instability remains as a plausible formation mechanism for giant planets. Subject headings: solar system: formation – planetary systems 1. Introduction The disk instability mechanism for giant planet formation is based on the formation and survival of self-gravitating clumps of gas and dust in a marginally gravitationally un- stable protoplanetary disk (Boss 1997; reviewed by Durisen et al. 2007). In order for a disk instability to succeed, the disk must be able to cool its midplane as the clumps form, allowing them to continue to contract to higher densities, and the clumps must be able to survive indefinitely in the face of Keplerian shear, tidal forces, and internal thermal pressure. Considering the complicated physical processes involved during the time evolution of a three dimensional disk instability, it is perhaps not surprising that the theoretical basis for the disk instability hypothesis remains unclear even after a decade of work on the subject. The Indiana University (IU) group has been active in studying disk instabilities, and has generally found that disk instabilities are unable to lead to the formation of self-gravitating, http://arxiv.org/abs/0704.1138v1 – 2 – dense clumps that could go on to form gas giant protoplanets (e.g., Pickett et al. 2000; Mej́ıa 2004; Cai et al. 2006a,b; Boley et al. 2006, 2007a,b). On the other hand, the Washington-Zurich group (e.g., Mayer et al. 2002, 2004, 2007) has presented models that support the hypothesis that disk instabilities can lead to the formation of long-lived giant gaseous protoplanets. We present here several new calculations that attempt to understand the reasons for some of these different outcomes regarding disk instability. 2. Energy Equation of State Boley et al. (2007a) pointed out that uncertainties about the ortho/para ratio of molec- ular hydrogen at low temperatures might lead to differences in the outcome of disk insta- bility models. They suggested that a mixture intermediate between pure parahydrogen and a 3:1 ortho/para ratio might be the most appropriate choice. Boley et al. (2007a) also noted that discontinuities in the specific internal energy equation for molecular hydrogen could lead to artificially low values of the first adiabatic exponent for a simple perfect gas (Γ1 = γ = 1 + Rg/(µcV ), where Rg is the gas constant, µ is the mean molecular weight, and cV is the specific heat at constant volume; Cox & Giuli 1968). Values of γ ≤ 4/3 can lead to dynamical instabilities (either expansion or contraction) away from a configuration of hydrostatic equilibrium (Cox & Giuli 1968). Boley et al. (2007a) suggested that γ ≤ 4/3 could artificially lead to clump formation. Boley et al. (2007a) stated that the energy equation of state (EOS) used in disk instabil- ity models by Boss (2001, 2002b, 2005) contained a discontinuity that might be responsible for artificially lowering γ below 4/3, leading to spurious fragmentation. Boley et al. (2007a) based this assertion on the equations of state described by Boss (1984). However, all of the hydrodynamical models run by the present author since Boss (1989) have been based on a different energy equation than that reported by Boss (1984), as a result of a direct com- parison with the equation of state routines employed by Werner Tscharnuter. The models of Boss (1989) revised the energy equation treatment to include an interpolation between temperatures of 100 K and 200 K, but this revision was not explicitly stated in the Boss (1989) paper as it seemed insignificant at the time. Using the notation of Boss (1984), the specific internal energy for molecular hydrogen for temperatures T < 100K is taken to be E∗H2 = 3/2 RgT/µ, while for temperatures between 100K and 600K it is E∗H2 = 5/2 RgT/µ. For intermediate temperatures, 100K < T < 200K, the internal energy is interpolated according to the equation E∗H2 = 3/2 RgT/µ [1+2/3 (T − 100)/100]. The Erratum by Boley et al. (2007b) used this revised EOS. – 3 – Boley et al. (2007a,b) showed that for a pure parahydrogen or 3:1 ortho:para mix, γ decreases significantly at ∼ 100K, but does not fall below 4/3. Figure 1 depicts the behavior of γ calculated with the revised Boss equation of state and shows that γ drops below 4/3 for 135K < T < 200K. The densest clumps found in Boss’s disk instability models generally have maximum temperatures below 135K, and even lower mean temperatures. Boss (2005) presented a disk instability model with the highest spatial resolution computed to date, and found that the densest clump that formed had a maximum temperature of 120K and a mean temperature of 94K. Boss (2006c) found that clumps formed in disk instabilities around M dwarf protostars had maximum temperatures less than 100K. Boss’s (2002a) Table 1 showed maximum clump temperatures of 115K to 126K for a range of models with varied opacities. Perhaps the most important point raised by Boley et al. (2007a, b) is that γ is likely to decrease significantly around 100K, and this softening of the pressure EOS will enhance the formation of dense clumps, as happens in locally isothermal disks with an effective γ = 1. In fact, Figure 1 in Boley et al. (2007b) shows that the Boss γ is higher than that of either of the preferred hydrogen mixtures for T < 100K, implying that clump formation is suppressed somewhat in the Boss models as a result. 3. Pressure Equation of State Several new models have explored using the same pressure EOS as is used in the IU group models (e.g., Cai et al. 2006a,b; Boley et al. 2006): the gas pressure p is given by (γ − 1)ρE, where γ = 5/3, ρ is the gas density, and E is the specific internal energy of the gas. Three models were run with this pressure EOS, starting from the same initial disk model as model HR in Boss (2001) – a 0.091M⊙ disk orbiting a 1M⊙ protostar with an outer disk radius of 20 AU. The calculations were made with the same three dimensional, gravitational, radiative hydrodynamics code as in Boss (2001) and in all subsequent Boss disk instability calculations (see these earlier papers for more details about the calculational techniques and initial conditions). The models had Nφ = 256 and NY lm = 32, though, compared to Nφ = 512 and NY lm = 48 for model HR (Boss 2001). The three new models varied the choice for the critical disk density (ρcr = 10 −13, 10−12, or 10−11 g cm−3) below which the disk temperature was forced back to its initial value (typically 40K in the outer disk). This artifice was employed in Boss (2001) and subsequent models in order to maintain a reasonably large time step when low density regions develop in the disk that are undergoing decompressional cooling. In all three models, spiral arms and transient clumps form within 200 yrs of evolution, – 4 – similar to the behavior with the usual (Boss 1984) pressure EOS (e.g., Boss 2001). Relatively high density clumps form, with maximum densities similar to those in a calculation with the same spatial resolution but the usual pressure EOS. Evidently using the γ = 5/3 EOS does not alter the results in a significant way because the Boss EOS also has γ = 5/3 for T < 100K, the regime where clumps form. The choice of ρcr makes little difference as well, as was found by Boss (2006b) for disk instability models in binary star systems. These results suggest that the reason for differing outcomes must be sought elsewhere (see Discussion). 4. Varied Disk Temperature Parameters We now present a set of four models varying several of the parameters that could affect the temperature distribution in the disk models. Table 1 summarizes the four models, which are all variations on model HR of Boss (2001). However, these models all had the same spatial resolution (Nφ = 512) and number of terms in the spherical harmonic expansion for the gravitational potential (NY lm = 48) as model HR in Boss (2001). The parameters that were varied included: the temperature of the thermal bath (i.e., the envelope temperature Te), the critical density in the disk below which the temperature was reset to the initial disk temperature at that orbital radius (as in the previous section), the critical density in the envelope below which the gas was assumed to be at a temperature equal to the envelope temperature Te (for grid points at least 8 degrees above the disk midplane), and finally, whether the temperature was forced to decline monotonically (Boss 2002a) with vertical height inside the disk (mono) or not (free). The former constraint errs on the side of artificially cooling the disk by removing local temperature maxima in the vertical direction. All models started at a time of 322 yrs of evolution in model HR, and continued for at least another 17 yrs of evolution (∼ 1 clump orbital period). The results of these variations on the standard assumptions are shown in Figures 2 and 3. The variation that produced the largest deviation from the standard assumptions (model H in Figure 2) was relaxing the constraint on the monotonic vertical (more precisely, in the θ angle) decline of temperatures within the disk (model TZ in Figure 3), though even this model led to the formation of well-defined clumps that were no more than a factor of two less dense than in model H. Models T and TE led to evolutions that were very similar to that of model H and so are not shown. The models show that these three variations in the details of how the disk thermodynamics is treated in the Boss models are not particularly significant for the outcome of a disk instability, presumably because of the thermal bath assumption. – 5 – 5. Discussion There are a number of possible reasons for different outcomes compared to other groups: Spatial resolution – Clump formation is strongly enhanced as the spatial resolution in the critical azimuthal direction is increased from Nφ = 64 to 512 (Boss 2000). Boss (2005) presented a model with Nφ = 1024 and a locally refined radial grid (equivalent to a calculation with over 8 × 106 grid points) that implied that in the continuum limit, the outcome of a disk instability is the formation of dense, self-gravitating clumps. Cai et al. (2006a,b) calculated models with Nφ = 128, increasing Nφ to 512 for two models only after those models had entered a phase of evolution when nonaxisymmetry was no longer growing. Boley et al. (2006) similarly calculated models with Nφ = 128, increasing Nφ for some models to 512 for the earliest phase of evolution, leading to the formation of dense clumps at the intersections of spiral structures. The clumps disappeared in a fraction of an orbital period. Clumps typically last no more than an orbital period in even the highest spatial resolution models of Boss (2005). Boss (2005) thus used virtual protoplanets to allow the orbital evolution of these dense clumps to be followed further than is possible with even a high spatial resolution calculation with a fixed Eulerian grid code. Gravitational potential solver – The Boss models use a spherical harmonic (Ylm(θ, φ)) expansion to solve Poisson’s equation for the gravitational potential, with the accuracy of the resulting gravitational potential being strongly dependent on the number of terms (NY lm) carried along in the expansion. As we have seen, model HR in Boss (2001) used NY lm = 48. Boss (2000, 2001) found that increasing NY lm led to the formation of significantly denser clumps. Boss (2005) further explored the effects of using an enhanced gravitational potential solver by replacing some of the mass in the densest regions of a clump with a point mass at the center of the relevant grid cell, finding that this led to even better defined, higher density clumps. In comparison, the IU group uses a direct solution of Poisson’s equation, with a boundary potential employing terms up to l = m = 10 (Pickett et al. 2000), implying a limited ability to depict small-scale gravitational forces in a strongly nonaxisymmetric disk. Mej́ıa (2004) considered Fourier analysis of her disk models for m ≤ 6, while Boley et al. (2006) considered m ≤ 63, finding increasingly little power in modes with m > 10, possibly consistent with the cutoff at l = m = 10 in their boundary potential. Artificial viscosity – Pickett & Durisen (2007) found that the inclusion of certain artificial viscosity (AV) terms could enhance the survival of clumps formed in a disk instability, and suggested that AV could thus explain the long-lived clumps found in SPH calculations by Mayer et al. (2002, 2004, 2007). Pickett & Durisen (2007) further noted that even calculations without any explicit AV (e.g., the Boss models) should be considered suspect, given the intrinsic numerical viscosity of any numerical code. Boss (2006b) showed that with – 6 – a large amount of explicit AV, clump formation is suppressed, as was also found by Pickett & Durisen (2007). However, the level of intrinsic numerical viscosity in Boss code models with Nφ = 256 appears to be equivalent to an α-viscosity of ∼ 10 −4 or smaller (Boss 2004), a level that appears to be negligible in comparison to typical explicit AV levels. Considering that the virtual protoplanet (VP) models of Boss (2005) had Nφ = 256, the continued survival of the VP for at least 30 orbital periods in these models is not likely to have been affected by the intrinsic numerical viscosity of the Boss code. Stellar irradiation – Mej́ıa (2004) considered the effects of stellar irradiation on the surface of the disk as a means of heating the disk surface and thereby possibly suppressing clump formation. The Boss models assume that the disk is immersed in a thermal bath appropriate for backscattering from infalling dust grains in the protostellar envelope, with an envelope temperature appropriate for a protostar that is not undergoing an FU Orionis outburst (Chick & Cassen 1997). While the Boss models thus do not include the effects of direct irradiation by the central protostar, the dynamical evolution of a three dimensional disk leads to strongly variable vertical oscillations and structures (Mej́ıa 2004; Boley & Durisen 2006; Jang-Condell & Boss 2007) that are not considered in simple theoretical models of flared accretion disks. The highly corrugated inner disk surfaces (stretching at least 29 degrees above the disk midplane; Jang-Condell & Boss 2007) will shield the outer disk surfaces from the central protostar, eliminating this source of heating for much of the outer disk. Radiative transfer – Boley et al. (2006) have calculated disk instability models using the flux-limited diffusion approximation (FLDA) along with a detailed treatment of the transi- tion from the optically thick disk to the optically thin atmosphere of the disk. They suggest using a plane-parallel (one dimensional) atmosphere as a test case. Boss (2001) investigated the effects of using the FLDA instead of the standard diffusion approximation (DA) coupled with a thermal bath for low optical depths, but did not find any significant differences. My- hill & Boss (1993) showed the results of the fully three dimensional, standard nonisothermal test case for protostellar collapse, calculated with their two independent Eddington approx- imation (EA) codes, finding good agreement. Whitehouse & Bate (2006) found that their FLDA models of the standard nonisothermal collapse problem led to temperature profiles similar to those found by Myhill & Boss (1993), though with appreciably hotter gas tem- peratures where the optical depth was ∼ 2/3. They attribute this difference to the FLDA retarding the loss of radiation in these layers compared to the EA. Given that the choice of the flux limiter can have an effect on the outcome (Bodenheimer et al. 1990), it is unclear whether any particular implementation of the FLDA is superior to the EA. The standard Boss models have used the DA coupled with a thermal bath to force the DA models to mimic an EA calculation. – 7 – Numerical heating – In calculations by the IU group, nonaxisymmetric perturbations tend to grow rapidly for a certain period of time and then begin to damp out (e.g., Cai et al. 2006a,b). In the Boley et al. (2006) calculations, the disk starts out with a mass of 0.07 M⊙ and a radius of 40 AU, but then expands outward to a radius of ∼ 80 AU, leading to the formation of rings inside 20 AU and a gravitationally stable region outside 20AU with spiral arms that do not fragment. The latter behavior is roughly consistent with the models by Boss (2003), who studied disks extending from 10 AU to 30 AU, and found fragments to form at 20 AU but not at 30 AU. Similarly, Boss (2006a) studied disks extending from 100 AU to 200 AU, and found no evidence for fragmentation. Thus on scales larger than ∼ 20 AU, the results of Boley et al. (2006) and Boss (2003, 2006a) are in basic agreement. The disagreement arises about what happens in the inner disks. Fragmentation typically occurs at 8 to 10 AU in the Boss models, whereas the inner disk rings do not fragment in the Boley et al. (2006) models. Boley et al. (2006) state that their inner disk is stable to ring fragmentation because of “numerical heating” at distances out to ∼ 7 AU. This non-physical heating appears to have affected the models of Cai et al. (2006a,b) as well as those of Boley et al. (2006), making the IU results for inner disks difficult to accept. 6. Conclusions While there are a number of potential reasons for the differences in disk instability models calculated by the IU group and the present author, at this time the major sources of these differences would appear to be some combination of several effects, namely spatial resolution, gravitational potential solver accuracy, and numerical heating in the inner disk of the IU models. Handling of the boundary conditions for the disk’s radiative energy losses is another possibility that is still under investigation by the author, though the models presented here suggest that this may not be a dominant effect. Given the current state of knowledge, and the new results presented herein, it appears that reports of the death of the disk instability model for giant planet formation have been greatly exaggerated. I thank Aaron Boley, Kai Cai, and Megan Pickett for working with me to understand the differences between their group’s results and my results, and the referees for their remarks. This research was supported in part by NASA Planetary Geology and Geophysics grant NNG05GH30G and is contributed in part to NASA Astrobiology Institute grant NCC2-1056. The calculations were performed on the Carnegie Alpha Cluster, the purchase of which was partially supported by NSF Major Research Instrumentation grant MRI-9976645. – 8 – REFERENCES Bodenheimer, P., Yorke, H. W., Róz̀yczka, M., & Tohline, J. E. 1990, ApJ, 355, 651 Boley, A. C., & Durisen, R. H. 2006, ApJ, 641, 534 Boley, A. C., et al. 2006, ApJ, 651, 517 Boley, A. C., Hartquist, T. W., Durisen, R. H., & Michael, S. 2007a, ApJL, in press ——. 2007b, ApJL (Erratum), submitted Boss, A. P. 1984, ApJ, 277, 768 ——. 1989, ApJ, 346, 336 ——. 1997, Science, 276, 1836 ——. 2000, ApJL, 536, L101 ——. 2001, ApJ, 563, 367 ——. 2002a, ApJ, 567, L149 ——. 2002b, ApJ, 576, 462 ——. 2003, ApJ, 599, 577 ——. 2004, ApJ, 616, 1265 ——. 2005, ApJ, 629, 535 ——. 2006a, ApJL, 637, L137 ——. 2006b, ApJ, 641, 1148 ——. 2006c, ApJL, 643, 501 Cai, K., et al. 2006a, ApJL, 636, L149 ——. 2006b, ApJL (Erratum), 642, L173 Chick, K. M., & Cassen, P. 1997, ApJ, 477, 398 Cox, J. P., & Giuli, R. T. 1968, Principles of Stellar Structure (New York, Gordon and Breach) Durisen, R. H., et al. 2007, in Protostars and Planets V, B. Reipurth, D. Jewitt, & K. Keil, eds. (Tucson, Univ. Arizona Press), p. 607 Jang-Condell, H., & Boss, A. P. 2007, ApJL, in press Mayer, L, Quinn, T., Wadsley, J., & Stadel, J. 2002, Science, 298, 1756 ——. 2004, ApJ, 609, 1045 – 9 – ——. 2007, astro-ph/0606361 Mej́ıa, A. C. 2004, PhD thesis, Indiana Univ. Myhill, E. A., & Boss, A. P. 1993, ApJS, 89, 345 Pickett, B. K., Cassen, P., Durisen, R. H., & Link, R. 2000, ApJ, 529, 1034 Pickett, M. K., & Durisen, R. H. 2007, ApJL, 654, L155 Whitehouse, S. C., & Bate, M. R. 2006, MNRAS, 367, 32 This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0606361 – 10 – Table 1. Models with varied disk temperature parameters. model Te (K) ρcr (disk) ρcr (envelope) T (θ) H 50K 10−11 10−11 mono T 100K 10−13 10−11 mono TZ 50K 10−13 10−11 free TE 50K 10−13 10−13 mono – 11 – 1 1.5 2 2.5 3 Fig. 1.— Adiabatic exponent Γ1 = γ used in Boss (2001) and in all subsequent Boss disk instability models (solid line). The short-dashed line shows γ = 4/3, while the long-dashed line shows γ = 5/3, as used by Cai et al. (2006a,b) and Boley et al. (2006). – 12 – -1 -0.5 0 0.5 1 Fig. 2.— Equatorial density contours for model H after 339 yrs of evolution. The disk has an outer radius of 20 AU and an inner radius of 4 AU. Hashed regions denote clumps and spiral arms with densities higher than 10−10 g cm−3. Density contours represent factors of two change in density. – 13 – -1 -0.5 0 0.5 1 Fig. 3.— Same as Figure 2, but for model TZ. Introduction Energy Equation of State Pressure Equation of State Varied Disk Temperature Parameters Discussion Conclusions

0704.1139

High-dimensional variable selection

High-dimensional variable selection The Annals of Statistics 2009, Vol. 37, No. 5A, 2178–2201 DOI: 10.1214/08-AOS646 c© Institute of Mathematical Statistics, 2009 HIGH-DIMENSIONAL VARIABLE SELECTION By Larry Wasserman and Kathryn Roeder1 Carnegie Mellon University This paper explores the following question: what kind of statis- tical guarantees can be given when doing variable selection in high- dimensional models? In particular, we look at the error rates and power of some multi-stage regression methods. In the first stage we fit a set of candidate models. In the second stage we select one model by cross-validation. In the third stage we use hypothesis testing to eliminate some variables. We refer to the first two stages as “screen- ing” and the last stage as “cleaning.” We consider three screening methods: the lasso, marginal regression, and forward stepwise regres- sion. Our method gives consistent variable selection under certain conditions. 1. Introduction. Several methods have been developed lately for high- dimensional linear regression such as the lasso [Tibshirani (1996)], Lars [Efron et al. (2004)] and boosting [Bühlmann (2006)]. There are at least two different goals when using these methods. The first is to find models with good prediction error. The second is to estimate the true “sparsity pattern,” that is, the set of covariates with nonzero regression coefficients. These goals are quite different and this paper will deal with the second goal. (Some discussion of prediction is in the Appendix.) Other papers on this topic include Meinshausen and Bühlmann (2006), Candes and Tao (2007), Wainwright (2006), Zhao and Yu (2006), Zou (2006), Fan and Lv (2008), Meinshausen and Yu (2008), Tropp (2004, 2006), Donoho (2006) and Zhang and Huang (2006). In particular, the current paper builds on ideas in Mein- shausen and Yu (2008) and Meinshausen (2007). Let (X1, Y1), . . . , (Xn, Yn) be i.i.d. observations from the regression model Yi =X i β + εi,(1) Received June 2008; revised August 2008. 1Supported by NIH Grant MH057881. AMS 2000 subject classifications. Primary 62J05; secondary 62J07. Key words and phrases. Lasso, stepwise regression, sparsity. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2009, Vol. 37, No. 5A, 2178–2201. This reprint differs from the original in pagination and typographic detail. http://arxiv.org/abs/0704.1139v2 http://www.imstat.org/aos/ http://dx.doi.org/10.1214/08-AOS646 http://www.imstat.org http://www.ams.org/msc/ http://www.imstat.org http://www.imstat.org/aos/ http://dx.doi.org/10.1214/08-AOS646 2 L. WASSERMAN AND K. ROEDER where ε ∼ N(0, σ2), Xi = (Xi1, . . . ,Xip)T ∈ Rp and p = pn > n. Let X be the n × p design matrix with jth column X•j = (X1j , . . . ,Xnj)T and let Y = (Y1, . . . , Yn) T . Let D= {j :βj 6= 0} be the set of covariates with nonzero regression coefficients. Without loss of generality, assume that D = {1, . . . , s} for some s. A variable selection procedure D̂n maps the data into subsets of S = {1, . . . , p}. The main goal of this paper is to derive a procedure D̂n such that lim sup P(D̂n ⊂D)≥ 1− α,(2) that is, the asymptotic type I error is no more than α. Note that throughout the paper we use ⊂ to denote nonstrict set-inclusion. Moreover, we want D̂n to have nontrivial power. Meinshausen and Bühlmann (2006) control a different error measure. Their method guarantees lim supn→∞P(D̂n ∩ V 6= ∅)≤ α where V is the set of variables not connected to Y by any path in an undirected graph. Our procedure involves three stages. In stage I we fit a suite of candidate models, each model depending on a tuning parameter λ, S = {Ŝn(λ) :λ ∈ Λ}. In stage II we select one of those models Ŝn using cross-validation to select λ̂. In stage III we eliminate some variables by hypothesis testing. Schematically, stage I−→ S stage II−→︸ ︷︷ ︸ screen stage III−→ D̂n︸ ︷︷ ︸ clean Genetic epidemiology provides a natural setting for applying screen and clean. Typically, the number of subjects, n, is in the thousands, while p ranges from tens of thousands to hundereds of thousands of genetic fea- tures. The number of genes exhibiting a detectable association with a trait is extremely small. Indeed, for type I diabetes only ten genes have exhibited a reproducible signal [Wellcome Trust (2007)]. Hence, it is natural to assume that the true model is sparse. A common experimental design involves a 2- stage sampling of data, with stages 1 and 2 corresponding to the screening and cleaning processes, respectively. In stage 1 of a genetic association study, n1 subjects are sampled and one or more traits such as bone mineral density are recorded. Each subject is also measured at p locations on the chromosomes. These genetic covariates usually have two forms in the population due to variability at a single nu- cleotide and hence are called single nucleotide polymorphisms (SNPs). The distinct forms are called alleles. Each covariate takes on a value (0, 1 or 2) VARIABLE SELECTION 3 indicating the number of copies of the less common allele observed. For a well-designed genetic study, individual SNPs are nearly uncorrelated unless they are physically located in very close proximity. This feature makes it much easier to draw causal inferences about the relationship between SNPs and quantitative traits. It is standard in the field to infer that an association discovered between a SNP and a quantitative trait implies a causal genetic variant is physically located near the one exhibiting association. In stage 2, n2 subjects are sampled at a subset of the SNPs assessed in stage 1. SNPs measured in stage 2 are often those that achieved a test statistic that ex- ceeded a predetermined threshold of significance in stage 1. In essence, the two stage design pairs naturally with a screen and clean procedure. For the screen and clean procedure, it is essential that Ŝn has two prop- erties as n→∞ as follows: P(D ⊂ Ŝn)→ 1(3) |Ŝn|= oP (n),(4) where |M | denotes the number of elements in a setM . Condition (3) ensures the validity of the test in stage III while condition (4) ensures that the power of the test is not too small. Without condition (3), the hypothesis test in stage III would be biased. We will see that the power goes to 1, so taking α = αn→ 0 implies consistency: P(D̂n =D)→ 1. For fixed α, the method also produces a confidence sandwich for D, namely, lim inf P(D̂n ⊂D ⊂ Ŝn)≥ 1−α. To fit the suite of candidate models, we consider three methods. In method Ŝn(λ) = {j : β̃j(λ) 6= 0}, where β̃j(λ) is the lasso estimator, the value of β that minimizes (Yi −XTi β)2 + λ |βj |. In method 2, take Ŝn(λ) to be the set of variables chosen by forward stepwise regression after λ steps. In method 3, marginal regression, we take Ŝn = {j : |µ̂j|> λ}, where µ̂j is the marginal regression coefficient from regressing Y onXj . (This is equivalent to ordering by the absolute t-statistics since we will assume that the covariates are standardized.) These three methods are very similar to basis pursuit, orthogonal matching pursuit and thresholding [see, e.g., Tropp (2004, 2006) and Donoho (2006)]. 4 L. WASSERMAN AND K. ROEDER Notation. Let ψ =minj∈D |βj |. Define the loss of any estimator β̂ by L(β̂) = (β̂ − β)TXTX(β̂ − β) = (β̂ − β)T Σ̂n(β̂ − β),(5) where Σ̂n = n −1XTX . For convenience, when β̂ ≡ β̂(λ) depends on λ we write L(λ) instead of L(β̂(λ)). If M ⊂ S, let XM be the design matrix with columns (X•j : j ∈M) and let β̂M = (XTMXM )−1XTMY denote the least- squares estimator, assuming it is well defined. Note that our use ofX•j differs from standard ANOVA notation. Write Xλ instead of XM whenM = Ŝn(λ). When convenient, we extend β̂M to length p by setting β̂M (j) = 0, for j /∈M . We use the norms v2j , ‖v‖1 = |vj | and ‖v‖∞ =max |vj |. If C is any square matrix, let φ(C) and Φ(C) denote the smallest and largest eigenvalues of C. Also, if k is an integer, define φn(k) = min M : |M |=k XTMXM and Φn(k) = max M : |M |=k XTMXM We will write zu for the upper quantile of a standard normal, so that P(Z > zu) = u where Z ∼N(0,1). Our method will involve splitting the data randomly into three groups D1, D2 and D3. For ease of notation, assume the total sample size is 3n and that the sample size of each group is n. Summary of assumptions. We will use the following assumptions through- out except in Section 8: (A1) Yi =X i β + εi where εi ∼N(0, σ2), for i= 1, . . . , n. (A2) The dimension pn of X satisfies pn→∞ and pn ≤ c1en , for some c1 > 0 and 0≤ c2 < 1. (A3) s≡ |{j :βj 6= 0}|=O(1) and ψ =min{|βj | :βj 6= 0}> 0. (A4) There exist positive constants C0,C1 and κ such that P(lim supn→∞Φn(n)≤C0) = 1 and P(lim infn→∞ φn(C1 logn)≥ κ) = 1. Also, P(φn(n)> 0) = 1, for all n. (A5) The covariates are standardized: E(Xij) = 0 and E(X ij) = 1. Also, there exists 0<B <∞ such that P(|Xjk| ≤B) = 1. For simplicity, we include no intercepts in the regressions. The assump- tions can be weakened at the expense of more complicated proofs. In par- ticular, we can let s increase with n and ψ decrease with n. Similarly, the normality and constant variance assumptions can be relaxed. VARIABLE SELECTION 5 2. Error control. Define the type I error rate q(D̂n) = P(D̂n ∩ Dc 6= ∅) and the asymptotic error rate lim supn→∞ q(D̂n). We define the power π(D̂n) = P(D ⊂ D̂n) and the average power πav = P(j ∈ D̂n). It is well known that controlling the error rate is difficult for at least three reasons: correlation of covariates, high-dimensionality of the covariate and unfaithfulness (cancellations of correlations due to confounding). Let us briefly review these issues. It is easy to construct examples where q(D̂n)≤ α implies that π(D̂n)≈ α. Consider the following two models for random variables Z = (Y,X1,X2): Model 1 X1 ∼N(0,1), Y = ψX1 +N(0,1), X2 = ρX1 +N(0, τ Model 2 X2 ∼N(0,1), Y = ψX2 +N(0,1), X1 = ρX2 +N(0, τ Under models 1 and 2, the marginal distribution of Z is P1 =N(0,Σ1) and P2 =N(0,Σ2), where ψ2 + 1 ψ ρψ ψ 1 ρ ρψ ρ ρ2 + τ2  , Σ2 = ψ2 + 1 ρψ ψ ρψ ρ2 + τ2 ρ ψ ρ 1 Given any ε > 0, we can choose ρ sufficiently close to 1 and τ sufficiently close to 0 such that Σ1 and Σ2 are as close as we like, and hence, d(P 1 , P 2 )< ε where d is total variation distance. It follows that P2(2 /∈ D̂)≥ P1(2 /∈ D̂)− ε≥ 1− α− ε. Thus, if q ≤ α, then the power is less than α+ ε. Dimensionality is less of an issue thanks to recent methods. Most methods, including those in this paper, allow pn to grow exponentially. But all the methods require some restrictions on the number s of nonzero βj ’s. In other words, some sparsity assumption is required. In this paper we take s fixed and allow pn to grow. False negatives can occur during screening due to cancellations of cor- relations. For example, the correlation between Y and X1 can be 0 even when β1 is huge. This problem is called unfaithfulness in the causality liter- ature [see Spirtes, Glymour and Scheines (2001) and Robins et al. (2003)]. False negatives during screening can lead to false positives during the second stage. 6 L. WASSERMAN AND K. ROEDER Let µ̂j denote the regression coefficient from regressing Y on Xj . Fix j ≤ s and note that µj ≡ E(µ̂j) = βj + k 6=j 1≤k≤s βkρkj, where ρkj = corr(Xk,Xj). If k 6=j 1≤k≤s βkρkj ≈−βj, then µj ≈ 0 no matter how large βj is. This problem can occur even when n is large and p is small. For example, suppose that β = (10,−10,0,0) and that ρ(Xi,Xj) = 0 ex- cept that ρ(X1,X2) = ρ(X1,X3) = ρ(X2,X4) = 1− ε, where ε > 0 is small. Then, β = (10,−10,0,0), but µ≈ (0,0,10,−10). Marginal regression is extremely susceptible to unfaithfulness. The lasso and forward stepwise, less so. However, unobserved covariates can induce unfaithfulness in all the methods. 3. Loss and cross-validation. Let Xλ = (X•j : j ∈ Ŝn(λ)) denote the de- sign matrix corresponding to the covariates in Ŝn(λ) and let β̂(λ) be the least-squares estimator for the regression restricted to Ŝn(λ), assuming the estimator is well defined. Hence, β̂(λ) = (XTλXλ) −1XTλ Y . More generally, β̂M is the least-squares estimator for any subset of variables M . When con- venient, we extend β̂(λ) to length p by setting β̂j(λ) = 0, for j /∈ Ŝn(λ). 3.1. Loss. Now we record some properties of the loss function. The first part of the following lemma is essentially Lemma 3 of Meinshausen and Yu (2008). Lemma 3.1. Let M+m = {M ⊂ S : |M | ≤m,D⊂M}. Then, L(β̂M )≤ 4m log p nφn(m) → 1.(6) Let M−m = {M ⊂ S : |M | ≤m,D 6⊂M}. Then, L(β̂M )≥ ψ2φn(m+ s) → 1.(7) VARIABLE SELECTION 7 3.2. Cross-validation. Recall that the data have been split into groups D1, D2 and D3 each of size n. Construct β̂(λ) from D1 and let L̂(λ) = Xi∈D2 (Yi −XTi β̂(λ)) We would like L̂(λ) to order the models the same way as the true loss L(λ) [defined after (5)]. This requires that, asymptotically, L̂(λ)−L(λ)≈ δn, where δn does not involve λ. The following bounds will be useful. Note that L(λ) and L̂(λ) are both step functions that only change value when a variable enters or leaves the model. Theorem 3.2. Suppose that maxλ∈Λn |Ŝn(λ)| ≤ kn. Then, there exists a sequence of random variables δn =OP (1) that do not depend on λ or X, such that, with probability tending to 1, |L(λ)− L̂(λ)− δn|=OP n1−c2 4. Multi-stage methods. Themulti-stage methods use the following steps. As mentioned earlier, we randomly split the data into three parts, D1, D2 and D3, which we take to be of equal size: 1. Stage I. Use D1 to find Ŝn(λ), for each λ. 2. Stage II. Use D2 to find λ̂ by cross-validation, and let Ŝn = Ŝn(λ̂). 3. Stage III. Use D3 to find the least-squares estimate β̂ for the model Ŝn. D̂n = {j ∈ Ŝn : |Tj |> cn}, where Tj is the usual t-statistic, cn = zα/2m and m= |Ŝn|. 4.1. The lasso. The lasso estimator [Tibshirani (1996)] β̃(λ) minimizes Mλ(λ) = (Yi −XTi β)2 + λ |βj | and let Ŝn(λ) = {j : β̃j(λ) 6= 0}. Recall that β̂(λ) is the least-squares estima- tor using the covariates in Ŝn(λ). Let kn =A logn where A> 0 is a positive constant. Theorem 4.1. Assume that (A1)–(A5) hold. Let Λn = {λ : |Ŝn(λ)| ≤ kn}. Then: 1. The true loss overfits: P(D ⊂ Ŝn(λ∗))→ 1 where λ∗ = argminλ∈Λn L(λ). 8 L. WASSERMAN AND K. ROEDER 2. Cross-validation also overfits: P(D⊂ Ŝn(λ̂))→ 1 where λ̂= argminλ∈Λn L̂(λ). 3. Type I error is controlled: lim supn→∞ P(D c ∩ D̂n 6=∅) ≤ α. If we let α= αn→ 0, then D̂n is consistent for variable selection. Theorem 4.2. Assume that (A1)–(A5) hold. Let αn→ 0 and ∞. Then, the multi-stage lasso is consistent, P(D̂n =D)→ 1.(10) The next result follows directly. The proof is thus omitted. Theorem 4.3. Assume that (A1)–(A5) hold. Let α be fixed. Then, (D̂n, Ŝn) forms a confidence sandwich lim inf P(D̂n ⊂D ⊂ Ŝn)≥ 1−α.(11) Remark 4.4. This confidence sandwich is expected to be conservative in the sense that the coverage can be much larger than 1− α. 4.2. Stepwise regression. Let kn = A logn for some A > 0. The version of stepwise regression we consider is as follows: 1. Initialize: Res = Y , λ= 0, Ŷ = 0 and Ŝn(λ) =∅. 2. Let λ← λ+1. Compute µ̂j = n−1〈Xj ,Res〉 for j = 1, . . . , p. 3. Let J = argmaxj |µ̂j |. Set Ŝn(λ) = {Ŝn(λ−1), J}. Set Ŷ =Xλβ̂(λ) where β̂λ = (X −1XTλ Y , and let Res = Y − Ŷ . 4. If λ= kn, stop. Otherwise, go to step 2. For technical reasons, we assume that the final estimator xT β̂ is truncated to be no larger than B. Note that λ is discrete and Λn = {0,1, . . . , kn}. Theorem 4.5. With Ŝn(λ) defined as above, the statements of Theorems 4.1, 4.2 and 4.3 hold. 4.3. Marginal regression. This is probably the oldest, simplest and most common method. It is quite popular in gene expression analysis. It used to be regarded with some derision but has enjoyed a revival. A version appears in a recent paper by Fan and Lv (2008). Let Ŝn(λ) = {j : |µ̂j | ≥ λ} where µ̂j = n −1〈Y,X•j〉. Let µj = E(µ̂j), and let µ(j) denote the value of µ ordered by their absolute values, |µ(1)| ≥ |µ(2)| ≥ · · · . VARIABLE SELECTION 9 Theorem 4.6. Let kn →∞ with kn = o( n). Let Λn = {λ : |Ŝn(λ)| ≤ kn}. Assume that |µj|> |µ(kn)|.(12) Then, the statements of Theorems 4.1, 4.2 and 4.3 hold. Assumption (12) limits the degree of unfaithfulness (small partial corre- lations induced by cancellation of parameters). Large values of kn weaken assumption (12), thus making the method more robust to unfaithfulness, but at the expense of lower power. Fan and Lv (2008) make similar assumptions. They assume that there is a C > 0 such that |µj| ≥ C|βj | for all j, which rules out unfaithfulness. However, they do not explicitly relate the values of µj for j ∈D to the values outside D as we have done. On the other hand, they assume that Z =Σ−1/2X has a spherically symmetric distribution. Un- der this assumption and their faithfulness assumption, they deduce that the µj ’s outside D cannot strongly dominate the µj ’s within D. We prefer to simply make this an explicit assumption without placing distributional as- sumptions on X . At any rate, any method that uses marginal regressions as a starting point must make some sort of faithfulness assumptions to succeed. 4.4. Modifications. Let us now discuss a few modifications of the basic method. First, consider splitting the data only into two groups, D1 and D2. Then do the following steps: 1. Stage I. Find Ŝn(λ) for λ ∈Λn, where |Ŝn(λ)| ≤ kn for each λ ∈ Λn using 2. Stage II. Find λ̂ by cross-validation, and let Ŝn = Ŝn(λ̂) using D2. 3. Stage III. Find the least-squares estimate β̂ using D2. Let D̂n = {j ∈ Ŝn : |Tj |> cn}, where Tj is the usual t-statistic. Theorem 4.7. Choosing log logn 2kn log(2pn) controls asymptotic type I error. The critical value in (13) is hopelessly large and it does not appear it can be substantially reduced. We present this mainly to show the value of the extra data-splitting step. It is tempting to use the same critical value as in the tri-split case, namely cn = zα/2m where m = |Ŝn|, but we suspect this will not work in general. However, it may work under extra conditions. 10 L. WASSERMAN AND K. ROEDER 5. Application. As an example, we illustrate an analysis based on part of the osteoporotic fractures in men study [MrOS, Orwoll et al. (2005)]. A sample of 860 men were measured at a large number of genes and outcome measures. We consider only 296 SNPs which span 30 candidate genes for bone mineral density. An aim of the study was to identify genes associated with bone mineral density that could help in understanding the genetic basis of osteoporosis in men. Initial analyses of this subset of the data revealed no SNPs with a clear pattern of association with the phenotype; however, three SNPs, numbered (67), (277) and (289), exhibited some association in the screening of the data. To further explore the effacacy of the lasso screen and clean procedure, we modified the phenotype to enhance this weak signal and then reanalyzed the data to see if we could detect this planted signal. We were interested in testing for main effects and pairwise interactions in these data; however, including all interactions results in a model with 43,660 additional terms, which is not practical for this sample size. As a compromise, we selected 2 SNPs per gene to model potential interaction effects. This resulted in a model with a total of 2066 potential coefficients, including 296 main effects and 1770 interaction terms. With this model, our initial screen detected 10 terms, including the 3 enhanced signals, 2 other main effects and 5 interactions. After cleaning, the final model detected the 3 enhanced signals and no other terms. 6. Simulations. To further explore the screen and clean procedures, we conducted simulation experiments with four models. For each model Yi = XTi β + εi where the measurement errors, εi and ε ij , are i.i.d. normal(0,1) and the covariates Xij ’s are normal(0,1) (except for model D). Models differ in how Yi is linked to Xi and the dependence structure of the Xi’s. Models A, B and C explore scenarios with moderate and large p, while model D focuses on confounding and unfaithfullness, as follows: (A) Null model: β = (0, . . . ,0) and the Xij ’s are i.i.d. (B) Triangle model: βj = δ(10− j), j = 1, . . . ,10, βj = 0, j > 10 and Xij ’s are i.i.d. (C) Correlated Triangle model: as B, but withXij(+1) = ρXij+(1−ρ2)1/2ε∗ij , for j > 1, and ρ= 0.5. (D) Unfaithful model: Yi = β1Xi1 + β2Xi2 + εi, for β1 = −β2 = 10, where the Xij ’s are i.i.d. for j = {1,5,6,7,8,9,10}, but Xi2 = ρXi1 + τε∗i2, Xi3 = ρXi1 + τε i10, and Xi4 = ρXi2 + τε i11, for τ = 0.01 and ρ= 0.95. We used a maximum model size of kn = n 1/2 which technically goes be- yond the theory but works well in practice. Prior to analysis, the covariates are scaled so that each has mean 0 and variance 1. The tests were initially performed using a third of the data for each of the 3 stages of the proce- dure (Table 1, top half, 3 splits). For models A, B and C, each approach VARIABLE SELECTION 11 Table 1 Size and power of screen and clean procedures using lasso, stepwise and marginal regression for the screening step. For all procedures α= 0.05. For p= 100, δ = 0.5 and for p= 1000, δ = 1.5. Reported power is πav. The top 8 rows of simulations were conducted using three stages as described in Section 4, with a third of the data used for each stage. The bottom 8 rows of simulations were conducted splitting the data in half, using the first portion with leave-one-out cross validation for stages 1 and 2 and the second portion for cleaning Size Power Splits n p Model Lasso Step Marg Lasso Step Marg 2 100 100 A 0.005 0.001 0.004 0 0 0 2 100 100 B 0.01 0.02 0.03 0.62 0.62 0.31 2 100 100 C 0.001 0.01 0.01 0.77 0.57 0.21 2 100 10 D 0.291 0.283 0.143 0.08 0.08 0.04 2 100 1000 A 0.001 0.002 0.010 0 0 0 2 100 1000 B 0.002 0.020 0.010 0.17 0.09 0.11 2 100 1000 C 0.02 0.14 0.01 0.27 0.15 0.11 2 1000 10 D 0.291 0.283 0.143 0.08 0.08 0.04 3 100 100 A 0.040 0.050 0.030 0 0 0 3 100 100 B 0.02 0.01 0.02 0.91 0.90 0.56 3 100 100 C 0.03 0.04 0.03 0.91 0.88 0.41 3 100 10 D 0.382 0.343 0.183 0.16 0.18 0.09 3 100 1000 A 0.035 0.045 0.040 0 0 0 3 100 1000 B 0.045 0.020 0.035 0.57 0.66 0.29 3 100 1000 C 0.06 0.070 0.020 0.74 0.65 0.19 3 1000 10 D 0.481 0.486 0.187 0.17 0.17 0.13 has type I error less than α, except the stepwise procedure which has trou- ble with model C when n = p = 100. We also calculated the false positive rate and found it to be very low (about 10−4 when p= 100 and 10−5 when p= 1000) indicating that even when a type I error occurs, only a very small number of terms are included erroneously. The lasso screening procedure exhibited a slight power advantage over the stepwise procedure. Both meth- ods dominated the marginal approach. The Markov dependence structure in model C clearly challenged the marginal approach. For model D, none of the approaches controlled the type I error. To determine the sensitivity of the approach to using distinct data for each stage of the analysis, simulations were conducted screening on the first half of the data and cleaning on the second half (2 splits). The tuning parameter was selected using leave-one-out cross validation (Table 1, bottom half). As expected, this approach lead to a dramatic increase in the power of all the procedures. More surprising is the fact that the type I error was near α or 12 L. WASSERMAN AND K. ROEDER below for models A, B and C. Clearly this approach has advantages over data splitting and merits further investigation. A natural competitor to screen and clean procedure is a two-stage adap- tive lasso [Zou (2006)]. In our implementation, we split the data and used one half for each stage of the analysis. At stage one, leave-one-out cross val- idation lasso screens the data. In stage two, the adaptive lasso, with weights wj = |β̂j |−1, cleans the data. The tuning parameter for the lasso was again chosen using leave-one-out cross validation. Table 2 provides the size, power and false positive rate (FPR) for this procedure. Naturally, the adaptive lasso does not control the size of the test, but the FPR is small. The power of the test is greater than we found for our lasso screen and clean procedure, but this extra power comes at the cost of a much higher type I error rate. 7. Proofs. Recall that if A is a square matrix, then φ(A) and Φ(A) denote the smallest and largest eigenvalues of A. Throughout the proofs we make use of the following fact. If v is a vector and A is a square matrix, φ(A)‖v‖2 ≤ vTAv ≤Φ(A)‖v‖2.(14) We use the following standard tail bound: if Z ∼N(0,1), then P(|Z|> t)≤ t−1e−t 2/2. We will also use the following results about the lasso from Mein- shausen and Yu (2008). Their results are stated and proved for fixed X but, under the conditions (A1)–(A5), it is easy to see that their conditions hold with probability tending to one and so their results hold for random X as well. Theorem 7.1 [Meinshausen and Yu (2008)]. Let β̃(λ) be the lasso esti- mator. Table 2 Size, power and false positive rate (FPR) of two-stage adaptive lasso procedure n p Model Size Power FPR 100 100 A 0.93 0 0.032 100 100 B 0.84 0.97 0.034 100 100 C 0.81 0.96 0.031 100 10 D 0.67 0.21 0.114 100 1000 A 0.96 0 0.004 100 1000 B 0.89 0.65 0.004 100 1000 C 0.76 0.77 0.002 1000 10 D 0.73 0.24 0.013 VARIABLE SELECTION 13 1. The squared error satisfies ‖β̃(λ)− β‖22 ≤ cm log pn nφ2n(m) → 1,(15) where m= |Ŝn(λ)| and c > 0 is a constant. 2. The size of Ŝn(λ) satisfies |Ŝn(λ)| ≤ τ2Cn2 → 1,(16) where τ2 = E(Y 2i ). Proof of Lemma 3.1. Let D ⊂M and φ= φ(n−1XTMXM ). Then, L(β̂M ) = εTXM (X MXM ) −1XTMε≤ ‖XTMε‖2 = Z2j , where Zj = n −1/2XT•jε. Conditional on X , Zi ∼N(0, a2j ) where a2j = n−1 ×∑n ij . Let A n =max1≤j≤pn a j . By Hoeffding’s inequality, (A2) and (A5), P(En)→ 1 where En = {An ≤ 2}. So 1≤j≤pn |Zj |> 4 log pn 1≤j≤pn |Zj |> 4 log pn,En 1≤j≤pn |Zj |> 4 log pn,E 1≤j≤pn |Zj |> 4 log pn,En + P(Ecn) An max 1≤j≤pn |Zj | 4 log pn,En + o(1) 1≤j≤pn |Zj | 2 log pn + o(1) 1≤j≤pn |Zj | 2 log pn )∣∣∣X + o(1) 2 log pn + o(1) = o(1). j∈M Z j ≤mmax1≤j≤pn Z2j and (6) follows. Now we lower bound L(β̂M ). Let M be such that D 6⊂ M . Let A = {j : β̂M (j) 6= 0} ∪D. Then, |A| ≤m + s. Therefore, with probability tend- ing to 1, L(β̂M ) = (β̂M − β)TXTX(β̂M − β) = (β̂M − β)TXTAXA(β̂M − β) 14 L. WASSERMAN AND K. ROEDER ≥ φn(m+ s)‖β̂M − β‖2 = φn(m+ s) (β̂M (j)− β(j))2 ≥ φn(m+ s) j∈D∩Mc (0− β(j))2 ≥ φn(m+ s)ψ2. Proof of Theorem 3.2. Let Ỹ denote the responses, and X̃ the design matrix, for the second half of the data. Then, Ỹ = X̃β + ε̃. Now L(λ) = (β̂(λ)− β)TXTX(β̂(λ)− β) = (β̂(λ)− β)T Σ̂n(β̂(λ)− β) L̂(λ) = n−1‖Ỹ −X̃β̂(λ)‖2 = (β̂(λ)−β)T Σ̃n(β̂(λ)−β)+δn+ 〈ε̃, X̃(β̂(λ)−β)〉, where δn = ‖ε̃‖2/n, and Σ̂n = n−11 XTX and Σ̃n = n−1X̃T X̃ . By Hoeffding’s inequality P(|Σ̂n(j, k)− Σ̃n(j, k)|> ε)≤ e−ncε for some c > 0, and so |Σ̂n(j, k)− Σ̃n(j, k)|> ε ≤ p2ne−ncε Choose εn = 4/(cn 1−c2). It follows that |Σ̂n(j, k)− Σ̃n(j, k)|> cn1−c2 ≤ e−2nc2 → 0. Note that |{j : β̂j(λ) 6= 0} ∪ {j :βj 6= 0}| ≤ kn + s. Hence, with probability tending to 1 |L(λ)− L̂(λ)− δn| ≤ cn1−c2 ‖β̂(λ)− β‖21 + 2ξn(λ) for all λ ∈Λn, where ξn(λ) = ε̃iµi(λ) and µi(λ) = X̃ i (β̂(λ)−β). Now ‖β̂(λ)−β‖21 =OP ((kn+s)2) since ‖β̂(λ)‖2 = OP (kn/φ(kn)). Thus, ‖β̂(λ)−β‖1 ≤C(kn+s) with probability tending to 1, for some C > 0. Also, |µi(λ)| ≤B‖β̂(λ)−β‖1 ≤BC(kn+ s) with probability tending to 1. Let W ∼N(0,1). Conditional on D1, |ξn(λ)| µ2i (λ)|W | ≤ BC(kn + s)|W |, VARIABLE SELECTION 15 so supλ∈Λn |ξn(λ)|=OP (kn/ n). � Proof of Theorem 4.1. 1. Let λn = τn C/kn,M = Ŝn(λn) and m= |M |. Then, P(m≤ kn)→ 1 due to (16). Hence, P(λn ∈ Λn)→ 1. From (15), ‖β̃(λn)− β‖22 ≤O kn log pn = oP (1). Hence, ‖β̃(λn)− β‖2∞ = oP (1). So, for each j ∈D, |β̃j(λn)| ≥ |βj | − |β̃j(λn)− βj | ≥ ψ+ oP (1) and hence, P(minj∈D |β̃j(λn)|> 0)→ 1. Therefore, Γn = {λ ∈Λn :D ⊂ Ŝn(λ)} is nonempty. By Lemma 3.1, L(λn)≤ cm log pn/(nφ(m)) =OP (kn log pn/n).(17) On the other hand, from Lemma 3.1, λ∈Λn∩Γcn L(β̂λ)>ψ 2φ(kn) → 1.(18) Now, nφn(kn)/(kn log pn)→∞, and so (17) and (18) imply that λ∈Λn∩Γcn L(β̂λ)>L(λn) Thus, if λ∗ denotes the minimizer of L(λ) over Λn, we conclude that P(λ∗ ∈ Γn)→ 1, and hence, P(D ⊂ Ŝn(λ∗))→ 1. 2. This follows from part 1 and Theorem 3.2. 3. Let A= Ŝn ∩Dc. We want to show that |Tj|> cn ≤ α+ o(1). |Tj |> cn |Tj |> cn,D ⊂ Ŝn |Tj |> cn,D 6⊂ Ŝn |Tj |> cn,D ⊂ Ŝn + P(D 6⊂ Ŝn) |Tj |> cn,D ⊂ Ŝn + o(1). Conditional on (D1,D2), β̂A is normally distributed with mean 0 and vari- ance matrix σ2(XTAXA) −1 when D⊂ Ŝn. Recall that Tj(M) = eTj (X MXM ) −1XTMY eTj (X MXM ) β̂M,j 16 L. WASSERMAN AND K. ROEDER whereM = Ŝn, s j = σ̂ 2eTj (X MXM ) −1ej and ej = (0, . . . ,0,1,0, . . . ,0) T , where the 1 is in the jth coordinate. When D ⊂ Ŝn, each Tj , for j ∈ A, has a t- distribution with n−m degrees of freedom wherem= |Ŝn|. Also, cn/tα/2m→ 1 where tu denotes the upper tail critical value for the t-distribution. Hence, |Tj |> cn,D ⊂ Ŝn|D1,D2 |Tj |> tα/2m,D ⊂ Ŝn|D1,D2 ≤ α+ an, where an = o(1), since |A| ≤m. It follows that |Tj |> cn,D ⊂ Ŝn ≤ α+ o(1). Proof of Theorem 4.2. From Theorem 4.1, P(D̂n ∩Dc 6= ∅) ≤ αn and so P(D̂n ∩Dc 6=∅)→ 0. Hence, P(D̂n ⊂D)→ 1. It remains to be shown P(D⊂ D̂n)→ 1.(19) The test statistic for testing βj = 0 when Ŝn =M is Tj(M) = eTj (X MXM ) −1XTMY eTj (X MXM ) For simplicity in the proof, let us take σ̂ = σ, the extension to unknown σ being straightforward. Let j ∈D,M= {M : |M | ≤ kn,D⊂M}. Then, P(j /∈ D̂n) = P(j /∈ D̂n,D ⊂ Ŝn) + P(j /∈ D̂n,D 6⊂ Ŝn) ≤ P(j /∈ D̂n,D ⊂ Ŝn) + P(D 6⊂ Ŝn) = P(j /∈ D̂n,D ⊂ Ŝn) + o(1) P(j /∈ D̂n, Ŝn =M) + o(1) P(|Tj(M)|< cn, Ŝn =M) + o(1) P(|Tj(M)|< cn) + o(1). Conditional on D1 ∪D2, for each M ∈M, Tj(M) = (βj/sj) +Z, where Z ∼ N(0,1). Without loss of generality, assume that βj > 0. Hence, P(|Tj(M)|< cn|D1 ∪D2) = P −cn − <Z < cn − VARIABLE SELECTION 17 Fix a small ε > 0. Note that s2j ≤ σ2/(nκ). It follows that, for all large n, cn − βj/sj <−ε n. So, P(|Tj(M)|< cn|D1 ∪D2)≤ P(Z <−ε n)≤ e−nε2/2. The number of models inM is pn − s j − s pn − s kn − s (pn − s)e kn − s )kn−s ≤ knpknn , where we used the inequality P(|Tj(M)|< cn|D1 ∪D2)≤ knpknn e−nε 2 → 0 by (A2). We have thus shown that P(j /∈ D̂n)→ 0, for each j ∈D. Since |D| is finite, it follows that P(j /∈ D̂n for some j ∈D)→ 0 and hence (19). � Proof of Theorem 4.5. A simple modification of Theorem 3.1 of Barron et al. (2008) shows that L(kn) = ‖Ŷkn −Xβ‖2 = oP (1). [The modification is needed because Barron et al. (2008) require Y to be bounded while we have assumed that Y is normal. By a truncation argument, we can still derive the bound on L(kn).] So ‖β̂kn − β‖2 ≤ L(kn) φn(kn + s) L(kn) = oP (1). Hence, for any ε > 0, with probability tending to 1, ‖β̂(kn)−β‖2 < ε so that |β̂j | > ψ/2 > 0, for all j ∈D. Thus, P(D ⊂ Ŝn(kn))→ 1. The remainder of the proof of part 1 is the same as in Theorem 4.1. Part 2 follows from the previous result together with Theorem 3.2. The proof of part 3 is the same as for Theorem 4.1. � Proof of Theorem 4.6. Note that µ̂j − µj = n−1 i=1Xijεi. Hence, µ̂j − µj ∼N(0,1/n). So, for any δ > 0, |µ̂j − µj |> δ P(|µ̂j − µj|> δ) 2/2 ≤ c1e 2/2→ 0. 18 L. WASSERMAN AND K. ROEDER By (12), conclude that D ⊂ Ŝn(λ) when λ = µ̂(kn). The remainder of the proof is the same as the proof of Theorem 4.5. � Proof of Theorem 4.7. Let A= Ŝn ∩Dc. We want to show that |Tj|> cn ≤ α+ o(1). For fixed A, β̂A is normal with mean 0 but this is not true for random A. Instead we need to bound Tj . Recall that Tj(M) = eTj (X MXM ) −1XTMY eTj (X MXM ) β̂M,j whereM = Ŝn, s j = σ̂ 2eTj (X MXM ) −1ej and ej = (0, . . . ,0,1,0, . . . ,0) T where the 1 is in the jth coordinate. The probabilities that follow are conditional on D1 but this is supressed for notational convenience. First, write |Tj |> cn |Tj |> cn,D ⊂ Ŝn |Tj |> cn,D 6⊂ Ŝn |Tj |> cn,D ⊂ Ŝn + P(D 6⊂ Ŝn) |Tj |> cn,D ⊂ Ŝn + o(1). When D ⊂ Ŝn, )−1 1 where Q = ((1/n)XT )−1, γ = n−1XT ε, and β (j) = 0, for j ∈A. Now, s2j ≥ σ̂2/(nC) so that |Tj | ≤ nC|β̂ Ŝn,j n log logn|β̂ Ŝn,j for j ∈ Ŝn. Therefore, |Tj |> cn,D ⊂ Ŝn Ŝn,j σ̂cn√ ,D ⊂ Ŝn Let γ = n−1XT ε. Then, ‖β̂A‖2 ≤ γT knmax1≤j≤pn γ VARIABLE SELECTION 19 It follows that Ŝn,j knmax1≤j≤pn |γj | kn log logn max 1≤j≤pn |γj |, since κ > 0. So, Ŝn,j |> σ̂cn√ n log logn ,D ⊂ Ŝn 1≤j≤pn |γj |> log logn Note that γj ∼N(0, σ2/n), and hence, |γj | 2σ2 log(2pn) There exists εn → 0 such that P(Bn)→ 1 where Bn = {(1 − εn) ≤ σ̂/σ ≤ (1 + ε)}. So, 1≤j≤pn |γj|> log logn 1≤j≤pn |γj |> σcn(1− εn) log logn σ(1− εn)cn log logn |γj | ≤ α+ o(1). � 8. Discussion. The multi-stage method presented in this paper success- fully controls type I error while giving reasonable power. The lasso and step- wise have similar performance. Although theoretical results assume indepen- dent data for each of the three stages, simulations suggest that leave-one-out cross-validation leads to valid type I error rates and greater power. Screening the data in one phase of the experiment and cleaning in a followup phase leads to an efficient experimental design. Certainly this approach deserves further theoretical investigation. In particular, the question of optimality is an open question. The literature on high-dimensional variable selection is growing quickly. The most important deficiency in much of this work, including this paper, is the assumption that the model Y = XTβ + ε is correct. In reality, the model is at best an approximation. It is possible to study linear procedures when the linear model is not assumed to hold as in Greenshtein and Ritov (2004). We discuss this point in the Appendix. Nevertheless, it seems useful to study the problem under the assumption of linearity to gain insight into these methods. Future work should be directed at exploring the robustness of the results when the model is wrong. Other possible extensions include: dropping the normality of the errors, permitting nonconstant variance, investigating the optimal sample sizes for each stage and considering other screening methods besides cross-validation. 20 L. WASSERMAN AND K. ROEDER Finally, let us note that the example involving unfaithfulness, that is, cancellations of parameters to make the marginal correlation much different than the regression coefficient, pose a challenge for all the methods and deserve more attention even in cases of small p. APPENDIX: PREDICTION Realistically, there is little reason to believe that the linear model is cor- rect. Even if we drop the assumption that the linear model is correct, sparse methods like the lasso can still have good properties as shown in Greenshtein and Ritov (2004). In particular, they showed that the lasso satisfies a risk consistency property. In this appendix we show that this property continues to hold if λ is chosen by cross-validation. The lasso estimator is the minimizer of i=1(Yi −XTi β)2 + λ‖β‖1. This is equivalent to minimizing i=1(Yi−XTi β)2 subject to ‖β‖1 ≤Ω, for some Ω. (More precisely, the set of estimators as λ varies is the same as the set of estimators as Ω varies.) We use this second version throughout this section. The predictive risk of a linear predictor ℓ(x) = xTβ is R(β) = E(Y −ℓ(x))2 where (X,Y ) denotes a new observation. Let γ = γ(β) = (−1, β1, . . . , βp)T and let Γ = E(ZZT ) where Z = (Y,X1, . . . ,Xp). Then, we can write R(β) = γTΓγ. The lasso estimator can now be written as β̂(Ωn) = argminβ∈B(Ωn) R̂(β) where R̂(β) = γT Γ̂γ and Γ̂ = n−1 i=1ZiZ Define β∗ = argmin β∈B(Ωn) R(β), where B(Ωn) = {β :‖β‖1 ≤Ωn}. Thus, ℓ∗(x) = x Tβ∗ is the best linear predictor in the set B(Ωn). The best linear predictor is well defined even though E(Y |X) is no longer assumed to be linear. Greenshtein and Ritov (2004) call an estimator β̂n persistent, or predictive risk consistent, if R(β̂n)−R(β∗) as n→∞. The assumptions we make in this section are: (B1) pn ≤ en , for some 0≤ ξ < 1; (B2) the elements of Γ̂ satisfy an exponential inequality P(|Γ̂jk − Γjk|> ε)≤ c3e−nc4ε for some c3, c4 > 0; VARIABLE SELECTION 21 (B3) there exists B0 <∞ such that, for all n, maxj,kE(|ZjZk|)≤B0. Condition (A2) can easily be deduced from more primitive assumptions as in Greenshtein and Ritov (2004), but for simplicity we take (A2) as an assumption. Let us review one of the results in Greenshtein and Ritov (2004). For the moment, replace (A1) with the assumption that pn ≤ nb, for some b. Under these conditions, it follows that ∆n ≡max |Γ̂jk − Γjk|=OP Hence, β∈B(Ωn) |R(β)− R̂(β)|= sup β∈B(Ωn) |γT (Γ− Γ̂)γ| ≤∆n sup β∈B(Ωn) ‖γ‖21 =Ω2nOP The latter term is oP (1) as long as Ωn = o((n/ logn) 1/4). Thus, we have the following. Theorem A.1 [Greenshtein and Ritov (2004)]. If Ωn = o((n/ logn) 1/4), then the lasso estimator is persistent. For future reference, let us state a slightly different version of their result that we will need. We omit the proof. Theorem A.2. Let γ > 0 be such that ξ+γ < 1. Let Ωn =O(n (1−ξ−γ)/4). Then, under (B1) and (B2), β∈B(Ωn) |R̂(β)−R(β)|> 1 =O(e−cn )(20) for some c > 0. The estimator β̂(Ωn) lies on the boundary of the ball B(Ωn) and is very sensitive to the exact choice of Ωn. A potential improvement—and something that reflects actual practice—is to compute the set of lasso estimators β̂(ℓ), for 0≤ ℓ≤ Ωn and then select from that set based on cross validation. We now confirm that the resulting estimator preserves persistence. As before, we split the data into D1 and D2. Construct the lasso estimators {β̂(ℓ) : 0≤ ℓ≤Ωn}. Choose ℓ̂ by cross validation using D2. Let β̂ = β̂(ℓ̂). 22 L. WASSERMAN AND K. ROEDER Theorem A.3. Let γ > 0 be such that ξ + γ < 1. Under (A1), (A2) and (A3), if Ωn =O(n (1−ξ−γ)/4), then the cross-validated lasso estimator β̂ is persistent. Moreover, R(β̂)− inf 0≤ℓ≤Ωn R(β̂(ℓ)) P→ 0.(21) Proof. Let β∗(ℓ) = argminβ∈B(ℓ)R(β). Define h(ℓ) = R(β∗(ℓ)), g(ℓ) = R(β̂(ℓ)) and c(ℓ) = L̂(β̂(ℓ)). Note that, for any vector b, we can write R(b) = τ2 + bTΣb− 2bTρ where ρ= (E(Y X1), . . . ,E(Y Xp))T . Clearly, h is monotone nonincreasing on [0,Ωn]. We claim that |h(ℓ + δ) − h(ℓ)| ≤ cΩnδ where c depends only on Γ. To see this, let u = β∗(ℓ), v = β∗(ℓ+ δ) and a= ℓβ∗(ℓ+ δ)/(ℓ+ δ) so that a ∈B(ℓ). Then, h(ℓ+ δ)≤ h(ℓ) =R(u)≤R(a) =R(v) +R(a)−R(v) = h(ℓ+ δ) + ρT v− δ(2ℓ+ δ) (ℓ+ δ)2 ≤ h(ℓ+ δ) + 2δC + δ(2Ωn + δ)C, where C =maxj,k |Γj,k|=O(1). Next, we claim that g(ℓ) is Lipschitz on [0,Ωn] with probability tending to 1. Let β̂(ℓ) = argminβ∈B(ℓ) R̂(β) denote the lasso estimator and set û= β̂(ℓ) and v̂ = β̂(ℓ+δ). Let εn = n −γ/4. From (20), the following chain of equations hold except on a set of exponentially small probability: g(ℓ+ δ) =R(v̂)≤ R̂(v̂) + εn ≤ R̂(v) + εn ≤R(v) + 2εn = h(ℓ+ δ) + 2εn ≤ h(ℓ) + cΩnδ +2εn =R(u) + cΩnδ+ 2εn ≤R(û) + cΩnδ+2εn = g(ℓ) + cΩnδ +2εn. A similar argument can be applied in the other direction. Conclude that |g(ℓ+ δ)− g(ℓ)| ≤ cΩnδ +2εn except on a set of small probability. Now let A= {0, δ,2δ, . . . ,mδ} where m is the smallest integer such that mδ ≥ Ωn. Thus, m∼ Ωn/δn. Choose δ = δn = n−3(1−ξ−γ)/8. Then, Ωnδn→ 0 and Ωn/δn ≤ n3(1−ξ−γ)/4. Using the same argument as in the proof of Theorem 3.2, |L̂(β̂(ℓ))−R(β̂(ℓ))|= σn, VARIABLE SELECTION 23 where σn = oP (1). Then, R(β∗(Ωn))≤R(β̂)≤ L̂(β̂(ℓ̂)) + σn ≤ L̂(mδn) + σn ≤ g(mδn) + 2σn ≤ g(Ωn) + 2σn + cΩnδn ≤ h(Ωn) + 2σn + εn + cΩnδn =R(β∗(Ωn)) + 2σn + εn + cΩnδn and persistence follows. To show the second result, let β̃ = argmin0≤ℓ≤Ωn g(ℓ) and β = argminℓ∈A g(ℓ). Then, R(β̃)≤ L̂(β̃) + σn ≤ L̂(β) + σn ≤R(β) + 2σn ≤R(β̃) + 2σn + cδnΩn and the claim follows. � Acknowledgments. The authors are grateful for the use of a portion of the sample from the Osteoporotic Fractures in Men (MrOS) Study to il- lustrate their methodology. MrOs is supported by the National Institute of Arthritis and Musculoskeletal and Skin Diseases (NIAMS), the National In- stitute on Aging (NIA) and the National Cancer Institute (NCI) through Grants U01 AR45580, U01 AR45614, U01 AR45632, U01 AR45647, U01 AR45654, U01 AR45583, U01 AG18197 and M01 RR000334. Genetic analy- ses in MrOS were supported by R01-AR051124. We also thank two referees and an AE for helpful suggestions. REFERENCES Barron, A., Cohen, A., Dahmen, W. and DeVore, R. (2008). Approximation and learning by greedy algorithms. Ann. Statist. 36 64–94. MR2387964 Bühlmann, P. (2006). Boosting for high-dimensional linear models. Ann. Statist. 34 559– 583. MR2281878 Candes, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Statist. 35 2313–2351. MR2382644 Donoho, D. (2006). For most large underdetermined systems of linear equations, the minimal l1-norm near-solution approximates the sparsest near-solution. Comm. Pure Appl. Math. 59 797–829. MR2217606 Efron, B., Hastie, T., Johnstone, I. and Tibshirani, R. (2004). Least angle regres- sion. Ann. Statist. 32 407–499. MR2060166 Fan, J. and Lv, J. (2008). Sure independence screening for ultra-high-dimensional feature space. J. Roy. Statist. Soc. Ser. B 70 849–911. Greenshtein, E. and Ritov, Y. (2004). Persistence in high-dimensional linear predictor selection and the virtue of overparametrization. Bernoulli 10 971–988. MR2108039 Meinshausen, N. (2007). Relaxed lasso. Comput. Statist. Data Anal. 52 374–393. MR2409990 Meinshausen, N. and Bühlmann, P. (2006). High dimensional graphs and variable se- lection with the lasso. Ann. Statist. 34 1436–1462. MR2278363 http://www.ams.org/mathscinet-getitem?mr=2387964 http://www.ams.org/mathscinet-getitem?mr=2281878 http://www.ams.org/mathscinet-getitem?mr=2382644 http://www.ams.org/mathscinet-getitem?mr=2217606 http://www.ams.org/mathscinet-getitem?mr=2060166 http://www.ams.org/mathscinet-getitem?mr=2108039 http://www.ams.org/mathscinet-getitem?mr=2409990 http://www.ams.org/mathscinet-getitem?mr=2278363 24 L. WASSERMAN AND K. ROEDER Meinshausen, N. and Yu, B. (2009). Lasso-type recovery of sparse representations of high-dimensional data. Ann. Statist. To appear. Orwoll, E., Blank, J. B., Barrett-Connor, E., Cauley, J., Cummings, S., En- srud, K., Lewis, C., Cawthon, P. M., Marcus, R., Marshall, L. M., McGowan, J., Phipps, K., Sherman, S., Stefanick, M. L. and Stone, K. (2005). Design and baseline characteristics of the osteoporotic fractures in men (MrOS) study—a large ob- servational study of the determinants of fracture in older men. Contemp. Clin. Trials. 26 569–585. Robins, J., Scheines, R., Spirtes, P. and Wasserman, L. (2003). Uniform consistency in causal inference. Biometrika 90 491–515. MR2006831 Spirtes, P., Glymour, C. and Scheines, R. (2001). Causation, Prediction, and Search. MIT Press. MR1815675 Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288. MR1379242 Tropp, J. A. (2004). Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theory 50 2231–2242. MR2097044 Tropp, J. A. (2006). Just relax: Convex programming methods for identifying sparse signals in noise. IEEE Trans. Inform. Theory 52 1030–1051. MR2238069 Wainwright, M. (2006). Sharp thresholds for high-dimensional and noisy recovery of sparsity. Available at arxiv.org/math.ST/0605740. Wellcome Trust (2007). Genome-wide association study of 14,000 cases of seven com- mon diseases and 3000 shared controls. Nature 447 661–678. Zhao, P. and Yu, B. (2006). On model selection consistency of lasso. J. Mach. Learn. Res. 7 2541–2563. MR2274449 Zhang, C. H. and Huang, J. (2006). Model selection consistency of the lasso in high- dimensional linear regression. Ann. Statist. 36 1567–1594. MR2435448 Zou, H. (2006). The adaptive lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418–1429. MR2279469 Department of Statistics Carnegie Mellon University Pittsburgh E-mail: [email protected] [email protected] http://www.ams.org/mathscinet-getitem?mr=2006831 http://www.ams.org/mathscinet-getitem?mr=1815675 http://www.ams.org/mathscinet-getitem?mr=1379242 http://www.ams.org/mathscinet-getitem?mr=2097044 http://www.ams.org/mathscinet-getitem?mr=2238069 http://arxiv.org/math.ST/0605740 http://www.ams.org/mathscinet-getitem?mr=2274449 http://www.ams.org/mathscinet-getitem?mr=2435448 http://www.ams.org/mathscinet-getitem?mr=2279469 mailto:[email protected] mailto:[email protected] Introduction Error control Loss and cross-validation Loss Cross-validation Multi-stage methods The lasso Stepwise regression Marginal regression Modifications Application Simulations Proofs Discussion Appendix: Prediction Acknowledgments References Author's addresses

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Depletion effects in smectic phases of hard rod--hard sphere mixtures

Depletion effects in smectic phases of hard rod–hard sphere mixtures Yuri Mart́ınez-Ratón∗ Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad 30, E-28911 Leganés, Madrid, Spain. Giorgio Cinacchi† Dipartimento di Chimica, Università di Pisa Via Risorgimento 35, I–56126, Pisa, ITALY Enrique Velasco‡ Departamento de F́ısica Teórica de la Materia Condensada and Instituto de Ciencia de Materiales Nicolás Cabrera, Universidad Autónoma de Madrid, E-28049 Madrid, Spain. Luis Mederos§ Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones Cient́ıficas, E-28049 Cantoblanco, Madrid, Spain. (Dated: November 3, 2018) It is known that when hard spheres are added to a pure system of hard rods the stability of the smectic phase may be greatly enhanced, and that this effect can be rationalised in terms of depletion forces. In the present paper we first study the effect of orientational order on depletion forces in this particular binary system, comparing our results with those obtained adopting the usual approximation of considering the rods parallel and their orientations frozen. We consider mixtures with rods of different aspect ratios and spheres of different diameters, and we treat them within Onsager theory. Our results indicate that depletion effects, and consequently smectic stability, decrease significantly as a result of orientational disorder in the smectic phase when compared with corresponding data based on the frozen–orientation approximation. These results are discussed in terms of the τ parameter, which has been proposed as a convenient measure of depletion strength. We present closed expressions for τ , and show that it is intimately connected with the depletion potential. We then analyse the effect of particle geometry by comparing results pertaining to systems of parallel rods of different shapes (spherocylinders, cylinders and parallelepipeds). We finally provide results based on the Zwanzig approximation of a Fundamental–Measure density– functional theory applied to mixtures of parallelepipeds and cubes of different sizes. In this case, we show that the τ parameter exhibits a linear asymptotic behaviour in the limit of large values of the hard–rod aspect ratio, in conformity with Onsager theory, as well as in the limit of large values of the ratio of rod breadth to cube side length, d, in contrast to Onsager approximation, which predicts τ ∼ d3. Based on both this result and the Percus–Yevick approximation for the direct correlation function for a hard sphere binary mixture in the same limit of infinite asymmetry, we speculate that, for spherocylinders and spheres, the τ parameter should be of order unity as d tends to infinity. I. INTRODUCTION In recent years experimental mixtures that closely re- semble a hard rod–hard sphere system have been studied; typically the rods are represented by tobacco mosaic or fd–virus particles, while the spheres are represented by polystyrene latex particles or globular proteins [1, 2, 3]. The phase diagrams found include the isotropic phase, as well as phases with liquid-crystalline symmetry. Among these, nematic (N) and smectic (Sm) phases have been found. In addition, bulk demixing transitions, as well as microsegregated phases of various symmetries, have been observed[2]. Intensive theoretical effort has been ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] §Electronic address: [email protected] devoted to the understanding of these systems. A useful concept in this effort is that of a depletion force. Struc- tural and thermodynamic stability of systems consisting of hard particles can be understood solely in terms of entropic effects which, in mixtures, can be reinterpreted as an attractive depletion force. Some studies have been done recently in an attempt to quantify attractive deple- tion interactions between solute particles, of anisotropic shape, mediated by solvent particles that can be isotropic or possess themselves liquid-crystalline order [4, 5, 6]. One of the microsegregated phases that has been ob- served is the lamellar phase. This phase, having smectic symmetry, consists of alternate pure layers of rods and spheres. The lamellar phase can be greatly stabilised with respect to the corresponding smectic phase in the pure–rod fluid, as predicted in Ref. [7], and shown in Refs. [8, 9, 10, 11, 12, 13]. In the lamellar phase, deple- tion forces result from the fact that it is entropically more favourable for spheres to occupy the interstitial space, which creates a large effectively attractive force between http://arxiv.org/abs/0704.1140v1 mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] adjacent layers of rods; enhanced smectic–phase stability ensues. Thus far, theoretical studies of the lamellar phase have mostly used, among others, the following approxima- tions: parallel rodlike particles with frozen orientational order, specific particle geometry, Onsager theory [14] and neglect of particle flexibility. Cinacchi et al.[15] stud- ied the hard-spherocylinder (HSPC)–hard-sphere (HS) mixture using Onsager theory and free particle orienta- tions, but presented phase diagrams including isotropic, nematic and the lamellar phase, without specifically ad- dressing any stability issue. In this paper we lift stepwise the first three of these restrictions and assess their impact on the enhancement of smectic stability in the mixture with respect to the pure rod system. Density–functional theory is a convenient theoretical tool to study the structure and phase behaviour of soft– condensed matter. Onsager theory is a most success- ful version of density–functional theory, intended for bodies interacting through hard potentials. Some of the shortcomings of Onsager theory are improved by Fundamental–Measure theory [16] (FMT), which intro- duces the fundamental particle measures in the theory and should therefore be quite general. Our theoretical analysis is based on both these density–functional ap- proximations. Specifically, we first use the HSPC model within Onsager theory to include the orientational de- grees of freedom of the rods by means of an orientational distribution function and the associated order parame- ter. The results will be discussed in terms of a parame- ter, τ , which is directly related to the depletion strength in smectic phases [10]. Then, we discuss the effect of particle geometry by considering various shapes for the hard particles in the frozen-orientation approximation. Finally, we consider a FMT approximation for mixtures of hard parallelepipeds (HPAR) and cubes (HC), and analyse it in the context of the Zwanzig approximation [17]; the latter consists of considering particle orienta- tions to be restricted to three mutually perpendicular axes. Recent advances on the application of FMT to freely rotating particles have focused on anisotropic par- ticles with infinitely narrow breadths (needles)[18]. Since one of our aims is to investigate how the depletion mech- anism changes as the breadth ratio of the particles is varied, these recent developments are not appropriate in our context, and the use of the Zwanzig approximation seems to be justified for lack of a better approach. FMT provides a different theoretical viewpoint with respect to which the predictions of Onsager theory can be assessed. In fact, the main purpose of this part of the work is to assess the impact of a proper inclusion in the theory of pair correlations on the depletion mechanism in smectic phases. In the framework of FMT, we additionally study in detail the asymptotic limit of τ with respect to dif- ferent parameters relating to the aspect ratio of the rod and the relative sizes of HPARs and HCs. With a view to completing this study, and based on some expressions used to calculate the τ parameter, which we evaluate for hard spheres, we speculate that, for HSPC–HS mixtures, τ ∼ 1 as the ratio of rod breadth to sphere diameter tends to infinity. In the following section the general procedure used to locate the nematic–smectic spinodal line is illustrated; for the sake of clarity, this is illustrated in the context of Onsager theory for a binary mixture of hard rods and HS’s. In Section III the τ parameter, which is used to quantify the stability of the smectic phase when HS’s are added, is defined, and a closed expression, valid in the context of Onsager theory and parallel rods, is pre- sented. The results are presented in Section IV, which has three subsections. The first is devoted to the effect of free particle orientations, the second addresses the ef- fect of particle shape and the third contains the results for mixtures of HPAR’s and HC’s, analysed by means of FMT in the Zwanzig approximation. This last subsection is in turn divided into two parts. The first is devoted to the results obtained by applying Onsager theory to the Zwanzig HPAR–HC system, while in the second we de- scribe the results from FMT, together with a detailed discussion on the asymptotic limits of the τ parameter. The conclusions are presented in Section V. The Ap- pendices collect expressions for the Fourier transforms of the overlap functions for the various particle geome- tries explored in the paper, together with a few details on the numerical minimisation of the functional in the context of the frozen–orientation approximation. An ex- act expression for τ in terms of the correlation functions, valid for a general mixture of freely rotating particles, is also included. Explicit expressions for these functions in the Zwanzig–Onsager and Zwanzig–FMT approaches for HPAR–HC and HS binary mixtures are provided. Finally, the asymptotic behaviour of the τ parameter, within the framework of FMT, and for the HPAR–HC mixture, is described. II. ONSAGER THEORY FOR HARD ROD–HARD SPHERE MIXTURES AND STABILITY ANALYSIS The first theory we employ to describe hard rod–hard sphere mixtures is Onsager theory, first proposed by Onsager[14] for a pure system of hard rods and exten- sively used in theoretical approaches to orientational or- dering in hard–rod fluids. It can be regarded as a trun- cated virial expansion, up to second order in density, of the excess part of the free energy. Thus, it contains the orientational–dependent second–virial coefficient exactly. It is a density–functional theory, since the free energy de- pends functionally on the orientational distribution func- tion. The extension of Onsager theory to mixtures is straightforward; the reader is referred to Ref.[15] for a detailed account on its implementation for the present model fluids. We consider a two-component mixture of hard rods and hard spheres labelled with 1 and 2, respectively. The density functional in the Onsager approximation, assum- ing uniaxial smectic symmetry and taking z as the coor- dinate along the smectic layer normal, is written as ρi(z) ρi(z) − ρ1(z)Sr(Q) i,j=1 dz′ρi(z)ρj(z ′)fij(z − z , (1) where Φ ≡ βF/V is the Helmholtz free-energy density per unit thermal energy, ds the smectic layer spacing, Sr(Q) is the orientational-entropy density of the hard rods, Q their orientational order parameter, ρ1(z) and ρ2(z) are local densities for the two components, and fij(z) are angular averages of the overlap functions be- tween species i and j, already integrated in the xy plane. The procedure used to calculate these averages has been described in detail elsewhere for the case of mixtures of spherocylinders [15]. In the case where one adopts the popular approximation of considering that particles pos- sess perfect orientational order (Q = 1), an approxima- tion used in all previous theoretical analyses of the hard- rod–hard-sphere mixture [7, 8, 9, 10, 13], the functions fij(z) can usually be written exactly, depending on the particle geometry. Note that f11 and f12 both depend on Q, but certainly not f22. In the nematic phase, ρi(z) is a constant. Since we are interested in searching for the nematic- smectic spinodal, we consider the following perturbations on the constant nematic densities: ρi(z) = ρxi (1 + λi cos kz) , i = 1, 2 (2) where ρ is the total mean density, x2 ≡ x, x1 = 1−x are the molar fractions of the two components, and k is the smectic wave number. Note that λi can be positive or negative; in the case that λ1λ2 < 0, smectic layers rich in one of the components will alternately appear, giving rise to a microsegregated smectic phase. The free energy difference between smectic and nematic states, to second order in the λi’s, is: i + ρ i,j=1 xixjλiλj f̃ij(k;Q) where f̃ij(k) are the following cosine Fourier transforms: f̃ij(k) = dtfij(t) cos kt, i, j = 1, 2. (4) The Hessian matrix, ∂2(β∆F/N)/∂λi∂λj , necessary to study the stability of the nematic phase against smectic fluctuations, is 1 + ρx1f̃11(k;Q) ] ρx1x2 f̃12(k;Q) ρx1x2 f̃12(k;Q) 1 + ρx2f̃22(k) The spinodal line is then obtained by solving the equa- tions ∆(k0, ρ0, Q0) ≡ 1 + ρ0x1f̃11(k0;Q0) 1 + ρ0x2f̃22(k0) −ρ20x1x2 f̃12(k0;Q0) ρ0,k0,Q0 ρ0,k0,Q0 = 0 (6) for ρ0, k0 and Q0, the values of the density, wave-vector and order parameter of the unstable nematic at the spin- odal, respectively; note that these quantities will depend on the value of the composition x. The second equation ensures that instability will occur at some particular k vector for the first time, as the density is decreased from above down to the value at the spinodal. For the pure case (x = 0) the instability equations reduce to 1 + ρ0f̃11(k0;Q0) = 0, (7) ∂f̃11 ρ0,k0,Q0 ρ0,k0,Q0 = 0. (8) These are the equations defining a spinodal to an ordered phase in a pure system: 1 − ρ0c̃(k0) = 0, where c̃(k) is the Fourier transform of the direct correlation function (in Onsager theory c̃(k) = −f̃(k)). III. THE τ PARAMETER The stability of the nematic phase against smectic-type fluctuations can be quantified in different ways. In this paper we use the parameter τ proposed by Dogic et al.[10] in their analysis of the hard-rod-hard-sphere mixture: τ = lim . (9) In this expression η is the total packing fraction of the mixture, η2 the partial packing fraction of spheres, and the derivative is evaluated at the nematic-smectic spin- odal line. Other definitions are possible; for example, one could use the spinodal line p(η2), with p the pressure, in- stead of η(η2). Here we adhere to the τ parameter as a measure of smectic stability. It turns out, as shown be- low, that the τ parameter can be directly related to the depletion potential, so that τ contains basic information on depletion forces in the system. We now obtain a compact expression for τ in the frame- work of Onsager theory, considering the case of perfect orientational order (Q = 1). The generalization to free orientational order and a general density functional is given in the Appendix C. Our strategy consists of ob- taining the nematic–smectic spinodal line η = η(η2) per- turbatively in the packing fraction η2 (i.e. for small val- ues of η2), and then extracting the value of τ from the first-order term. We begin by first expanding Eqns. (6) at small x. We assume the expansions ρ = ρ0 + ρ 0x+ ..., k = k0 + k 0x+ ..., (10) where now ρ0 and k0 are the density and wave-vector, at the x = 0 spinodal, and ρ′0 and k 0 are the derivatives of ρ and k with respect to the molar fraction x, at the x = 0 spinodal, respectively. Inserting these equations into the first two of Eqns. (6) we obtain, at zeroth order in x: 1 + ρ0f̃11(k0) = 0, f̃ 11(k0) = 0. (11) To first order in x we get = 1− ρ20f̃ 12(k0), k ′ = −2 f̃12(k0)f̃ 12(k0) f̃11(k0)f̃ 11(k0) . (12) Now, differentiating the equations η = η1 + η2 = ρ(1 − x)v1 + η2 and η2 = ρxv2, and taking the limit x→ 0, we obtain the relation τ = lim where v1, v2 are the volumes of the two particle species, rods and spheres respectively. This equation, along with Eqn. (11), gives η0 = −v1/f̃11(k0), and τ = 1− η20 f̃212(k0) f̃12(k0) f̃11(k0) . (14) In fact, this expression remains valid for binary mixtures composed of any convex bodies, but can only be used in the approximation of frozen particle orientations (Q = 1) and in the framework of the Onsager approximation. The τ parameter is calculated by solving Eqns. (11) and then evaluating Eqn. (14). We now proceed to show the direct relation existing between the parameter τ and the depletion interaction between two parallel rods mediated by the solvent parti- cles (hard spheres), at least in the limit where the density of the solvent particles ρ2 and their diameter, relative to the breadth of the rods, are very small. These are the conditions under which the Asakura–Oosawa approxima- tion [19] is valid, and the depletion potential becomes Vdep(r, ρ2) = −ρ2 [f12 ∗ f12] (r) (the asterisk standing for convolution). Now ∂Ṽdep(k0) = −f̃212(k0), (15) and Eqn. (14) gives τ = 1 + ∂Ṽdep f̃211(k0) (this equation can also be obtained from a mean–field perturbative treatment of the fluid, by considering par- allel hard rods that interact via a depletion potential in mean–field approximation). Eqn. (16) relates the τ co- efficient with the depletion potential, and shows that τ contains basic information on depletion forces. In turn, it provides a condition under which τ may be negative, and thus directly links the depletion potential to the en- hancement of smectic–phase stability in fluid mixtures of hard rods and spheres. IV. RESULTS A. Mixtures of HSPC and HS: frozen versus free orientations All previous theoretical analyses of the lamellar phase in the hard-rod–hard-sphere mixtures [7, 10, 13] have relied on the approximation of considering perfect ori- entational order. In this section we assess the impact of this severe approximation on the formation of smec- tic phases in the HSPC/HS mixture. We have investi- gated a number of HSPC/HS mixtures using the orig- inal Onsager theory, changing the HSPC aspect ratio κ1 ≡ (L1 + D1)/D1 and the HSPC breadth relative to the HS diameter, d ≡ D1/D2. Fig. 1 shows the spin- odal line η–η2 of the nematic–smectic transition, for the cases of frozen and free orientations, and for κ1 = 8 and various values of the scaled HS diameter. We note that, in all cases shown, the smectic phase is stabilised with respect to the nematic phase (negative slope at η2 = 0) on adding HS to the pure HSPC fluid; this is always the case when d ≥ 1 (spheres smaller than the cylinder di- ameter). The effect is amplified when both the HSPC aspect ratio length (κ1) is increased and the sphere di- ameter is decreased (d increases). However, the effect is less pronounced when the HSPC are free to orient their main axes. The τ parameter, shown in Fig. (2), reflects this behaviour, which points to a less strong depletion effect due to orientational fluctuations. B. Other hard rod–hard sphere mixtures: effect of particle shape To investigate the effect of the shape of the rods on the depletion effect and, in turn, on the stability of the smec- tic phase, Vesely[13] has recently considered parallel- rod models consisting of linear overlapping hard-sphere chains, hard ellipsoids and hard spheroellipsoids. Here we have studied, still in the approximation of frozen ori- entations, two additional types of hard particles: cylin- ders (HCYL) and parallelepipeds with a square trans- verse shape (HPAR). To minimize the effect of the dif- ferent particle geometries, the particles were chosen to have the same volume (see later). In all cases the minor- ity component continues to consist of hard spheres. The functions fij(z), and also their Fourier transforms, can be calculated analytically in these cases. The relevant expressions can be found in the Appendix A. 0 0.005 0.010 0.015 0.020 d -1=1/3 0 0.005 0.010 0.015 0.020 d -1=1/3 FIG. 1: Nematic–smectic spinodal line η(η2), according to Onsager theory, for the case κ1 = 8, and d −1 = 1, 1/2 and 1/3 (indicated as labels). (a) Frozen–orientation approximation; (b) free orientations. Note that horizontal and vertical scales in both graphs are the same. Results for the τ parameter in the case of HCYL/HS and HPAR/HS mixtures, along with the previously con- sidered HSPC/HS mixtures, are shown in Figs. 3(a)-(c). The τ parameter is plotted against the aspect ratio of the rods κ1 in Fig. 3(a) for different values of the scaled in- verse HS diameter d = D1/D2 (D1 being the side length of the parallelepiped in the case of HPAR); Fig. 3(b) is a zoom of (a) in the region of small κ1. Finally, the τ parameter against d for different values κ1 is plotted in Fig. 3(c). We can see from Fig. 3(a) that τ tends to decrease linearly, becoming negative for sufficiently large aspect ratios. This effect is more and more pronounced as the diameter of the HS is decreased. An interesting feature is that, when the diameter of HS is greater than that of the rod, depletion is larger, and therefore smec- tic stability is enhanced more substantially, as one goes 0 1 2 3 4 5 6 κ 1=21 FIG. 2: Values of τ as a function of inverse HS diameter ratio d, for various values of HSPC length–to–breadth ratio κ1. Lines: Frozen–orientation approximation (Q = 0); from top to bottom: κ1 = 6, 8, 11 and 21. Symbols: free orientations. Squares: κ1 = 5; open circles: κ1 = 8; filled circles: κ1 = 11, and triangles: κ1 = 21. from HPAR to HCYL or HSPC (note that these two are very similar). The opposite behaviour (i.e. depletion in HPAR is enhanced with respect to HCYL and HSPC) results when the HS diameter is equal or less than that of the HSPC [Fig. 3(a)]. For short rods τ exhibits oscil- latory behaviour [see Fig. 3(b)]. Finally, fixing κ1 and increasing d, we find that the τ parameter decreases as a cubic power-law in d and that its value for HPAR is much less than that for HSPC or HCYL. The linear and cubic dependences of τ with respect to κ1 and d, respectively, can be elucidated from Eqn. (14) using the fact that the volume ratio can be expressed as v1/v2 = κ1d 3 for the HCYL case (the other cases being similar), while the squared ratio of Fourier transforms of the overlap functions tends asymptotically, for large κ1 and d, to a constant. Simple calculations give, in this limit, τHC = τHSPC = − 3 (17) with k∗0 = k0L1 the reduced wave number of the N–Sm spinodal instability in the one–component system. To study the effect of a varying particle shape on the phase behaviour of the rod-sphere mixtures, we have per- formed a full minimization of the Helmholtz free-energy functional with respect to smectic-like density profiles of the rods [ρ1(z)] and spheres [ρ2(z)]. The minimization was carried out with respect to the Fourier amplitudes of the truncated Fourier expansion of the density pro- files and smectic wave number (details are provided in Appendix B). The choice of a Fourier expansion is justi- fied by the fact that the HS equilibrium density profiles 0 10 20 30 40 50 60 2 3 4 5 6 7 8 0 2 4 6 8 10 κ1=10 FIG. 3: (a) The τ parameter vs. κ1 for different values of the parameter d (indicated in the graph). (b) A zoom of (a) for small κ1’s. (c) τ vs. d for different values of κ1 as indicated in the figure. (c) τ parameter as a function of d for different values of κ1, as indicated. Results for HSPC, HCYL, and HPAR are plotted with solid, dashed, and dotted lines, respectively. obtained with this minimization differ very significantly from those obtained via the usual single–amplitude ex- ponential parametrization, as shown in Fig. 4. We can see that the parametrization produces too sharp density peaks in the HS profile, which enhances the depletion effect. 0 0.2 0.4 0.6 0.8 1 FIG. 4: Equilibrium density profiles of the equimolar HSPC– HS mixture (different components are labelled in the figure) resulting from minimization with respect to the Fourier am- plitudes (solid lines) and with respect to a single–amplitude exponential parametrization (dashed lines). The HSPC has an aspect ratio κ1 = 10 and its breadth coincides with the HS diameter, taken as unity. The pressure in reduced units was chosen to be 0.4. Since we would like to compare the results from the density–functional minimization of mixtures composed of rods of different geometries (HSPC, HCYL and HPAR), a sensible criterion is required to choose the sizes of the particles so as to make sure that these differences arise from the shape and not from different sizes and volumes. For this purpose we have used three different criteria: (i) all particles have the same diameter and the same length, (ii) all particles have the same length and the same volume, and (iii) all particles have the same aspect ratio and the same volume. Comparing the equilibrium density profiles resulting from use of these three criteria, we have concluded that differences between particles are minimised using the third criterion (these results are not shown here). We have calculated, using the free, i.e. Fourier–based, minimization, the phase diagrams of binary mixtures of rods (HSPC, HCYL and HPAR) and HS. The HSPC and HS diameters were taken to be equal and adopted as unit of length. The aspect ratios of all the rods were set to 10, all particle volumes being the same. The following relations result: DHC = 1− (3κHSPC) ]−1/3 DHSPC, DHP = DHC, Lµ = κHSPCDµ, (18) with µ=HSPC, HC or HP. The resulting phase diagrams are shown in Fig. 5. The first interesting feature of all phase diagrams is the presence of a tricritical point, where the nature of the N–Sm transition changes from 0 0.2 0.4 0.6 0.8 1 HCYLHPAR FIG. 5: Phase diagrams in the reduced pressure–composition plane of three different mixtures composed of rods of different geometries (HSPC, HCYL, and HPAR) with HS. Different lines corresponding to each mixture are labelled in the figure. Also indicated are the stability regions of the nematic and the smectic phases. Solid lines represent the binodals of the N– Sm coexistence, while the continuous N–Sm transitions have been indicated by a dashed line. second to first order[20]. For pressures higher than that of the tricritical point, the demixing instability region becomes wider. The width of this demixing window in- creases for mixtures going from HSPC to HCYL and then HPAR. This scenario clearly demonstrates that in- creasing differences in shape between spheres and rods enhance the depletion in smectics, resulting in a wider demixing gap. C. Zwanzig model: Onsager theory versus FMT In order to check whether the effects presented above are robust with respect to the theory and approxima- tions used, we have analysed a mixture of HPAR and HC using Fundamental–Measure theory in the Zwanzig ap- proximation. Zwanzig [21] introduced a model that con- siderably reduces the complexities associated with min- imising a density-functional that depends on particle ori- entations. The approximation consists of restricting the possible orientations of a rod to lie along the three or- thogonal axes x, y and z, treating a pure system as a three-component mixture (one component for each orien- tation). The approximation is very useful and has been used quite substantially. In our FMT context, the re- sulting theory treats spatial correlations very accurately so that it should better describe the onset of smectic or- der in the nematic fluid. We have used the HPAR-HC system and the Zwanzig approximation because in this combination a FMT can be rigorously formulated [17]. The Zwanzig approximation should not be terribly lim- iting, since the nematic fluid is well oriented. In order to elucidate the effect of restricted orientations separately from that inherent to the proper inclusion of higher–order correlations, a study of the HPAR-HC mixture using a Zwanzig–Onsager theory, together with that on mixtures of parallel HPAR and HC, is necessary. 1. Zwanzig model in the Onsager formulation In Appendix D we have written expressions for the Fourier transforms of the correlations functions c̃ij(k) as obtained from the Zwanzig–Onsager approach. These expressions are necessary to calculate the τ parameter using Eqn. (C.10). The results obtained are plotted in Fig. 6. In the same figure the parallel case is also plotted for the sake of comparison. The usual cubic power–law behaviour of τ with respect to d is obtained; this is typical of any Onsager approach, as discussed in Sec. IVB. The Zwanzig approach gives a lower value for τ as compared to the parallel case. This result is similar to that already obtained using the freely rotating Onsager approach for HSPC. For high values of κ1, both approaches collapse into a single line, which is due to the high value of the order parameter Q of a fluid composed of long particles. 0 2 4 6 8 10 FIG. 6: τ parameter as a function of d for three different values of κ1, as indicated. Solid and dashed lines correspond to the Zwanzig and parallel approaches, respectively. 2. Zwanzig model in the FMT formulation Details on this approximation can be found in Ref. [17]. Here we only give a brief sketch of the theory. We continue to use the notation introduced in the previous section for the dimensions of the particles, i.e. the length of the HPAR is L1, the side length is D1 (these particles are assumed to have a square section), and D2 is the side length of the cubes. Since the unit vector Ω̂ only has three possible orientations, the one–particle distribution functions ρs(r, Ω̂) can be expressed as ρs(r, Ω̂) = ρsµ(r)δ(Ω̂ − êµ) (19) where êµ, µ = 1, 2, 3, are unit vectors along the three perpendicular directions xyz, respectively, and ρsµ(r) are the local density of species s parallel to the µ-axis. The excess part of the free-energy density, in reduced thermal units, is obtained in [17], and has the form Φex(r; {ρs}) = −n0 ln(1− n3) + n1 · n2 1− n3 n2xn2yn2z (1− n3)2 with the functions {nα} (α = {0, 1x, 1y, 1z, 2x, 2y, 2z, 3}) being weighted densities obtained as nα(r) = dr′ρsµ(r ′)ω(α)sµ (r− r ′), (21) where ω sµ are characteristic functions whose spatial in- tegrals give the fundamental measures of the particles (edge length, surface and volume). The ideal part of the free energy density in reduced thermal units for this model is Φid(z) = ρsµ(z) [log ρsµ(z)− 1] , (22) so that the total free energy per unit volume and unit thermal energy can be calculated as dz [Φid(z) + Φexc(z)] . (23) Density and order–parameter profiles can be defined; in particular, the density profile is ρs(z) = ρsµ(z). (24) The nematic–smectic spinodal line can be obtained in this theory using the same kind of arguments used for the Onsager theory (Sec. II). Note, however, that the expression (14) for the τ parameter is not valid in this context, and we need to use the more general formula (C.10), valid for any mixture of freely rotating particles. In the Appendix C we describe in detail how this formula is obtained. An alternative way to calculate this param- eter is to implement a numerical differentiation scheme on the spinodal line η(η2). The nematic–smectic spinodal lines for the case κ1 = 8 and various values of the scaled cube side length d−1 = D2/D1 are shown in Fig. 7, whereas the corresponding τ parameters for different values of κ1 are shown in Fig. 8. As was the case in the mixtures analysed using Onsager theory, τ decreases as the size of the cubes is diminished, 0 0.005 0.01 0.015 0.02 0.025 d -1=0.8 FIG. 7: Nematic–smectic spinodal line η–η2, as obtained from Fundamental–Measure theory for the HPAR-HC mixture, for κ1 = 8 and various values of d −1, as indicated in the graph. indicating a stronger depletion effect and a corresponding enhancement of the smectic–phase stability. An interest- ing feature of these results is that they appear to exhibit linear behaviour for large values of d (i.e. as the size of the cubes becomes smaller). Fig. 8 shows this by means of least–square fits to linear functions. The coefficients of the fit are functions of the aspect ratio κ1, and, in turn, appear to be linear in κ1, as shown in Fig. 9. 0 2 4 6 8 10 τ κ1=6 FIG. 8: τ parameter as a function of d for various values of the parallelepiped length–to–breadth κ1 = L1/D1 (indicated in the graph). The data are indicated by symbols; linear least- square fits τ = α(κ1) + β(κ1)d are included as lines for each value of κ1. To explain the linear behaviour obtained for the τ pa- rameter as a function of d, we have analytically calcu- lated the asymptotic behaviour of τ in the régime d≫ 1 for a mixture of parallel parallelepipeds and cubes in the 6 8 10 12 14 16 18 20 FIG. 9: Symbols: coefficients α (circles) and β (squares) as a function of κ1. Solid lines: linear fits α(κ1) = α1 + α2κ1 and β(κ1) = β1 + β2κ1, with β2 = −0.038. FMT formulation. In Fig. 10 we compare the τ param- eter, for various values of κ1, obtained from the Zwanzig and parallel FMT approaches. An interesting feature is that depletion is enhanced in the Zwanzig approach, a result opposite to that from the Onsager approach (Fig. 0 2 4 6 8 10 κ1 =6 FIG. 10: The τ parameter as a function of d calculated from the Zwanzig (solid line) and parallel (dashed line) FMT ap- proaches. Values of κ1 are indicated as labels. Details on the calculation of the asymptotic behaviour of τ with respect to d have been relegated to Appendix F. From the general expression of τ (Eqn. C.10), partic- ularized for the parallel case (Q = 1, which makes the third terms in the numerator and denominator to van- ish), the linear behaviour can be explained if the sum of the two terms in the numerator are of order d−2. Since the volume ratio is v1/v2 = κ1d 3, we obtain the asymp- totic behaviour τ ∼ κ1d = L1/D2. This result can also be obtained using the Zwanzig model (Figs. 8 and 9), which shows that the third term of the numerator in Eqn. (C.10) is also of order ∼ d−2. The coefficient β(κ1) of the linear fits of τ with respect to κ1 (Fig. 9) depends lin- early on κ1 and has a slope β2 = −0.038 (see caption of Fig. 9), which is in the range of values predicted by the asymptotic limit ofW (κ1) ≡ τ/(κ1d) as a function of κ1, shown in Fig. (11). The function (∂∆/∂ρ2) [Eqn. (F.1)] is of order d−2 only within the FMT formulation, the Onsager approach giving a term ∼ d0. This, in turn, shows the importance of properly taking account of pair correlations between particles in order to adequately describe depletion inter- actions. The asymptotic limit coincides with the adhe- sive limit because, when the mixture is highly asymmet- ric, the attractive depletion potential between two big particles becomes narrower and deeper, tending in the limit to a Dirac delta function at contact. This result is confirmed by our calculations in Appendix F, which show that the inverse Fourier transform of one of the terms of the function (∂∆/∂ρ2) is a Dirac delta func- tion at contact. We have repeated the same analysis in Appendix F for the case of a HS binary mixture in the FMT formulation (PY approach). We have obtained that (∂∆/∂ρ2) ∼ O(d−3). Also, a Dirac delta func- tion at contact is present in the inverse Fourier trans- form of (∂∆/∂ρ2) . The difference in the limit of in- finite size asymmetry between the HPAR-HC and HS cases is due to the difference in particle shapes: when two big particles are closer enough the parallel sides of two parallelepipeds exclude many more small particles from their interstitial space than the curved surface of two big hard spheres. The discussion above forces us to speculate about the possible asymptotic behaviour of the τ parameter for the HSPC-HS mixture in the limit d≫ 1: since v1/v2 ∼ d 3, τ ∼ O(d0) [Eqns. (C.10) and (F.2)]; this is a consequence of the caps of the spherocylinders being spherical. This argument does not apply to the behaviour of τ in the limit of large κ1 for fixed d, which continues to be linear. V. CONCLUSIONS AND PERSPECTIVES The first result from this study on depletion effects in smectic phases is that, within Onsager theory, the in- clusion of orientational degrees of freedom significantly reduces the enhanced stability of the smectic phase over the nematic phase as compared to that obtained from the frozen-orientation approximation. A second conclu- sion that can be drawn is that when the surface curvature of the rods is very different from that of the other species (e.g. the case of HPAR and HS), depletion is enhanced and the smectic phase stabilizes at lower packing frac- tions as HS become more abundant. This effect is also reflected in the fact that the N–Sm demixing gap in the phase diagram widens as one goes from HSPC to HP particles through the HC geometry. Finally, the inclu- sion of more realistic pair correlations between particles, in the manner of FMT theory, changes the asymptotic behaviour of the τ parameter (a convenient measure of depletion in smectics) for large values of the ratio d, i.e. the asymmetry between the breadth of the rod and the sphere diameter; in particular, for HPAR, τ increases lin- early with d, while for HSPC we speculate that τ goes to a finite limit. The limiting case of large aspect ratios and finite d is similarly captured by both Onsager and Fundamental–Measure theories. In this work we have studied the depletion mecha- nism along the N–Sm spinodal line for different mixtures by explicit calculation of the τ parameter evaluated on this line in the pure fluid. An interesting task could be to extend this study to equilibrium smectic phases, consisting of well-developed density peaks, by defining some quantity evaluated at the smectic density profile. In this case depletion-based mechanisms may add inter- esting phenomenology, such as strong microsegregation between different species, which could be directly quan- tified using some suitably defined parameter. Work along this direction is currently in progress. Acknowledgments We are grateful to the referee for his/her comments, suggestions and careful reading of the manuscript. We thank MIUR (Italy) and Ministerio de Educación y Ciencia (Spain) for financial support under the 2005 binational integrated program. We gratefully ac- knowledge financial support from Ministerio de Edu- cación y Ciencia under grants Nos. FIS2005-05243- C02-01, FIS2005-05243-C02-02, FIS2004-05035-C03-02, BFM2003-0180 and from Comunidad Autónoma de Madrid (S-0505/ESP-0299). YMR was supported by a Ramón y Cajal research con- tract from the Ministerio de Educación y Ciencia. VI. APPENDIX Appendix A. Overlap functions: case of frozen orientations When particle orientations are parallel, the functions fij(z) and their Fourier transforms can be written exactly for a large class of particle shapes and their mixtures. We consider in turn each of the particle geometries analysed in the paper. In the case of mixtures of HSPC of length L1 and breadth D1, and HS of diameter D2, the Fourier trans- forms are f̃11(k) = {sin [k(D1 + L1)] − D1k cos [k(D1 + L1)]− sin (kL1)} , f̃12(k) = {sin [(D12 + L1)k/2] D12 cos [(D12 + L1)k/2]− sin (kL1/2) f̃22(k) = [sin (kD2)− kD2 cos (kD2)] , (A.1) where D12 = D1 +D2. For mixtures of HCYL of length L1 and breadth D1, and HS of diameter D2, the appropriate Fourier trans- forms: f̃11(k) = 2πD21 sin (kL1) , f̃12(k) = 2π k(L1 +D2) k(L1 +D2) k(L1 +D2) − sin − sin + sin f̃22(k) = [sin (kD2)− kD2 cos (kD2)] (A.2) where J1(x) and H1(x) are the Bessel and Struve func- tions of first order, respectively. For mixtures of HPAR with length and square side length L1, D1, respectively, and HS of diameter D2, the Fourier transforms are f̃11(k) = sin (kL1) , f̃12(k) = 8 sin [k(L1 +D2)/2] k(L1 +D2) − sin k(L1 +D2) − sin + sin f̃22(k) = [sin (kD2)− kD2 cos (kD2)] . (A.3) Appendix B. Minimisation of Onsager functional in the frozen-orientation approximation We assume a Fourier expansion for the density profiles ρi(z): ρi(z) ≡ ρxiψi(z) = ρxi s(i)n cosnkz , (B.1) with k the smectic wave number, and s n the n-th Fourier amplitude for species i. Introducing this expression into the free-energy funtional, Eqn. (1), we obtain Φ = ρ log ρ− 1 + xi lnxi + dzψi (z) lnψi (z) + ρW(x, k, {s , (B.2) Here we have expressed the excess part in terms of the function W(x, k, {s(i)n }) = f̃ij(0) + s(i)n s n f̃ij(kn) , (B.3) where kn = nk and the Fourier transforms f̃ij(k) were defined in Section II (since we are assuming perfect ori- entational order, we set Q = 1). From this, the pressure βP = ρ+ ρ2W(x, k, {s(i)n }) (B.4) To calculate phase equilibria in the mixture it is more convenient to work in conditions of constant pressure. Then, we minimize the Gibbs free energy per particle and unit thermal energy, (Φ + βP ) (B.5) with respect to the smectic wave number and the co- efficients {s n }. The corresponding derivatives can be written down explicitely but they have to be solved nu- merically using an iterative method. In practice, the Fourier expansions have been truncated; they include ca. 40 terms so as to satisfy a stringent convergence criterion in the iterative procedure (the number of Fourier ampli- tudes was chosen to guarantee an absolute error less than 10−7 in the density profiles). Appendix C. τ parameter: general case The strategy is to obtain the nematic–smectic spinodal line perturbatively in the composition x, since we only need to know the spinodal in the neighbourhood of x = 0 to obtain the τ parameter. The equations to solve are three: two defining the spinodal, the third giving the equilibrium state of the nematic fluid on the spinodal [Eqns. (6)]. These equations can be written as ∆(k, ρ1, ρ2, Q) = [1− ρ1c̃11(k, ρ1, ρ2, Q)] × [1− ρ2c̃22(k, ρ1, ρ2, Q)]− ρ1ρ2c̃ 12(k, ρ1, ρ2, Q) = 0 (C.1) ∂∆(k, ρ1, ρ2, Q) = 0 (C.2) ∂Φ(ρ1, ρ2, Q) = 0, (C.3) where c̃ij(k, ρ1, ρ2, Q) are the Fourier transforms of the direct correlation functions between species i and j. The relationship between infinitesimal changes in the vari- ables {k, ρ1, ρ2, Q} along the spinodal can be calculated from (C.1) as dρ1 + dρ2 + dk = 0. (C.4) Similarly, the relation between the changes in the order parameter Q and densities {ρ1, ρ2} can be obtained from (C.3) as ∂Q∂ρ1 dρ1 + ∂Q∂ρ2 dρ2 + dQ = 0. (C.5) Defining the coefficient υ ≡ dρ1/dρ2, we obtain from Eqns. (C.4) and (C.5): υ = − ∂2Φ/∂Q∂ρ2 (∂2Φ/∂Q2) ∂2Φ/∂Q∂ρ1 (∂2Φ/∂Q2) , (C.6) where the condition given in (C.2) of the absolute mini- mum of ∆(k, ρ1, ρ2, Q) with respect to the wave number k was used. Note that this expression is valid for any composition x of the mixture. Now, evaluating all the derivatives at ρ2 = 0 we obtain, from Eqn. (C.1) = −ρ0 ∂c̃11 , (C.7) = −ρ0 ∂c̃11 , (C.8) = −ρ0 ∂c̃11 , (C.9) where the superscripts mean that all derivatives are eval- uated at x = 0, k = k0, and Q = Q0, values correspond- ing to the one-component fluid spinodal [note that, from Eqn. (C.1), we get 1 − ρ0c̃11 = 0 at ρ2 = 0, which has been used to obtain the derivatives above]. Now, using the fact that, for the one-component fluid, 1−ρ0c̃ 11 = 0, and that the τ parameter can be expressed through the coefficient υ evaluated at x = 0 as τ = 1 + (v1/v2)υ (0), we finally obtain τ = 1− ∂c̃11 ∂c̃11 ∂2Φ/∂ρ2∂Q (∂2Φ/∂Q2) ∂c̃11 ∂c̃11 ∂2Φ/∂ρ1∂Q (∂2Φ/∂Q2) (C.10) which is a general expression for the τ parameter. In the Onsager approximation, we have c̃ij(k, ρ1, ρ2, Q) = f̃ij(k,Q) and the derivatives of c̃ij with respect to the densities ρi vanish, while for parallel rods (Q = 1) the derivative with respect to Q also vanishes, and we obtain expression (14). For a fluid mixture of particles without orientational degrees of freedom, such as a mixture of parallel paral- lelepipeds and cubes or a mixture of hard spheres, expres- sion (C.10) for τ can be reinterpreted as follows. From Eqn. (C.10), the τ parameter can be re-expressed as τ = 1− ∂S−1ef ∂S−1ef , (C.11) where S−1ef (k, ρ1, ρ2) is the inverse structure factor of an effective one-component fluid of particles, labelled as 1, with interactions between them being mediated by parti- cles labelled as 2. This structure factor can be calculated by evaluating the second functional derivative with re- spect to ρ1(r) of the semi-grand canonical free-energy functional Υ[ρ1] ≡ F [ρ1, ρ2]− µ2 drρ2(r), (C.12) at the bulk densities[22]. Here F [ρ1, ρ2] is the Helmholtz free-energy functional of the binary mixture and the den- sity profile ρ2(r) in Eqn. (C.12) is calculated from the condition of fixed chemical potential of species 2: δF [ρ1, ρ2] δρ2(r) = µ2. (C.13) The result for this effective structure factor is S−1eff (k, ρ1, ρ2) = 1− ρ1c̃11(k, ρ1, ρ2) ρ1ρ2c̃ 12(k, ρ1, ρ2) 1− ρ2c̃22(k, ρ1, ρ2) . (C.14) After the inclusion of its first derivative with respect to ρ1 and ρ2, evaluated at ρ2 = 0 in Eqn. (C.11), we exactly obtain Eqn. (C.10) without the third terms in both nu- merator and denominator (as they vanish for fluids with- out orientational degrees of freedom). Appendix D. Correlation functions for HPAR-HC mixture in the Zwanzig–Onsager approach In the Zwanzig approximation the rods point along one of the Cartesian axes x, y or z. For the binary HP-HC mixture in the Zwanzig approach the correlation func- tions evaluated at k = (0, 0, k) can be calculated as − c̃11(k) = 2x f̃1x,1x(k) + f̃1x,1y(k) + x2‖f̃1z,1z(k) + 4x⊥x‖f̃1x,1z(k), (D.1) −c̃12(k) = 2x⊥f̃1x,2(k) + x‖f̃1z,2(k), (D.2) −c̃22(k) = f̃22(k), (D.3) where the subindexes 1µ (µ = x, y, z) label particle 1 (rods), which point along the direction µ, while the subindex 2 labels cubes. x⊥ and x‖ are respectively the fraction of rods perpendicular and parallel to the nematic director (which is taken to be parallel to the z axis). These variables are functions of the nematic order pa- rameter Q [i.e. x⊥ = (1 − Q)/3, and x‖ = (1 + 2Q)/3], the equilibrium values of which should be calculated from the extremum condition of the free-energy density with respect to Q. This condition reads 1 + 2Q − 2ρ1 (L1 −D1) (D.4) The expressions for the Fourier-transformed overlap func- tions are f̃1x,1y(k) = 2(L1 +D1) 2 sin(kD1) f̃1x,1x(k) = 8L1D1 sin(kD1) f̃1x,1z(k) = 4D1(L1 +D1) sin [k(L1 +D1)/2] f̃1z,1z(k) = 8D sin(kL1) f̃1x,2(k) = 2(L1 +D2)(D1 +D2) sin [k(D1 +D2)/2] f̃22(k) = 8D sin(kD2) f̃1z,2(k) = 2(D1 +D2) 2 sin [k(L1 +D2)/2] , (D.5) Note that, due to the discrete axial symmetry of the par- ticles, some overlap functions can be expressed in terms of others; for instance, f̃1x,1x = f̃1y,1y, and f̃1z,1x = f̃1z,1y. Appendix E. Correlation functions for HPAR-HC and HS mixtures in the FMT approach To obtain the Fourier transforms of the correlation functions for HP-HC mixtures [17], and for HS mixtures [16], we have used the FMT functional, which has the same structure in both systems, namely: − c̃ij(k) = χ (0)f̃ij(k) + χ R̃ij(k) + χ S̃ij(k) + χ(3)Ṽij(k), (E.1) where f̃ij , R̃ij , S̃ij and Ṽij are Fourier transforms of the overlap function, and of the mean radius, surface and volume of the overlap region between particles i and j, respectively. For a HPAR-HC mixture the mean radius and the surface area of the overlap region are vectors oriented along the directions parallel to the edge lengths (mean radius) and perpendicular to the sides (surface area) of the parallelepipeds. For the HS mixture they are scalars, as are the quantities χ(1) and χ(2), which for the HPAR-HC mixture are vectors. Finally, the correlation functions depend on the wave vector k or on its mod- ule k in the HPAR-HC and HS mixtures, respectively. The expressions for χ(i) corresponding to the HPAR-HC mixture are χ(0) = , χ(1) = (1− η)2 , (E.2) (2) = (1− η)2 (1− η)3 , (E.3) χ(3) = (1− η)2 2ξ1ξ2 (1− η)3 6ξ2xξ2yξ2z (1 − η)4 (E.4) with ρ = i ρi and η = i ρivi the total density and packing fraction, respectively, ζ2 = (ξ2yξ2z , ξ2zξ2x, ξ2xξ2y), and ξ1x = ξ1y = ρiDi, ξ1z = ρ1L1 + ρ2D2, (E.5) ξ2x = ξ2y = ρ1L1D1 + ρ2D 2, ξ2z = i , (E.6) with L1 and D1 the length and breadth of the paral- lelepiped, while D2 is the edge-length of the cube.For the HS mixture we have χ(0) = , χ(1) = (1− η)2 , (E.7) χ(2) = (1 − η)2 (1 − η)3 , (E.8) χ(3) = (1 − η)2 2ξ1ξ2 (1− η)3 (1− η)4 , (E.9) where ρiDi, ξ2 = π i , (E.10) and Di (i = 1, 2) the particle diameters. To calculate the τ parameter we only need expressions for c̃11 and c̃12. The Fourier transforms of the geometric measures of overlapping bodies for the HPAR-HC fluid with a wave number k = (0, 0, k) (smectic symmetry) are f̃11(k) = 8D sin(kL1) , (E.11) R̃11(k) = 4D sin(kL1) sin(kL1) sin(kL1/2) , (E.12) S̃11(k) = 2D sin(kL1/2) sin(kL1/2) sin(kL1) , (E.13) Ṽ11(q) = 4D sin(kL1/2) (E.14) The expressions for particles 1 and 2 are f̃12(k) = 2D sin(kL12/2) (E.15) R̃12(k) = 2D12 sin(kL12/2) ,D1D2 sin(kL12/2) sin(kL1/2) sin(kD2/2) , (E.16) S̃12(q) = 2D1D2 sin(kL1/2) sin(kD2/2) sin(kL1/2) sin(kD2/2) sin(kL12/2) (E.17) Ṽ12(q) = 4(D1D2) 2 sin(kL1/2) sin(kD2/2) , (E.18) where D12 = D1 +D2 and L12 = L1 +D2. For HS mixtures these expressions have the form f̃ij(k) = sin(kDij/2) cos(kDij/2) , (E.19) R̃ij(k) = sin(kDi/2) sin(kDj/2) cos(kDij/2) , (E.20) S̃ij(k) = (4π)2 sin(kDi/2) sin(kDj/2) sin(kDij/2) , (E.21) Ṽij(k) = (4π)2 sin(kDi/2) sin(kDj/2) sin(kDij/2) + cos(kDi/2) cos(kDj/2)] , (E.22) for i, j = 1, 2. Appendix F. Asymptotic behaviour of τ The asymptotic behaviour of (∂∆/∂ρ2) for d = D1/D2 ≫ 1 can be calculated from (C.8) and (E.1), particularizing the latter equation to HPAR-HC and HS mixtures. After some algebra, we arrive at = −ρ0 ∂c̃11 y(1 + y) 2(1 + y)2T 21 (k 0) + 2yT1(k 0) + y 2T 21 (k ,(F.1) where κ1 = L1/D1, T0(x) = cosx, and T1(x) = sinx/x, while y = η0/(1 − η0), and k 0 = k0L1. The expression corresponding to the HS fluid reads y(1 + y) 2y2(1 + 3y)TV (k 0) + Tδ(k + 2yTf(k 0) + y(1 + 6y)TS(k 0)} , (F.2) where we have defined the dimensionless quantities TV (k 0) = 6v 1 Ṽ (k 0), Tδ(k 0) = 3s 1 δ̃(k 0), Tf (k 3v−11 f̃(k 0), and TS(k 0) = 6(v1s1) −1S̃(k∗0), with s1 and v1 the surface area and volume of particle 1, while δ̃(q) is the Fourier transform of the Dirac delta function δ(D1−|r|). The presence of a delta function indicates that, in the limit d → ∞, the one-component sticky-sphere limit of the fluid is obtained. We arrive at the same conclusion, in the same limit, for the HPAR/HC mixture, by compar- ing Eqns. (F.1) and (F.2). Each term in the right-hand side of Eqn. (F.1), from left to right, has the same mean- ing as in the HS mixture: they are related to the Fourier transform of the volume, adhesiveness, overlap function, and surface area of the overlap region between two parti- cles, respectively. For example, the inverse Fourier trans- form of T0(k ∗), with the smectic symmetry q = (0, 0, q), results in a term proportional to δ(L1 − |z|)/2, which reflects the stickiness of parallelepipeds along the z di- rection. A complete effective density functional for the infinite asymmetric limit was worked out for hard-cube mixtures in Ref.[23]. The direct correlation function re- sulting from this functional has a Dirac delta function located at contact of the sides of the cubes. However, there is an important difference between the HP-HC and HS mixtures, which is related to the square and cubic power dependence of the expressions (F.1) and (F.2) with respect to the asymmetric parameter d = D1/D2. This result is related to the fact that the planar geometry of the sides of parallelepipeds enhances the depletion interaction between two big particles. We should compare these results with those obtained using the Onsager approach, where c̃ij(k, ρ1, ρ2) = f̃ij(k) and then (∂∆/∂ρ2) ∼ O(d0). Thus, we can conclude that depletion interactions in this limit cannot be properly described by the Onsager approximation. Finally, to obtain the asymptotic behaviour of the τ parameter for the HP-HC mixture in the limit d → ∞ , we use Eqns. (C.10), (F.1) and (E.1), particularized for HP, together with the spinodal instability condition 1− ρ0c̃ 11 = 0, to obtain explicitly τ ∼ κ1dW (κ1), (F.3) W (κ1) = − 2(1 + y)2T 21 (k 0/2) + [T0(k 0) + 2yT1(k + y2T 21 (k 3 + y−1 + 2(3 + y)T1(k 3 + y + 6(1 + y)3 T 21 (k (F.4) In Fig. (11) the function W (κ1) is plotted. As can be seen from the figure, (i) W varies very little with respect to κ1, and (ii) depletion is maximum in the Onsager limit (κ1 → ∞). Note that, on taking the latter limit, the condition d = D1/D2 ≫ 1 must be fulfilled. It is well known that, when the PY approximation is used, the condition 1−ρ0c̃11(k, ρ0) > 0 is always fulfilled for all η and q in the physical parameter region. Because of the absence of a fluid-solid spinodal in this approxi- mation, we cannot calculate the τ parameter to estimate the effect of depletion at the freezing transition. 0 0.2 0.4 0.6 0.8 1 -0.04 -0.038 -0.036 -0.034 -0.032 FIG. 11: The function W (κ1). [1] M. Adams, S. Fraden, Biophys. J. 74, (1998) 669. [2] M. Adams, Z. Dogic, S.L. Keller, S. Fraden, Nature 393, (1998) 349. [3] Z. Dogic, S. Fraden, Current Opinion in Colloid & Inter- face Science 11, (2006) 47. [4] D. L. Cheung and M. P. Allen, Phys. Rev. E 74, (2006) 021701. [5] D. Andrienko, M. Tasinkevych, P. Patricio, M. P.Allen, and M. M. Telo da Gamma, Phys. Rev. E 68, (2003) 051702. [6] R. Roth, R.van Roij, D. Andrienko, K. R. Mecke, and S. Dietrich, Phys. Rev. Lett. 89, (2002) 088301. [7] T. Koda, M. Numajiri, S. Ikeda, J. Phys. Soc. Jap. 65, (1996) 3551. [8] T. Koda, S. Ikeda, Mol. Cryst. Liq. Cryst. 318, (1998) [9] T. Koda, Y. Sato, S. Ikeda, Prog. Theor. Phys. Suppl. 138, (2000) 476. [10] Z. Dogic, D. Frenkel, and S. Fraden, Phys. Rev. E 62, (2000) 3925. [11] G. Cinacchi, E. Velasco, L. Mederos, J. Phys. Cond. Mat- ter 16, (2004) S2003; [12] S. Lago, A. Cuetos, B. Mart́ınez–Haya, L. F. Rull, J. Molecular Recognition 17, (2004) 417. [13] F. J. Vesely, Mol. Phys. 103, (2005) 679. [14] L. Onsager, Ann. N.Y. Acad. Sci. 51, (1949) 627. [15] G. Cinacchi, L. Mederos, E. Velasco, J. Chem. Phys. 121, (2004) 3854. [16] Y. Rosenfeld, J. Chem. Phys. 89, 4272 (1988); Phys. Rev. Lett. 63, (1989) 980. [17] J. A. Cuesta and Y. Mart́ınez–Ratón, Phys. Rev. Lett. 78, (1997) 3691; J. Chem. Phys. 107, (1997) 6379. [18] A. Esztermann, H. Reich, M. Schmidt, Phys. Rev. E 73, (2006) 011409. [19] S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954); J. Polym. Sci. 33, 183 (1958). [20] Tricritical points are predicted by a large number of mod- els for liquid crystals and other systems, even in pure flu- ids. In particular, the present phase diagram is topolog- ically equivalent to that predicted by the Blume-Emery- Griffiths model [Phys. Rev. A 4, 1071 (1971)] for a He4– He3 mixture, identifying rods with He4, spheres with He3 and inverse temperature with reduced pressure. [21] R. Zwanzig, J. Chem. Phys. 39, 1714 (1963). [22] J. A. Cuesta and Y. Mart́ınez-Ratón, J. Phys: Condens. Matter 11, (1999) 10107. [23] Y. Mart́ınez-Ratón and J. A. Cuesta, Phys. Rev. E 58, (1998) R4080.

0704.1141

In Search of the Spacetime Torsion

7 IN SEARCH OF THE SPACETIME TORSION J. G. PEREIRA Instituto de F́ısica Teórica, Universidade Estadual Paulista Rua Pamplona 145, 01405-900 São Paulo, Brazil Whether torsion plays or not a role in the description of the gravitational interaction is a problem that can only be solved by experiment. This is, however, a difficult task: since there are different possible interpretations for torsion, there is no a model–independent way to look for it. In these notes, two different possibilities will be reviewed, their consistency analyzed, and the corresponding experimental outputs briefly discussed. 1 Gravitation and Universality Gravitation has a quite peculiar property: particles with different masses and different compo- sitions feel it in such a way that all of them acquire the same acceleration and, given the same initial conditions, follow the same path. Such universality of response — usually referred to as universality of free fall— is the most fundamental characteristic of the gravitational interaction.1 It is unique, peculiar to gravitation: no other basic interaction of nature has it. Universality of free fall is usually identified with the weak equivalence principle, which establishes the equality between inertial and gravitational masses. In fact, in order to move along the same trajectory, the motion of different particles must be independent of their masses, which have then to be canceled out from the equations of motion. Since this cancellation can only be made when the inertial and gravitational masses coincide, this coincidence naturally implies universality. Einstein’s general relativity is a theory fundamentally based on the weak equivalence prin- ciple. In fact, to comply with universality, the presence of a gravitational field is supposed to produce curvature in spacetime, the gravitational interaction being achieved by letting (spin- less) particles to follow the geodesics of the curved spacetime. In general relativity, therefore, geometry replaces the concept of gravitational force, and the trajectories are determined, not by force equations, but by geodesics. It is important to emphasize that only a universal inter- action can be described by such a geometrization, in which the responsibility for describing the interaction is transferred to spacetime. It is also important to remark that, in the eventual lack of universality, the geometrical description of general relativity would break down.2 The fundamental connection of general relativity is the Christoffel connection, a Lorentz- valued connection written in a coordinate basis.3 In terms of the spacetimea metric gµν , it is written as gρλ(∂µgλν + ∂νgλµ − ∂λgµν). (1) It is a connection with vanishing torsion, νµ = 0, but non- vanishing curvature, λνµ 6= 0. In terms of this connection, the equation of motion of a test particle is given by the geodesic We use the Greek alphabet µ, ν, ρ, . . . = 0, 1, 2, 3 to denote spacetime indices. http://arxiv.org/abs/0704.1141v1 equation µν uθ u ν = 0, (2) which says that the particle four-acceleration vanishes identically. This property reveals the absence of the concept of gravitational force, a basic characteristic of the geometric description. 2 What About Torsion? A general Lorentz connection has two fundamental properties: curvature and torsion.4 Why should then matter produce only curvature? Was Einstein wrong when he made this assumption by choosing the Christoffel connection? Does torsion play any role in gravitation? Cartan was the first to ask these questions, soon after the advent of general relativity. As a possible answer, a new theory was formulated, called Einstein–Cartan theory,5 in which the Christoffel connection was replaced by a general connection presenting both curvature and torsion. The main idea behind the Einstein–Cartan construction is the fact that, at the microscopic level, matter is represented by elementary particles, which in turn are characterized by mass (that is, energy and momentum) and spin. If one adopts the same geometrical spirit of general relativity, not only mass but also spin should be source of gravitation at this level. Like in general relativity, energy and momentum should appear as source of curvature, whereas spin should appear as source of torsion. This means essentially that, in this theory, curvature and torsion represent independent degrees of freedom of gravity. As a consequence, there might exist new physics associated to torsion. Of course, at the macroscopic level, where spins vanish, the Einstein–Cartan theory coincides with general relativity. At the microscopic level, however, where spins are relevant, it shows different results from general relativity. If this is interpretation is correct, Einstein made a mistake when he did not include torsion in general relativity. Because of the Einstein–Cartan theory, there is a widespread belief that torsion is intimately related to spin, and consequently important only at the microscopic level. This belief, however, is not fully justified: in addition to lack experimental support, it is based on a very particular model for gravitation, which is well known to present several consistency problems. For example, it is not consistent with the strong equivalence principle.6 Another relevant problem is that, when used to describe the interaction of the electromagnetic field with gravitation, the Einstein– Cartan coupling prescription violates the U(1) gauge invariance of Maxwell’s theory.7 This last problem is usually circumvented by postulating that the electromagnetic field does not couple to torsion.8 This “solution”, however, is quite unreasonable.9 The purpose of these notes is to call the attention to another possible interpretation for torsion,6 which is consistent and does not present the above mentioned problems. This solution is based on a different model for gravitation, known as the teleparallel gravity equivalent of general relativity,10 or simply teleparallel gravity. 3 A Glimpse on Teleparallel Gravity Teleparallel gravity corresponds to a gauge theory for the translation group, with the field strength given by the torsion tensor. Its main difference in relation to general relativity is the connection field: instead of Christoffel, the fundamental connection of teleparallel gravity is the so called Weitzenböck connection. In terms of the tetrad b haµ, it is written as µν = ha ρ ∂νh µ. (3) We use the Latin alphabet a, b, c, . . . = 0, 1, 2, 3 to denote algebraic indices related to the tangent Minkowski spaces. These indices are raised and lowered with the Minkowski metric ηab, whereas the spacetime indices are raised and lowered with the metric gµν = ηab h In contrast to Christoffel, it is a connection with non-vanishing torsion, νµ 6= 0, but vanishing curvature, νµ = 0. The Weitzenböck and the Christoffel connections are related by µν , (4) where µν) (5) is the contortion tensor. In teleparallel gravity, the equation of motion of a test particle is given by the force equa- tion 11 µν uθ u µν uθ u , (6) with torsion playing the role of gravitational force. It is similar to the Lorentz force equation of electrodynamics, a property related to the fact that, like Maxwell’s theory, teleparallel gravity is also a gauge theory. Using expression (4), the force equation (6) can be rewritten in terms of the Christoffel connection, in which case it reduces to the geodesic equation of general relativity: µν uθ u ν = 0. (7) The force equation (6) of teleparallel gravity and the geodesic equation (7) of general relativity, therefore, describe the same physical trajectory. This means that the gravitational interaction has two equivalent descriptions: one in terms of curvature, and another in terms of torsion.12 Although equivalent, however, there are conceptual differences between these two descriptions. In general relativity, curvature is used to geometrize the gravitational interaction. In teleparallel gravity, on the other hand, torsion accounts for gravitation, not by geometrizing the interaction, but by acting as a force. As a consequence, there are no geodesics in teleparallel gravity, but only force equations, quite analogous to the Lorentz force equation of electrodynamics. One may wonder why gravitation has two equivalent descriptions. This duplicity is related precisely to that peculiarity: universality. Like the other fundamental interactions of nature, gravitation can be described in terms of a gauge theory – - just teleparallel gravity. Universality of free fall, on the other hand, makes it possible a second, geometrized description, based on the weak equivalence principle — just general relativity. As the sole universal interaction, it is the only one to allow also a geometrical interpretation, and hence two alternative descriptions. From this point of view, curvature and torsion are simply alternative ways of describing the gravitational field,13 and consequently related to the same degrees of freedom of gravity. If this interpretation is correct, Einstein was right when he did not include torsion in general relativity. 4 Conclusions According to the Einstein–Cartan theory, as well as to other generalizations of general relativity, torsion represents additional degrees of freedom of gravity. As a consequence, new physical phenomena associated to its presence are predicted to exist. With this point of view in mind, there has been recently a proposal to look for these new phenomena using the Gravity Probe B data.14 The idea is that, if torsion is able to couple to spin, consistency arguments require that it might also be able to couple to rotation — that is, to orbital angular momentum. The data of Gravity Probe B could then be used to look for signs of this coupling. On the other hand, from the point of view of teleparallel gravity, torsion does not represent additional degrees of freedom, but simply an alternative to curvature in the description of gravitation. In this case, there are no new physical effects associated with torsion. According to teleparallel gravity, therefore, torsion has already been detected: it is the responsible for all known gravitational effects, including the physics of the solar system, which can be reinterpreted in terms of a force equation, with torsion playing the role of gravitational force. Which of these interpretations is the correct one? From the theoretical point of view, we can say that the teleparallel interpretation presents several conceptual advantages in relation to the Einstein–Cartan theory: it is consistent with the strong equivalence principle,6 and when applied to describe the interaction of the electromagnetic field with gravitation, it does not violate the U(1) gauge invariance of Maxwell’s theory.9 From the experimental point of view, on the other hand, at least up to now, there are no evidences for new physics associated with torsion. We could then say that the existing experimental data favor the teleparallel point of view, and consequently general relativity. However, due to the weakness of the gravitational interaction, no experimental data exist on the coupling of the spin of the fundamental particles to gravitation. For this reason, in spite of the conceptual soundness of the teleparallel interpretation, we prefer to say that a definitive answer can only be achieved by further experiments. Acknowledgments The author would like to thank R. Aldrovandi and H. I. Arcos for useful discussions. He would like to thank also FAPESP, CNPq and CAPES for partial financial support. References 1. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, New York 1973). 2. R. Aldrovandi, J. G. Pereira and K. H. Vu, Gen. Rel. Grav. 36, 101 (2004) [gr-qc/0304106]. 3. R. Aldrovandi and J. G. Pereira, An Introduction to Geometrical Physics (World Scientific, Singapore, 1995). 4. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1963). 5. For a textbook reference, see V. de Sabbata and M. Gasperini, Introduction to Gravitation (World Scientific, Singapore, 1985). 6. H. I. Arcos and J. G. Pereira, Class. Quantum Grav. 21, 5193 (2004) [gr-qc/0408096]. 7. F. W. Hehl, J. D. McCrea, E. W. Mielke and Y. Ne’emann, Phys. Rep. 258, 1 (1995). 8. This “postulate” is a commonplace in most texts about Einstein–Cartan theories. For a discussion, see V. de Sabbata and C. Sivaram, Spin and Torsion in Gravitation (World Scientific, Singapore, 1994). 9. V. C. de Andrade and J. G. Pereira, Int. J. Mod. Phys. D 8, 141 (1999) [gr-qc/9708051]. 10. For a review, see H. I. Arcos and J. G. Pereira, Int. J. Mod. Phys. D 13, 2193 (2004) [gr-qc/0501017]. 11. V. C. de Andrade and J. G. Pereira, Phys. Rev. D 56, 4689 (1997) [gr-qc/9703059]. 12. V. C. de Andrade, L. C. T. Guillen and J. G. Pereira, Phys. Rev. D 64, 027502 (2001) [gr-qc/0104102]. 13. H. I. Arcos, V. C. de Andrade and J. G. Pereira, Int. J. Mod. Phys. D 13, 807 (2004) [gr-qc/0403074]. 14. Y. Mao, M. Tegmark, A. Guth and S. Cabi, Constraining Torsion with Gravity Probe B [gr-qc/0608121]. http://arxiv.org/abs/gr-qc/0304106 http://arxiv.org/abs/gr-qc/0408096 http://arxiv.org/abs/gr-qc/9708051 http://arxiv.org/abs/gr-qc/0501017 http://arxiv.org/abs/gr-qc/9703059 http://arxiv.org/abs/gr-qc/0104102 http://arxiv.org/abs/gr-qc/0403074 http://arxiv.org/abs/gr-qc/0608121 Gravitation and Universality What About Torsion? A Glimpse on Teleparallel Gravity Conclusions

0704.1142

Observations of chemical differentiation in clumpy molecular clouds

Observations of chemical differentiation in clumpy molecular clouds Jane V. Buckle1, Steven D. Rodgers2, Eva S. Wirström3, Steven B. Charnley2, Andrew J. Markwick-Kemper2,4, Harold M. Butner5 and Shigehisa Takakuwa6,7 1 Astrophysics Group, Cavendish Laboratory, J J Thomson Avenue, Cambridge, CB3 0HE, UK. email: [email protected] 2 Space Science & Astrobiology Division, NASA Ames Research Center, Moffett Field, CA 94035, USA. 3 Onsala Space Observatory, Chalmers University of Technology, SE - 43992 Onsala, Sweden 4 Astronomy Department, University of Virginia, USA 5 Joint Astronomy Centre, 660 North A’ohoku Place, Hilo, HI 96720, USA 6Harvard-Smithsonian Center for Astrophysics, Submillimeter Array Project, 645 North A’ohoku, Hilo, HI 96720 USA. 7 ALMA Office, National Astronomical Observatory of Japan, Tokyo, 181-8588, Japan We have extensively mapped a sample of dense molecular clouds (L1512, TMC-1C, L1262, Per 7, L1389, L1251E) in lines of HC3N, CH3OH, SO and C18O. We demonstrate that a high degree of chemical differentiation is present in all of the observed clouds. We analyse the molecular maps for each cloud, demonstrating a systematic chemical differentiation across the sample, which we relate to the evolutionary state of the cloud. We relate our observations to the cloud physical, kinematical and evolutionary properties, and also compare them to the predictions of simple chemical models. The implications of this work for understanding the origin of the clumpy structures and chemical differentiation observed in dense clouds are discussed. 1 Introduction Low-mass stars form in dense cloud cores. Understanding how stars form thus requires detailed knowledge of the physical and chemical evolution of molecular clouds. The classic picture, in which star formation is magnetically-dominated and mediated by ambipolar diffusion [e.g. 1], posits long molecular cloud lifetimes (∼ 107 years). However, there is an increasing realisation that dissipation of turbulence, driven externally at large scales, plays a vital role [e.g. 2]. In this case, star formation is dynamic and rapid, and molecular cloud lifetimes are short (∼ 105−106 years). Dense cloud chemistry will follow different evolutionary paths depending upon the lifetime of molecular clouds, and hence on the time-scale of low-mass star formation [e.g. 3]. One may also expect that the spatial distribution of molecules should also reflect the underlying cloud dynamics. Observations of dense interstellar clouds in a variety of molecular tracers show that they contain a distribution of dense gaseous structures [4]. These structures exhibit a size distribution ranging from that of dense cores [∼ 0.05 − 0.1 pc, 5], to clumps [∼ 0.01 − 0.05 pc, 6, 7] down to so-called small clumpinos [∼ 0.005 pc, 8]. The spatial distributions of molecules in cold, apparently quiescent, clouds do appear to show chemical abundance gradients, i.e. chemical differentiation. Previously, the best studied sources have been the clouds TMC-1 and L134N [9, 10, 11]. These clouds show striking differences in the location of the emission peaks in several molecules. In TMC-1, for example, emission from the carbon chain molecules (e.g. the cyanopolyynes) is observed to be anticorrelated with that of ammonia and N2H+ and several deuterated species [e.g. 12, 13, 14]. These clouds also show distinct emission peaks from other specific molecules, such as methanol and SO. High resolution observations show that chemical differentiation is present at all spatial scales [8, 15, 16]. Although, in dense cores, depletion of CO, CS and other heavy molecules on to dust grains is undoubtedly important [17, 18], the reason for the spatial differentiation seen amongst other molecules is unclear, as is its prevalence in other molecular clouds. It is possible that there is a link between the observed chemical differentiation and the clumpy struc- ture, and hence to the physical origin of clumps through the cloud turbulence. There are currently two possibilities [see 19, 20]. First, numerical simulations show that dissipation of externally driven turbu- lence can produce structures, similar to the clumps and cores that are observed, through the process of ‘turbulent fragmentation’ [2]. Second, winds and outflows have a significant effect on the structure and Table 1: Core Properties Core RA(1950) DEC(1950) Class Vlsr D (km s−1) (pc) Per7 03:29:39.5 +30:49:50 I 6.81 350 L1389 04:00:38.0 +56:47:59 prestellar/0 -4.65 600 TMC-1C 04:38:34.5 +25:55:00 prestellar 5.27 140 L1512 05:00:54.4 +32:39:00 prestellar 7.11 140 L1251E 22:38:36.4 +74:55:50 mixed -3.93 350 L1262 23:23:32.2 +74:01:45 0 4.11 200 evolution of the surrounding protostellar envelope, containing energies sufficient to physically disrupt the envelope material and to alter the chemistry. These outflows begin very early in the star formation pro- cess, and cloud turbulence could be excited by them [21]. In this case, the cloud structure is regulated by the action of stars [e.g. 22]. The presence of parsec-scale flows [e.g. 23] and wind blown bubbles [24] are clear observational evidence that protostars affect their environment in this way, and that the influence of existing star formation can have a very long reach. In this picture, the turbulence should be a function of the local star-formation activity, however it is unclear if enough kinetic energy can be injected in all clouds to sustain the observed level of turbulence. As the nature of the turbulence should affect the formation and evolution of clumpy structure, we might therefore expect that there should be chemical differences between starless cores and those containing protostars at the Class 0 and/or Class I stages [25], with the former being the product solely of turbulent fragmentation. Models have previously been developed of the spatial and temporal chemical evolution in clumpy molecular clouds [e.g. 17, 26, 27, 28, 29, 30, 31], but with limited success. To quantify the chemical effects that embedded protostars may have on cloud structure and chemistry, we have previously mapped the Barnard 5 cloud (B5) in many molecular lines. B5 is an active region of star formation in Perseus and contains molecular outflows, wind-blown bubbles, and regions where the outflows appear to be interacting with dense clumps [32, 33, 34]. Most previous observations have focused on the region containing the Class I source, B5 IRS1, and the molecular distributions over the rest of the cloud were previously poorly known. These observations revealed a high degree of spatial differentiation, with similarities to the molecular distributions in TMC-1 [35], but also to some other starless cores in Taurus [36]. Comparison with dynamical-chemical models of B5 [26], based on the Norman & Silk [22] picture, appear to indicate that molecular ices are continually being formed and destroyed [7, 36], but that processes not included in these models must be responsible. However, B5 is only one such source; to constrain any chemical models multi-molecule maps of more sources are needed, for example, to see if similar molecular differentiation occurs in dense clouds with varying degrees of star formation activity and protostellar evolution. We have therefore increased our sample to observe dense core chemistry in different natal environments: starless cores, and those containing young stellar objects (YSOs) at the Class 0 and Class I phases. The sources were selected from the sample mapped by Caselli et al. [37] in N2H+ at 0.063 km s−1, 54′′ resolution; lower resolution ammonia maps also exist [5]. Caselli et al. [37] found that the position of the N2H+ peak emission tended to be close to the position of peak NH3 emission but generally not coincident with that of the star. From our B5 observations, we found that several molecules are strongly anticorrelated with N2H+ and NH3 emission at small scales, as well as with each other (e.g. methanol and the carbon chains). Hence, the N2H+ maps of Caselli et al. [37] can be useful to predict where other molecules could be expected to peak in a cloud, assuming the chemical differentiation has the same underlying cause. The six cores we observed have a range of star-forming activity (see Table 1). TMC-1C is a starless core, with a region of diffuse sub-mm emission to the south, which extends in a ridge beyond our map to the north [38]. This cloud has previously been mapped in carbon chain molecules [39] and in methanol and HC13O+ [8]. L1389 is associated with an IRAS source, IRAS 04005+5647, although it has been classified as pre-stellar, or an extremely young protostellar source [40]. L1262 contains both a stellar core, IRAS 23238+7401, which drives a protostellar outflow, and a starless core [41]. L1251E contains a YSO, IRAS 22385+7457, which drives an outflow, as well as two infrared sources likely to be T Tauri stars [42]. There are also several infrared sources in the immediate vicinity of our map. L1512 is a starless core, containing a north-south ridge of diffuse sub-mm emission [41]. Per7 contains a sub-mm peak in addition to the Class I source, IRAS 03295+3050 [23]. Table 2: Molecular Line Properties Molecule Transition Frequency Eupper Einstein A ncrit (GHz) (K) (s−1) (cm−3) C18O 1→0 109.782 5.27 6.30 ×10−08 2× 103 SO 23 →12 99.300 9.22 1.10 ×10−05 3× 105 HC3N 10→9 90.979 24.01 5.81 ×10−05 7× 105 CH3OH 21 →11(E) 96.739 12.55 2.56 ×10−06 1× 105 20 →10(A+) 96.741 6.97 3.38 ×10−06 5× 104 20 →10(E) 96.744 20.11 3.41 ×10−06 4× 104 21 →11(E) 96.755 28.04 2.62 ×10−06 5× 104 We chose to observe four molecular species (see Table 2). The isotope C18O serves to trace much of the low density (∼ 103 cm−3) material in the cloud. Methanol is formed solely on grains in dense clouds [e.g. 43]. Once desorbed into the gas, methanol molecules only have a finite lifetime until they are destroyed. Hence, the CH3OH distribution provides an excellent observational measure of the degree of gas-grain cycling in clumpy clouds. Cores showing weak methanol emission will tend to be older since any CH3OH liberated previously will gradually re-accrete back onto the dust. As we discovered a new ’cyanopolyyne-peak’ position in B5, similar to that in TMC-1 [35], we also mapped the cyanoacetylene emission in these clouds. The formation and destruction of cyanopolyynes can provide information on those molecules returned to the gas, and also define chemical time-scales in dynamically-evolving clouds [29]. Markwick et al. [29] concluded that such chemically interesting positions should appear to be offset from the position of any YSO present; as well as from the peak ammonia emission. The detection of these organic molecule ‘hotspots’ is thus only possible by making large-scale HC3N maps, to include positions distinct from the nominal (0,0) positions of previous searches [cf. 27]. The HC3N transition chosen traces gas at densities above ∼ 105 cm−3. Finally, to explain the large number of S-bearing molecules detected in dense clouds, chemical models infer that elemental sulphur is only modestly depleted in dense clouds, relative to more refractory elements [45]. Interstellar sulphur chemistry involves primarily neutral reactions, and so it defines yet another, albeit slow, chemical time-scale. Shock chemistry or grain-surface catalysis have been proposed to be important for S chemistry [30, 46]; the latter mechanism is supported by the detection of OCS in interstellar ices [47]. We therefore mapped the JK = 32 − 21 line of SO. The plan of the paper is as follows. In Section 2 we describe the observations and how the data analysis was performed. In Section 3, we detail the chemical morphology of each source and note the general trends we see in chemical differentiation. Simple chemical models of the chemical evolution in clumpy media are presented and compared to the observational data in Section 4. Conclusions from this study are given in Section 5. 2 Observations, Data Reduction and Analysis Single dish maps of HC3N, SO, CH3OH and C18O transitions at 3mm were made using the Onsala 20m telescope, Sweden, in April 2005. With the exception of CH3OH, the spectra were observed in frequency- switched mode, using 12.5 kHz channels to provide a velocity resolution of ∼0.04 km s−1. CH3OH was observed with the same resolution, but in position-switched mode due to the number of transitions in the band. The 20m beam size at these frequencies is approximately 40′′, and the main beam efficiency, ηmb, is 0.46. Table 2 lists molecular line details for the transitions that we observed. The cores observed were taken from Caselli et al. [37], and include cores at a range of evolutionary stages, from prestellar to Class I (Table 1). The areas mapped cover the N2H+ emission regions within each core [37], ranging in size from 3–20 sq. arcmin. The typical RMS noise level in the maps is 0.1–0.3 K. Contour plots and channel maps are shown in Figs 3.1–3.1. Data reduction was carried out using the radio data reduction packages XS (bespoke software for the Onsala 20m telescope), CLASS and SPECX. Single Gaussian fits provide a good fit to the line profiles. For CH3OH, the transitions detected in each spectra were fit with dependent Gaussians. The velocity and linewidth of each transition was fixed with respect to the 20 →10(A+) transition at 96.741 GHz, since the separation of the K transitions is fixed by the structure of the molecule. The results of the fitting procedures are given in Table 3, and for CH3OH, values are given for the 20 →10(A+) transition. For lines that were not detected, the 3σ error estimate is given as the upper limit to the integrated intensity. Since we have observed single transitions in C18O, SO and HC3N, and detect only one or two CH3OH transitions in a large fraction of the spectra, we determine a lower limit to the column density [48]. In this procedure, the rotation temperature can be approximated as Trot = for non-linear molecules (CH3OH), and as Trot = for linear molecules (C18O, SO and HC3N). A lower limit to the column density can then be calculated as: Nmin = 8πkν hc3Aulgu Tmbdv Q( )e3/2 non− linear (1) Nmin = 8πkν hc3Aulgu Tmbdv Q( )e linear (2) where Tmb has been calculated from T∗A using ηmb, partition functions (Q[T]) and statistical weights (gu) have been taken from the JPL molecular line database [49], energy levels and Einstein A values (Aul) have been taken from the Leiden Atomic and Molecular Database [50] and Cragg et al. [51], listed in Table 2. For CH3OH, the partition function Q(Trot)=1.28T1.5rot was adopted from Menten et al. [52], and takes into account equal populations of A and E species [53]. For sources where we have detected 3 or 4 CH3OH transitions, we have also carried out a rotation diagram analysis. The resulting column densities generally agree to within a factor of a few. However, as we have detected only one strong transitions in each A and E species towards most positions, we have based our CH3OH analysis in this paper upon the 20 → 10(A+) transition, in the same manner as for the other molecules. This provides a coherent analysis of this data that we can compare with simple chemical models. From our C18O observations, we can then derive abundances relative to H2 following Frerking et al. [54] for N(C18O) ≥ 3 ×1014 cm−2: N(H2) = N(C18O) 1.7× 1014 + 1.3 × 1021 cm−2 (3) We derive H2 column densities ∼ 2 × 1022 cm−2 within the clumpy regions of our data, leading to abundances on the order of x(CH3OH) ∼ 2 × 10−8, x(HC3N) ∼ 8 × 10−11 and x(SO) ∼ 4 × 10−10 in the molecular clumps, or ‘hotspots’. 3 Molecular Morphology We now describe the observed chemical morphology. We first discuss the trends between the four molecules in each individual source, focusing on the emission present in the position-velocity channel maps (Fig. 3.1–3.1). We then summarise the apparent general trends. 3.1 Individual Sources L1512 L1512 shows several extended clumps and ridges of emission in C18O. The southern part of the map shows two clumps ((-60′′,-30′′) and (0′′,-30′′)) which surround the compact HC3N clump (-30′′,-30′′). Emission from SO is also seen in extended ridges and clumps. The strongest clump is associated with the single HC3N clump (-30′′,-30′′), while the second strong clump is associated with the CH3OH peak to the north (-60′′,120′′). We see SO emission at 6.62 km s−1 with none apparent in either CH3OH or HC3N. The strongest SO emission clump appear in gas in which CH3OH is present, at 6.88 km s−1, the SO and CH3OH emission peaks are offset by approximately 30′′. The methanol peak is at 7.12 km s−1, where most of the weaker clumpy emission is found; at this velocity, there is no corresponding spatial SO emission in this region. There are a couple of weak CH3OH emission clumps present to the south, and these are embedded in the same gas as two SO clumps. An HC3N clump shows up at -6.88 km s−1 that is spatially distinct from the SO and CH3OH clumps but partially overlaps the weaker extended SO emission. The main HC3N clump is at 7.12 km s−1, overlaps the extended SO and CH3OH there, but is distinct from the weak SO and CH3OH clumps, and is located about 100′′ from the major SO-CH3OH clumps. In fact, we find that both these HC3N clumps coincide with minima in the C18O distributions, as well as with the N2H+ peak (VLSR = 5.27 km s−1), perhaps indicating that molecular depletion is important at these positions. However, the fact that a methanol clump is present at 7.12 km s−1 in a region with Figure 1: L1512. Top: Contour maps of integrated intensity, from left: C18O, SO, HC3N and CH3OH. Contours are 50%–100% in 10% intervals of the peak integrated intensity in K km s−1, and 1σ rms is ∼ 0.04 K km s−1. A cross marks the N2H+ peak [37]. Bottom: Channel maps of integrated intensity. Contours levels are marked on each plot. Clockwise from top left: C18O, SO, CH3OH and HC3N. little/no C18O emission somewhat confuses this interpretation. C18O emission peaks are anti-correlated with emission peaks in all other molecules. In summary, we see gas with only SO, gas with strong SO and CH3OH emission intermingled but with distinct emission peaks (i.e. clumps), as well as gas with strong CH3OH emission and no/weak SO. The HC3N clumps are markedly anticorrelated with SO and CH3OH clumps in position and are correlated with C18O troughs. These could be cyanopolyyne late-time depletion peaks [55]. TMC-1C TMC-1C shows several clumps of emission in C18O extending in a north-south ridge. The peak of emission is near the centre of the diffuse sub-mm emission (-30′′,-60′′), with several smaller clumps leading north to a more extended emission region at the edge of our map (-90′′,120′′). HC3N emission is confined to the southern half of our map, where it appears in a ring, part of which overlaps the C18O clump. SO emission appears in a relatively narrow ridge of emission extending to the north-west. Emission from CH3OH has a similar morphology to emission from SO, but is more extended, and peaks further to the Figure 2: TMC-1C. As Fig. 3.1. A triangle marks the approximate centre of the diffuse sub-mm emission to the south, which extends slightly to the east, and to the north beyond our map [38]. A cross marks the N2H+ peak [37]. north. In this source SO displays very clumpy emission at 5.12 and 5.38 km s−1; there is only weak/no CH3OH emission at 5.12 km s−1, where most of the SO molecular clumps are. At 5.38 km s−1, CH3OH has 3 clumps and there is evidence for coincident weak SO emission. There is an SO clump coincident with a minimum in C18O emission; this is approximately the location of N2H+ emission maximum [37], and suggests a depletion zone here. This is curious: in L1512 we have HC3N correlated with a possible depletion zone, whereas here it is SO and there is only weak/no HC3N present - or at least tending to avoid it. The N2H+ peak is at VLSR of 5.27 km s−1, at a similar velocity to emission from C18O, CH3OH and HC3N. The velocity of SO emission at this position is blue-shifted by 0.1 km s−1, which presumably means that the depleted material does not contain the SO peak. The structure of the C18O, HC3N and SO emission could be interpreted as that of a clumpy, fragmented torus in 3D. In this scenario, the SO emission is actually coming from a chemically differentiated fragment of the torus/ring. The HC3N peaks are spatially distinct from the CH3OH clumps at 5.38 km s−1 but there is weak SO emission in them; an exception is the SO clump near (0,+60′′) which appears to have no HC3N. Again, C18O emission peaks are anti-correlated with emission peaks in all other molecules. Thus, in the SO clumps, methanol is either deficient or absent. To some degree, SO and HC3N appear to be intermingled, although the clumps/peaks are physically distinct, and one SO clump has no HC3N. The CH3OH clumps probably have some SO in them but are spatially distinct from the Figure 3: L1262. As Fig. 3.1. The IRAS source is marked with a star, and sub-mm peaks with triangles [41]. A cross marks the N2H+ peak [37]. HC3N distribution. C18O emission minima correspond to SO maximum (0,-60′′) and to HC3N maximum (-60,-90). L1262 Emission from C18O towards L1262 is seen in two strong clumps, one just to the east of of the IRAS source (0′′,0′′), and one just to the east of the sub-mm source (60′′,-30′′). Emission from HC3N is seen in a clumped ridge, with one clump, like that of C18O, to the east of the sub-mm peak, and one associated with the IRAS source, west of the C18O clump (30′′,-30′′). SO emission peaks in a clump to the north of the sub-mm peak (0′′,30′′), and extends in a clumpy ridge to the north and south. A further clump is seen to the west, at the edge of our map, where the C18O emission also extends. Emission from CH3OH is seen in an extended clump associated with the sub-mm source (0′′,30′′), and also in a compact clump to the east of the IRAS source (60′′,-30′′). The main SO and CH3OH clumps are co-incident with the HC3N clump that is near the sub-mm peak. There is an SO clump at 3.62 km s−1 that appears to grow and merge into the major SO clump at 3.88 km s−1, where CH3OH emission becomes more evident. At 4.12 km s−1, the SO emission clearly shows up as ∼ 4 clumps. There are 2 CH3OH clumps and these are evident at 4.12 and 4.38 km s−1. In both cases, the CH3OH appears to be mixed with the enhanced SO emission, although at 4.38 km s−1 the SO emission is absent. The strongest SO and CH3OH emission are to the north - the clumps/peaks are spatially distinct and should have different SO/CH3OH ratios. The methanol clump at 4.38 km s−1 is almost ‘pure’ CH3OH - emission from other molecules is weak/absent. There are 2 HC3N clumps with the strongest to the north at 3.88 km s−1. This clump is associated with the strongest SO peak. Only the weaker HC3N clump to the south is also evident at 4.12 km s−1 which is located across a region of Figure 4: Per7. As Fig. 3.1. A star marks the IRAS source, a triangle marks the sub-mm peak [23], and a cross marks the N2H+ peak [37]. weak/absent SO and CH3OH emission. There are two C18O clumps, which are distinct both spatially and kinematically. Through both of these, we see a kinematic sequence of C18O and SO, then C18O and SO plus CH3OH, then just CH3OH as we move from the blue to the red in velocity channels, although the spatial peak in emission from these molecules are offset from each other. In L1262, we detected 4 SO clumps. The strongest one also contains quite strong HC3N and weak CH3OH emission at 3.88 km s−1; however, at 4.12 km s−1 the CH3OH clump grows stronger and more distinct while the HC3N emission disappears. This is similar to that seen previously, where the strongest SO and CH3OH clumps tend to be in gas where the molecules coexist. We note that the methanol clumps are spatially anticorrelated with the positions of the HC3N ones, and that the weaker CH3OH and HC3N clumps have little SO. However, the positional offset of the emission peaks of each molecule is only marginal (∼0.5 beamwidth). The SO, CH3OH and HC3N peaks are all anticorrelated with C18O peaks. These molecules also tend to avoid the region around the N2H+ emission peak (at VLSR = 4.11 km s−1), suggesting that molecular depletion on grains could be influencing the observed morphology. There may be a kinematical sequence through the clumpy structures around (-60, 0). Per 7 Per7 shows several regions of clumpy, asymmetric emission from all four molecules, most of it contained in ridge extending to the north-east and south-west from the sub-mm peak. Emission from C18O is relatively bright towards this source, while emission from HC3N, CH3OH and SO is fairly weak. C18O peaks in an extended region to the NE of the sub-mm peak (around (30′′,30′′)). HC3N peaks in the same place, but emission is much more compact, with a second emission peak to the SW (-90′′,-60′′). SO appears in a series of clumps along the NE-SW ridge, with the brighter clumps those to the SW (-90′′,- 60′′). CH3OH also appears in a series of clumps along the NE-SW ridge, but the brightest clump is to the far NE (90′′,90′′), where no SO emission is seen. Another peak occurs in between the two strongest SO emission clumps (-60′′,0′′). Very clumpy structure is evident in emission from SO. At 6.38 km s−1 there are 2 SO clumps to the north with no/little CH3OH or HC3N emission. However, the peak emission clumps of SO and CH3OH are in the same gas, but physically distinct (there is also a possible/weak HC3N clump at this velocity). At 6.62 km s−1, SO and CH3OH show several clumps in gas where they are comingled but with the peaks positionally offset from each other. There are 3 major CH3OH clumps and it seems that the northern one contains little or no SO and definitely no HC3N. At 6.62 km s−1, there is one clear HC3N clump that actually sits in the CH3OH minimum. In fact, once again, the HC3N and CH3OH emission is generally anticorrelated. The C18O does not show compact peaks in the contour map, but is also anticorrelated with HC3N and SO. All the molecules avoid the associated N2H+ peak. In this source we again see clumps with essentially only SO present, as well as that the peak emis- sion from SO and CH3OH emanates from gas in which they are comingled, but where their emission peaks/clumps are spatially distinct. There appear to be methanol clumps with no HC3N and little or no SO. Again, there appears to be a chemical gradient through the clumpy structures at around (+100, 0): 6.38 km s−1 : SO only - 6.62 km s−1 : strong CH3OH and unchanged SO : 5.88 km s−1 strong CH3OH no SO. L1389 L1389 shows relatively compact and weak emission from all four molecules. C18O peaks in a clump to the east of the IRAS source (-30′′,0′′). HC3N peaks in a clump south of the C18O emission. Emission from SO also peaks to the east of the IRAS source (-30′′,-30′′), but is more extended. CH3OH emission appears in a strong clump to the south of the IRAS source, extending to the west (-60′′,-30′′). Chemically, this is a morphologically simple cloud. The SO shows a single clump at -4.88 km s−1 with weak, diffuse emission at -4.62 km s−1; this clump appears to have very weak HC3N emission but no CH3OH. The CH3OH clump is at -4.62 km s−1 and this corresponds to a region of relatively weaker SO emission. The HC3N clump appears at -4.62 km s−1 and and is spatially distinct from the SO peak in velocity and the CH3OH peak in position. So, there is a clump in which SO is prevalent, although it does contain some HC3N. A distinct HC3N-CH3OH anticorrelation is also evident. All the molecular clumps (positions of peak emission) are located away from N2H+ peak, supporting the idea that molecular depletion is important here. L1251E Towards L1251E, emission from C18O is seen in several small clumps, the strongest of which is associated with the IRAS source (-210′′,0′′). Two clumps are also seen to the north and south of an infrared source, either side of a clump of HC3N emission (-150′′,0′′). HC3N emission is seen in a second clump associated with the IRAS source (-210′′,0′′). SO emission is seen in a ridge of emission extending to the north (-120′′,120′′), not clearly associated with emission from any of the other molecules, nor any of the infra- red/IRAS sources. The clumps near the second infra-red source are associated with the weaker CH3OH clumps. CH3OH emission is seen strongly peaked in a clump associated with both the IRAS source and the first infrared source (-180′′,0′′), with a second peak to the north, just south of the main SO clump (-90′′,60′′). The strong SO emission clump to the north, evident from -4.88 to 4.12 km s−1, contains little or no CH3OH, no HC3N, and is unconnected to the level of C18O emission present. The CH3OH emission is extended and fragmented and three main clumps are evident. There are 2 HC3N clumps present. The HC3N peak/clump at (-120, 0) appears quite extended, shows weak CH3OH emission but none from SO. In summary, there is an SO clump that contains a small amount of HC3N emission and only a small amount of CH3OH emission. The HC3N - CH3OH peaks are anti-correlated to the east, where the HC3N peak appears in a region with no CH3OH emission. To the west, HC3N and CH3OH emission are comingled, but the peaks are offset. Figure 5: L1389. As Fig. 3.1. A star marks the IRAS source, a triangle marks the sub-mm peak [40], and a cross marks the N2H+ peak [37]. 3.2 General Trends - Summary Several trends in the chemical differentiation are evident in our maps. The morphology throughout our sample ranges from one or two isolated clumps detected in each molecules, to extended emission containing a strongly-emitting clump and a several weaker ones. SO and CH3OH Clumps We find clumps of gas that contain significant emission only from SO (e.g. L1389, L1251E, PER 7, TMC-1C), with only one of these showing significant emission from HC3N (L1262). There is evidence that some SO emission maxima could correspond to depleted regions (TMC-1C). Strong extended SO emission is also evident where CH3OH emission becomes stronger; in this case we find that the dominant CH3OH and SO clumps can exist in the same gas but that they are always offset from each other. On the other hand, there are some methanol clumps from which SO emission appears to be absent. Based on comparing maps in velocity space, there is some kinematical evidence (e.g. in L1262, PER 7, L1512) for well-defined emission sequence in which there are : regions with only SO, then a region with SO and CH3OH intermingled - then a region with only CH3OH, suggesting we could be looking down a cylinder/filament. Figure 6: L1251E. As Fig. 3.1. The IRAS source is marked with a filled star, and unidentified IR sources with empty stars [42]. A cross marks the N2H+ peak [37]. The HC3N Clumps We find that the HC3N clumps are always spatially anticorrelated with the CH3OH clumps. Some of these clumps can contain some very weak CH3OH emission, and moderately strong SO emission (L1262), but there are others where the HC3N is the sole molecule present (L1251E). Depletion: C18O Distribution & N2H+ Cores C18O emission is tracing the less dense gas towards these sources, since the critical density of the transition we have observed is 1 or 2 orders of magnitude less than the transitions we observed in SO, HC3N and CH3OH. The emission generally has several clumps embedded within an extended, diffuse region, and emission from spatially distinct regions is generally seen to be kinematically distinct as well. For example, towards L1512, emission to the north-west is seen at 6.88 km s−1, while emission to the south-east is seen at 7.12 km s−1. There is a clear anti-correlation between emission from C18O and all other molecules towards the youngest sources, those classified as prestellar cores. Towards the older sources, where protostars have formed, emission becomes more intermingled, and C18O emission can be seen associated with SO and CH3OH peaks, although the spatial positions are offset (e.g. L1262). Another population of objects implicitly present in our maps are the N2H+ cores mapped by Caselli et al. [37]. These are probably the most stable, longest lived cores in the clouds, apart from the ones that apparently harbour protostars, and are almost certainly sites of extensive molecular depletion onto grains. All molecules in our sample tend to avoid these positions. An exception is in L1512 where the HC3N clump is coincident with the N2H+ peak in both position and velocity. 4 Chemical Differentiation in a Clumpy Medium: Theory We conclude that we are observing chemical morphologies which are produced by a time-dependent gas-grain chemistry which is evolving in a turbulent dense medium. Clumps form and dissipate, perhaps coalescing, and long-lived stable structures (i.e. dense cores) are produced in which star formation occurs. In this section we show how simple chemical models can help explain these observations. Clumps are formed out of the interclump medium (nH ∼ 103cm−3), perhaps by compression in shock waves [56], or in MHD waves [31]. Our observations indicate that molecular desorption/depletion from/onto dust grains plays an important role in determining the chemical differentiation. For a molecule, X say, of molecular weight µX, the depletion (accretion) time-scale at 10K is tX = 2.2× 109 yr (4) where a total grain surface area per H nucleon of 3 × 10−22 cm2 and unit sticking efficiency have been assumed [cf. 57]. In these dense regions (nH ∼ 104 − 105 cm−3) accretion and chemical reactions are driven more rapidly and dynamical fate of the clump becomes an issue. It is therefore useful to know the relevant physical time-scales that are important for clump evolution. 4.1 Clump Physics The fate of clumps can be related simply to parameters that can be derived from observations. From our observational data we can obtain estimates for the clump mass, M , its virial mass, Mvir, and radius R. With R known, the virial mass for a sphere of constant density can be estimated from [58] Mvir ∼ 210(∆v)2avR M� (5) where (∆v)av is the average line-width of the clump gas (H2 and He) in km s−1 and R is in parsec For clumps in molecular clouds, there are three possible evolutionary tracks [cf. 6]. A clump will be stable against gravitational collapse if (0.5Mvir < M < Mvir), whereas if it is gravitationally unstable (M > Mvir) it will collapse on the free-fall time-scale tff = 4.35× 107 yr (6) If (M < 0.5Mvir) then the clump is gravitationally unbound and so it will dissipate on a time-scale tdiss ∼ where Ceff is the effective sound speed in the gas. The fate of some clumps can also depend upon the local environment. A cloud volume containing a number of clumps of filling factor γ, and line-of-sight relative velocity vrel can collide and coalesce (i.e. merge) with a time-scale tmerge ∼ Clumps with any dynamical fate can grow when the merger time-scale is sufficiently short; unbound clumps can do so when this is comparable or less than the dissipation time-scale, i.e. when tmerge<∼ tdiss [6, 8, 15]. If we take the derived clump properties of the Taurus clouds (TMC-1C and L1512) as an example, we then, following Peng et al. [6], we estimate that typically R ∼ 0.02− 0.03 pc and the clump masses are in the range M ∼ 0.3− 1 M�. A detailed analysis of clump properties is beyond the scope of this paper, however, we note that throughout our sample we can identify clumps that are unbound, as well as some that are gravitationally stable and others that are unstable. Some clumps, such as CH3OH and SO clumps in TMC-1C and L1512, may actually show substructure at higher spatial resolution. In this case, such smaller clumps would be more likely to be unbound and, given their proximity, perhaps more prone to interaction. 4.2 Chemical Model We now show how a simple static model of clump chemical evolution, with accretion on to dust and selective injection of simple mantle molecules, can explain the observations. As discussed in section 1, methanol cannot be produced efficiently in the gas phase, and is thought to be formed on the surfaces of dust grains. Hence, the CH3OH-rich clumps in our sources must necessarily trace clumps in which grain ice mantles have recently been liberated into the gas phase. The exact mechanism that causes the desorption of the ices is unknown but we expect that it will be connected to clump evolution. Markwick et al. [29] proposed that the desorption is driven by grain-grain collisions, induced by MHD waves generated by clump motions. Alternatively, Dickens et al. [65] speculated that grain-grain streaming resulting from clump collisions may be responsible. In this paper, we make no assumptions about the dynamical mechanism, other than that it is related to clump formation, and so ice mantle liberation, if it occurs, happens at the same time that the clump is formed. This is obviously a simplification since there must be a period of CO accretion to provide the precursor ices for CH3OH production. We assume that a clump forms instantaneously from the interclump medium, and simply follow the chemistry until all the molecules are condensed on grains. In reality, the chemical evolution will be sensitive to the dynamical evolution of the clump (e.g. whether it is collapsing, coalescing or dissipating). We do not consider the clump physical evolution explicitly in this static model; this is discussed in detail elsewhere [59]. Based on the critical densities of the observed molecular tracers, we adopt a value of nH = 2×105 cm−3 as representative of the clump densities, which we consider to be approximately spatially constant. This choice yields tff ∼ 1 × 105 years, and tdiss ∼ 2 × 105 years, and clump masses similar to those derived from the observations, viz M = 0.35M� 0.02 pc 2× 105 cm−3 We assume a cosmic ray ionization rate of 3× 10−17 s−1, a temperature of 10 K, and a visual extinction of 10 magnitudes, which is sufficiently large to render photo-processes effectively irrelevant. Freeze-out of molecules onto grains occurs at a rate given by eqn. (4), except for N2 which is assumed to remain in the gas phase [e.g. 17]. The chemical model is based on the model of Rodgers & Charnley [60], and has been updated to incorporate recent laboratory measurements of the dissociative recombination channels for several ions, including CH3OH 2 [44, 61, 62, 63, 64]. The initial molecular abundances in the newly-formed clump are determined by those in the interclump medium, which are themselves controlled by both (i) the chemistry in these regions, and (ii) whether the material previously passed through a dense phase. This latter effect may be important, since if most clumps eventually dissipate, recycling dense cloud material back into the surrounding gas, the abundances in the interclump medium retain a memory of the higher density phase [cf. 26]. We assume that the material in the interclump medium is predominantly molecular, and that ≈ 99% of the carbon and nitrogen incorporated into a clump is initially present as CO and N2, with ≈ 1% in neutral atomic form. The oxygen not bound up in CO is assumed to be atomic, as is all the initial sulphur. To account for depletion, the gas-phase sulphur abundance is assumed to be 10−8 [45], and we assume complete depletion of all metals. The initial fractional ionization is 1.5× 10−7. 4.3 Results Based on the relative binding energies of species thought to be present on interstellar grains [67], we consider three types of clumps: • Type I. No sublimation of any ice species. The initial gas-phase composition is equal to that of the interclump medium. • Type II. Sublimation of the most volatile ice species. We assume injection of CO2 and H2S with abundances of 10−5 and 10−7 respectively. • Type III. Sublimation of tightly-bound species. We inject CO2, CH3OH, and H2O with abundances of 10−5, 2× 10−8, and 3× 10−5 respectively. tff tdiss 102 103 104 105 106 10-11 10-10 Time (yr) CO x 10-4 HC3N SO2 tff tdiss 102 103 104 105 106 10-11 10-10 Time (yr) CO x 10-4 HC3N SO N2H SO2 tff tdiss 102 103 104 105 106 10-11 10-10 Time (yr) CO x 10-4 HC3N CH3OH SO2 Figure 7: Abundances versus time for selected species. The vertical dotted lines mark the time-scales for gravitational collapse (tff) and clump dissipation (tdiss). Top: type I cores (no ice sublimation). Middle: type II cores (CO2 and H2S sublimation). Bottom: type III cores (CO2, H2O, and CH3OH sublimation). The abundances of sublimated CO2 and H2O are based on observations of interstellar ices in dark clouds [66], and the CH3OH abundance is chosen to match our derived column densities (see section 2). H2S has not been detected in interstellar ices, but based on the fact that many of the most abundant ice species are simple hydrides, it is likely to be present; our assumed abundance is consistent with the observed upper limits [68]. Hydrogen sulphide has a low sublimation enthalpy [67], and so it will be released along with CO2 in type II cores. We neglect H2S injection in type III cores in order to distinguish between the amount of SO that can be formed from H2S, and the amount that can be formed by simply boosting the OH abundance (as will occur when water is injected). Figure 4.3 shows the time evolution of each of the four observed species – CO, SO, HC3N, and CH3OH – in each clump. Also shown are N2H+ and SO2. The free-fall time, tff , and the clump dissipation time, tdiss are marked on each panel. In all cases, molecules are depleted from the gas on a freeze-out timescale of a few ×105 yr. For type I clumps, SO and HC3N are produced by chemical reactions in the dense gas, peaking between 5 × 104 and 105 yr. SO is formed via the reaction of atomic S with OH, and HC3N from CN + C2H2. Type I clumps younger than ∼ 105 yr can account for regions with both SO and HC3N emission but no methanol. At longer times, SO is destroyed faster than HC3N, so long-lived clumps of this type are able to explain those regions which are HC3N-rich but SO-poor. In type II clumps, the injection of H2S significantly enhances the amount of SO produced in the gas. Compared to type I clumps, the HC3N peak is slightly reduced, and shifted to later times. This is due to the reduced hydrocarbon abundances which result from the fact that the injected CO2 drives down the C+ abundance. Type II cores can explain regions with large SO abundances, together with small HC3N abundances and no CH3OH. Type III clumps are obviously the only ones which can explain CH3OH-rich clumps, but we find that the methanol only survives for ∼ 104 yr before it is destroyed by protonation followed by dissociative recombination. The HC3N in these clumps only peaks after all the CH3OH has been lost, which naturally explains the anti-correlation between methanol and cyanoacetylene. The SO abundances in type III clumps are intermediate between those of types I and II, and peaks occur at around 2 × 104 yr. At this time, much of the initial CH3OH has been destroyed, and no significant amounts of HC3N have been formed. Therefore, type III cores younger than ∼ 104 yr will contain both CH3OH and SO, whereas slightly older cores will only contain SO. Because the injected water leads to an increased OH abundance in the gas, type III cores eventually evolve to a state with SO2/SO > 1, so the SO/SO2 ratio may potentially be used to place more firm constraints on the ages of these clumps [c.f. 68, 69]. 4.4 Application to Observations We can compare our model results with the observational dataset, to try to place constraints on the ages and the dynamical states of specific objects. Considering the general trends described in section 3.2, we find that our model naturally accounts for the CH3OH–HC3N anti-correlation. Clumps with CH3OH emission must have ages of t<∼ 10 4 yr, and those with no SO emission must be even younger (t<∼ 5 × 10 3 yr). Because these ages are much less than those associated with the dynamical evolution of the clumps, the dynamical state of methanol clumps cannot be constrained from the observations. As these type III clumps evolve, they eventually lose all the initial methanol; for 3×104<∼ t<∼ 10 5 yr, they will appear only in SO, after which time the HC3N abundance rises. The evolution of type III clumps provides a simple explanation for the observed transition from pure-CH3OH to mixed-CH3OH-SO to pure-SO gas in different velocity channels. Clumps with both SO and HC3N emission could be old type II or III (t > 105 yr), or possibly younger type I clumps. Regions containing only HC3N emission are likely to be old type I clumps, with ages t ∼ 2× 105 yr ∼ tdiss. As these latter clumps have ages comparable to tdiss, it is probable that pure HC3N clumps represent long-lived, stable clumps: gravitationally bound against dissipation, but not collapsing. If so, then some of these regions should also be associated with enhanced N2H+ abundances, as we in fact see in L1512. We now briefly consider each source in turn: L1512 The strong CH3OH and SO clump in the northwest must have an age of ∼ 104 yr. In contrast, the SO cores toward the southeast have only weak methanol emission and must be older. This SO emission surrounds the HC3N peak which must be even older, as no SO is seen at this position, suggesting an age of ∼ 2× 105 yr. This is consistent with the CO depletion and N2H+ emission also seen toward this position [37] TMC-1C The methanol clumps in this source have only weak SO emission, and so must be t<∼ 5000 yr. The clumpy ring of material to the southeast has little or no methanol, but SO and HC3N are somewhat co-mingled. This is best explained by type I clumps, implying that the mechanism which formed the clumpy ring was not energetic enough to cause significant desorption of grain mantles. L1262 In this source, the channel velocity maps at 4.12 km s−1 show a clumpy ring of SO emission surrounding the HC3N peak. As is the case in L1512 these may represent evolved type III clumps which have lost the original methanol. The central HC3N clump will be older (t>∼ 10 5 yr) than the surrounding SO clumps (3× 104 < t < 105 yr). Both CO clumps in this source display the kinematical sequence of SO followed by SO plus CH3OH, followed by only CH3OH, that is typical of the time evolution of type II clumps. Per 7 Again, we see a transition in velocity space from pure SO to pure CH3OH via mixed gas. As in L1262, this indicates the evolution of clumps where CH3OH ice is liberated along the line of sight. The HC3N peak toward the northeast is associated with weak SO, and may represent evolved gas. However, the peak does not appear to be correlated with the N2H+ peak or with significant CO depletion. L1389 Again, we see SO-rich gas surrounding a HC3N-rich region. As in L1512, this suggests an older, stable clump in the center surrounded by regions in which clump formation and/or ice mantle sublimation occurred more recently. This source also shows a clear HC3N-CH3OH anti-correlation. L1251E SO is only visible in this map toward the north and east, and appears to be associated with some CH3OH, implying an age for this gas of ∼ 104 yr. Towards the south and west, CH3OH and HC3N clumps are prevalent yet SO is absent, which requires the presence of both young and old cores. This is the only area in which we see co-mingling of HC3N and CH3OH emission – although the peak positions are offset – and is the only region whose chemistry is not readily explained by our simple model. 5 Conclusions We have mapped several prestellar and protostellar cores in CH3OH, HC3N, SO and C18O. We find that the emission from these molecules traces clumpy structure across all of the cores. The molecular clumps show striking differences in the location and velocity of emission peaks between molecules. We see a similar degree of chemical differentiation in all of the cores that we have observed, despite the differences in known star forming activity. The HC3N clumps are generally anti-correlated with N2H+ peaks, and with CH3OH clumps. In L1512, a HC3N clump does appear along with a N2H+ peak, perhaps indicative of ongoing molecular depletion. CH3OH clumps and SO clumps also show distinct emission peaks. C18O clump peaks are not generally correlated with emission peaks in any of the other molecules that we observed. These morphological differences in molecular peaks towards sources of different evolutionary stage suggest that depletion is important, but that other processes must also be driving the chemical differentiation that we see. Our observations suggest there is a kinematic sequence through the clumps of different molecular species within the core. There is a well defined emission sequence in velocity space where we see only clumps of SO, then clumps where SO and CH3OH are intermingled, and then a region with CH3OH only. This may indicate that we are looking down a cylinder, or filament. We have investigated the origin of the chemical differentiation using a simple chemical model, in which clumps form rapidly and in which sublimation of ice mantles can occur as the clump is formed. Despite its simplicity, this model is able to account for a wide variety of the chemical differentiation observed in our sources. For clumps in which CH3OH is injected, the methanol survives for ∼ 104 yr, so the observed CH3OH-rich clumps are young compared to the time-scale for dynamical evolution. In these clumps, HC3N is not produced until all the CH3OH has been destroyed, which explains the observed anti-correlation of methanol and cyanoacetylene. The model also predicts some some regions will have HC3N/SO > 1 whereas others will have HC3N/SO < 1, depending on the age of the source and/or the degree of mantle sublimation. HC3N peaks at late times, and the observed regions with only HC3N emission are most likely tracing older clumps which are dynamically stable against both collapse and dissipation. Such regions should also be associated with marked CO depletion and elevated N2H+ abundances. In conclusion, it appears that molecular clouds containing protostars at different epochs of star for- mation exhibit similar trends in their chemical spatial differentiation. The origin of this differentiation can be understood through simple models of the molecular gas-grain interaction, and suggests that peri- odic removal of ice mantles is occurring in dark clouds. Mantle disruption and dark cloud chemistry are probably closely connected to the physics of clump evolution, dynamical-chemical models incorporating of these processes are currently being developed [59]. Acknowledgements SBC and SDR acknowledge support by the NASA Goddard Center for Astrobiology and the NASA Long Term Space Astrophysics Program, with partial support from the NASA Origins of Solar Systems Program. References [1] Shu F.H., Adams F.C. & Lizano S., 1987 ARA&A, 25,23 [2] Mac Low M.-M.& Klessen R. S. 2004, Rev. Mod. Phys., 76, 125 [3] Hartmann L., Ballesteros-Paredes J, & Bergin E. A. 2001, ApJ, 562, 852 [4] Evans N.J. II, 1999, ARA&A 37, 311 [5] Benson P. J., & Myers P.C., 1989 ApJS 71, 89 [6] Peng R., Langer W.D., Velusamy T., Kuiper T.B.H. & Levin S., 1998, ApJ 497, 842 [7] Takakuwa, S., Mikami, H., Saito, M., & Hirano, N., 2000, ApJ 542, 367 [8] Takakuwa S., Kamazaki T., Saito M. & Hirano N., 2003, ApJ 584, 818 [9] Swade D.A, 1989, ApJ 345 828 [10] Pratap P., Dickens J.E., Snell R.L., Miralles M.P., Bergin E.A., Irvine W.M. & Schloerb F.P., 1997, ApJ 486, 862 [11] Dickens J.E., Irvine W.M., Snell R.L., Bergin E.A., Schloerb F.P., Pratap P. & Miralles M.P., 2000, ApJ 542, 870 [12] Olano, C., Walmsley, C.M. & Wilson, T.L. 1988, A&A 196, 194 [13] Turner B.E., 2001, ApJS 136, 579 [14] Hirota T., Ikeda M.& Yamamoto S., 2001, ApJ 547, 814 [15] Morata O., Girart J.M., Estalella R., 2003, A&A 397, 181 [16] Tafalla M., Santiago J. 2004, A&A 414, L53 [17] Bergin E. A., Alves J., Huard T., Lada, C.J., 2002, ApJ 570, L101 [18] Caselli P., van der Tak F.F.S., Ceccarelli, C. & Bacmann A., 2003, A&A 403, L37 [19] Scalo J. & Elmegreen B.G., 2004, ARA&A 42, 275 [20] Elmegreen B.G. & Scalo J., 2004, ARA&A 42, 275 [21] Reipurth B. & Bally J., 2001, ARA&A 39, 403 [22] Norman C. & Silk J. 1980, ApJ, 238, 138 [23] Walawender J., Bally J. & Reipurth B., 2005, AJ 129, 2308 [24] Quillen A.C., Thorndike S.L., Cunningham A., Frank A., Gutermuth R.A., Blackman E.G., Pipher J.L., Ridge N., 2005, ApJ 632, 941 [25] Motte F., Andre P. & Neri, R., 1998, A&A 336, 150 [26] Charnley S.B., Dyson J.E., Hartquist T.W. & Williams D.A., 1988, MNRAS 235, 1257 [27] Suzuki H., Yamamoto S., Ohishi M., Kaifu N., Ishikawa S., Hirahara Y. & Takano S., 1992, ApJ 392, 551 [28] Aikawa Y., Herbst E., Roberts H., Caselli P., 2005, ApJ 620, 330 [29] Markwick, A.J., Millar, T.J. & Charnley S.B., 2000, ApJ, 535, 256 [30] Charnley S.B., Rodgers S.D., Ehrenfreund P., 2001, A&A 378, 1024 [31] Garrod R.T., Williams D.A., Hartquist T.W., Rawlings J.M.C., Viti S., 2005, MNRAS 356, 654 (Erratum : 362, 749) [32] Goldsmith P.F., Langer W.D., Wilson R.W., 1986, ApJ 303, L11 Langer, W. D., & Wilson, R. W., 1986, ApJ 303, L11 [33] Fuller, G. A., Myers, P. C., Welch, W. J., Goldsmith, P. F., Langer, W. D., Campbell, B. G., Guilloteau, S. & Wilson, R. W., 1991, ApJ 376, 135 [34] Kelly M.L., MacDonald G.H., Millar T.J., 1996, MNRAS 279, 1210 [35] Charnley S.B. et al. 2006, ApJ submitted [36] Butner H.M. et al. 2006, ApJ, submitted [37] Caselli P., Bensen P.J., Myers P.C. & Tafalla M., 2002, ApJ 572, 238 [38] Schnee S. & Goodman A., 2005, ApJ 632, 1168 [39] Cernicharo J., Guelin M., Askne J., 1984, A&A 138, 371 [40] Laundhardt R. & Henning T. 1997, A&A 326, 329 [41] Shirley, Y.L., Evans Neal J., II, Rawlings J.M.C., Gregersen E.M., 2000, ApJS 131, 249 [42] Kun M. & Prusti T., 1993, A&A 272, 235 [43] Ehrenfreund P. & Charnley S.B., 2000, ARA&A 38, 427 [44] Geppert W.D. et al., 2006, in Astrochemistry Throughout the Universe: Recent Successes and Current Challenges, eds. D.C. Lis, G.A. Blake & E. Herbst, Cambridge University Press, in press [45] Ruffle, D. P., Hartquist, T. W., Caselli, P. & Williams, D. A., 1999, MNRAS 306, 691 [46] Millar T.J. & Herbst E., 1990, A&A, 231, 466 [47] Palumbo, M.E., Geballe, T.R. & Tielens A.G.G.M., 1997, ApJ 479, 839 [48] Thompson M.A., MacDonald G.H. & Millar T.J., 1999, A&A 342, 809 [49] Pickett H.M., Poynter R.L., Cohen E.A., Delitsky M.L., Pearson J.C., and Muller H.S.P., 1998, J. Quant. Spectrosc. & Rad. Transfer 60, 883 [50] Schöier F.L., van der Tak F.F.S., van Dishoeck E.F., Black J.H., 2005, A&A 432, 369 [51] Cragg, D.M., Mikhtiev, M.A., Bettens, R.P.A., Godfrey, P.D., & Brown, R.D., 1993, MNRAS 264, [52] Menten K.M., Walmsley C.M., Henkel C., & Wilson T.L., 1986, A&A 157, 318 [53] Kalenskii S.V., Dzura, A.M., Booth R.S., Winnberg A., Alakoz A.V., 1997, A&A 321, 311 [54] Frerking M.A., Langer W.D. & Wilson R.W., 1982, ApJ 262, 590 [55] Ruffle D.P., Hartquist T.W., Taylor S.D. & Williams D.A., 1997, MNRAS 291, 235 [56] Klessen R.S., 2001, ApJ, 556, 837 [57] Brown P.D. & Charnley S.B., 1990, MNRAS 244, 432 [58] MacLaren I., Richardson K.M. & Wolfendale A.W., 1988, ApJ 333, 821 [59] Rodgers, S.D. et al. 2006, in preparation. [60] Rodgers S.D. & Charnley S.B., 2001, ApJ, 546, 324 [61] Geppert W.D. et al., 2004, ApJ, 609, 459 [62] Geppert W.D. et al., 2004, ApJ, 610, 1228 [63] Geppert W.D. et al., 2005, ApJ, 631, 653 [64] Kalhori S. et al., 2002, A&A, 391, 1159 [65] Dickens J.E., Langer W.D., Velusamy T., 2001, ApJ, 558, 693 [66] Knez C. et al., 2005, ApJL, 635, L145 [67] Aikawa Y., Umebayashi T., Nakano T., Miyama S.M., 1997, ApJ, 486, L51 [68] Charnley S.B., 1997, ApJ, 481, 396 [69] Buckle J.V. & Fuller G.A., 2003, A&A, 399, 1047 ∫ T∗ A ∫ T∗ A ∫ T∗ A ∫ T∗ A Introduction Observations, Data Reduction and Analysis Molecular Morphology Individual Sources General Trends - Summary Chemical Differentiation in a Clumpy Medium: Theory Clump Physics Chemical Model Results Application to Observations Conclusions

0704.1143

Challenges for MSSM Higgs searches at Hadron Colliders

ANL-HEP-PR-07-19 EFI-07-07 FERMILAB-PUB-07-074-T Challenges for MSSM Higgs searches at Hadron Colliders M. Carenaa, A. Menonb,c and C.E.M. Wagnerb,c,d aTheoretical Physics Dept., Fermi National Laboratory, Batavia, IL 60510 bHEP Division, Argonne National Laboratory, 9700 Cass Ave., Argonne, IL 60439, USA cEnrico Fermi Inst., Univ. of Chicago, 5640 S. Ellis Ave., Chicago, IL 60637, USA dKICP and Dept. of Physics, Univ. of Chicago, 5640 S. Ellis Ave.,Chicago IL 60637, USA November 7, 2018 Abstract In this article we analyze the impact of B-physics and Higgs physics at LEP on standard and non-standard Higgs bosons searches at the Tevatron and the LHC, within the framework of minimal flavor violating supersymmetric models. The B-physics constraints we consider come from the experimental measurements of the rare B-decays b → sγ and Bu → τν and the experimental limit on the Bs → µ+µ− branching ratio. We show that these constraints are severe for large values of the trilinear soft breaking parameter At, rendering the non-standard Higgs searches at hadron colliders less promising. On the contrary these bounds are relaxed for small values of At and large values of the Higgsino mass parameter µ, enhancing the prospects for the direct detection of non-standard Higgs bosons at both colliders. We also consider the available ATLAS and CMS projected sensitivities in the standard model Higgs search channels, and we discuss the LHC’s ability in probing the whole MSSM parameter space. In addition we also consider the expected Tevatron collider sensitivities in the standard model Higgs h → bb̄ channel to show that it may be able to find 3 σ evidence in the B-physics allowed regions for small or moderate values of the stop mixing parameter. http://arxiv.org/abs/0704.1143v2 1 Introduction Over the last twenty years, the Standard Model (SM) has provided an exceptionally accurate description of all high energy physics experiments – whether they be electroweak precision or flavor physics observables. The only part of the Standard Model that remains to be tested is the mechanism for electroweak symmetry breaking. In the Standard Model, electroweak symmetry breaking is achieved by the scalar Higgs field acquiring a vacuum expectation value (vev), thereby giving mass to the quarks, leptons and gauge bosons. However, this mechanism for electroweak symmetry breaking has a problem in that the Higgs potential is unstable with respect to radiative corrections, that is the scalar Higgs mass gets radiative corrections proportional to the cutoff due to fermion and boson loops. A number of extensions of the Standard Model have been suggested to try to alleviate this problem. Supersymmetry is one of the most promising of these extensions of the SM, in which every SM fermion (boson) has a spin-0 (spin-1/2) super-partner. The minimal supersymmetric extension of the Standard Model or MSSM, with gauge invariant SUSY breaking masses of the order of 1 TeV, predicts an extended Higgs sector with a light SM-like Higgs boson of mass lower than about 130 GeV [1]–[12] that is in good agreement with precision electroweak measurements. However the flavor structure of these SUSY breaking masses is not well understood. If there are no tree-level flavor changing neutral currents associated with the gauge and super-gauge interactions, the deviations from SM predictions are small. Such small deviations can be achieved if the quark and squark mass matrices are block diagonalizable in the same basis (an example is flavor blind squark and slepton masses). The flavor violating effects in these minimal flavor violating models are induced by loop factors proportional to CKM matrix elements as in the Standard Model. The B-physics properties of these kinds of supersymmetric extensions of the SM have been studied in great detail in Refs. [13]–[20]. The recent improvements in our understanding of B-physics observables have put inter- esting constraints on Higgs searches in the MSSM at the Tevatron and LHC colliders. In Ref. [21] we analyzed the constraints that the non-observation of the Bs → µ+µ− rare decay and the measurement of the b → sγ rare decay put on non-standard model Higgs searches at hadron colliders. In this article, we additionally explore the regions of SUSY parameter space that can be probed in SM-like Higgs searches for different benchmark scenarios. We also extend our analysis in the B-physics sector to include the additional information coming from the recent measurement of BR(Bu → τν) at Belle [22] and Babar [23]. We find an interesting region of parameter space (i.e. large values of the Higgsino mass parameter µ and moderate values of the stop mixing parameter Xt) for which non-standard Higgs searches are not strongly constrained by B-physics. In particular, we find that scenarios with small stop mixing, like the so called minimal mixing scenario [24], and large Higgsino parameter µ look very promising for the Tevatron and the LHC. B-physics constraints in these scenarios seem to allow the region around a CP-odd Higgs mass MA ∼ 160 GeV and tanβ ∼ 50 (where tanβ = v2/v1 is the ratio of the two Higgs vev’s), which can be easily probed at the Tevatron in the near future. For non-standard Higgs searches we show the present D0 [25] and CDF [26] excluded regions in the MA − tan β plane with 1 fb −1 of data in the ττ inclu- sive channel and the Tevatron and LHC available projections for 4 fb−1 and 30 fb−1 [27, 28] respectively, that depend only slightly on the other low energy SUSY parameters. Small to moderate MSSM Higgs masses are also interesting from the point of view of direct dark matter detection experiments, since in that case t-channel Higgs exchange contributes im- portantly to neutralino dark matter scattering off nuclei. This contribution implies a strong connection between the constraints on SUSY parameters from direct dark matter searches and non-standard MSSM Higgs searches at colliders. In particular, the present direct de- tection limits on neutralino dark matter within the MSSM puts strong constraints on Higgs searches unless the Higgsino component of the neutralino is quite small (i.e. large values of µ), independent of the stop sector parameters [29]. This article is organized as follows. In section 2, we define our theoretical setup for both the B-physics constraints and Higgs searches within the MSSM. In section 3, we discuss representative benchmark scenarios that have different properties for B-physics and Higgs searches. We show that within the MSSM there is a strong complementarity between the constraints coming from non-standard Higgs searches and rare B-decays. Taking into account these constraints we study the potential for standard model like Higgs boson discovery at the Tevatron and the LHC [27, 28]. For the Tevatron Higgs searches we assumed, conservatively, a final Tevatron luminosity of 4 fb−1, while for Higgs searches at LHC, in the early phase, we used the expected 30 fb−1 luminosity estimates. Finally we conclude in section 4. 2 Theoretical Setup 2.1 Higgs Searches and Benchmark Scenarios 2.1.1 Couplings and Masses of the Higgs Sector in the MSSM In the MSSM there are three neutral scalar Higgs fields. Assuming no extra sources of CP violation in the MSSM beyond that of the SM, there are two CP-even Higgs bosons which are admixtures of the real neutral H01 and H 2 components − sinα cosα cosα sinα and an additional CP-odd Higgs field A, where α is the mixing angle that diagonalizes the CP-even Higgs mass matrix. The tree-level Higgs couplings to the SM fermions and gauge bosons are given by [30, 31] (φdd̄)SM ((φuū)SM) (hdd̄)MSSM ((huū)MSSM) (Hdd̄)MSSM ((Huū)MSSM) (Add̄)MSSM ((Auū)MSSM) − sinα/ cos β (cosα/ sinβ) cosα/ cosβ (sinα/ sinβ) tan β (cotβ) (φV V )SM (hV V )MSSM (HV V )MSSM (AV V )MSSM sin(β − α) cos(β − α)  (2) where V can be either the Z or W vector boson. At moderate or large values of tanβ, one of the two CP-even Higgs bosons tends to couple strongly to the gauge bosons while the other one only couples weakly. We will denote the Higgs boson that couples to the gauge bosons the strongest as SM-like. The CP-odd and the other CP-even Higgs bosons are denoted as non-standard and have tanβ enhanced couplings to the down quarks and leptons (see Eq. 2). The identification of the SM-like Higgs depends critically on the size of the pole mass of the pseudo-scalar Higgs MA. For large values of MA, the lighter Higgs becomes SM-like and its mass has the approximate analytic form [1, 2, 3] (Mmaxh ) 2 = M2Z cos 2(2β)(1− 8π2v2 4π2v2 X̃t + t + − 32πα3 (X̃tt+ t , (3) where X̃t = , Xt = At − µ/ tanβ, t = log and MSUSY is the geometric mean of the stop masses. In Eq. (3), we have included the leading two-loop radiative corrections from the stop sector but we have not included the two-loop corrections associated with the relation between the top quark mass and the top Yukawa coupling at the stop mass scale, that depends on the relative sign of the gluino mass and Xt [6]. At values of the CP-odd Higgs boson mass MA less than m h and large values of tanβ, α ∼ β and the heavier CP-even Higgs is SM-like with mass given approximately by Eq. (3). 2.1.2 SM-like Higgs Boson Searches The CMS and ATLAS collaborations have calculated the signal significance curves for stan- dard model Higgs detection at the LHC. Due to the modified Higgs couplings in the MSSM, for the same Higgs masses, these estimates can change significantly with changes in the su- persymmetric mass parameters. To quantify when the significance will be either enhanced or reduced we consider the quantity [30, 31] σ(PP̄ → Xφ)MSSMBR(φ → Y )MSSM σ(PP̄ → Xφ)SMBR(φ → Y )SM where X are particles produced in association with the Higgs and Y are SM decay products of the Higgs1. As the predicted SM-like Higgs mass range within the MSSM is less than or about 130 GeV, we only consider the light Higgs production and decay channels qq̄φ → qq̄τ τ̄ and φ → γγ at the LHC and W/Zφ(φ → bb̄) at the Tevatron. At a luminosity larger than 30 fb−1 at the LHC, the tt̄φ will become effective. However as we are considering only the early phase of the LHC we will not study this process. For the qq̄φ → qq̄τ τ̄ channel the Higgs is produced dominantly by weak-boson fusion. Hence, the tree-level production cross-section is proportional to the square of the (φV V )SM 1For the region of parameter space we study only standard model decays are open. coupling in Eq. (2), which implies that the ratio of production cross-sections in Eq. (4) is proportional to sin2(β − α)(cos2(β − α)) when MA is larger (smaller) than Mmaxh . At large tan β and MA > M h (MA < M h ) the Higgs mixing angle sinα ∼ −1/ tanβ (cosα ∼ 1/ tanβ). Hence, in this region of the MA − tan β plane the (hV V )MSSM ((HV V )MSSM) couplings are very close to their SM values. Therefore at large tan β and small or large values of MA, compared to M h , the ratio σ(PP̄ → Xφ)MSSM/σ(PP̄ → Xφ)SM is close to one. For φ → γγ channel the Higgs is mainly produced through gluon fusion which is induced by third generation quark and squark loops. For squark masses greater than 500 GeV, like those we are considering in this paper, the squark contributions are small and the SM-like Higgs has a production cross-section similar to that of the standard model Higgs. Whenever MA is comparable to the SM-like Higgs mass, |MA −mmaxh | ∼< 10 GeV, both the CP-even Higgs bosons acquire similar masses and have non-standard gauge and yukawa couplings. Hence for each of these channels we follow the prescription given in Ref. [30] and sum the contributions from both the CP-even Higgs states so that σ(PP̄ → Xh)MSSMBR(h → Y )MSSM + σ(PP̄ → XH)MSSMBR(H → Y )MSSM σ(PP̄ → Xφ)SMBR(φ → Y )SM , (5) because we assume that the two signals cannot be separated. If MA is larger (smaller) than M h and the loop corrections to the off-diagonal elements of the CP-even Higgs mass matrix are small, then the large tan β induced corrections do not enhance or reduce the hbb̄ (Hbb̄) or hτ τ̄ (Hττ̄ ) couplings and they remain Standard Model like. Hence, in these regions of parameter space the branching ratios into either b’s or τ ’s are close to their Standard Model values. The φγγ coupling is induced through quark loops and hence is generally small. However, in scenarios where the φbb̄ and φττ̄ couplings are suppressed, like for example if there is a cancellation of the off-diagonal CP-even mass Higgs matrix element due to radiative effects, the φ → γγ branching ratio can be relatively enhanced. We shall discuss this case in section 3.3. 2.1.3 Non-standard Higgs Boson Searches At large tan β the non-standard Higgs bosons are produced in association with bottom quarks or through gluon fusion. For both of these processes, at large tan β, the relevant coupling is the bottom Yukawa coupling [24, 32]. Therefore including the relevant large tanβ radiative correction we find the production cross-section is proportional to the square of the bottom Yukawa y2b = (y 2 tan2 β/(1 + ǫ3 tanβ) 2, where the precise definition of this loop induced correction is given in Eq. (15). In addition, at large tan β [24, 32] the branching ratio of the decay of the non-standard Higgs boson into ττ is approximately given by Br(A,H → τ+τ−) ≃ (1 + ǫ3 tanβ) (1 + ǫ3 tan β)2 + 9 . (6) Hence the total production rate of the CP-odd Higgs boson at large tan β is σ(gg, bb̄ → A)× BR(A → τ+τ−) ∼ σ(gg, bb̄ → A)SM tan2 β (1 + ǫ3 tanβ)2 + 9 . (7) Therefore we can define a ratio similar to Eq. (4) σ(gg, bb̄ → A)MSSMBR(A → τ+τ−)MSSM σ(gg, bb̄ → φ)SMBR(φ → τ+τ−)SM ∼ tan (1 + ǫ3 tan β)2 + 9 and a analogous expression holds for the CP-even non-standard Higgs boson production and decay rates. 2.2 B Physics Observables and Limits We will consider the four B physics observables: BR(Bs → µ+µ−), ∆Ms, BR(b → sγ) and BR(Bu → τν) within the minimal flavor violating MSSM. 2.2.1 BR(Bs → µ+µ−) In the Standard Model the relevant contribution to the Bs → µ+µ− process comes through the Z-penguin and the W-box diagrams which have the analytic form [18, 33] BR(Bs → µ+µ−)SM = MBsτBsF |VtbVts|2 10(xt) (9) where τBs is the mean lifetime, FBs is the decay constant of the Bs meson, xt = mt/MW and C10(x) = b0(x)− c0(x) (10) c0(x) = 3x+ 2 (x− 1)2 ln(x) b0(x) = (x− 1)2 ln(x) . (12) Therefore the predicted SM value comes out to be [18, 33] BR(Bs → µ+µ−)SM = (3.8± 0.1)× 10−9. (13) However in the presence of supersymmetry at large tanβ, there are significant contributions from Higgs mediated neutral currents, which have the form [15, 16] BR(Bs → µ+µ−) = 3.5× 10−5 1.5ps 230MeV ]2 [ |Vts| 0.040 (16π2ǫY ) (1 + ǫ3 tanβ)2(1 + ǫ0 tanβ)2 where ǫ3 = ǫ0 + y t ǫY . (15) The gluino loop factor ǫ0 and the chargino-stop loop factor ǫY are given by M3µC0(m ,M23 ) (16) AtµC0(m , µ2) (17) respectively, where m is the ith sbottom mass, mt̃i is the i th stop mass, M3 is the gluino mass, µ is the higgsino mass parameter, At is the soft SUSY breaking stop trilinear parameter C0(x, y, z) = (x− y)(z − y) log(y/x) + (x− z)(y − z) log(z/x). (18) The present experimental exclusion limit at 95% C.L. from CDF [34] is BR(Bs → µ+µ−) ≤ 1× 10−7, (19) which puts strong restrictions on possible flavor changing neutral currents in the MSSM at large tanβ. Additionally, if no signal is observed, the projected exclusion limit, at 95% C.L., on this process for 4 fb−1 at the Tevatron is [27] BR(Bs → µ+µ−) ≤ 2.8× 10−8. (20) Similarly, if no signal is observed at the LHC, the projected ATLAS bound at 10 fb−1 is [35] BR(Bs → µ+µ−) ≤ 5.5× 10−9. (21) Therefore considering Eq. (14) in the absence of a signal, these experiments will put very strong constraints on the allowed MSSM parameter space. In addition, LHCb has the po- tential to claim a 3σ (5σ) evidence (discovery) of a standard model signature with as little as ∼ 2fb−1(6fb−1) of data [36]. 2.2.2 ∆Ms In the Standard Model the dominant contribution to ∆Ms comes from W-top box diagrams that have the analytical form [15, 16] ∆Ms = MBsη2F B̂Bs|Vts|2S0(mt) (22) where MBs is the Bs meson mass, B̂Bs is the Bs bag parameter, η2 is the NLO QCD factor S0(mt) ≃ 2.39 167GeV )1.52 . (23) The updated theoretical predictions from the CKMfitter and UTFit groups are slightly different. The UTFit group finds the 95 % C.L. range [37] (∆Ms) SM = (20.9± 2.6)ps−1 (24) which is consistent with the CKMfitter groups’ 2σ range [38] 13.4 ps−1 ≤ (∆Ms)SM ≤ 31.1 ps−1 (25) and central value of 18.9 ps−1. About a year ago, the D0 collaboration reported a signal consistent with values of ∆Ms in the range 21 (ps)−1 > ∆Ms > 17 ps −1 (26) at the 90 % C.L. [39]. More recently, the CDF collaboration has made a measurement, with the result [40] ∆Ms = (17.77± 0.10(stat)± 0.07(syst))ps−1. (27) The large theoretical uncertainties and the precise experimental value suggest that small or moderate negative contributions to ∆Ms may be easily accommodated. As shown in Refs. [14, 15, 16, 18] for large tan β and uniform squark masses one obtains negative contri- butions to ∆Ms that are well approximated by (∆Ms) DP = −12.0ps−1 230MeV m̄b(µs) 3.0GeV m̄s(µs) 0.06GeV m̄4t (µs) (16π2ǫ2Y ) (1 + ǫ3 tan β)2(1 + ǫ0 tanβ) . (28) In the next section we will discuss the interplay between the BR(Bs → µ+µ−) in Eq. (14) and ∆Ms in Eq. (28) within the framework of minimal flavor violating MSSM. 2.2.3 BR(b → sγ) The next B-physics process of interest is the rare decay b → sγ. The world experimental average of the branching of this rare decay is [41, 42] BR(b → sγ)exp = (3.55± 0.24+0.09 −0.10 ± 0.03)× 10−4. (29) This experimental result is close to the SM central value and so puts constraints on flavor violation in any extension of the Standard Model. However, the theoretical uncertainties in the Standard Model for this process are quite large [42] BR(b → sγ)SM = (2.98± 0.26)× 10−4. (30) Using the experimental and SM ranges for the BR(b → sγ) we find the 2σ allowed range is 0.92 ≤ BR(b → sγ) BR(b → sγ)SM ≤ 1.46. (31) This bound is appropriate for constraining new physics contributions due to the cancellation of the dominant uncertainties coming from infrared physics effects. In minimal flavor violating MSSM there are two new contributions from the charged Higgs and the chargino-stops diagrams. The charged Higgs amplitude, including the stop induced two-loop effects, is proportional to the factor [43, 44] AH+ ∝ 1− 2αs µM3 tan β cos2 θt̃C0(m ,M23 ) + sin 2 θt̃C0(m ,M23 ) 1 + ǫ3 tanβ , (32) where θt̃ is the stop mixing angle. The chargino-stop amplitude has the form [43, 44] Aχ− ∝ µAt tan β 1 + ǫ3 tan β , m2χ−). (33) where f(m2 , mχ̃−) ∼ 1/max(m2t̃1 , m ) is the one-loop factor that depends on the stop masses and the chargino mass. The specific dependences of these amplitudes on MSSM parameters are important in understanding the constraints on the SUSY contributions to BR(b → sγ), which will be discussed below. 2.2.4 BR(Bu → τν) The final B-physics observable of interest is the process Bu → τν which the Belle experi- mental collaboration finds to be [22] BR(Bu → τν)Belle = (1.79+0.56 −0.49(stat) +0.46 −0.51(syst))× 10−4, (34) while the Babar collaboration finds a value [23] BR(Bu → τν)Babar = (0.88+0.68 −0.67(stat)± 0.11(syst))× 10−4. (35) The two values are within 2σ of each other and both of them are consistent with the standard model prediction. The average of these two experiments is [37] BR(Bu → τν)Exp = (1.31± 0.48)× 10−4. (36) The Standard Model contribution is mediated by the W-boson and has the generic form [45] BR(Bu → τν)SM = G2FmBm F 2B|Vub|2τB (37) βtan( ) M = 150 GeVA X = 1 TeVt M = 150 GeVA X = 0 TeVt M = 250 GeVA X = 1 TeVt M = 250 GeVA X = 0 TeVt ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� Allowed by Inclusive V Allowed by UTfit V 10 20 30 40 50 60 70 80 90 100 Figure 1: The green(grey) hatched area is the 2σ allowed region of the ratio RBτν if the fitted value of |Vub| is used to calculate the standard model prediction of the Bu → τν decay rate. The yellow(light grey) is corresponding region if the inclusive determination of |Vub| is used instead of the fitted value. The solid (dashed) lines show the variation of RBτν with respect to tanβ for MA = 150GeV(250GeV), while the red (grey) color and blue (dark grey) color correspond to Xt = 0 and Xt = 1 TeV respectively. and using the UTFit fitted value for |Vub| = (3.68 ± 0.14) × 10−3 (which is also in good agreement with the CKMfitter value [38]), τB and the extracted value of FB = 0.237 ± 0.037 GeV leads to the value [37] BR(Bu → τν)SM = (0.85± 0.13)× 10−4. (38) Observe, however that the value of |Vub| = (4.49 ± 0.33) × 10−9, extracted from inclusive semileptonic decays is higher and leads to the standard model prediction BR(Bu → τν)SM = (1.39± 0.44)× 10−4 [37]. In the MSSM there is an extra contribution due to the charged Higgs which interferes destructively with the SM contribution, so that at large tanβ the ratio of the two is [45, 46, RBτν = BR(Bu → τν)MSSM BR(Bu → τν)SM tan2 β 1 + ǫ0 tanβ . (39) Now assuming a 2σ deviation in Eq. (36) and Eq. (38) that is due to the charged Higgs contribution, we find the allowed range of values for this ratio to be 0.32 ≤ RBτν ≤ 2.77. (40) However as discussed above, if the inclusive determination of |Vub| is used instead of the fitted value we get a different range of allowed values for RBτν . In Fig. 1 we show the effect of choosing the |Vub| inclusive value over the fitted value. The green (grey) hatched region is allowed if we use the fitted value of |Vub| while the yellow (light grey) region is allowed if we use the extracted value of |Vub| from inclusive semileptonic b-decays. From Fig.1 we can see that if MA = 150 GeV and Xt = 0 the allowed values are tanβ ∼ 10 − 25 and tan β ∼ 53 − 70 using the fitted value of |Vub|, while using the inclusive value of |Vub| we find 10 ∼< tan β ∼< 37 or 43 ∼< tan β ∼< 63. Therefore, when we project this constraint onto the MA − tan β plane the allowed regions are significantly different, especially at larger values of MA. In particular the region of intermediate tan β that is excluded by the Bu → τν constraint is much smaller if we use the inclusive value of |Vub| instead of the fitted value because the lower bound on RBτν is smaller for the value extract from inclusive b-decays. Whenever we consider the constraint on the Bu → τν rate in this paper we will use the fitted values, so expect our bounds to be quite conservative and one could enlarge the B physics allowed region by going to larger values of |Vub|. 3 B physics constraints and Higgs searches at hadron Colliders In this section we shall use the above B physics limits and Higgs search capabilities to put constraints on the allowed regions of MSSM parameter space. In particular we project these constraints onto the MA − tan β plane. We also assume that all the squark masses are uniform and denoted by MSUSY , 2M1 = M2 = 500 GeV and we use the central value for the top-quark measured, at the Tevatron to bemt = 170.9±1.8 GeV [48]. Within this framework we study four benchmark scenarios by varying the parameters µ, Xt = At−µ/ tanβ, MSUSY and M3. We numerically calculate the ratio r, defined for non-standard Higgs searches in Eq. (8), using the CPsuperH program [49]. To estimate the present excluded region and the projected Tevatron reach we used the 1 fb−1 CDF results presented in Ref. [26], the projected 4 fb−1 curves from Ref. [27] and the 1 fb−1 D0 results from Ref. [25] for the maximal mixing scenario with µ ∼ −200 GeV. To estimate the LHC reach we used the results for the maximal mixing scenario with µ ∼ −200 GeV in Fig. 6 of Ref. [24], which is based on the study in Ref. [50]. Using Eq. (8), each of these curves are rescaled for each of the different parametric scenarios we consider in this paper. Let us stress that the results of Ref. [50], we are using, are in reasonably good agreement with the latest CMS studies for different τ decay final states, which include a full detector simulation [51, 52, 53, 54]. For the SM-like Higgs searches at 30 fb−1, we used the CMS and the ATLAS studies shown in Ref. [28, 50] to estimate the signal significance in the h → ττ and h → γγ channel. We used CPsuperH [49] to calculate the relevant branching ratios and couplings needed to estimate the value of R in Eq. (4). For the Tevatron searches we used the updated values of the luminosity needed to discover a Standard Model Higgs, from Ref. [55], to estimate the variation of signal significance with respect to SM Higgs mass at 4 fb−1 for each experiment. The projections at the Tevatron assume an improvement in the sensitivity of detectors along with a basic increase in the luminosity [55]. Before presenting our analysis, let us stress that, from the form of the double penguin contribution to ∆Ms in Eq. (28) and the large tanβ contribution to BR(Bs → µ+µ−) in AM (GeV) ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 AM (GeV) 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 AM (GeV) ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� 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������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 AM (GeV) ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 Figure 2: The red (grey) region, in all four figures, is excluded by the CDF experiment’s search for non-standard Higgs bosons in the inclusive A → τ+τ− channel at 1 fb−1 luminosity. The dotted line shows the corresponding D0 excluded region at 1 fb−1. (a) The solid and dashed lines represent the future reach for the Tevatron (at 4 fb−1)and LHC (at 10 fb−1 for Bs → µ+µ− and at 30 fb−1 for A → τ+τ−) respectively, where the red (dark gray) lines correspond to the non-standard Higgs search reaches in the H → ττ channel while the black lines are the projected BR(Bs → µ+µ−) bounds for µ = −100 GeV, Xt = 2.4 TeV, MSUSY = 1 TeV and M3 = 0.8 TeV. The green (gray) hatched regions are those allowed by the present B-physics constraints on the Bu → τν b → sγ and Bs → µ+µ− branching ratios. (b) and (c) For the same SUSY mass parameters the yellow (light gray) area is the 5σ discovery region in the h → γγ channel, while the green (gray) hatched area is the same for the h → ττ channel for the CMS and ATLAS experiments respectively at 30 fb−1. (d) Green (gray) hatched region is the 3σ evidence region for the SM-like Higgs searches (at 4 fb−1) at the Tevatron. (b)–(d) The areas surrounded by the dashed black curves correspond to the regions allowed by present B-physics constraints. Eq. (14), it is clear that the two quantities are greatly correlated. As we shown in Ref. [15, 16] for the case of uniform squark masses, Eq. (14) and Eq. (28) imply that |(∆Ms)SUSYDP | BR(Bs → µ+µ−)SUSY ∼ 0.034(ps) tan β . (41) Notice that the only SUSY parameters this ratio depends on are MA and tan β. Considering the present experimental limit on BR(Bs → µ+µ−) in Eq. (19), we showed in Ref. [21] that, as is apparent in Eq. (41), the double penguin contributions to ∆Ms can be at most a few ps−1 for MA < 1 TeV. As these corrections are negative with respect to the SM contribution, they make the theoretical predictions agree slightly better with the experimentally measured value. However given that the theoretical errors in Eq. (24) and Eq. (25) are large and the SUSY contributions are small, the ∆Ms measurement only puts a very weak constraint on Higgs searches once the Bs → µ+µ− bound is imposed. 3.1 Large to moderate Xt and small µ This scenario is a modified version of the one called maximal mixing because we chose the sign of AtM3 to be negative. This choice of sign tends to reduce the value of the SM-like Higgs mass making it easier for the Tevatron collider to possibly probe this scenario. On the other hand the change in the sign of M3 with respect to that in the maximal mixing scenario [24] does not significantly affect B-physics constraints and the non-standard Higgs boson search limits, as can be seen in Fig.9(a) of Ref. [21]. The SM-like Higgs mass depends strongly on the stop mixing parameter Xt, and it attains its maximum value for Xt ∼ 6MSUSY = 2.4 TeV. For these values of Xt, small µ and small MA, which can be probed at the Tevatron, we need the sign of µAt to be negative so that the stop-chargino contribution to b → sγ amplitude in Eq. (33) cancels against that of the charged Higgs in Eq. (32) [21]. The Bs → µ+µ− constraint in this scenario is quite strong because the Bs → µ+µ− branching ratio in Eq. (14) is proportional to At, which is large, and in the denominator the factor 1 + ǫ3 tan β ∼ 1, as the ǫ3 loop-factor is small. The Bu → τν constraint has two allowed regions related to the two possible signs of the amplitude, as can be seen in Eq. (39). At low values of tan β and large values of MA the SM contribution dominates, while at complementary values of MA and tanβ the SUSY contribution dominates. In Fig. 2 (a) the present limit on the Bs → µ+µ−, and the measurements of the b → sγ and Bu → τν decay rates allow the green (gray) hatched region for Xt = 2.4 TeV, M3 = −800 GeV, MSUSY = 1 TeV and µ = −100 GeV. The red (dark gray) region is excluded by the CDF experiment’s non-standard Higgs search in the inclusive τ+τ− decay mode. The dotted red (dark grey) is the corresponding excluded region according to the D0 collaboration. The red (dark gray) solid and dashed curves show the regions that can be excluded by non-standard Higgs searches at the Tevatron for a future luminosity of 4 fb−1 and at the LHC for a luminosity of 30 fb−1 respectively. The black solid and dashed curves corresponds to the future Bs → µ+µ− limits for the Tevatron at a luminosity of 4 fb−1 and the LHC at a luminosity of 10 fb−1 shown in Eq. (20) and Eq. (21) respectively. A AM (GeV) ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 AM (GeV) 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 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150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 Figure 3: (a)–(d) The lines and the colors correspond to the same quantities as in Fig. (2), where the SUSY parameters are the same except for Xt = 1 TeV. reach similar to Eq.(21) and comparable to the standard model prediction is expected at LHCb with only a few fb−1 of data [36]. As the B-physics allowed region corresponds to large values of MA and small values of tanβ, the SM contribution to the amplitude of the Bu → τν process is larger than the SUSY contribution to the same amplitude. The region where the SUSY contribution to the amplitude of the Bu → τν process is larger than the SM contribution is excluded by the present bounds on the Bs → µ+µ− branching ratio in Eq. (19). As we found in Ref. [21] the maximal mixing scenario is strongly constrained by B- physics and the addition of the Bu → τν limit makes these constraints even stronger. For these values of SUSY parameters B-physics constraints prefer low to moderate values of tan β. In addition the Tevatron will find it difficult to discover a non-standard Higgs boson for this scenario. Moreover, the LHC at a luminosity of 30 fb−1 will only be able to probe a very small portion of the B-physics allowed parameter space in the A/H → ττ channel. In Fig. 2 (b and c) we show the parts of the MA − tanβ that can be probed in Standard Model Higgs searches at the CMS and ATLAS experiments, respectively. The yellow (light gray) regions are those that can be probed in h → γγ channel while the green (dark gray) hatched regions can be probed in h → ττ channel with a luminosity of 30 fb−1 at 5 σ. Present available studies with the ATLAS detector show that it will be able to probe all of the B-physics allowed region. According to the new analysis shown in Ref. [28], the CMS detector may not be able to probe the region of moderate MA in the h → ττ channel. However due to a significant improvement in the CMS sensitivity in the γγ channel a large portion of the B-physics allowed region can still be probed. If the sign of AtM3 were positive the qualitative features of the CMS reach and ATLAS reach would remain the same. In Fig. 2 (d) we show the region of the MA − tan β plane that the Tevatron can probe in the h → bb̄ channel with a luminosity of 4 fb−1 per experiment and a signal significance of 3 standard deviations. For the modified maximal mixing scenario the region that can be probed is relatively large compared to the standard one [24, 32], because the sign of AtM3 is negative. For negative AtM3 the maximum SM-like Higgs boson mass is approximately ∼ 125 GeV compared to the standard maximal mixing scenario which has 130 GeV as the maximum Higgs mass [49]. In Fig. 3 we show the effect of going to a lower value of stop mixing parameterXt = 1 TeV. There are two disconnected B-physics allowed regions for these SUSY parameters shown in Fig. 3 (a). There is a tiny upper region at around (MA, tanβ) ∼ (150 GeV, 43) and a much larger lower tan β region where all the B physics constraints are just satisfied. In the upper region the SUSY contribution to the amplitude of the Bu → τν rate is larger than the SM contribution to the same process, while in the lower region the opposite is true. The area between these two regions is excluded because the ratio RBτν in Eq. (40) is below the 2σ bound. The reach via SM-like Higgs searches for these SUSY parameters, are similar to the maximal mixing scenario. CMS has difficulties seeing the SM-like Higgs in part of the regions allowed by B-physics constraints, but the ATLAS experiment will cover all of MA − tanβ plane. The Tevatron experiments may now cover the whole allowed region of the MA− tanβ plane at 3σ. 3.2 Large µ and small or negligible Xt For the minimal mixing scenario, Xt is equal to zero and the chargino-stop contribution to the b → sγ process is small. Due to a reasonable agreement between the Standard Model prediction and the experimental measurement of the b → sγ rate, we need the charged Higgs contribution in Eq. (32) to be small. For a light charged Higgs, this requirement can be achieved by going to large values of µ, M3 and tan β because of a cancellation between the tree-level term and the loop induced term in Eq. (32). Since At is small, the Bs → µ+µ− limit puts a weak constraint on the MA − tan β plane. Additionally, for these values of parameters the usual bound on tanβ that comes from requiring that yb be perturbative up to the GUT scale may be relaxed: Since the bottom Yukawa has the form 2mb tan β v(1 + ǫ3 tanβ) and as ǫ3 tanβ needs to be real, positive and of order one, for the above cancellation in the charged Higgs amplitude to occur2, the denominator suppresses the bottom Yukawa coupling for large values of tan β. The SM-like Higgs searches put an interesting constraint on scenarios with large values of |µ| and small values of Xt, since unless MSUSY is sufficiently large the SM-like Higgs mass tends to be below the LEP bound of 114.4 GeV. The impact of the LEP bound on the excluded region in the MA− tanβ plane is very sensitive to µ, MSUSY and the top mass. For instance, for MSUSY ∼ 1 TeV this scenario is highly constrained by the LEP bounds on the SM-like Higgs mass, but increasing MSUSY to 2 TeV is sufficient to avoid this constraint [56]. The corresponding results for MSUSY = 2 TeV are shown in Fig. 4. We have pre- viously analyzed this scenario in Ref. [21] without adding the Bu → τν constraints. In Fig. 4 we see that the addition of this new constraint excludes the diagonal region with corners (100 GeV, 38), (155 GeV, 28), (450 GeV, 80) and (190 GeV, 65) for the parameters µ = 1.5 MSUSY and M3 = 0.8 MSUSY . In Fig. 4 (a) we show the effect of the LEP bound on the B-physics allowed regions. The region below the blue (black) solid line shows the area excluded by the LEP bound in the MA − tan β plane. From Fig. 4 (b) and (c) it is clear that the CMS and ATLAS experiment can probe most of the allowed B-physics regions of the MA − tanβ plane, using SM-like Higgs searches in the h → γγ and the h → ττ channels. CMS has an inaccessible region at large MA in the ττ -channel because in this region the τ Yukawa coupling is only slightly above the standard model value and according to Ref. [28] CMS does not have a 5σ signal significance with 30 fb−1 of data for any standard model Higgs mass. However, given that the Higgs mass and the h → ττ coupling vary smoothly with MA and tan β the discovery potential is also above 4σ for most of the region that appears inaccessible in Fig. 4 (b). Again, at 4 fb−1 the Tevatron could have a 3σ evidence over most of the parameter space allowed by B-physics and the LEP Higgs mass bound. We would like to stress that the B physics and the LEP excluded regions, for the minimal mixing scenario, allow a clear region of MA = 130 − 170 GeV and tan β = 50 − 70. These values are easily within the Tevatron’s sensitivity region for non-standard Higgs searches in the ττ channel. In addition, the SM-like Higgs boson mass is close to the current limit and therefore should be visible at the Tevatron at the 3 σ level with an increase in sensitivity and luminosity. Both CDF and D0 collaborations have recently made public their findings in the inclusive A → ττ channel at a luminosity of 1 fb −1. The CDF experiment finds a slight excess [26] while the D0 experiment [25] finds a reduction in the signal for the same values of the the ττ visible mass. The D0 limit further limits the upper B-physics allowed region to values of MA = 130− 150 GeV and tan β ∼ 55. This scenario can be relatively insensitive to small changes in the value of Xt. It would seem that increasing the value ofXt would make the Bs → µ+µ− constraint extremely strong. However, there is a 1/µ2 dependence from the (1 + ǫ30)(1 + ǫ3) factor in the denominator of Eq. (14) and only a linear µ dependence in its numerator. 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150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 AM (GeV) ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 AM (GeV) ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 Figure 4: (a)–(d) The lines and the colors correspond to the same quantities as in Fig. (2), where the SUSY parameters are the same except for Xt = 0 GeV, µ = 1.5 MSUSY and MSUSY = 2 TeV. The region below the blue (black) solid line corresponds to the area excluded by the LEP bound on the SM-like Higgs boson for mt = 170.9 GeV. ǫ are positive and µ is large, even moderate values of Xt do not strengthen the Bs → µ+µ− constraint. Additionally at large values of µ, M3 and tanβ the charged Higgs contribution to the b → sγ amplitude in Eq. (32) may have the opposite sign to the SM one, a novel result that only occurs for this range of parameters. In this region of parameter space, to cancel this negative charged Higgs amplitude we need the chargino-stop contribution in Eq. (33) to be positive or the sign of µAt to be positive. 3.3 Small αeff This scenario was studied in Ref. [56] in which the off-diagonal components of the CP-even Higgs mass matrix are approximately zero. This approximate cancellation can be achieved by making, for instance, the following choice of parameters µ = 2.5 TeV, Xt = −1200.0 TeV, MSUSY = 800GeV, M3 = 500GeV. (43) A consequence of this cancellation is that the couplings of the SM-like Higgs boson to the b-quarks and τ -leptons are suppressed. AM (GeV) ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 ������ ������ ������ ������ ������ ������ ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� AM (GeV) 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 ����������� ����������� ����������� ����������� ����������� ����������� ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ������������ ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� AM (GeV) 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� AM (GeV) ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 Figure 5: (a)–(d) The lines and the colors correspond to the same quantities as in Fig. (2), where the SUSY parameters are the same except for M3 = 500 GeV, MSUSY = 800 GeV, Xt = −1.2 TeV and µ = 2.5 TeV. In Fig. 5 we present the effect of this choice of parameters on the B-physics allowed region and on Higgs searches at the LHC and Tevatron. The B-physics constraints are quite severe and similar to the large Xt scenario we discussed above. The h → γγ channel for SM-like Higgs searches is enhanced because the h → τ τ̄ and h → bb̄ branching ratios are suppressed, leading to an enhancement of the h → γγ branching ratio. Therefore the CMS and ALTAS experiments will be able to probe a large part of the MA−tan β plane in the h → γγ channel. The Tevatron will not be able to probe most of the B-physics allowed region because of the suppression of the h → bb̄ branching ratio. 4 Conclusions In this article we have studied the inter-play between B-physics constraints and Higgs searches at hadron colliders in the framework of minimal flavor violating SUSY models. The results we present here depend on the projected sensitivities of the CMS and ATLAS experiments and the Tevatron collider in the different SM-like and non-standard Higgs boson channels. The Tevatron projections assumed in this work [55] need to be further solidified by improvements in the analyses that CDF and D0 are performing. Both CMS and ATLAS have recently performed improvements in their projections in the γγ inclusive channel and CMS has also recently updated their h → ττ vector boson fusion study [28]. We have illus- trated this interplay between Higgs searches at hadron colliders and B-physics constraints using four benchmark senarios. In particular the B-physics constraints are extremely severe for SUSY parameters which have large values of Xt and small values of µ. For SM-like Higgs boson searches the LHC experiments should be able to probe all of the allowed region of parameter space with 30 fb−1, but the Tevatron collider will have difficulties doing this with 4 fb−1 of data. Discovering a SM-like Higgs boson at the CMS experiment with 30 fb−1 of data will be challenging in this scenario, since CMS has a better sensitivity in the h → γγ rather than in the h → ττ channel and as the hbb̄ and the hτ τ̄ couplings are somewhat enhanced for moderate or small MA, the h → γγ branching ratio is smaller than in the SM. On the other hand, the ATLAS experiment will easily probe the allowed region of parameter space because the h → ττ branching ratio is enhanced for these values of SUSY parameters. The Tevatron will find it very difficult to detect a SM-like Higgs in this scenario because the SM-like Higgs is heavy and the signal significance, in the h → bb̄ channel, drops sharply with increasing Higgs mass. Additionally, in this scenario the B-physics constraints favor regions which have large values of MA and low values of tanβ while the non-standard Higgs boson searches at hadron colliders are less efficient in these regions. Therefore at a luminosity of 30 fb−1 the LHC will be able to observe the SM-like Higgs, but may find it difficult to discover non-standard Higgs bosons. The B-physics constraints are far weaker for large values of µ and small values of Xt due to a suppression of SUSY contributions to the Bs → µ+µ− and the b → sγ rates. At the same time the present LEP bounds on the SM-like Higgs mass put strong constraints on the allowed regions of parameter space, in particular for MSUSY ≤ 1 Tev. For the minimal mixing scenario with MSUSY = 2 TeV we have studied, the LHC will be able to probe most of the B-physics allowed region in non-standard Higgs searches, for values of MA < 500 GeV. For SM-like Higgs searches, with 30 fb−1 of data, the CMS collaboration should be able to probe most of the allowed regions, while the ATLAS collaboration will be able to probe all of them. In addition, this scenario is the most promising for the Tevatron to detect both the SM-like Higgs and the non-standard Higgs bosons in the near future. The final benchmark scenario we studied was that of small αeff . Due to the suppression of SM-like Higgs couplings to b-quarks and τ ’s, the γγ channel is enhanced. Due to this enhancement both the LHC experiments will be able to discover the SM-like Higgs over most of the B-physics allowed parameter space. The Tevatron will find it difficult to detect a SM-like Higgs due its mass and suppressed couplings to bb̄. In conclusion, scenarios with lower values of stop mixing parameter Xt and larger values of higgsino mass parameter µ will be easier to probe at hadron colliders through direct higgs searches of both standard and non-standard Higgs bosons. At larger values of Xt, direct non-standard Higgs boson searches are strongly constrained by present bounds on B-physics observables. On the other hand, the SM-like Higgs boson mass is enhanced through radiative corrections, rendering it more easily detectable at the LHC. Finally, the observation of a SM- like Higgs in the h → ττ channel and not in the h → γγ or vice versa, may be used to obtain additional information on the values of the supersymmetry breaking parameters. Acknowledgements: M.C. and C.W. would like to thank the Aspen Center for Physics, where part of this work was done. We wish to thank Patricia Ball, Thomas Becher, Avto Kharchilava, Enrico Lunghi, Matthias Nuebert and Frederic Teubert. Work at ANL is supported in part by the US DOE, Div. of HEP, Contract DE-AC02-06CH11357. Fermilab is operated by Universities Research Association Inc. under contract no. DE-AC02-76CH02000 with the DOE. This work was also supported in part by the U.S. Department of Energy through Grant No. DE-FG02-90ER40560. References [1] J. R. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B 257, 83 (1991). J. R. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B 262, 477 (1991). Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys. 85, 1 (1991). M. A. Diaz and H. E. Haber, Phys. Rev. D 45, 4246 (1992). [2] M. Carena, J. Espinosa, M. Quirós and C. Wagner, Phys. Lett. B 355 (1995) 209, hep-ph/9504316; M. Carena, M. Quirós and C. Wagner, Nucl. Phys. B 461 (1996) 407, hep-ph/9508343. [3] M. Carena, M. Quiros and C. E. M. Wagner, Nucl. Phys. B 461, 407 (1996) [arXiv:hep-ph/9508343]. [4] H.E. Haber, R. Hempfling and A.H. Hoang, Z. Phys. C 75, 539 (1997), [arXiv:hep-ph/9609331]. [5] S. Heinemeyer, W. Hollik and G. Weiglein, Eur. Phys. J. C 9, 343 (1999) [arXiv:hep-ph/9812472]. [6] M. Carena, H. Haber, S. Heinemeyer, W. Hollik, C. Wagner, and G. Weiglein, Nucl. Phys. B 580 (2000) 29, hep-ph/0001002. J. R. Espinosa and R. J. Zhang, JHEP 0003, 026 (2000) [arXiv:hep-ph/9912236]. [7] J. Espinosa and R. Zhang, Nucl. Phys. B 586 (2000) 3, hep-ph/0003246. [8] A. Brignole, G. Degrassi, P. Slavich and F. Zwirner, Nucl. Phys. B 631 (2002) 195, hep-ph/0112177; Nucl. Phys. B 643 (2002) 79, hep-ph/0206101. http://arxiv.org/abs/hep-ph/9504316 http://arxiv.org/abs/hep-ph/9508343 http://arxiv.org/abs/hep-ph/9508343 http://arxiv.org/abs/hep-ph/9609331 http://arxiv.org/abs/hep-ph/9812472 http://arxiv.org/abs/hep-ph/0001002 http://arxiv.org/abs/hep-ph/9912236 http://arxiv.org/abs/hep-ph/0003246 http://arxiv.org/abs/hep-ph/0112177 http://arxiv.org/abs/hep-ph/0206101 [9] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich and G. Weiglein, Eur. Phys. J. C 28 (2003) 133, hep-ph/0212020. [10] M. Carena and H. E. Haber, Prog. Part. Nucl. Phys. 50, 63 (2003) [arXiv:hep-ph/0208209]. [11] S. Heinemeyer, W. Hollik, H. Rzehak and G. Weiglein, Eur. Phys. J. C 39 (2005) 465, hep-ph/0411114. [12] S. Martin, Phys. Rev. D 67 (2003) 095012, hep-ph/0211366; Phys. Rev. D 71 (2005) 016012, hep-ph/0405022; [13] S. Bertolini, F. Borzumati, A. Masiero and G. Ridolfi, Nucl. Phys. B 353, 591 (1991). [14] G. Isidori and A. Retico, JHEP 0111, 001 (2001) [arXiv:hep-ph/0110121]. [15] A. J. Buras, P. H. Chankowski, J. Rosiek and L. Slawianowska, Phys. Lett. B 546, 96 (2002) [arXiv:hep-ph/0207241]. [16] A. J. Buras, P. H. Chankowski, J. Rosiek and L. Slawianowska, Nucl. Phys. B 659, 3 (2003) [arXiv:hep-ph/0210145]. [17] K. S. Babu and C. F. Kolda, Phys. Rev. Lett. 84, 228 (2000) [arXiv:hep-ph/9909476]. [18] A. Dedes and A. Pilaftsis, Phys. Rev. D 67, 015012 (2003) [arXiv:hep-ph/0209306]. [19] D. A. Demir, Phys. Lett. B 571, 193 (2003) [arXiv:hep-ph/0303249]. [20] J. Foster, K. i. Okumura and L. Roszkowski, JHEP 0508, 094 (2005) [arXiv:hep-ph/0506146]. [21] M. Carena, A. Menon, R. Noriega-Papaqui, A. Szynkman and C. E. M. Wagner, Phys. Rev. D 74, 015009 (2006) [arXiv:hep-ph/0603106]. [22] K. Ikado et al., arXiv:hep-ex/0604018. [23] B. Aubert et al. [BABAR Collaboration], arXiv:hep-ex/0608019. [24] M. Carena, S. Heinemeyer, C. E. M. Wagner and G. Weiglein, Eur. Phys. J. C 45, 797 (2006) [arXiv:hep-ph/0511023]. [25] http://www-d0.fnal.gov/Run2Physics/WWW/results/prelim/HIGGS/H29/ D0 Note 5331-CONF [26] http://www.physics.ucdavis.edu/%7Econway/talks/ConwayAspen2007.pdf [27] See,for example, B. Heinemann presentation to the P5 Committee, Fermilab, Septem- ber 2006, http://hep.ph.liv.ac.uk/∼beate/homepage/p5-discovery.pdf http://arxiv.org/abs/hep-ph/0212020 http://arxiv.org/abs/hep-ph/0208209 http://arxiv.org/abs/hep-ph/0411114 http://arxiv.org/abs/hep-ph/0211366 http://arxiv.org/abs/hep-ph/0405022 http://arxiv.org/abs/hep-ph/0110121 http://arxiv.org/abs/hep-ph/0207241 http://arxiv.org/abs/hep-ph/0210145 http://arxiv.org/abs/hep-ph/9909476 http://arxiv.org/abs/hep-ph/0209306 http://arxiv.org/abs/hep-ph/0303249 http://arxiv.org/abs/hep-ph/0506146 http://arxiv.org/abs/hep-ph/0603106 http://arxiv.org/abs/hep-ex/0604018 http://arxiv.org/abs/hep-ex/0608019 http://arxiv.org/abs/hep-ph/0511023 http://www-d0.fnal.gov/Run2Physics/WWW/results/prelim/HIGGS/H29/ http://www.physics.ucdavis.edu/ http://hep.ph.liv.ac.uk/~beate/homepage/p5-discovery.pdf [28] Talk by A. Nikitenko, ICHEP conference 2006, Moscow, Russia, July 2006, http://ichep06.jinr.ru/reports/327 7s6 17p10 nikitenko.ppt ATLAS note: SN-ATLAS-2003-024, Prospects for the search for a Standard Model Higgs boson in Atlas using Vector Boson Fusion Talk by F Gianotti ICHEP conference 2006, Moscow, Russia, July 2006, http://ichep06.jinr.ru/programme.asp http://documents.cern.ch/cgi-bin/setlink?base=atlnot&categ=CONF&id=phys-conf- 2006-018 [29] M. Carena, D. Hooper and P. Skands, Phys. Rev. Lett. 97, 051801 (2006) [arXiv:hep-ph/0603180]. M. Carena, D. Hooper and A. Vallinotto, Phys. Rev. D 75, 055010 (2007) [arXiv:hep-ph/0611065]. [30] M. Carena, S. Mrenna and C. E. M. Wagner, Phys. Rev. D 60, 075010 (1999) [arXiv:hep-ph/9808312]. M. Carena, S. Mrenna and C. E. M. Wagner, Phys. Rev. D 62, 055008 (2000) [arXiv:hep-ph/9907422]. [31] L. Roszkowski, R. R. de Austri and R. Trotta, arXiv:hep-ph/0611173. [32] U. Aglietti et al., arXiv:hep-ph/0612172. [33] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996) [arXiv:hep-ph/9512380]. [34] R. Bernhard et al. [CDF Collaboration], arXiv:hep-ex/0508058. [35] Nikolai Nikitine, talk given at ”FLAVOUR IN THE ERA OF THE LHC” Opening plenary meeting: CERN, November 7-10 2005; R. McPherson, talk given at the Aspen Winter Conference, Aspen, CO, February 12–18, 2006, http://www.aspenphys.org [36] http://teubert.web.cern.ch/teubert/BsmumuLHCb.ppt [37] M. Bona et al. [UTfit Collaboration], “The unitarity triangle fit in the standard model and hadronic parameters from lattice QCD: A reappraisal after the measurements of Delta(m(s)) and JHEP 0610, 081 (2006) [arXiv:hep-ph/0606167]. [38] J. Charles et al. [CKMfitter Group], Eur. Phys. J. C 41, 1 (2005) [arXiv:hep-ph/0406184]; http://www.slac.standford.edu/xorg/ckmfitter/plots eps2005/ckmEval results eps 05.ps [39] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. 97, 021802 (2006) [arXiv:hep-ex/0603029]. http://ichep06.jinr.ru/reports/327_7s6_17p10_nikitenko.ppt http://ichep06.jinr.ru/programme.asp http://documents.cern.ch/cgi-bin/setlink?base=atlnot http://arxiv.org/abs/hep-ph/0603180 http://arxiv.org/abs/hep-ph/0611065 http://arxiv.org/abs/hep-ph/9808312 http://arxiv.org/abs/hep-ph/9907422 http://arxiv.org/abs/hep-ph/0611173 http://arxiv.org/abs/hep-ph/0612172 http://arxiv.org/abs/hep-ph/9512380 http://arxiv.org/abs/hep-ex/0508058 http://www.aspenphys.org http://teubert.web.cern.ch/teubert/BsmumuLHCb.ppt http://arxiv.org/abs/hep-ph/0606167 http://arxiv.org/abs/hep-ph/0406184 http://www.slac.standford.edu/xorg/ckmfitter/plots_eps2005/ckmEval_results_eps_05.ps http://arxiv.org/abs/hep-ex/0603029 [40] S. Giagu [CDF Collaboration], arXiv:hep-ex/0610044. [41] [Heavy Flavor Averaging Group (HFAG)], arXiv:hep-ex/0603003. [42] T. Becher and M. Neubert, Phys. Rev. Lett. 98, 022003 (2007) [arXiv:hep-ph/0610067]. [43] G. Degrassi, P. Gambino and G. F. Giudice, JHEP 0012, 009 (2000) [arXiv:hep-ph/0009337]. [44] M. Carena, D. Garcia, U. Nierste and C. E. M. Wagner, Phys. Lett. B 499, 141 (2001) [arXiv:hep-ph/0010003]. [45] G. Isidori and P. Paradisi, Phys. Lett. B 639, 499 (2006) [arXiv:hep-ph/0605012]. [46] W. S. Hou, Phys. Rev. D 48, 2342 (1993). [47] A. G. Akeroyd and S. Recksiegel, J. Phys. G 29, 2311 (2003) [arXiv:hep-ph/0306037]. [48] E. Brubaker et al. [Tevatron Electroweak Working Group], arXiv:hep-ex/0608032. [49] J. S. Lee, A. Pilaftsis, M. Carena, S. Y. Choi, M. Drees, J. R. Ellis and C. E. M. Wagner, Comput. Phys. Commun. 156, 283 (2004) [arXiv:hep-ph/0307377]. [50] S. Abdullin et al., Eur. Phys. J. C 39S2, 41 (2005). [51] R. Kinnunen and S. Lehti, CMS Note 2006/075. [52] A. Kalinowski, M. Konecki and D. Kotlinski, CMS Note 2006/105. [53] S. Lehti, CMS Note 2006/101. [54] S. Gennai, S. Heinemeyer, A. Kalinowski, R. Kinnunen, S. Lethi, A. Nikitenko, G. Weiglein, arXiv:0704.0619v1 [hep-ph] [55] L. Babukhadia et al. [CDF and D0 Working Group Members], FERMILAB-PUB-03- 320-E [56] M. Carena, S. Heinemeyer, C. E. M. Wagner and G. Weiglein, Eur. Phys. J. C 26 (2003) 601 [arXiv:hep-ph/0202167]. http://arxiv.org/abs/hep-ex/0610044 http://arxiv.org/abs/hep-ex/0603003 http://arxiv.org/abs/hep-ph/0610067 http://arxiv.org/abs/hep-ph/0009337 http://arxiv.org/abs/hep-ph/0010003 http://arxiv.org/abs/hep-ph/0605012 http://arxiv.org/abs/hep-ph/0306037 http://arxiv.org/abs/hep-ex/0608032 http://arxiv.org/abs/hep-ph/0307377 http://arxiv.org/abs/0704.0619 http://arxiv.org/abs/hep-ph/0202167 Introduction Theoretical Setup Higgs Searches and Benchmark Scenarios Couplings and Masses of the Higgs Sector in the MSSM SM-like Higgs Boson Searches Non-standard Higgs Boson Searches B Physics Observables and Limits BR(Bs + -) BR(b s ) BR(Bu ) B physics constraints and Higgs searches at hadron Colliders Large to moderate Xt and small Large and small or negligible Xt Small eff Conclusions

0704.1144

Optimization in Gradient Networks

Optimization in Gradient Networks Natali Gulbahce∗ Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, MS B284, Los Alamos, NM 87545, USA (Dated: November 5, 2018) Gradient networks can be used to model the dominant structure of complex networks. Previous works have focused on random gradient networks. Here we study gradient networks that minimize jamming on substrate networks with scale-free and Erdős-Rényi structure. We introduce structural correlations and strongly reduce congestion occurring on the network by using a Monte Carlo opti- mization scheme. This optimization alters the degree distribution and other structural properties of the resulting gradient networks. These results are expected to be relevant for transport and other dynamical processes in real network systems. PACS numbers: 89.75.k, 05.45.Xt, 87.18.Sn Complex networks have been shown to offer a powerful framework for the study of dynamical processes in complex systems [1, 2]. A fundamen- tal question in this context is to identify how the structure of dominant connections influences the dynamical properties of the entire system. Gradi- ent networks have been introduced in references [4, 5] precisely to characterize the structure gov- erning transport in complex networks. Remark- ably, similar structures have been identified re- cently in the optimization of synchronizability in oscillator networks [6, 7]. Here, we consider the problem of optimization of transport and show that by using a Monte Carlo approach congestion can be reduced in complex networks. We discuss the structural correlations that emerge with op- timization and its implications for real systems. I. INTRODUCTION The efficiency of transport systems has been of interest in various fields, including physics, biology and engineer- ing. In transport processes, the item being transported usually follows the steepest descent of the underlying sur- face, e.g., water flowing down the slopes of a mountain. Flow in networks has been modeled by Toroczkai et al. [4, 5] by the introduction of local gradients on a sub- strate network. The gradient network defined on this network has provided significant insights into the domi- nant structures that provide transport efficiency. They have considered a fixed network of N nodes with a scalar potential, Vi at each node i. The gradient ∇Vi of the po- tential at each node i is a directed edge which points from i to the neighbor with the minimum potential among all the neighbors of i. Toroczkai et al have shown several properties of gradi- ent networks which we briefly summarize. An interest- ∗E-mail address:[email protected] ing topological property of the gradient network is that its in-degree distribution is scale-free for both scale-free (SF) and Erdős-Rényi (ER) substrate networks [2]. A gradient network with a non-degenerate potential distri- bution, is a group of trees, hence no loops exist in this network other than self-loops. Only this property makes the gradient networks very common in seemingly unre- lated problems, i.e., synchronization in oscillatory net- works [6, 7]. The relationship between network topology and congestion has also been investigated [4, 5] by intro- ducing a measure of congestion, the jamming coefficient. This measure involves the ratio of the number of nodes that receive at least one gradient link, Nreceive and the number of nodes that send a link. By definition, every node sends one out link, therefore the number of senders in the network, Nsend = N . The jamming coefficient is J = 1− 〈〈Nreceive/Nsend〉V 〉network . (1) The operations 〈. . .〉V and 〈. . .〉network denote the sta- tistical averaging over local potentials and networks re- spectively. Maximal congestion occurs at J = 1, and no congestion occurs when every link receives a gradient link, corresponding to J = 0. It was also found that J is independent of number of nodes N for scale free substrate network, and these networks are not prone to maximal jamming. The jamming coefficient was exten- sively studied by Park et al. [8] where they compared it for ER and SF networks with the same average degree, 〈k〉, for 2 < 〈k〉 < 200. With randomly assigned poten- tials on each node, below 〈k〉 ≈ 10, they found that ER networks are less congested than SF networks. Here we introduce a Monte Carlo optimization scheme that reduces jamming significantly and introduces struc- tural correlations into the system that are not built in. The remainder of the paper is organized as follows. In Section II we introduce the algorithm for optimizing jam- ming coefficient that is initially calculated from randomly assigned scalar values at each node. We compare the op- timized jamming values of a scale-free network and an Erdős-Rényi network for various values of average de- grees. In Section III we investigate the structure of the http://arxiv.org/abs/0704.1144v1 mailto:[email protected] optimal gradient network. In particular we study the de- gree distribution and the correlations between the degree and the potential of each node. Finally in Section IV we discuss the implications and the possible extensions of optimal gradient networks. II. OPTIMIZATION OF JAMMING Previous work [4, 5] has focused on random gradient networks where the potential on each node has a ran- domly assigned value. More generally, the potentials can be a dynamic quantity evolving in time due to perturba- tions, sources and sinks internal and external to the sys- tem of interest. Alternatively, the potentials can evolve to become correlated to the network properties such as its degree distribution. For example, consider the net- works of routers where every router has a capacity. If a router is central and highly connected, it usually has a higher capacity in order to handle the traffic en-route ef- fectively. Recently, a congestion aware routing algorithm has been introduced where the transport on the network of routers is driven by congestion-gradients [9]. Here we develop a Monte Carlo algorithm to achieve two goals: reduce jamming in the network, and observe the emerging optimal correlation between in-degree and potential of each node. For a given network and potential distribution, we redefine J in Eq. 1 as J = 1−Nreceive/N where Nsend = N by definition. The initial potential dis- tribution is chosen from a Gaussian distribution, and at each iteration the potential of a random node is mod- ified such that global congestion is reduced. We use a Metropolis algorithm [11] with the following steps: 1. Pick a node, i at random. 2. Vary Vi by δV , i.e., Vinew = Vi + δV where δV is a Gaussian random variable with variance σ2 = 1. 3. Recalculate J with Vinew. 4. Accept Vinew with probability p ∼ exp[−∆J/T ] where ∆J = Jnew − Jold. 5. Go to step 1, and repeat. The fictitious temperature, T is chosen to adjust the ac- ceptance ratio to about 40%. We perform the optimiza- tion procedure until J(t) equilibrates, i.e., at large t the time autocorrelation function of J(t) [10], CJ (t) = [〈J(t)J(0)〉 − 〈J〉2] [〈J2〉 − 〈J〉2] , (2) goes to zero. Using the described optimization algorithm, we can significantly reduce the jamming coefficient in both SF and ER networks. Following previous work [4, 8] we choose as the SF network a Barabási-Albert model [2, 3]. For this network the average connectivity is 〈k〉 = 2m where each node has at minimum m links. The network size throughout the paper is chosen to be N = 10000. For the ER networks, 〈k〉 = pN where p is the probability of having a link between any pair of nodes. We use the same 〈k〉 when comparing the two types of network by adjusting m and p. The evolution of jamming coefficient during optimization is shown in Fig. 1 as a function of algorithmic time in units of Monte Carlo steps (mcs) for 〈k〉 = 4. At t = 0 (see also inset of Fig. 1), the SF network has a higher jamming coefficient, but after roughly 50000 mcs, the SF network becomes less congested compared to the ER network. The initial and final jamming coeffi- cients for the two networks are JiSF = 0.57, JiER = 0.52, JfSF = 0.31, and JfER = 0.36, respectively. 0 40000 80000 4 10 6 10 8 10 1 10 2 10 t(mcs) 5. 5. 5. . 65. FIG. 1: Optimized jamming, J(t), as a function of time in Monte Carlo steps (mcs) for SF (•) and ER (◦) networks with the same average degree, < k >= 4 and N = 10000. As shown in the inset, SF network with random potential distribution has a higher jamming coefficient at t = 0 compared to the ER network. After optimization is completed, however, optimal SF network has a lower jamming coefficient. We define ∆Jr and ∆Jo, the difference in jamming be- tween SF and ER networks for random and optimal net- works respectively. For a given network with 〈k〉, ∆Jr is calculated initially at t = 0 and ∆Jo at topt after opti- mization is completed, for 2 ≤ 〈k〉 ≤ 126. As shown in Fig. 2, ∆Jo < 0 for 〈k〉 > 2 whereas ∆Jr > 0 indicating that SF networks have a lower jamming coefficient after optimization, a result significantly different than those for random gradient networks [8]. III. STRUCTURAL PROPERTIES OF OPTIMAL NETWORKS As shown in Fig. 1 congestion can be reduced in gra- dient networks by varying the potentials at each node. It is reasonable to expect that the obtained optimal po- tentials may also alter the structure of the gradient net- work. Next, we analyze the structural properties of op- 0 20 40 60 80 100 120 140 random optimal FIG. 2: Difference between the jamming values of SF and ER networks, ∆J = JSF − JER for various 〈k〉 for random (◦) and optimal (•) gradient networks. ∆J < 0 for all simulated 〈k〉 > 2 indicating that optimal SF networks have a lower jamming coefficient than optimal ER networks. timal gradient networks such as the degree distribution. Previously, the random gradient network of an ER sub- strate network was shown numerically and analytically [4, 5] to have an in-degree distribution of R(l) ∼ l−1. However, the SF substrate network with degree distribu- tion P (k) ∼ k−3, was shown to have a gradient degree distribution of R(l) ∼ l−3. The degree distribution of the random and optimal ER and SF networks are shown for p = 0.001 (〈k〉 = 10) and m = 3 (〈k〉 = 6) respectively in Figure 3 along with the expected scaling exponents for the random gradient networks of −1 and −3. The statistical averaging is ob- tained over 100 networks. For ER networks, the scaling region extends with higher average connectivity, however we chose to use a small 〈k〉 = 10 for which optimization is more efficient. The optimization is performed for 1 mil- lion Monte Carlo steps. The jamming values initially are 0.62, 0.71 and after optimization reduce to 0.36, 0.60 for SF and ER respectively. The in-degree distribution, R(l), for the SF network varies significantly with the optimiza- tion. The cut-off degree is reduced an order of magnitude (from roughly 100 to 10) compared to the random one, and the scaling is now steeper. On the other hand, the ER network does not show a major change in the degree distribution. Next, we analyze the probability distribution of poten- tials for nodes in the substrate networks with degree k, before and after the optimization to observe any degree- potential correlations. The results are shown in Fig. 4 for the SF network (m = 3). To reduce noise in the data especially for large values of k, the degrees are binned into four groups: 3 < k ≤ 10, 10 < k ≤ 30, 30 < k ≤ 50, 50 < k ≤ 100. Before optimization with initial random Gaussian potentials within range [-4, 4] , each set has the same probability distribution P (Vi) as shown in the inset of Fig 4. This behavior is expected as no correlation was 0.01 0.01 1000 FIG. 3: In-degree distributions, R(l), for gradient networks of (a) SF and (b) ER networks with m = 3 and p = 0.001 respec- tively before (◦) and after (•) optimization of jamming. The dashed lined indicates the scaling of R(l) ∼ lα from Ref. [4, 5], where α = −3 and −1 for random SF and ER networks re- spectively. The scaling exponent α is not a good fit for opti- mal SF data but a good fit for optimal ER network. built in between the degree of a node and its potential. After the optimization however, the range of the poten- tials has broadened significantly, and the nodes with high degree have accumulated very large potentials. With the improved jamming coefficient, it is natural to expect some correlation to emerge between the poten- tial of the node and its degree. If the node has a large degree and a small potential, this node will be preferred by most of its neighbors for sending an out link, and thus this will contribute to higher jamming. However if the potential is large, the neighboring nodes will not pre- fer this highly connected node and thus not increase the jamming. This intuitive observation implies the possi- bility of obtaining a reduced congestion by starting with potentials that are inversely correlated with the degree of each node on the substrate network. We tested this case for the SF network (m = 3) with a correlated po- tential distribution, Vi = r/ki at node i with degree ki V (k)i V (k) 0.04 0.08 0.12 −40 −20 0 20 40 0.00 −4 −2 0 2 4 FIG. 4: Probability distribution of the potentials of nodes with degree k for SF network with m = 3 before (inset) and after optimization. The degrees are binned to indicate the correlation between degree of a node and its potential: 3 < k ≤ 10 (+), 10 < k ≤ 30 (�), 30 < k ≤ 50 (•), 50 < k ≤ 100 (◦). Before the optimization P (Vi) do not show any correlations with k, however after the optimization, nodes with high degree, get the large values of potential which facilitates reduced jamming. where r is a random number chosen from a uniform dis- tribution. This assignment with correlations built-in did not make the jamming coefficient lower. On the contrary it was higher, J = 0.77, than the value without degree correlation, J = 0.62. An interesting observation that Fig. 4 provides is that nodes with small degree carry potentials distributed over a large range [−40, 40]. For example, nodes with degrees, 3 < k < 10 have a roughly Gaussian potential distribu- tion. For higher k, the distribution narrows down and shifts toward larger values. For nodes with 50 < k < 100, all nodes have large potentials, within range [0, 40]. This observation might explain why in the test case the jam- ming was actually higher when the degree was correlated with the potential. In that case, we only assigned low po- tentials to low degree nodes which still yields congestion much higher than the optimal one. An analytical for- mulation that distributes the potentials according to its degree mimicking the transitive behavior in Fig. 4 seems possible, but is beyond the scope of this paper. IV. CONCLUSION We have introduced a Monte Carlo method to optimize congestion in random gradient networks. Previously the potentials have been assigned randomly and was shown that ER networks had lower jamming coefficient below 〈k〉 = 10 than SF networks with the same connectivity [8]. This was puzzling as the connectivity commonly ob- served in natural and man-made networks [2] is usually in this range, but tends to be scale-free, and in scale-free networks jamming is independent of N . With the Monte Carlo based optimization scheme we optimized jamming by varying the potentials so that optimal congestion was achieved. We found that optimal SF networks have lower congestion factor for 〈k〉 > 2. This reduced congestion is the result of a complex correlation between the degree and the potential of a node. We found that nodes with large degrees in the substrate network get large positive values whereas nodes with small degrees get a Gaussian like distribution of potentials. Throughout the paper we have used the definition of jamming introduced in Ref. [4] for a substrate network with unweighted links. A natural extension of this work for generality is to assign weights to links and redefine the jamming coefficient accordingly. A possible definition is wij−ci ci where i = 1 · · ·N , j is the number of neighboring links node i has, ci is the capacity of node i, wij is the weight of the incoming link from j to i, and the operation [x] = 0 if x < 0. If the weights and capacity of all nodes are 1, then this definition of J reduces to the one in Eq. 1 without the averaging. With this definition and the optimization method introduced in the paper, it is possible to study real world networks and get insights to the dominant structures of transport for these systems. Acknowledgments The author thanks Adilson E. Motter for useful dis- cussions and suggestions, Gregory Johnson and Frank Alexander for the careful reading of the manuscript. This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No.DE-AC52-06NA25396 and supported by the DOE Office of Science ASCR Program in Applied Math- ematics Research. [1] A. E. Motter, M. A. Matias, J. Kurths, and E. Ott, Phys- ica D 224, vii (2006). [2] M. E. J. Newman, A.-L. Barabási and D. J. Watts (Edi- tors), The Structure and Dynamics of Networks (Prince- ton University Press, 2006). [3] A.-L. Barabási and R. Albert, Science 286, 509 (1999). [4] Z. Toroczkai and K. E. Bassler, Nature 428, 716 (2004). [5] Z. Toroczkai, B. Kozma, K. E. Bassler, N. W. Hengart- ner, and G. Korniss, e-print cond-mat/0408262. [6] T. Nishikawa and A. E. Motter Phys. Rev. E 73, 065106 (2006). [7] T. Nishikawa and A. E. Motter, Physica D 224, 77 (2006). [8] K. Park, Y.-C. Lai, L. Zhao, and N. Ye, Phys. Rev. E 71, 065105 (2005). [9] B. Danila , Y. Yu, S. Earl, J. A. Marsh, Z. Toroczkai, and K. E. Bassler, Phys. Rev. E 74, 046114 (2006). [10] H. Gould, J. Tobochnik and W. Christian, Introduc- tion to Computer Simulations Methods, Addison-Wesley (2006). [11] M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press (1999). http://arxiv.org/abs/cond-mat/0408262

0704.1145

Fermionic construction of partition function for multi-matrix models and multi-component TL hierarchy

CRM-xxxx (2005) nlin.SI/05xxxxxx Fermionic construction of partition function for multi-matrix models and multi-component TL hierarchy1 J. Harnad†‡2 and A. Yu. Orlov⋆3 † Centre de recherches mathématiques, Université de Montréal C. P. 6128, succ. centre ville, Montréal, Québec, Canada H3C 3J7 ‡ Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montréal, Québec, Canada H4B 1R6 ⋆ Nonlinear Wave Processes Laboratory, Oceanology Institute, 36 Nakhimovskii Prospect Moscow 117851, Russia Abstract We use p-component fermions (p = 2, 3, . . . ) to present (2p−2)N -fold integrals as a fermionic expectation value. This yields fermionic representation for various (2p−2)-matrix models. Links with the p-component KP hierarchy and also with the p-component TL hierarchy are discussed. We show that the set of all (but two) flows of p-component TL changes standard matrix models to new ones. 1Work of (J.H.) supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds FCAR du Québec; that of (A.O.) by the Russian Academy of Science program “Fundamental Methods in Nonlinear Dynamics” and RFBR grant No 05-01-00498. [email protected] 3 [email protected] http://arxiv.org/abs/0704.1145v1 1 Introduction Let dµα(x, y) be a set of measures (in general, complex), supported on a finite set of products of curves in the complex x and y planes. Let ρα, α = 2, . . . , p− 1, be a set of functions in two variables. Let x(α) = (x 1 , . . . , x N ) and y (α) = (y 1 , . . . , y N ), α = 1, . . . , p are two sets of variables, where x(p) and y(1) are fixed by i = y i = N − i, i = 1, . . . , N (1.1) We shall use the following notation dµα(x (α), y(α+1)) := dµα(x i , y (α+1) i ) (1.2) We consider the following integral over (2p− 2)N variables x(α) = (x 1 , . . . , x N ) and y(α+1) = (y (α+1) 1 , . . . , y (α+1) N ), α = 1, . . . , p− 1: (α), x(α)) dµα(x (α), y(α+1)) (1.3) where (1), x(1)) = det i,j=1,...,N i − x j ) =: ∆N (x (1)), (p), x(p)) = det i,j=1,...,N i − y j ) =: ∆N(y are Vandermonde determinants, and where (α), x(α)) = det i , x i,j=1,...,N , α = 2, . . . , p− 1 (1.4) Developing each detρα into N ! monomial terms (each is labeled by an element of the permutation group SN), and, for given choice of the element of the permutation group, say σ, using the change of variables inside of each N -fold integral to the left (namely, x(β) → σ(x(β)), y(β) → σ(y(β)) for all β ≤ α), then, using the anti-symmetry of ∆N (x (which is the integrand of the very left N -fold integral), one finds that each term of the mentioned development yields the same contribution. This is a standard way to re-write (1.3) as (N !)2p−2 . . . ∆N (x dµ1(x i , y i )ρ2(y i , x . . . i , x i )dµ2(x i , y · · · . . . ρp−1(y (p−1) i , x (p−1) i )dµp−2(x (p−2) i , y (p−1) . . . dµp−1(x (p−1) i , y i )∆N(y Integrals (1.3) may be related to the so-called determinantal ensembles [6]. For special choice of measures dµα and functions ρα, integrals (1.3) arose in the study of multi-matrix models, where matrices M1,M2,M3, . . . ,M2p−2 with eigenvalues respectively equal to the sets {x i , i = 1, . . . , N},{y i , i = 1, . . . , N},{x i , i = 1, . . . , N},. . . ,{y i , i = 1, . . . , N}, are coupled in an open chain. It occurs in case when one can reduce the integration over matrix entries to the integrals over eigenvalues of each matrix (for these topic see [16], [17] and Appendices to [18], [1]). Depending on dµα and functions ρα, these are models of normal matrices, and certain models of ran- dom Hermitian (anti-Hermitian) matrices and certain models of random unitary matrices, see [4], [5], [11], [3], [8], [16], [17], together with discrete versions of these matrix mod- els [18]. For instance, to obtain the partition function for the model of random N by N Her- mitian matrices, M1, . . . ,M2p−2, coupled in a chain, P2p−2 Vk(Mk)+Tr(c1M1M2+···+c2p−3M2p−3M2p−2) one takes dµα(x, y) = e c2α−1xy+V2α−1(x)+V2α(y), α = 1, . . . , p− 1 (1.5) ρα(x, y) = e c2αxy, α = 2, . . . , p− 1 (1.6) Then x i , i = 1, . . . , N , are eigenvalues of Hermitian matrices with odd numbers, say M2α−1, while y i , i = 1, . . . , N , are eigenvalues of M2α, α = 1, . . . , p − 1. For future purpose, let us use the obvious freedom to re-write dµα and ρα in form dµ1(x, y) → e c1xy+V1(x), dµp−1(x, y) → e c2p−3xy+V2p−2(x), (1.7) dµα(x, y) → e c2α−1xy, α = 2, . . . , p− 2, ρα(x, y) → e c2αxy+V(x)+V(y), α = 2, . . . , p− 1 (1.8) In the present paper we have two tasks. First, we equate the integral (1.3) to the fermionic vacuum expectation value. Here we use the so-called p-component fermions. This may be considered as a continuation of of the work [11]. Second, as a continuation of [7], we relate ZN to the coupled p-component KP hier- archies, or, the same to the p component TL hierarchy. For this purpose we consider the following deformation of the first and the last measures dµ1(x, y) → dµ1(x, y|t (1), n, t̄(1)) := xn1eV (x,t (α))+V (x−1,t̄(1))dµ1(x, y), (1.9) dµp−1(x, y) → dµp−1(x, y|t (p), n, t̄(p)) := ynpeV (y,t (p))+V (y−1,t̄(p))dµp−1(x, y) (1.10) V (x, t(α)) = xmt(α)m , V (x −1, t̄(α)) = x−mt̄(α)m , α = 1, p, (1.11) and also the following deformations of functions ρα, α = 2, . . . , p− 1, ρα(x, y) → τn(α)(t (α) + [x], t̄(α) + [y]), α = 2, . . . , p− 1, (1.12) where in the right hand side we have tau functions (labeled by α = 2, . . . , p − 1) of the one-component TL hierarchy, and where +[x] and +[y] denote the so-called Miwa shift of a TL (a one-component TL) higher times, details are written down below. The deformation (1.9)-(1.12) relates integrals (1.3) to the coupled p-component KP hierarchies. If in (1.3) we take the deformed measures and the deformed functions ρα as described above, then, ZN turns out to be a certain tau function of coupled p-component KP, or the same, p-component TL hierarchy, where the sets of complex numbers t(α) = 1 , t 2 , . . . ), t̄ (α) = (t̄ 1 , t̄ 2 , . . . ), and the set of integers n (α), α = 1, . . . , p, play the role of higher p-component TL times. For the sake of brevity we shall also use the notations t = (t(1), . . . , t(p)) and t̄ = (t̄(1), . . . , t̄(p)). Important to mark, that the deformation (1.9), (1.10) and (1.12) seems do not keep the form (1.7)-(1.8). In our case the interaction ec2αM2αM2α+1+V2α(M2α)+V2α+1(M2α+1) is replaced by arbitrary chosen one-component TL tau function (1.12) where x = x(α) is the collection of eigenvalues of the matrixM2α while y = y (α) is the collection of eigenvalues of the matrix M2α+1. Let us note that one can consider (p− 1)N -fold integrals if he specifies the measures dµα(x, y) to be proportional to Dirac delta function which equate x to a function of y (it may be δ(x− y)). The present paper is a part of series of papers devoted to fermionic approaches to multi-fold integrals, see [12], [1], [2]. Let us mark that our fermionic constructions of papers [1], [2] and of the present paper are different from what was considered in [11] and also different of [12]. 1.1 Free fermions LetA be the complex Clifford algebra over C generated by charged free fermions {fi, f̄i}i∈Z, satisfying the anticommutation relations [fi, fj]+ = [f̄i, f̄j]+ = 0, [fi, f̄j ]+ = δij . (1.13) Any element of the linear part W := (⊕m∈ZCfm)⊕ ⊕m∈ZCf̄m (1.14) will be referred to as a free fermion. We also introduce the fermionic free fields f(x) := k, f̄(y) := −k−1, (1.15) which may be viewed as generating functions for the fj , f̄j’s. This Clifford algebra has a standard Fock space representation defined as follows. Define the complementary, totally null (with respect to the underlying quadratic form) and mutually dual subspaces Wan := (⊕m<0Cfm)⊕ ⊕m≥0Cf̄m , Wcr := (⊕m≥0Cfm)⊕ ⊕m<0Cf̄m , (1.16) and consider the left and right A-modules F := A/AWan, F̄ := WcrA\A. (1.17) These are cyclic A-modules generated by the vectors |0〉 = 1 mod AWan, 〈0| = 1 mod WcrA, (1.18) respectively, with the properties fm|0〉 = 0 (m < 0), f̄m|0〉 = 0 (m ≥ 0), 〈0|fm = 0 (m ≥ 0), 〈0|f̄m = 0 (m < 0). (1.19) The Fock spaces F and F̄ are mutually dual, with the hermitian pairing defined via the linear form 〈0||0〉 on A called the vacuum expectation value. This is determined by 〈0|1|0〉 = 1; 〈0|fmf̄m|0〉 = 1, m < 0; 〈0|f̄mfm|0〉 = 1, m ≥ 0, (1.20) 〈0|fn|0〉 = 〈0|f̄n|0〉 = 〈0|fmfn|0〉 = 〈0|f̄mf̄n|0〉 = 0; 〈0|fmf̄n|0〉 = 0, m 6= n, (1.21) together with the Wick theorem which implies, for any finite set of elements {wk ∈ W}, 〈0|w1 · · ·w2n+1|0〉 = 0, 〈0|w1 · · ·w2n|0〉 = σ∈S2n sgnσ〈0|wσ(1)wσ(2)|0〉 · · · 〈0|wσ(2n−1)wσ(2n)|0〉. (1.22) Here σ runs over permutations for which σ(1) < σ(2), . . . , σ(2n− 1) < σ(2n) and σ(1) < σ(3) < · · · < σ(2n− 1). Now let {wi}i=1,...,N , be linear combinations of the fj’s only, j ∈ Z, and {w̄i}i=1,...,N linear combinations of the f̄j ’s, j ∈ Z. Then(1.22) implies 〈0|w1 · · ·wN w̄N · · · w̄1|0〉 = det (〈0|wiw̄j|0〉) |i,j=1,...,N (1.23) Following refs. [9], [10], for all N ∈ Z, we also introduce the states 〈N | := 〈0|CN (1.24) where CN := f̄0 · · · f̄ N−1 if N > 0 (1.25) CN := f−1 · · · fN if N < 0 (1.26) CN := 1 if N = 0 (1.27) |N〉 := C̄N |0〉 (1.28) where C̄N := fN−1 · · · f0 if N > 0 (1.29) C̄N := f̄N · · · f̄−1 if N < 0 (1.30) C̄N := 1 if N = 0 (1.31) The states (1.24) and (1.28) are referred to as the left and right charged vacuum vectors, respectively, with charge N . In what follows we use the notational convention ∆N(x) = det (x i )|i,k=1,...,N (N > 0), ∆0(x) = 1, ∆N(x) = 0 (N < 0). (1.32) From the relations 〈0|f̄N−kf(xi)|0〉 = x i , 〈0|f−N+k−1f̄(yi)|0〉 = y i , k = 1, 2, . . . , (1.33) and (1.23), it follows that 〈N |f(x1) · · · f(xn)|0〉 = δn,N∆N(x), N ∈ Z, (1.34) 〈−N |f̄(y1) · · · f̄(yn)|0〉 = δn,N∆N(y), N ∈ Z. (1.35) Following [9], [10] we consider ĜL∞ element g = eh, h = hi,jfif̄j , hi,j ∈ C (1.36) Via the conjugation, (·) → g(·)g−1 , each g ∈ ĜL∞ acts on the spaces (⊕m∈ZCfm) and ⊕m∈ZCf̄m as linear transformations [9], [10]. We suppose that the following factorization condition is valid: g = g+g−, 〈0|g + = 〈0|, g−|0〉 = |0〉, (1.37) where g+, g− ∈ ĜL∞. Remark. Though, the property (1.37) is valid for a rather wide class of (1.36) (which includes all cases when the sum in (1.36) is finite) , however, we do not know the general theorem providing sufficient and necessary conditions to have this property in case the sum in (1.36) is infinite. Consider 〈0|vN · · · v1gv̄1 · · · v̄N |0〉, where each vi ∈ (⊕m∈ZCfm) and each v̄i ∈ ⊕m∈ZCf̄m , i = 1, . . . , N . Denoting wi = −1vig+ ∈ (⊕m∈ZCfm) and w̄i = (g−)v̄i(g−) ⊕m∈ZCf̄m we have 〈0|vN · · · v1gv̄1 · · · v̄N |0〉 = 〈0|wN · · ·w1w̄1 · · · w̄N |0〉 = det〈0|wiw̄j|0〉|i,j=1,...,N where the second equality is due to the Wick theorem (1.23). Thus 〈0|vN · · · v1gv̄1 · · · v̄N |0〉 = det〈0|vigv̄j · · · |0〉|i,j=1,...,N (1.38) 1.2 Multi-component fermions One obtains the so-called p-component fermion formalism by re-numerating the above free fermions (1.13) as follows f (α)n := fpn+α−1 , f̄ n := f̄pn+α−1 , (1.39) f (α)(z) := k , f̄ (α)(z) := z−k−1f̄ k , (1.40) where α = 1, . . . , p. From (1.13) we obviously have [f (α)n , f m ]+ = [f̄ n , f̄ m ]+ = 0, [f n , f̄ m ]+ = δα,βδn,m. (1.41) Right and left vacuum vectors are respectively defined |0, . . . , 0 ︸ ︷︷ ︸ 〉 := |0〉, 〈0, . . . , 0 ︸ ︷︷ ︸ | := 〈0| (1.42) where |0〉 and 〈0| were introduced in (1.19). As it follows from (1.19) f (α)m |0, . . . , 0〉 = 0 (m < 0), f̄ m |0, . . . , 0〉 = 0 (m ≥ 0), (1.43) 〈0, . . . , 0|f (α)m = 0 (m ≥ 0), 〈0, . . . , 0|f̄ m = 0 (m < 0). (1.44) We also introduce the states 〈n(1), . . . , n(p)| := 〈0, 0|Cn(1) · · ·Cn(p) (1.45) where Cn(α) := f̄ 0 · · · f̄ n(α)−1 if n(α) > 0 (1.46) Cn(α) := f −1 · · · f if n(α) < 0 (1.47) Cn(α) := 1 if n (α) = 0 (1.48) |n(1), . . . , n(p)〉 := C̄n(p) · · · C̄n(1)|0, 0〉 (1.49) where C̄n(α) := f n(α)−1 · · · f 0 if n (α) > 0 (1.50) C̄n(α) := f̄ · · · f̄ −1 if n (α) < 0 (1.51) C̄n(α) := 1 if n (α) = 0 (1.52) Let us call (1.45) and (1.49) respectively left and right charged vacuum vectors with the charge (n(1), . . . , n(p)). We easily verify that f (α)m |∗, n (α), ∗〉 = 0 (m < n(α)), f̄ (1)m |∗, n (α), ∗〉 = 0 (m ≥ n(α)), (1.53) 〈∗, n(α), ∗|f (α)m = 0 (m ≥ n (α)), 〈∗, n(α), ∗|f̄ (α)m = 0 (m < n (α)), (1.54) where ∗ serve for irrelevant components in vacuum vectors. Remark 1.1. For calculations we use the Wick theorem in form (1.23). There are two ways to do it: (1) The first one is to use (1.23) just remembering that p-component fermions are composed of usual ones, see (1.39). (2) The second way is to use formula (1.23) separately for each component. Namely, to calculate the vacuum expectation value of an operator O, first, we present it in form i · · ·O i (1.55) 〈0|O|0〉 = i · · ·O i |0〉 = i |0〉 · · · 〈0|O i |0〉 (1.56) where the Wick theorem in form (1.23) is applied to each of 〈0, . . . , 0|O i |0, . . . , 0〉. 2 Fermionic representation for ZN Consider the element of the Clifford algebra of the following form g = eA1g2e A2g3 · · · e Ap−2gp−1e Ap−1 , (2.1) where f (α)(x)f̄ (α+1)(y)dµα(x, y), α = 1, . . . , p− 1, (2.2) with measure dµα(x, y), which we do not specify. In (2.1) gα = e hα , hα = i,j f j , h i,j ∈ C, (2.3) so that we have i gα = gαf i , α 6= β, i ∈ Z (2.4) We also suppose that each gα = e h(α), α = 2, . . . , p − 1, may be factorized into ĜL elements g+α and g α as follows (see (1.37)) gα = g α , 〈∗, 0̂, ∗|g+α = 〈∗, 0̂, ∗|, g−α |∗, 0̂, ∗〉 = |∗, 0̂, ∗〉 (2.5) where by ∗ we denote irrelevant components of a vacuum vector (different from the com- ponent α marked by hats). Now, let us notice that by (1.38) we have 〈0|f̄ (α)(y1) · · · f̄ (α)(yN)gαf (α)(xN ) · · ·f (α)(x1)|0〉 = det 〈0|f̄ (α)(yi)gαf (α)(xj)|0〉 i,j=1,...,N (2.6) Now let us prove, that for special choice of functions ρα, α = 2, . . . , p− 1, namely, for 〈0|f̄ (α)(y)gαf (α)(x)|0〉 = ρα(y, x) (2.7) we have (N !)p−1〈N, 0, . . . , 0,−N |g|0, 0, . . . , 0, 0〉 = ZN (2.8) Indeed, to get a non-vanishing expectation value in the left hand side, we have to pick up only N -th term, , in the Taylor series for eA1 (this is because eA1 is the only factor of g which contains the first component fermions, and the matrix element 〈N, ∗|An1 |0, ∗〉 ≡ 0 until n = N). Using the known formula (1.34), we obtain, that the left hand side of (2.8) is equal to the integral dµ1(x 1 , y 1 ) . . . dµ1(x N , y N )∆N (x (1))R1, R1 = 〈∗, 0, . . . , 0,−N |f̄ (2)(y 1 ) · · · f̄ (2)(y N )g2 · · · |∗, 0, . . . , 0〉 where we put ∗ on the first place of the left and right vacuum vectors to show that we forget about the first component fermions. This is the first step. Then, we have to pick up only N -th term, , when developing the next factor eA2. Otherwise, the vacuum expectation values of the second component fermions vanishes. This is because the second component fermions are in presence only in eA1 , g2 and in e factors of g, and the g2 is a sum of monomials, each of which contains equal number of f (2) and f̄ (2) fermions, while eA1 contains only f (2) , and eA2 contains only f̄ (2) fermions. Thus, second component fermions yields the expression f̄ (2)(y 1 ) · · · f̄ (2)(y N )g2f (2)(x 1 ) · · · f (2)(x which should be integrated with measures i=1 dµ(∗, y i )dµ(x i , ∗), and then substituted inside 〈N, 0, . . . , 0,−N | and |0, 0, . . . , 0, 0〉. Denoting 〈0|f̄ (2)(y 1 ) · · · f̄ (2)(y N )g2f (2)(x 1 ) · · · f (2)(x N )|0〉 = ̺2(y (2), x(2)) (2.9) (which, by (2.6), is equal to detρ2(y i , x j )) we obtain that the l.h.s of (2.8) is equal to the integral ∫ dµ1(x 1 , y 1 ) . . . dµ1(x N , y N )∆N (x dµ2(x 1 , y 1 ) . . . dµ2(x N , y N )̺3(y (3), x(3))R2, R2 = 〈∗, ∗, 0, . . . , 0,−N |f̄ (2)(y 1 ) · · · f̄ (2)(y N )g2 · · · |∗, ∗, 0, . . . , 0〉 where we put ∗ on the first and second places of the left and right vacuum vectors to show that we forget about the first and the second component fermions. This is the second step. Then, it is easy to see that each exponential eAα should be replaced by their N -th Taylor term, otherwise the l.h.s. of (2.8) vanishes, it means we have 〈N, 0, . . . , 0,−N |g|0, 0, . . . , 0, 0〉 = (N !)p−1 〈N, 0, . . . , 0,−N |AN1 g2A 2 · · · gp−1A p−1|0, 0, . . . , 0, 0〉 (2.10) Continuing excluding step by step third- forth- and so on component fermions, and, on the last step, using the known formula (1.35), we obtain that (2.10) is equal to (1.3). At last we want to make the following remark Remark 2.1. Insert additional factors to (2.1) as follows g = eA1g2e A2g3 · · · e Ap−2gp−1e Ap−1 → g := e A1g2e A2g3 · · · e Ap−2gp−1e Ap−1gpg1e Ap (2.11) where gα = e i,j h j , Aα = f (α)(x)f̄ (α+1)(y)dµα(x, y), α = 1, . . . , p (2.12) where f̄ (p+1)(y) ≡ f̄ (1)(y). (Thus we add g1, gp and dµp(x, y) to our collection of data, gα, α = 2, . . . , p− 1 and dµα(x, y), α = 1, . . . , p− 1). Then 〈0, 0, . . . , 0, 0|g |0, 0, . . . , 0, 0〉 = (2.13) where cN are certain numbers and each Z is the following integral over 2pN variables x(α) = 1 , . . . , x ) and y(α) = (y 1 , . . . , y ), α = 1, . . . , p: (α), x(α)) dµα(x (α), y(α+1)), y(p+1) ≡ y(1) (2.14) (notice that variables y(1) and x(p) are not fixed by (1.1)) In (2.14) dµα(x (α), y(α+1)) (α = 1, . . . , p − 1) are defined by (1.2) and dµp(x (p), y(1)) := dµp(x i , y i ), (2.15) and each ̺α(y (α), x(α)) is defined by (2.6)-(2.7), where now α = 1, . . . , p. Sums (2.13) and their relation to the grand partition function of closed chains of coupled random matrices and to integrable equations will be considered in a forthcoming paper. 3 Deformation of measure and relations to integrable hierarchies The described deformation dµ1(x, y) → dµ1(x, y|t (1), n, t̄(1)) := xn1eV (x,t (α))+V (x−1,t̄(1)))dµ1(x, y), (3.1) dµp−1(x, y) → dµp−1(x, y|t (p), n, t̄(p)) := ynpeV (y,t (p))+V (y−1,t̄(p)))dµp−1(x, y) (3.2) V (x, t(α)) = xmt(α)m , V (x −1, t̄(α)) = x−mt̄(α)m , α = 1, p, (3.3) Then, it is quite known fact that in this case ZN = τN (t (1), t̄(p)), where τN (t (1), t̄(p)) is a tau function of the (one-component) TL hierarchy. Indeed, one just re-writes (1.3) as 2N -fold integral with a modified measure dµmod (the latter depends on the choice of ρα) : dµmod(x i , y ,t̄(1))+V (y ,t̄(p))eV (x ,t(1))+V (y ,t(p))∆N(x (1))∆N(y Moreover, as a function of t(1), t(p), t̄(1), t̄(p), the integral ZN(t (1), t(p), n1, np, t̄ (1), t̄(p)) is a tau function of the coupled two-component KP, or, the same, a tau function of the two-component TL hierarchy, see [1]. Now, in addition, we consider the following deformations of functions ρα, α = 2, . . . , p− ρα(x, y) → ρα(x, y|t (α), n(α), t̄(α)) := (3.4) 〈n(α)|eH (α)(t(α))f̄ (α)(y 1 ) · · · f̄ (α)(y N )gαf (α)(x 1 ) · · ·f (α)(x H̄(α)(t̄(α))|n(α)〉 (3.5) where t(α) = (t 1 , t 1 , . . . ) and t̄ (α) = (t̄ 1 , t̄ 2 , . . . ) are the deformation parameters, and where the “Hamiltonians” H , k = ±1,±2, . . . , are defined by H(α)(t(α)) = , H̄(α)(t̄(α)) = f (α)n f̄ n+k (3.6) (for future purpose we define them for α = 1, . . . , p range). Let us note that the expectation value (3.5), by definition [9], [10], [15], is a tau function of one component TL and in our case may be denoted by τn(α)(t (α) + [x(α)], t̄(α) + [y(α)]) where [x] := , . . . Now let us prove that the combination of deformations (3.1)-(3.2) and (3.4) is equiv- alent to the replacement 〈N, 0, . . . , 0,−N |g|0, 0, . . . , 0, 0〉 → τN(t,n, t̄) := 〈N + n (1), n(2), . . . , n(p−1),−N − n(p)|eH(t)geH̄(t̄)|n(1), n(2), . . . , n(p−1),−n(p)〉, (3.7) where H(t) = , H̄(t̄) = the “Hamiltonians” H k were defined earlier by (3.6). Proof. Indeed, we have that each terms of type (2.9), namely, each 〈0|f̄ (α)(y 1 ) · · · f̄ (α)(y N )gαf (α)(x 1 ) · · ·f (α)(x N )|0〉 = ̺(y (α), x(α)), α = 2, . . . , p− 1, (3.8) is now replaced by 〈n(α)|eH (α)(t(α))f̄ (α)(y 1 ) · · · f̄ (α)(y N )gαf (α)(x 1 ) · · ·f (α)(x H̄(α)(t̄(α))|n(α)〉 (3.9) which is, by definition [9], [10], [15], a tau function of one component TL. Due to (1.38) it is equal to detρα(xi, yj|t,n, t̄) where ρα(xi, yj|t,n, t̄) := 〈n (α)|eH (α)(t(α))f̄ (α)(y i )gαf (α)(x H̄(α)(t̄(α))|n(α)〉 As for α = 1, p we have 〈N, ∗|eH (1)(t(1))f(x1) · · ·f(xN)e H̄(1)(t̄(1))|0, ∗〉 = a1∆N (x)e i=1 V (xi,t (1))+V (x ,t̄(1)), (3.10) 〈∗, 0|eH̄ (p)(t̄(p))f̄(yN) · · · f̄(y1)e H̄(p)(t̄(p))|∗,−N〉 = ap∆N (y)e i=1 V (yi,t (p))+V (y ,t̄(p)), (3.11) where aα = e k=1 kt k , which contribute to the deformation respectively of dµ1 and of dµp−1. The end of proof. Thus, we obtain that the deformation of functions ρα, α = 2, . . . , p − 1, and also of dµ1, dµp−1 reduce to the fact that ZN is equal to τN(t,n, t̄). It is known [9], [10], [14] that thus constructed τN (t,n, t̄) is a tau function of the coupled p-component KP hierarchy, or, the same, p-component TL hierarchy. 4 Conclusion We equate the multi-integral (1.3) to the fermionic expectation value (2.8). On the one hand we hope that the fermionic representation allows to evaluate different magnitudes re- lated to the matrix models, like spectral determinants (compare with [2]), or perturbative series generalizing [12], [13]. On the other hand it allows to incorporate the study of these integrals and related multi-matrix models to the study of multi-component integrable hierarchies Acknowledgements The authors would like to thank T. Shiota and J. van de Leur for helpful discussions, and (A.O.) thanks A. Odzijevicz for kind hospitality during his stay in Bialystok in June 2005, which helped stimulate ideas leading to this work. References [1] J. Harnad and A. Yu. Orlov, ”Fermionic construction of partition functions for two- matrix models and perturbative Schur function expansions”, ”Random Matrices, Random Processes and Integrable Systems”, Special Issue of J. Phys. A, based on the Centre de recherches mathematiques short program, Montreal, June 20-July 8, 2005; Journal of Physics A39 (2006) 8783-8810; arXiv math-phys/0512056 [2] J. Harnad and A.Yu. Orlov, “Fermionic approach to the evaluation of integrals of rational symmetric functions”, submitted to Theoretical and Mathematical Physics [3] B. Eynard, M.L. Mehta, “Matrices coupled in a chain: eigenvalue correlations”, J. Phys. A: Math. Gen. 31, 4449 (1998). [4] Mehta, M. L., Random Matrices, 2nd edition (Academic, San Diego, 1991). [5] C. Itzykson and J.-B. Zuber, “The planar approximation. II”, J. Math. Phys. 21, 411–421 (1980). [6] A. Borodin, A. Soshnikov, ”Janossy Densities I. Determinantal Ensembles” math-ph/0212063 [7] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A.Yu.Orlov, “Matrix Models of 2D Gravity and Toda Theory”, Nucl. Phys. B 357, 565–618 (1991). [8] M. Bertola, B. Eynard and J. Harnad, “Duality, Biorthogonal Polynomials and Multi–Matrix Models”, Commun. Math. Phys. 229, 73–120 (2002). [9] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, “Transformation groups for soliton equations”, in Nonlinear integrable systems-classical theory and quantum theory, eds. M. Jimbo, and T. Miwa, World Scientific, pp. 39–120, (1983). http://arxiv.org/abs/math-phys/0512056 http://arxiv.org/abs/math-ph/0212063 [10] Jimbo,M and Miwa,T,: Solitons and Infinite Dimensional Lie Algebras, Publ. RIMS Kyoto Univ. 19, 943–1001 (1983) [11] Zabrodin, A., Kharchev, S., Mironov, A., Marshakov, A. and Orlov, A.: Matrix Models among Integrable Theories: Forced Hierarchies and Operator Formalism, Nuclear Physics B 366 (1991) 569–601 [12] J. Harnad and A.Yu. Orlov, “Scalar products of symmetric functions and matrix integrals”, Theoretical and Mathematical Physics 137, 1676–1690, (2003). [13] J. Harnad and A. Yu. Orlov, “Matrix integrals as Borel sums of Schur function expansions”, In: Symmetries and Perturbation theory SPT2002, eds. S. Abenda and G. Gaeta, World Scientific, Singapore, (2003); arXiv nlin.SI/0209035 [14] V.G. Kac and J.W. van de Leur. The n-component KP hierarchy and representa- tion theory, in Important developments in soliton theory, eds. A.S. Fokas and V. E. Zakharov, Springer Series in Nonlinear Dynamics (1993), 302 – 343 [15] K. Ueno and K. Takasaki, Adv. Stud. Pure Math. 4, 1-95, (1984). [16] Zinn-Justin,P. and Zuber,J.-B.: On some integrals over the U(N) unitary group and their large limit, arXiv math-ph/0209019 [17] A. Yu. Orlov, ”New Solvable Matrix Integrals”, International Journal of Mathemat- ical Physycs A, Vol. 19, Supplement (2004) 276-293; arXiv nlin. SI/0209063 [18] A.Yu.Orlov and T.Shiota, ”Schur function expansion for normal matrix model and associated discrete matrix models”, Physics Letters A 343 (2005) 384-396; more detailed version: arXiv math-ph/0501017 http://arxiv.org/abs/nlin/0209035 http://arxiv.org/abs/math-ph/0209019 http://arxiv.org/abs/math-ph/0501017 Introduction Free fermions Multi-component fermions Fermionic representation for ZN Deformation of measure and relations to integrable hierarchies Conclusion

0704.1146

The Plasma Puddle as a Perturbative Black Hole

arXiv:0704.1146v2 [hep-th] 7 Jun 2007 Preprint typeset in JHEP style - HYPER VERSION The Plasma Puddle as a Perturbative Black Hole Clifford Cheung and Jared Kaplan Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138 E-mail: [email protected], [email protected] Abstract:We argue that the weak coupling regime of a large N gauge theory in the Higgs phase contains black hole-like objects. These so-called “plasma puddles” are meta-stable lumps of hot plasma lying in locally un-Higgsed regions of space. They decay via O(1/N) thermal radiation and, perhaps surprisingly, absorb all incident matter. We show that an incident particle of energy E striking the plasma puddle will shower into an enormous number of decay products whose multiplicity grows linearly with E, and whose average energy is independent of E. Once these ultra-soft particles reach the interior they are thermalized by the plasma within, and so the object appears “black.” We determine some gross properties like the size and temperature of the the plasma puddle in terms of funda- mental parameters in the gauge theory. Interestingly, demanding that the plasma puddle emit thermal Hawking radiation implies that the object is black (i.e. absorbs all incident particles), which implies classical stability, which implies satisfaction of the Bekenstein en- tropy bound. Because of the AdS/CFT duality and the many similarities between plasma puddles and black holes, we conjecture that black objects are a robust feature of quantum gravity. http://arxiv.org/abs/0704.1146v2 mailto:[email protected] mailto:[email protected] 1. Introduction The AdS/CFT correspondence [1], [2] [3] has greatly improved our understanding of both gravity and gauge theory by providing a concrete realization of the holographic principle. For example, much work has been devoted to studying strongly coupled quasi-CFT dynam- ics using perturbative gravity. Conversely, CFTs have been useful for illuminating aspects of black hole physics, including the unitarity of Hawking evaporation. It has been argued [4] that there exist black holes that can be localized in the IR of asymptotically AdS geometries, and that these solutions are dual to “plasma balls” in a confining CFT (for related work, see [5], [6], [7]). These plasma balls are meta-stable lumps of hot gluon plasma, and like black holes, they absorb all incoming matter and radiate thermally. Interestingly, since the AdS/CFT duality maps quantum effects to classical effects and vice versa, Hawking radiation is nontrivial from the gravitational point of view but straightforward in terms of the dual plasma ball description. Conversely, the “blackness” of black holes – their ability to absorb all incoming particles – is not obvious in the confining CFT picture. In particular, consider the CFT dual of a particle thrown into a black hole: a glue ball thrown into a plasma ball. Naively, it seems that with sufficiently high energy such a glue ball would blast through, in the same way that an extremely high energy proton might barrel through the RHIC fireball. From this point of view, the most fundamental property of black holes – that they absorb all incoming particles – appears to be violated. In [4], this problem was beautifully solved by taking into account the parton substruc- ture of the incident particles [8], in accordance with Susskind’s ideas [9, 10]. In the dual gauge theory, the only available objects outside the plasma ball are mesons and glue balls, and at large ’t Hooft coupling these highly boosted hadrons contain a huge number of soft partons. Thus it is simply impossible to fire a high energy parton into the plasma ball – instead, an incoming glue ball fragments into many low energy partons which are promptly absorbed. The purpose of the present paper is to explore the possibility that a weakly coupled gauge theory might furnish a perturbative dual to a black hole. At first this might seem like an unlikely prospect, particularly since the dual of a weakly coupled CFT is a strongly coupled gravitational theory. Indeed, it is not unreasonable to expect a phase transition between the weak and strong coupling regimes of the gravitational theory, corresponding to immensely different CFT physics in the two regimes. Despite these expectations, however, we will argue that “plasma puddles” in a weakly coupled gauge theory are qualitatively very similar to plasma balls/black holes. Our setup is similar to [4], except that we consider a perturbative gauge theory in the Higgs phase, rather than a strongly coupled gauge theory in the confining phase. The low energy theory is comprised of photons and W bosons, rather than mesons and glue balls. Specifically, we will study an N = 4 SU(N) SYM theory Higgsed down to U(1)N−1. By heating up a region of space we can locally un-Higgs the gauge group (see figure 1), creating a spatially varying Higgs vev. This means that W bosons have a position dependent mass that will act as an effective potential for the enclosed plasma. We will see – 1 – that for sufficiently large puddles, the plasma has a temperature much less than the height of the enclosing potential, and so it is kinematically trapped. Thus, classical stability is ensured. That said, the plasma puddle does emit radiation in a thermal spectrum, but we find that its lifetime is τ ∼ NR, (1.1) where R is the radius of the puddle, so plasma puddles are infinitely long lived in the large N (classical) limit. Since our theory is weakly coupled, it is possible to analyze the absorptive properties of a plasma puddle using standard perturbation theory. Unlike glue balls, photons and W bosons have no partonic substructure – since they are elementary point particles, arbitrarily large boosts involve no partonic subtleties. Thus it would seem that at sufficiently high en- ergies, a plasma puddle can be penetrated. However, incident particles actually experience something very similar to parton showering – they decay near the boundary of the plasma puddle, bifurcating into a large number of ultra-soft daughter particles whose ensemble energy is smeared over the surface of the puddle and eventually thermalized. Qualitatively, this matches the process of gravitational “hair removal” in which an infalling particle is delocalized over the surface of a black hole as it is absorbed. In fact, this showering can only occur at the plasma puddle boundary (the effect requires momentum nonconserva- tion), which dovetails nicely with the notion that black hole absorption is a local effect at the event horizon, independent of the interior. In order for a plasma puddle to absorb even the highest energy particles, an incident particle of energy E has to shower into a large number of decay products that are too soft to escape the enclosing potential. In particular, it is necessary that the average energy of the final decay products, Eavg, does not increase with increasing E. If this is not the case, then an arbitrarily large E implies a commensurately large Eavg, allowing the decay products to blast through the plasma unharmed. Thus, total absorption is only possible if the rate1 of showering increases at least linearly with energy. For E less than a particular (large) threshold we show that this is the case: the rate of decay of a gauge boson to two W ’s is ΓA→WW ∼ λE, (1.2) where λ is the ’t Hooft coupling. Moreover, even though the decay rate goes to a constant for incident particles above this threshold energy, we find that they are still absorbed. In particular, these high energy particles shower promptly inside the plasma puddle into a large number of decay products at the threshold energy, which in turn decay at a rate λE. Thus, a high energy incident particle will shower into O(λER) decay products each of energy Eavg ∼ (λR)−1. We confirm this reasoning with a more precise argument in section 3.2. As long as Eavg is less than the height of the enclosing potential, these decay products are trapped inside the plasma puddle, and so the puddle appears black. 1For a detailed calculation of the dimensionless probability of showering, see appendix B – 2 – Figure 1: A plasma puddle cross section (above) and energy profile (below). The latter depicts the W mass as it asymptotes to a value of m0 at infinity and vanishes within an un-Higgsed region of radius R. The plasma within settles into a hot puddle of temperature T . Both diagrams show a high energy incident particle on the left showering in the atmosphere, as well as some Hawking radiation escaping on the right. Interestingly, as we show in section 4, if we demand that the plasma puddle emit ther- mal Hawking radiation, then it automatically also absorbs all incident matter, is classically stable, and satisfies the Bekenstein entropy bound [11]. During the process of Hawking evaporation, a stable plasma puddle will lose each of these properties sequentially, until it eventually becomes a free gas of gauge bosons. Since the plasma puddle is so similar to a black hole, our hope is that the large λ plasma ball/black hole duality established in [4] interpolates at small λ to a correspondence between the plasma puddle and some black object of a strongly coupled gravitational theory. Note that for our purposes we will focus entirely on the gauge degrees of freedom of the N = 4 SYM, ignoring gaugino partners and scalar moduli. Moreover, as we are interested only in parametric scalings, we will be largely ignoring numerical factors. The outline of the paper is as follows. In section 2 we determine the general properties of our – 3 – setup: gross properties of the plasma puddle, Hawking radiation, and classical stability. In section 3, we give a nice physical estimate of the rate of particle showering in the plasma puddle atmosphere. In section 4 we show that demanding thermal Hawking radiation immediately implies other black hole-like properties, and we conclude in section 5. The appendices contain a more formal derivation of the probability of decay. 2. The Plasma Puddle In this section we give a detailed account of what a plasma puddle is and how it forms. Our setup is as follows. Consider an N = 4 SU(N) SYM at large N and weak ’t Hooft coupling λ that is Higgsed down to U(1)N−1 (note that due to the Higgsing, the theory is not conformal and so particles and S-matrices are well defined). The spectrum of the theory consists of N − 1 massless photons and N2−N massive W bosons. Now, let us fire an ensemble of high energy W ’s into a small region of space2. The influx of W ’s heats up the region and locally un-Higgses the gauge group. Once the W ’s thermalize, the resulting meta-stable object is a plasma puddle. The local un-Higgsing can be parameterized by a spatially varying Higgs vev which induces a spatially varying mass, m(x), for the W bosons. The W mass profile vanishes inside the plasma puddle but asymptotes to some nonzero mass m0 outside (the precise mass of individual W s depends on the Higgsing pattern, but the details will not be impor- tant). Let us define the atmosphere to be the region near the plasma puddle boundary in which the mass profile varies. Again, we emphasize that the W ’s are confined by the m(x) potential, while the photons are actually massless everywhere. If we ignore all interactions, it is straightforward to see what happens to free streaming W ’s as they collapse into a plasma puddle. Those W ’s with E > m0 escape to infinity while those with E < m0 settle at the basin of the m(x) potential. Due to the potential barrier, this puddle of W ’s can never escape. The story is similar if we include gauge interactions, except that the puddle of W ’s thermalizes into a puddle of photons and W bosons. To see this, we compute the mean free path d of a gauge boson Aij traversing the hot plasma. From color conservation, Aij can only scatter off of some Ajk, leaving Ail and Alk in the final state, where k and l are free indices. Summing the amplitude squared, g4, over phase space, k and l contribute N2 to the cross section, yielding σ ∼ λ2T−2, (2.1) since the temperature sets the energy scale of the interaction. The number density n of a gauge boson with a given pair of color indices is T 3, so the mean free path is d ∼ (nσ)−1 ∼ (λ2T )−1. (2.2) 2Note that an ensemble of W bosons that are initially at rest will naturally collapse due to the attractive force of dilaton gravity, as in described in [12]. In the nonrelativistic limit, this behaves exactly like true gravitational collapse. – 4 – For sufficiently large ’t Hooft coupling the mean free path is smaller than the size of the plasma puddle and thus gauge bosons cannot go very far without scattering (see figure 2). Let us denote this as the highly thermalized regime. In this case the plasma puddle quickly thermalizes into a hot, homogenous soup of photons and W bosons at a temperature T . The plasma puddle is classically stable as long as its temperature is less than m0, so that the interior plasma is kinematically trapped. Without this condition, nothing prevents the plasma from simply escaping to infinity, and so the relation T < m0, (2.3) implies classical stability of the plasma puddle. In addition, since the plasma puddle is highly thermalized, only particles at the very surface have any hope of escape. We argue in section 2.2 that this phenomenon is an O(1/N) area effect that is thermal and dual to Hawking radiation. For these reasons (along with their absorptive properties, which we discuss in section 3) we claim that plasma puddles in the d < R regime are black hole-like objects. On the other hand, if λ is sufficiently small, then d > R and a typical gauge boson can traverse the extent of the plasma puddle without ever scattering (see figure 2). However, given a time of order d, the plasma will eventually thermalize, yielding a collection of nearly free photons and W ’s at a temperature T . The W bosons travel in straight lines through the puddle until they reach the potential barrier from m(x), after which they roll back towards the interior, and repeat. As a result the W ’s form a relativistic plasma at the bottom of the m(x) potential, but they become nonrelativistic in near the m(x) barrier wall, simply because they have less kinetic energy there. In contrast, the photons are massless everywhere and can free stream outwards. As we discuss in section 2.2, since plasma puddles in the d > R regime do not Hawking radiate in the traditional sense, we do not identify them as dual black holes. 2.1 Gross Properties and Relations Thus far our discussion has included T and R as a priori attributes of the plasma puddle. However, as we show in this section, these two variables are fixed in terms of the “universal” quantities N , λ, m0, and the total energy of the plasma puddle, M . To begin we note that unlike Schwarzchild black holes, plasma balls/puddles do not obey the relation T ∼ 1/R for the following reason. Since the plasma puddle has an entropy S = N2T 3R3, the relation T ∼ 1/R would imply that the entropy is independent of the size of the object. However, this is not the case – the entropy of a plasma puddle/ball increases with size. In the case of the strongly coupled plasma balls of [4], the temperature of large plasma balls is set by the confinement scale. Analogously, one might expect a similar situation for plasma puddles, i.e. that the temperature is given by the Higgs vev. We will see that this is not the case. Once formed, a plasma puddle is an isolated system, so its total energy is conserved. However, its entropy should be maximized subject to this constraint, so we treat the plasma 3Here we use the symbol M in anticipation of matching this total energy of the plasma puddle to the mass of a dual black hole. – 5 – puddle with the micro-canonical ensemble. In N = 4 SYM, the scalar fields Φij are in the adjoint representation of the gauge group. When these fields Higgs SU(N) to U(1)N−1, their vevs can be written as diagonal N ×N matrices with spatially varying components. The total energy of a plasma puddle arises from two contributions: the gradient kinetic energy from the (spatially varying) Higgs mechanism, and the thermal energy of the plasma within. Because these contributions are boundary and volume effects, respectively, we are essentially balancing the pressure against the surface tension. Neglecting O(1) factors, the total energy of the plasma puddle is thus d3xTr(∇Φ)2 +N2T 4. (2.4) The second term only accounts for the relativistic (T ≫ m(x)) region of the plasma. The nonrelativistic (T ≪ m(x)) region has an energy density N2T 2m(x)2 exp(−m(x)/T ), and this is always less than or equal to N2T 4. Thus neglecting this contribution merely amounts to a rescaling of the plasma energy by an O(1) factor. Each diagonal entry of Φ gets a vev that asymptotes to ∼ m0/g at infinity but vanishes in some region of size R. Moreover, let us define L to be the thickness of the atmosphere, i.e. the region in which the Higgs vev varies. Estimating the energy we find that M = N2 m20(R+ L) + T 4R3 . (2.5) Before maximizing the entropy at fixed energy, let us motivate why this fixes the radius of the plasma puddle. Consider what happens as we increase the radius of the puddle at fixed m0. This increases the gradient Φ energy, but since the total energy is fixed, this forces the temperature to diminish. Since the entropy goes as T 3R3, the increase in radius and decrease in temperature are competing effects. Hence, there is an extremal value for R such that the entropy is maximized. Note that this is what fixes the size of a balloon full of gas. Fixing M allows us to solve for T , yielding an entropy S = N2T 3R3 (2.6) NR3/4 M − N 2m20(R+ L) . (2.7) The entropy is maximized for L ∼ R, because this maximizes the term in large parentheses. Maximizing the entropy with respect to R, we find that R ∼ λM N2m20 , (2.8) and also that T ∼ Nm λ3/4M1/2 . (2.9) Thus we see that while strongly coupled plasma balls obey the relation R ∼ M1/3 [4], perturbative plasma puddles have that R ∼ M . – 6 – Finally, let us add some remarks about the precise shape of the Higgs vev, Φ. Near the boundary of the plasma puddle it is nontrivial to determine Φ, because the plasma back-reacts on the Higgs vev and vice versa. However, outside R the Φ field is free, so it obeys Laplace’s equation, and thus for r > R, (2.10) obtains. Thus the scalar field atmosphere has a 1/r tail, so again we see that the thickness L of the plasma puddle is of order R. In d ≥ 4 spacetime dimensions (in the gauge theory), the Higgs profile would go as Φ ∼ 1− , (2.11) so in higher dimensions there are no IR subtleties. For this reason we do not expect that this tail is qualitatively important for black hole/plasma puddle physics. 2.2 Hawking Radiation In the gravitational picture, the total absorption of incident particles by a black hole is obvious at the classical level, while Hawking radiation is a nontrivial quantum effect. In contrast, in our CFT setup we will see that plasma puddle absorption is nontrivial but Hawking radiation is manifest. In this section we consider the latter. To begin, let us consider the regime in which d < R and the enclosed plasma is highly thermalized. In this case the gauge bosons cannot go very far without scattering, and thus a particle has no hope of escape unless it is at the very surface of the plasma puddle. Moreover, since the m(x) potential kinematically bounds W bosons, a surface particle can only escape if it is a photon. Since only 1/N of the particles is a photon (due to the Higgsing pattern SU(N) → U(1)N−1), the plasma puddle radiates photons in a thermal spectrum at temperature T . Since this is a O(1/N) effect, it is natural to associate this radiation to O(~) Hawking radiation in a gravitational dual. Next, let us compute the rate at which a plasma puddle loses energy as the result of Hawking radiation. In an infinitesimal time interval dt, an order one fraction of photons that are within dt of the surface of the plasma puddle will exit. If nγ is the number density of photons in the plasma, then there are nγR 2dt photons in this region near the surface. Since each has an energy of order T , = −nγR2T. (2.12) Assuming that these photons are in thermal equilibrium, we find = −NR2T 4 = −Nm , (2.13) – 7 – so we see that at fixed temperature the energy loss is an area effect, while at fixed m0 the energy loss is completely independent of size and temperature. The lifetime is given by the time it takes for the total energy M to be radiated τ = −M ∼ NR, (2.14) so plasma puddles are completely stable in the classical limit N → ∞. The situation is more complicated if d > R, so let us be more careful. We consider scattering inside the plasma puddle for arbitrary d and R. If the overall number density of particles is ntot, then there is a total of ntotR 3 particles, and so the total rate of scattering processes is ntotR 3/d. Of these processes, an O(1/N) fraction create additional photons. Likewise, there are nγR 3 photons in the plasma, so the rate for scattering events involving photons is nγR 3/d. Almost all of these scattering events destroy photons, because AW → AW is suppressed by a factor N compared to AW → WW . Thus scattering events change the number of photons in the plasma at a rate d(nγR scattering ntot − nγ . (2.15) However, radiation leaving the plasma puddle also decreases the number of photons at the d(nγR radiation = −nγR2. (2.16) These two processes balance when . (2.17) Thus we see that as d becomes larger than R, the number of photons becomes less than 1/N of the total number of particles, and so we leave thermal equilibrium. In particular, once d > R, the rate of Hawking radiation begins to decrease by the significant factor R/d. Thus, in the d > R regime the plasma puddle does not radiate a thermal spectrum, and exiting photons can free stream from anywhere within the plasma, not just the surface. For this reason we do not consider such a plasma puddle to be the dual of a black hole. We have glossed over a subtlety in our treatment of Hawking radiation. As we argue in section 3, a high energy photon striking the plasma puddle atmosphere will shower into many other gauge bosons, until all of the decay products have a tiny energy of order (λR)−1. But since it is possible that T > (λR)−1 (in fact, an even stronger condition is required to be in the highly thermalized regime), we might worry that an outgoing photon will simply shower in the plasma puddle atmosphere. If this happens, its low energy decay products are likely to be reabsorbed by the plasma puddle, leaving no reason to expect Hawking radiation, let alone a thermal distribution of Hawking radiation. The resolution to this puzzle is that once an outgoing photon makes it to the outer atmosphere of the plasma, it does not have the 2m(x) of energy necessary to shower into – 8 – Figure 2: Here we show the typical trajectory of a gauge boson in the highly thermalized regime, d < R, and otherwise. Hawking radiation is only thermal when d < R. two W bosons, since it only has energy T . Thus, kinematical constraints allow photons at temperature T to escape from the plasma puddle as thermal Hawking radiation, even though high energy incident photons typically shower as they fall in. Finally, let us discuss briefly the possibility of W boson Hawking radiation. While most of the W bosons are of course kinematically trapped, there is a Boltzmann tail in the thermal distribution which allows an e−m(x)/T fraction of the W ’s to escape. While this effect is exponentially suppressed, it is not suppressed by powers of 1/N . This is puzzling from the standpoint of the gravity dual, in which all Hawking radiation should be an O(~) effect. Understanding the true N dependence of W boson Hawking radiation will require further study. 3. Plasma Puddle Absorption The primary feature of the plasma puddle that distinguishes it from more conventional objects that are hot and stable (such as stars) is its ability to absorb all incident matter. In this section we verify that the plasma puddle is black by showing that a high energy incident particle will shower into numerous soft decay products that thermalize with the interior plasma. While a formal calculation of the decay probability is given in appendix A, we present a more direct physical argument in the following section. 3.1 Estimation of the Showering Rate To begin, let us consider the propagation of a single photon through free space. The photon is absolutely stable due to kinematics and phase space – momentum conservation – 9 – Figure 3: An incoming gauge boson is surrounded by a cloud of virtual particles. Only when the gauge boson strikes the plasma puddle atmosphere can these virtual particles become real. The characteristic length scale of quantum fluctuations is given by (λE)−1. only allows decays into exactly collinear, massless decay products and since this is a set of measure zero in phase space the decay rate vanishes. However, in the presence of momentum violation (such as the plasma puddle atmosphere), noncollinearities are allowed and so the decay rate is nonzero. To estimate this rate, we invoke the uncertainty principle, which tells us that a photon in the vacuum is immersed in a O(λ) cloud of virtual gauge bosons constantly coming in and out of existence. Since the only relevant time scale for a massless particle in free space is its energy E, the photon produces virtual particles of energy ∼ E at a rate Γ ∼ λE. Indeed, once the photon strikes the plasma puddle atmosphere, the resulting momentum violation allows for these virtual particles to become real4. Since Γ is the rate at which photons are produced, and in the momentum violating background they can simply escape to infinity, we expect that ΓA→WW ∼ λE. (3.1) The reader may suspect that this answer breaks down at large energy, and this is in fact the case. There are several reasons why we might expect this to happen. First of all, if the photon is arbitrarily boosted (relative to the rest frame of the plasma puddle), it eventually reaches the threshold energy for plasma puddle creation, and so the correct 4This is a common effect, and is detailed in standard textbooks on quantum field theory [14]. For example, consider the electron, which carries with it a number density of virtual photons given by Nγ ∼ α. Bremsstrahlung effects from the scattering of an electron off of a charged target can be understood as virtual photons becoming real due to a nontrivial background. – 10 – calculation would involve plasma puddle/plasma puddle scattering. This is akin to firing a particle into a black hole at trans-Planckian energies, which is really the same as black hole/black hole scattering. Secondly, at very high energies theW decay products of the photon are so collinear that their splitting is completely unmeasurable (by the plasma puddle) and is thus unphysical. Let us illustrate this soft collinear divergence more explicitly. As noted previously, the photon cannot decay into two exactly collinear W bosons due to phase space, but if it receives of a momentum “kick”, then it can decay into W s which split in the transverse direction. However, it can only decay in this way in the momentum violating background. In the plasma puddle atmosphere, where the plasma itself is negligible, the only source of momentum violation (position dependence) is the gradient of the Higgs field, so we expect that the kick is proportional to 1/R. The angle subtended by the outgoing W ’s is approximately k⊥/E, where k ∼ E/R is the transverse momentum difference between the W ’s5. Demanding that the “cone” traced out by the outgoing particles grows to a size greater than their Compton wavelength ∼ 1/m0, we find that ⇒ E < m20R. (soft collinear bound) (3.2) If this bound is violated, then there is a soft collinear divergence and the splitting is unmeasurable. Note that if the energies of the decay products are different, only the lesser of the two need satisfy this bound. In appendix A.4 we address these high energy issues directly by calculating the rate of decay across the extent of the plasma puddle, yielding ΓA→WW ∼ λE, (E < m20R) (3.3) ∼ λm20R, (E > m20R,Bremsstrahlung) (3.4) , (E > m20R, symmetric decay), (3.5) where “Bremsstrahlung” denotes the region in phase space in which one of the outgoing particles is much softer than the other (that is, with energies of order m20R and E −m20R) and “symmetric decay” denotes the regime in which they are comparably energetic (with energies of order E). Since the probability of a symmetric decay is suppressed by a factor of m20R/E relative to Bremsstrahlung, soft emission totally dominates showering at energies above the threshold m20R. For example, consider an incoming particle boosted to some high energy a hundred times greater than m20R. Emission of a soft particle of energy m20R is a hundred times more likely than decay into two particles of comparable energy. Moreover, once this Bremsstrahlung occurs, the harder particle only loses m20R energy, and so a consecutive soft emission is still ninety nine times more likely than a symmetric decay, and so on. Before long, the particle emits enough Bremstrahlung that its energy drops to m20R and its decay rate begins to scale linearly with energy. In the following section we consider the physical consequences of these rates. 5We estimate k⊥ as follows. We begin with an incident photon with 4-momentum (E, 0, 0, E) and assume that it receives a momentum “kick” (0, 0, 0,−1/R). This allows it to split into two massless particles with ≈ E2 − (E − 1/R)2 ∼ E/R – 11 – 3.2 Absorption and Eavg In section 3 we saw that PA→WW increases linearly with E up to the scale m 0R, after which it remains constant. Next, we show that this implies that an incident particle, no matter how energetic, will always be absorbed by the plasma puddle. To begin, let us consider showering in the E < m20R regime. An incident particle of energy E will decay in the plasma puddle atmosphere at a rate λE. In turn, its decay products have a smaller rate for decaying simply because they have less energy. In fact, after a long sequence of decays the average energy of the final decay products eventually diminishes to Eavg ∼ (λR)−1. At this point the decay rate dips below 1/R, and the ultra- soft daughter particles can traverse the extent of the plasma puddle without showering. Thus, showering completely terminates once the decay products reach an energy of (λR)−1. Since the final decay products have a Compton wavelength proportional to the radius of the plasma puddle, showering effectively de-localizes the incident particle over the entire puddle. This picture matches nicely with Susskind’s ideas about black hole absorption [9, 10]. In addition, as long as this terminal energy is less than m0, the daughter particles are kinematically trapped by the Higgs profile, and they will eventually be thermalized. This translates into a blackness bound < m0, (3.6) which must be satisfied if the plasma puddle is to absorb all incident matter. We now argue that the above conclusions are still valid even above the threshold energy, m20R. As we argued earlier, such a high energy particle favors the emission of soft particles of energy m20R. However, because the decay rate for Bremsstrahlung remains constant at high energies, one might wonder whether a sufficiently energetic particle might blast through the plasma puddle. A careful analysis shows this is not the case. In particular, let us very roughly estimate the rate of 1 → n showering for an incident particle of energy E = nm20R, where n is some larger integer. Noting that a 1 → n decay includes a sequence of i → i+ 1 sub-processes, and is often dominated by on-shell regions of phase space, we can estimate the rate of 1 → n decay by Γ1→n = P1→2 × P2→3 × . . .× Pn−1→n, (3.7) (λm20R 2)n, (3.8) where schematically P ∼ ΓR (see appendix A, where we calculate these probabilities explicitly). As we show in section 4, classical stability immediately implies that the quantity in parentheses is greater than unity. Thus, we argue that an incident particle of energy nm20R will decay into n particles of energy m 0R within a region of size R. After this sequence of Bremsstrahlung events, the decay products all have energies of order m20R, and so we can apply our analysis for the E < m20R regime to these decay products. Since these particles decay to a multitude of ultra-soft particles of energy (λR)−1, we find that total absorption occurs, no matter what the incident energy. – 12 – Interestingly, we are finding that large multiplicity events dominate at high energies, which is reminiscent of Hawking evaporation – after all, the probability that a large black hole will decay to two particles is extremely small, because it is suppressed by e−S , while the decay to a huge number of soft particles is virtually guaranteed. Thus the necessity of including 1 → n decays at high energies seems to indicate that high energy particles carry a great deal of entropy. 4. Implications of d < R In section 2.1 we argued that in the highly thermalized regime (d < R) the plasma puddle emits thermal Hawking radiation at the surface. In this section we show that d < R also implies total absorption, which implies classical stability, which implies satisfaction of the Bekenstein entropy bound. 6 For this reason, we identify d < R plasma puddles as black hole-like objects. Applying our relations from section 2.1, we can rewrite the d < R bound, the blackness bound (Eq. (3.6)), the classical stability bound (Eq. (2.3)), and the Bekenstein bound as m0R > λ −7/2, (highly thermalized) (4.1) > λ−1, (total absorption) (4.2) > λ−1/2, (classical stability) (4.3) > λ1/2, (Bekenstein bound). (4.4) The above sequence of bounds gives an interesting depiction of what happens during plasma puddle evaporation. As the energy of the plasma puddle diminishes via Hawking radiation, its size shrinks commensurately. First, the plasma puddle leaves the highly thermalized regime and stops emitting thermal Hawking radiation. Next, the plasma puddle becomes transparent to incident matter, and finally the object becomes classically unstable. After this point, the enclosed photons and W bosons in the plasma simply stream outwards, leaving a free gas of gauge bosons. It would be interesting to explicitly connect these processes with the Horowitz-Polchinski transition [15], [16], [17]. It is also noteworthy that the Bekenstein bound is precisely equivalent to the condition , (4.5) which is necessary for the application of classical thermodynamics. As this bound is ap- proached, the energy/size of the system begins to saturate the uncertainty principle, and we are forced to count individual quantum mechanical microstates. 5. Discussion In this paper we have argued for the existence of black hole-like objects living in large N gauge theories at weak ’t Hooft coupling. Since these theories are completely pertur- bative, we can calculate much of the physics. In particular, in the regime in which the 6The Bekenstein entropy bound gives an upper bound on the entropy of a system in terms of its gravi- tational mass M and its size R: S < MR. – 13 – mean free path d is smaller than R, we find that these meta-stable puddles of plasma are classically stable and emit radiation in a thermal spectrum. While these properties are of course common to any hot star-like object, we moreover find that the plasma puddle absorbs all incident matter, no matter how energetic. This occurs because high energy particles invariably shower into ultra-soft decay products that are kinematically bound by the effective potential from the spatially varying Higgs vev. All in all, we find this to be compelling evidence that the plasma puddle is dual to a black object in a strongly coupled gravitational theory. In addition, our work gives a particularly nice picture of plasma puddle evaporation, which may be connected to the Horowitz-Polchinski transition [15], [16], [17]. Indeed from section 4 we saw that as the object radiates away energy, it eventually leaves the d < R regime and stops emitting thermal Hawking radiation. After even more energy loss, the plasma puddle stops being black, and eventually becomes classically unstable. Interestingly, we also find that the d < R bound gives a nice lower bound on the total energy of a plasma puddle, given by M > λ−9/2N2m0. (5.1) Since a plasma puddle must have at least this much energy to form a black hole-like object, we might interpret the left-hand side as the CFT dual to the Planck mass. Acknowledgements We would like to thank Ofer Aharony and Nima Arkani-Hamed for helpful discussions, and Toby Wiseman for correspondence. We also thank Jonathan Heckman for collaboration when this work was at an early stage. Jared Kaplan is supported by a Hertz Foundation Graduate Fellowship. A. Detailed Absorption Computation In this section we give a detailed derivation of the probability for the decay of a photon to two W bosons as it passes through the plasma puddle atmosphere. Although we are formally calculating a probability, we can re-interpret it as a decay rate via Γ ∼ P/R. To begin, we consider a toy scalar model for simplicity. This scalar field theory has an action ∂φ2 + ∂χ2 − (m2(x3) + gφ)χ2. (A.1) In this theory, a massless scalar field φ is coupled cubically to a scalar χ whose mass m2 is a nontrivial function of x3. Here φ and χ are scalar analogs of the photon and the W boson, respectively, and m2 mimics the effects of a space varying Higgs vev on the W mass. Notice that we have made the simplification that the plasma puddle is infinite in the x1 and x2 directions, which is a very good approximation if the plasma puddle is large and the incoming particles approach from the x3 direction. – 14 – A.1 Propagator Next, let us compute the χ propagator in the regime in which the incoming and outgoing energies are much larger than the characteristic energy scale 1/R set by the mass profile. To do this we use the WKB approximation to solve the wave equation7. The wave equation is given by χ = 0, (A.2) where the space dependent mass is m2(x3) = m 0(1−B(x3)), (A.3) and B(x3) is a “bump” function which vanishes at infinity and peaks to unity in a compact region of size R. From section 2.1 we know that B(r) goes as 1/r for large r, but for now lets us take B(x3) to be a general function. Consider the following WKB ansatz solution: χ(x) = exp(ip̃(x)x) (A.4) p̃(x) ≡ p0, p1, p2, p3 + f(x3) , (A.5) where f(x3) is a function that will be determined by plugging χ into the wave equation. From now on, a tilde on a momentum variable will represent a nontrivial space dependence of this kind. If we plug this ansatz back into the wave equation, we find −p2 +m20 +m20 −B + f + x3f ′ − − ix3f χ. (A.6) If the quantity in curved parentheses vanishes, then χ is an eigenstate of the wave equation. In the regime where p3 ≫ 1/R, the terms containing 1/p3 are small, since they go as inverse powers of p3R, so we drop them. Now it is easy to solve for f in the resulting differential equation −B + f + x3f ′ = 0, (A.7) yielding p̃(x) = p0, p1, p2, p3 + 2p3x3 B(x′3)dx . (A.8) Thus in the WKB approximation, the χ propagator is Gχ(x, y) = p2 −m20 × ei(p̃(x)x−p̃(y)y), (A.9) to zeroth order in 1/(ER) and all orders in m20. 7If we include m0 as a mass insertion in Feynman diagrams, then our answer is necessarily a series expansion in m0. Using the WKB approximation, we will be neglecting terms suppressed by higher powers of 1/(ER), but keeping terms to all orders in m0. – 15 – A.2 Amplitude Next, let us consider the amplitude for a φ particle of momentum p to decay into two χ particles of momenta k and q in the spatially varying background. To do so we write down the corresponding time ordered 3-point correlator in coordinate space 〈χ(y)χ(z)φ(w)〉 ∼ g Gχ(y − x)Gχ(z − x)Gφ(x− w)d4x. (A.10) Next, we Fourier transform to the variables p, k and q. In accordance with the LSZ reduction formula, a scattering amplitude corresponds to the Fourier transform of the appropriate time ordered correlator with the pole from each external leg stripped off. Amputating the legs, we find that the Feynman amplitude is M(k, q, p) = g × d4x ei(k̃(x)+q̃(x)−p)x, (A.11) = g × δ(012)(k + q − p)F (k3, q3, p3) (A.12) F (k3, q3, p3) = dx3 e i(k̃3(x3)+q̃3(x3)−p3)x3 . (A.13) where g is the dimensionful coupling strength. Notice that in the m20 → 0 limit, F reverts to a full 4-momentum conserving delta function, as expected. Next, let us evaluate F . By separating a factor of exp i(k3 + q3 − p3)x in the integrand of F , it is clear that F is simply the Fourier transform of the quantity B(x′3)dx . (A.14) Let us consider the case in which the Higgs profile is a square bump function and so B is unity for |x3| < R and 0 otherwise. Given this simplification the integral is easy to evaluate piece-wise and F takes the simple form F ∼ lim sin(a/ǫ+Rb) sinR(a+ b) (a+ b) , (A.15) ∼ πδ(a) − b sinR(a+ b) (a+ b) , (A.16) where we are using ǫ to regulate the δ function, and for convenience we have defined a = k3 + q3 − p3, (A.17) b = m20 . (A.18) Since R is the largest length scale in the problem, we can actually simplify F even further, writing F ∼ δ(a) − b δR(a+ b), (A.19) where δR denotes an R-regulated delta function with width 1/R and height R. Physically, F takes this form because it receives two contributions, corresponding to exact momentum conservation and deviations from momentum conservation, set by the scale 1/R. – 16 – Also, note that our results are parametrically correct even if the Higgs profile differs from the square bump function form which we have assumed. Looking at other forms for B, we find it is only really necessary that B ≈ 1 in the region x3 ∈ [−R,R]. Next, let us integrate over phase space and compute the total probability of decay using this amplitude. A.3 Probability The decay rate for a 1 → 2 process is given by Γ ∼ 1 d3kd3q |M|2. (A.20) In translationally invariant theories, M is proportional to a 4-momentum conserving delta function, so |M|2 is a product of squares of delta functions. While naively this introduces a divergent δ(0) term, we normally divide Γ by the volume of spacetime, hence removing these factors. However, in our case, F has a component that exactly conserves 4-momentum and a component that violates momentum in the x3 direction by an amount 1/R. From basic kinematics, we know that the contribution to Γ from the 4-momentum conserving piece will not contribute, since a massless particle cannot decay into two massive particles in free space. For this reason, we will only need to compute the momentum violating contribution to |F |2. Since we are not dividing by the size of the x3 direction, we are actually computing the decay rate integrated over all of x3, i.e. the total probability of decay (see appendix B for details). For an incoming momentum p = (E, 0, 0, E), (A.21) the (dimensionless) decay probability in our toy model is Pφ→χχ ∼ δ(k0 + q0 − E)δ(k1 + q1)δ(k2 + q2)|F (k3, q3, E)|2, (A.22) where the energies are ~k2 +m20, (A.23) ~q2 +m20. (A.24) It is trivial to integrate over the transverse q momenta, after which we parameterize the two transverse k momenta in polar coordinates as (k1, k2) = k⊥(cos θ, sin θ) (A.25) d3k = k⊥dk⊥dθdk3. (A.26) From the energy conservation delta function we obtain the useful expressions (E2 + k23 − q23), (A.27) (E2 + q23 − k23). (A.28) – 17 – Applying these formulae, the delta function becomes δ(k0 + q0 − E) = × δ(k⊥ −K⊥), (A.29) where K⊥ is defined as (k3 + q3 − E)(k3 − q3 − E)(k3 + q3 + E)(k3 − q3 + E)−m2. (A.30) The factor multiplying the delta function cancels with most of the integral, eliminating all k⊥ dependence except for the delta function! Note that for a completely 4-momentum conserving interaction, k3+ q3 = E, and so K = −m2 < 0 and the k⊥ integral yields zero, corresponding to the fact that a massless particle cannot decay into two massive particles if 4-momentum is conserved. Since we lack momentum conservation in the 3-direction, > 0 and the k⊥ integral instead gives unity. Consequently, we obtain the probability of a φ particle to decay into two χ particles Pφ→χχ ∼ dk3dq3|F (k3, q3, E)|2. (A.31) The domain of integration is the compact region k23 +m q23 +m 0 ≤ E, (A.32) as derived from energy conservation. This inequality is saturated when k⊥ is zero. Note that we have made no approximations in evaluating this phase space integral, so the result, written in terms of F , is correct up to numerical coefficients. A.4 From Toy Scalars to Gauge Bosons Given Pφ→χχ, it is straightforward to obtain PA→WW , the probability of a photon to decay to two W ’s. The only parametric difference in the two calculations is that in the gauge theory calculation, g becomes a dimensionless coupling and the three gauge boson interaction has an extra factor of momentum due to the derivative coupling. This introduces an extra factor of E2 into the decay probability. Moreover, since the outgoing W ’s can be any of N gauge bosons, we also get a factor of N , yielding PA→WW ∼ λ dk3dq3|F (k3, q3, E)|2. (A.33) Before we plug in for F , let us pause to note a high energy subtlety which was mentioned earlier in terms of soft collinear divergences. Naively, we would be tempted to simply set b = −a in F because of the delta function of a + b. However, since b is only fixed to be equal −a up to a 1/R width, this replacement is only valid if b is of order 1/R or more, i.e. if , (A.34) – 18 – which is precisely the soft collinear bound derived in Eq. (3.2). As we will show, the energy scaling of the decay probability is quite different above and below this bound. First, let us consider the E < m20R regime. Setting b = −a and integrating |F |2 over k3 and q3 (ignoring contributions from the 4-momentum conserving delta function), we find that PA→WW ∼ λ dk3dq3 |δR (a+ b)|2 (A.35) ∼ λδR(0) dk3, (A.36) ∼ λER, (E < m20R) (A.37) which is our final answer for the probability of a photon to decay to two W bosons at energies E < m20R. Next, let us determine the probability of decay in the regime E > m20R, which we will divide into two phenomenologically distinct regions of phase space. First, consider the regime where k3 is less than m 0R but q3 is large enough that their sum, E, exceeds m20R (obviously our result is symmetric under k3 ↔ q3). We will denote this as the “Bremsstrahlung” regime because one of the outgoing particles is much softer than the other. Since Eq. (A.34) obtains in this limit, the entire discussion of the previous para- graphs applies except that the k3 integral is bounded by k3 < m 0R, and so PA→WW ∼ λm20R2. (E > m20R, Bremsstrahlung) (A.38) If we are instead interested in a “symmetric decay,” defined by k3, q3 > m 0R, then it becomes necessary to include the b/a in our expression for F , and the answer becomes PA→WW ∼ λm40R . (E > m20R, symmetric decay) (A.39) Note the crucial difference between these two regions of phase space – the first corre- sponds to decay products of energy m20R and E − m20R while the second corresponds to two decay products with energies roughly of order E. A.5 Higher Order Effects and Approximations Thus far we have only taken into account the role of three gauge boson interactions mediat- ing binary decays in the plasma puddle atmosphere. However, it is fair to ask whether these contribution necessarily dominate over the four gauge boson interactions, which mediate trinary decays. In this section compare the relative sizes of PA→WWW and PA→WW→WWW, and argue that the former contribution is subdominant. Repeating our procedure from section A.2, we find the contribution to the decay am- plitude from the four gauge boson interaction, M(k, q, r, p) = g2 × d4x ei(k̃(x)+q̃(x)+r̃(x)−p)x. (A.40) – 19 – Again, this amplitude simplifies to the form δR(a+ b), where this time a = k3 + q3 + r3 − p3, (A.41) b = m20 . (A.42) Summing over N2 possible final states, we find that PA→WWW ∼ λ2ER. (A.43) Despite multiple phase space integrals, the form of this answer is expected because |F |2 ∼ |δR(a + b)|2 yields precisely one factor of R, and factors of E make up the rest of the expression. Comparing, we see that the probability of one trinary decay PA→WWW is much less than that of two consecutive binary decays PA→WW→WWW ∼ (λER)2, (A.44) so it is the latter that dominates the 1 → 3 decay. Thus, the dominant contribution to the decay is binary and our expression from section 3.2 is valid. Similarly, loop effects are not large even though the 1 → n rate is large at high energies, because only initial and final states involve propagators that are almost on-shell. B. Rates and Cross Sections Normally, we are interested quantities such as the cross section and decay rate. However, in our setup the “decay rate” is position dependent, and so to avoid this complication we simply compute the total probability that the incident particle will decay. This requires a slight revision of the standard formulas for converting S-matrix elements into physical observables. Let us begin with Weinberg’s [13] formula Eq. 3.4.9. Putting the universe in a box of size L and in a time interval of size T , we have that dP (α → β) ∼ 1 |Sβα|2dβ. (B.1) In a 4-momentum conserving theory, the S-matrix element is defined as Sβα ∼ iδ0123(pβ − pα)Mβα, (B.2) where we identify δ(0) ∼ L. However, in our case momentum is violated in the x3 direction, so we have that Sβα ∼ iδ012(pβ − pα)F (pβ3 − pα3). (B.3) Thus the cancellation of volume factors will not be complete. Instead dP (α → β) ∼ T |Mβα|2δ012(pβ − pα) |F (pβ3 − pα3)|2 dβ. (B.4) – 20 – Normally we would divide by T and take the L → ∞ limit, but in our case this sends the probability to zero. This is just the statement that a particle in infinite volume will, on average, take an infinite time to hit the domain wall. Similarly, given a finite volume, if T → ∞ then probability will blow up. Thus we should take T,L → ∞ with T/L held constant. It seems that this leads to an arbitrary factor, but physically this factor should be one, since it just corresponds to the number of times the particle crosses the domain wall. References [1] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]. [2] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150]. [3] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]. [4] O. Aharony, S. Minwalla and T. Wiseman, Class. Quant. Grav. 23, 2171 (2006) [arXiv:hep-th/0507219]. [5] S. B. Giddings, Phys. Rev. D 67, 126001 (2003) [arXiv:hep-th/0203004]. [6] K. Kang and H. Nastase, Phys. Rev. D 72, 106003 (2005) [arXiv:hep-th/0410173]. K. Kang and H. Nastase, Phys. Lett. B 624, 125 (2005) [arXiv:hep-th/0501038]. [7] H. Nastase, [arXiv:hep-th/0501068]. H. Nastase, [arXiv:hep-th/0603176]. [8] J. Polchinski and M. J. Strassler, JHEP 0305 (2003) 012 [arXiv:hep-th/0209211]. [9] L. Susskind and P. Griffin, [arXiv:hep-ph/9410306]. [10] L. Susskind, J. Math. Phys. 36, 6377 (1995) [arXiv:hep-th/9409089]. [11] J. D. Bekenstein, Phys. Rev. D 23, 287 (1981). [12] R. Sundrum, [arXiv:hep-th/0312212]. [13] S. Weinberg, The Quantum Theory of Fields, Vol. 1. [14] M. Peskin and D. Schroeder, An Introduction To Quantum Field Theory. [15] G. T. Horowitz and J. Polchinski, Phys. Rev. D 55, 6189 (1997) [arXiv:hep-th/9612146]. [16] L. Alvarez-Gaume, C. Gomez, H. Liu and S. Wadia, Phys. Rev. D 71, 124023 (2005) [arXiv:hep-th/0502227]. [17] L. Alvarez-Gaume, P. Basu, M. Marino and S. R. Wadia, Eur. Phys. J. C 48, 647 (2006) [arXiv:hep-th/0605041]. – 21 –

0704.1147

Information, complexity, brains and reality (Kolmogorov Manifesto)

Information, complexity, brains and reality (Kolmogorov Manifesto) Starlab Technical Note TN00054 Status: Public Initiated: 06-2001 Revisions: 10-2002, 12-2002, 12-2003, 09-2004, 12-2004, 02-2005, 08-2005, 10-2005, 07-2006, 12-2006, 04-2007 Giulio Ruffini1 Starlab Edifici de l’Observatori Fabra, C. de l’Observatori s.n. Muntanya del Tibidabo, 08035 Barcelona, Spain 1email: [email protected] Contents 1. INFORMATION 6 1.1. Information and Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2. Kolmogorov Complexity, Science and Evolution . . . . . . . . . . . . 14 1.3. An example: compressing GPS signals . . . . . . . . . . . . . . . . . 19 1.4. On Gödel’s theorem and Algorithmic Complexity . . . . . . . . . . . 20 1.5. Turing machines in dynamical systems: towards absolute complexity . 21 1.6. Using Rule 110 for GA based compression . . . . . . . . . . . . . . . 24 1.7. Kolmogorov Complexity and the Value of Scientific Theories . . . . . 26 2. THE BRAIN 28 2.1. The closed Loop: self-modeling . . . . . . . . . . . . . . . . . . . . . 28 2.2. Pain and Consciousness . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3. Sleep and Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4. Neuroscience as part of the program to understanding Science and the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5. Music and dripping faucets . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6. Handling information . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.7. Oscillations and resonance . . . . . . . . . . . . . . . . . . . . . . . . 37 2.8. Presence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3. LIFE 42 3.1. The Theory of Interacting Turing Machines, Aging and Cancer . . . . 47 3.2. Natural selection as Computation . . . . . . . . . . . . . . . . . . . . 49 3.3. Maximum Entropy Production . . . . . . . . . . . . . . . . . . . . . . 50 4. PHYSICS 52 4.1. Time and Kolmogorov Complexity . . . . . . . . . . . . . . . . . . . 52 4.2. Time capsules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3. Kolmogorov Complexity and the Wheeler-DeWitt equation . . . . . . 55 4.4. The arrow of time: from simple to complex . . . . . . . . . . . . . . . 55 4.5. What is computation in Barbour’s Platonia? . . . . . . . . . . . . . . 56 4.6. Data, program, computer . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.7. Working with partial information: Statistical and Quantum Mechanics 56 5. CLOSING 62 ABSTRACT In these notes I discuss some aspects of information theory and its relationship to physics and neuroscience. Information has a rather central role in all human activ- ities, including science, and it is a well defined concept associated to the number of degrees of freedom of a state. The unifying thread of this somewhat chaotic essay is the concept of Kolmogorov or algorithmic complexity (Kolmogorov Complex- ity, for short). Another important notion in the discussion is that of a Self-Entity or Agent, which is defined here as a semi-isolated physical (or informational) sys- tem capable of controlling its physical interfaces or couplings with the rest of the universe. This working definition may be thought of as the conceptual primitive for the brain. I analyze implications of the assertion that the brain is a modeling tool exchang- ing information with the rest of the Universe and conclude that, ultimately, all questions about reality should be framed in an information theoretic framework with the brain at the center. Indeed, by “Universe” here I mean the universe of sensory input to our brains, the only acceptable starting point for the definition of Reality, and that for all practically purposes one can state that there is only infor- mation. As a consequence, any theory of everything must start from information theory. Put in other words, information theory provides the conceptual framework to understand the relation of brains with the Universe. In particular, one of the building blocks of physics is the concept of state. Information is the currency of the brain to talk about states. As a consequence of this view, I will try to show that it is natural to interpret cognition, in general, and science, in particular, as the art of finding algorithms that apprach the Solomonoff-Kolmogorov-Chaitin (algorithmic) Complexity limit with appropriate tradeoffs. In addition, I argue that what we call the universe is an interpreted abstraction—a mental construct—based on the observed coherence between multiple sensory input streams and our own interactions (output streams). Hence, the notion of Universe is itself a model. Rules or regularities are the build- ing blocks of what we call Reality—an emerging concept. I highlight that physical concepts such as “mass”, “spacetime” or mathematical notions such as “set” and “number” are models (rules) to simplify the sensory information stream, typically in the form of invariants. The notion of repetition is in this context a fundamental modelling building block. Although the discussion is quite high level and verging on philosophy, I try to illustrate these notions with examples from EEG experi- mental data analysis with subjects realizing low level pattern learning tasks, where repetition plays a central role. As it turns out, these ideas can be framed naturally in the context of an emerg- ing field called Presence. Presence studies how the human brain constructs the model of reality and self using replacement/augmentation of sensorial data. Pres- ence belongs to a wider class studying how cognitive systems build models of their environment and interact with it. In Presence, the technological goal is to place a brain in a controlled and convincing and interactive information bath. In the later sections I consider the fact that our brains have access to incomplete information about the state of the universe, and discuss its implications in the theories of statistical and quantum mechanics, both of which have this much in common: they deal with incomplete knowledge about the classical microstate in some sense. I emphasize that the so-called macrostate is not a state at all. In relation to biology and the origins of the brain and in the same vein, I argue that life and, in particular, nervous systems evolved to find methods to simplify and encapsulate the apparent complexity of the universe, the context being the usual one: permanence of the fittest (natural selection). Thus, biological evolution is Nature’s algorithm for the optimal compression problem, a fact that we can seek to exploit both in research and in practical applications. Finally, I try to address the problem of time in the same light. I discuss briefly the role of time in computation and evolution, and consider a timeless, computer-less description of reality in which all that exists is a program. 1. INFORMATION In these leisurely written notes, I analyze and discuss the implications of a real- ization I had while thinking about a particular experiment in a well-known and fastinating neuroscience experimental paradigm. In this experiment1, further dis- cussed below, subjects were presented so-called “standard” and “deviant” auditory patterns (i.e., controlled sound sequences delivered through headphones), and their electroencephalograpic responses recorded on key with the presentation of the pat- terns. This is a standard experimental set up to study the response of the brain and nervous processing to such stimuli using electroencephalography (EEG). EEG is a fascinating measurement technique exploiting the fact that our nervous system relies on electrical signals to communicate and process information. EEG is a tool provid- ing fast temporal sampling of brain activity (as opposed to others such as functional magnetic resonance imaging). The experiment, which I describe below, and related discussions led me to think about the brain as a pattern discovery machine seeking to model and therefore compress the input information stream. What we call mod- eling is in fact directly related to the concept of algorithmic complexity—a concept I had not explicitly encountered at the time but which intuitively I was looking for. I came across a book which discussed the issue of algorithmic complexity and, more importantly, defined Kolmogorov complexity, a very beautiful concept which is the unifying thread in the following sections [7]. Putting the brain in center stage together with the idea that brains seek to model and simplify the incoming flux of information has important consequences. I will try in these notes to place this perspective in some sort of a theoretical context. The motivation is the deep connection between physics, information science and cognitive systems (of which brains are a prime example). For example, consider the following question: what is Science? Surely, we all have an idea of that. Science is what scientists do...Physics, Biology are clear examples of scientific disciplines. Science is about observing phenomena and understanding nature, about extracting the rules of reality in different disguises. Science can and should be used to improve our lives. These are all terms of reference for science. Yet, we can give a more concise definition of science. A simple and intriguing answer is that science is what brains do. This should be taken as a definition of science—not a definition of what brains do, although in a sense they are equivalent. What this says, in essence, is that we are all scientists, whether we know it or not. As I discuss 1Experiment by the team of Neurodynamics at the University of Barcelona (UB-NL) led by Dr. Carles Grau. 101101010101 100101001010 Figure 1: The central concept in this paper is this: the brain creates the model of reality (including self) through information interaction (in and out) with the universe “outside”. The universe outside includes any intermediary body. The universe outside is the “information bath” below, brains try to simplify the incoming information flow gathered by our senses. Simplicity is power—this much should be rather clear. And I plan to walk further down this conceptual road and convince the reader that in some sense “simplicity” is actually “reality”. Science, in the traditional sense of the word, is of course also about seeking simplicity and compression. When a scientist develops a model of what is observed, the result is a tool for data compression. A good model can be used to literally compress the observed data. Just compute the difference between data and model and compress the result. A good model will give result in a difference file with mostly zeroes. Although it might be true that we all are scientists, science requires further discipline and a rather strict methodology. It could be called “professional thinking”. But in essence, the difference between a Scientist and a scientist is quantitative, not qualitative. What does this all have to do with reality? What is the essence of reality? This is a deep and very old question, asked for the first time a long time ago. If you are reading this, you definitely exist. What is reality for you? I would like to argue that the concept of reality can be firmly anchored in the concept of information—and that our perception of reality is the model we build to simplify our existence based on that information. Reality is a construct of our brains—a model. Reality is the model brains build to sustain existence. How does the brain create this construct? If we don’t understand this it will be very hard to progress in fundamental science. We have already encountered stum- bling blocks that elude us, such as the role of the observer in Quantum Mechanics. I think these issues are profoundly related. Here are some somewhat provocative statements related to the discussion and to be further ananalyzed below: 1. Reality is a byproduct of information processing. Reality is a model our brains, swimming in an information bath, build to sustain existence. 2. Mathematics is the only really fundamental discipline and its currency is in- formation. 3. The nervous system is an information processing tool. Therefore, Information Science is crucial to understand how brains work. 4. The brain compression efforts provide the substrate for reality (the brain is the “reality machine”). 5. The brain is a pattern lock loop machine. It discovers and locks into useful regularities in the incoming information flux, and based on those it constructs a model for reality. 6. Definition: A self-entity or agent is a physical semi-isolated system capable of selecting and partially controlling physical interfaces (“couplings”) with the environment. It is the physical version of a Turing Machine. 7. Definition (alternative version): A self-entity or agent is an informational semi-isolated system capable of selecting and partially controlling information interfaces (“couplings”) with the environment. 8. A self-entity has only access to partial information about the universe. 9. Information flows between self-entities only through (physical or informa- tional) interactions which we call sensors and actuators (interfaces). 10. The mutual information between coupled systems is nonzero (this is a conse- quence of physical law, e.g., quantum mechanics). 11. Pleasure and Pain (to be further defined) are the sole drivers of action, the goal functions. Our “firmware”. 12. Genes contain (mutual)-information about their targeted habitats. 13. A successful genome will have large mutual information with respect to its environment. 14. The information flux into the human brain is of the order of2 1 GB/s The first two statements, for example, provide a tentative answer to the problem of the unreasonable effectiveness of Mathematics in Physics: Mathematics is effec- tive at describing reality simply because reality is information, and mathematics is basically dealing with the management of information and modelling. In what follows we will suppose that we are physical instantiations of Turing machines, that all we really know about the universe comes through our senses and interaction via actuators in the form of information, and that brains seek to compress this information in a useful manner—and see where this leads us. Another aspect I will bring to the discussion is the problem of Time. What is Time? Why do we ”feel” time the way we do? For example, why do we have memories of the past, but no memories of the future if physical laws are time-reversal invariant? Where does the arrow of time come from? Information and computation seem to require both dynamics and therefore, Time. Computation and time flow go hand in hand, and it is hard to picture one without the other. Can we learn 2This is just a wild guess: I want to bring the questions to life, but the answer is far from obvious. something by looking at this old problem from an information theoretic point of view? Finally, let me clarify that the ideas in this paper are mostly original (to the extent that anything can be original), although by now I am fully aware that other people are working along similar lines and I am sure thinking very similar thoughts. Since I started working on this notes, books like “On Intelligence” [19] and more recently “The Computational Universe” [25] have come out, both of which deal with the issue of algorithmic complexity, and even a movie (“What the Bleep”). This is probably a consequence of the fact that we have now entered in full the Information Era, where information and brains have come into sharp focus. As I mentioned above, I encountered the notion of Kolmogorov Complexity when looking for a mathematical definition to describe the capacity of the brain to detect and learn patterns. I was amazed and excited by how well this powerful concept fit the notion I was looking for. I am sure these ideas will resonate strongly in other minds. Finally, I would like to clarify that the nature of this work is rather exploratory, a personal tour of some of the implications of information science and algorithmic complexity. I would have never started writing it if I’d waited for the moment in which it was all clear. Alas, this moment may never come! For all practical pursposes, then, this is a live document and I intend to continue working on it and revising it. Hopefully, in the meanwhile the reader will find something useful in it and take it along. 1.1. Information and Sensors Information is that which allows us to fix, completely or partially, the “state” of a system. By “state” and “system” we refer here to the standard physical concepts. Information theory was born many centuries ago (think of cryptography) but came of age with the advent of the telegraph and electromagnetic communications. I think that physicists have been using this concept for centuries, because the notion of ”state” necessitates a conceptual framework using information. In physical terms, information is what is needed to constrain and ultimately fix a microstate, and it is easy to see that one can do that in a succint or redundant manner. Let me give an example of what concretely a state is and how you would go about describing it with information. Consider a 2D universe with 3 undistinguishable particles. Then, the classical state of the system could be described by the following expression: S = {[p1x, p y ], [p y], [p You can read this as saying, “there is a particle at position such and such and velocity such and such, another one at...” etc. Each one of the entries is a number, and you have to decide what precision you want in each. Classically, we can just say that we need to provide 9 degrees of freedom. The classical concept of degree of freedom was closely related to that o information. In order to fix the state, a classical observer is allowed to make as many measurements as she likes. Measurements will disturb a bit the state, but in principle the error can be made as small as desired. So a classical observer in a classical world can get all the information describing the state of the observed system. In quantum mechanics the universe is also described by a state, and in this case the state is a function—the wavefunction. Each pair of classical degrees of freedom becomes a variable in this function. So in case above, you would have to specify a state of the form Ψ = Ψ(x1, x2, x3). Furthermore, this function is symmetric in the three variables if the particles are undistinguishable. In this case, however, measurements will disturb the state. The observer can access some aspects of the state (technically, she has to choose some basis), but not all. A quantum measurement provides information about a compo- nent of the state in some chosen basis, and destroys the rest. Quantum mechanics rules are really about what information an observer can access on the universe. In this light, one could say that classical and quantum mechanics theories reg- ulate how an agent can capture information from the universe or observed system, and that they do so in a very different manner. But information remains at the center stage in both. In fact, I would say that the limitations imposed by quantum mechanics highlight even more the role of information in physical theories. And perhaps more significantly, the central role of the observer in the story. Imagine an agent (human, machine) which is inmersed into an environment. The agent has programs which allow it to cope with the environment and thus survive. In order to put them to work, though, it needs information. It needs to assess, to some extent, the state of the environment. Through sensors, it recovers information from the environment, including the state of its own ‘”interfaces” (“In what position are my arms?”). In order to discuss states and information exchange, we need to carefully define what a sensor is. For the purposes of this work, I would like to propose the following information theoretic definition of a sensor: A sensor is a device that can capture and relay information from a sensed system to a commanding system. Note that according to this definition, certain objects or systems which we may not expect to have such a status will also fit the definition of sensor. For instance, a USB flash memory stick is a sensor, according to this definition, as is a cellphone! In my view, these are perfectly good examples of a sensor. Another interesting way to state the same is that there is an increase in the mutual information between the sensor and the sensed system3. When we say that a sensor transduces all we are saying is tha the information is encoded in the state of some physical system first, then in another. But the bottom line is that a sensor is there to capture information. Thus, sensors here are just information interfaces between the universe and the entity. This interfaces may be passive or active (sensing can be a rather active process, using computational feedback to focus on different information streams). The universe may be in any of many states. In general, a sensor provides only partial information about what state that may be, especially in quantum mechanics (more on this topic below). The combination of several sensors provides the means for a sharper identification of the state (through data fusion). It is more than conceivable that the evolution of sensors by entities is closely related to the ability for model building. Some sensors seem to work better than others, capturing relevant information for survival. From the evolutionary point of view, if some information streams add little value, the associated sensors must be ultimately discarded. In fact, this can also happen at the organism level. Sensor selection, sensor focus, is already part of the compression problem as approached evolutively. Sensor selection is equivalent to a particular choice of lossy compression. Of course, sensors are necessary but not sufficient. In order to have an interesting, in- teracting entity, sensors, “computers”, operating systems and programs are needed. Last, but not least, actuators are highly desirable! They close the loop with the universe, and they allow us to alter it as well as place our sensors in appropriate places (by, e.g., locomotion). One way to look at locomotion (e.g., walking) from 3Is this the Heisenberg Principle in disguise? When we measure something we disturb it because the sensed is also “tainted” by the sensor through this information exchange. This process seem rather fundamental and unavoidable. environment Figure 2: The brain creates the model of reality through information exchange (in and out) with the universe “outside”. In this case we show a brain interacting with the normal universe by bi-directional information exchange, using sensors and actuators. the point of view of actuators, is that when we walk, our actuators are sending information to the universe so that it moves and we can access other data streams. Finally, it is worthwhile developing new sensors. By this we mean devices which can capture aand constrain the state in different and more precise ways than our senses. A microscope is an example of this, as is a voltmeter. A more interesting concept is “sense synthesis”. By this I mean the creation of direct interfaces from the universe to the brain, bypassing our traditional senses and even the peripheral nervous system. Some of this have already been developed and tested, e.g., brain implants into the visual cortex to provide a sense akin to sight4. However, one can 4http://en.wikipedia.org/wiki/Brain-computer interface image the creation of totally new senses and perceptions by such direct access to the central nervous system, e.g., to perceive the state of a network, or of a mathematical matrix, or of a community. 1.2. Kolmogorov Complexity, Science and Evolution Here is a naive definition of science: science is what brains do. As scientists, we seek algorithms that lower the apparent complexity of our universe of sensory input. I apologize for the repetition, but I want to emphasize that here by “Universe” I just mean the universe of sensory input to our brains. The brain is some sort of a box into which a myriad signals arrive from all the sensory input (at what rate?) our body generates after interaction with the universe and from which information is “transmitted back” to the universe. And I would like to argue that the brain evolved to find methods to simplify and encapsulate the apparent complexity of the universe, the arena being the usual one: survival of the fittest. Simple “truthful” models empower agents with better chances for survival and greater control of our environment. I order to clarify this somewhat radical information-brain centered position, I would like to consider the following thought experiment. A person is placed into an enviroment in which computers prepare and control her sensorial input for the person (via sophisticated 3D displays, audio, haptics, smells, vestibular stimuation, etc.) and in which the person brain commands are intercepted at the level of peripheral nerves. If this experiment is done correctly, the person can be embedded or inmersed into a Virtual Environment. In this experiment, it suffices to describe the universe in terms of bits, because the universe is really a program in a computer managing the sensorial input—see Figure 2.. This experiment has been realized with simple organisms and is the central paradigm in the field of Presence, further discussed below. The reader familiar with the movie Matrix will easily recognize the concept. But I want to emphasize here the central role played by information. In fact we can recognize three main aspects: • a human agent • bydirectional human-machine interfaces • a machine agent All these have to be there in order to create the subject-universe loop. We recall here the definition of Kolmogorov Complexity. The Kolmogorov complexity of a data set is the length of the shortest program which is able to generate it. Solomonoff, Kolmogorov and Chaitin showed during the second half of the 20th century that this is a good, meaningful definition. In particular, although the precise length of the minimizing program depends on the programming language used, it does so only up to a constant. We must keep an eye on this aspect also: some programming languages may be more efficient than others, and easier to implement! How is the programming language itself selected? Here, aspects other than simplicity will no doubt play a role, such as the practical issues of implementation in “wetware”. As a first example of this concept, consider the sequence 1212121212121212121212121212121212121212 1212121212121212121212121212121212121212 It is easy to see that its Kolmogorov Complexity is rather small. Here is a simple algorithm to describe it: “Repeat 12 forty times” The fundamental property of the definition of Kolmogorov Complexity is that if we were to write this algorithm in another programming language, it would only differ in length by a constant. By constant, we specifically mean that this number will not depend on the algorithm we wish to translate. In effect, this constant is the length of the translating program (the one translating one language into another). Intuitively, we expect that the above sequence can be coded in a similar manner in different languages. As was mentioned in the introduction and the above example illustrates, the con- cept of “repetition” is a fundamental block in “compression science”. This concept is rather important and leads brains naturally to the notion of number and count- ing (and probably mathematics itself). Even simple programs like a file compressor (e.g., gzip), will detect this repetitive aspect in the data and compact a large file consisting of repeated 12s rather efficiently. Some very basic blocks of mathematics can be seen arising from the branch of information theory that deals with repetitive phenomena. Come to think of it, the notion of repetition leads to set theory, and counting leads naturally to the concept of number, primes, etc. Moreover, repe- tition is the building block of Recursive Functions, which are indeed known to be equivalent to Turing machines. As a familiar illustration of the fundamental nature of repetition consider this definition: a number is said to be prime if it cannot be (nontrivially) obtained by the repeated sum of other numbers. That is, if it cannot be expressed as the repeated sum of any other number other than 1 and itself (trivially once). That is, it is not possible to generate a prime number by repeating another number many times and summing. This implies that prime numbers generate a clone of Z embedded in Z (the structure of the set of integers) in the same way that 1 generates Z, and in an isormophic way. That is, Z ∼ p · Z. The prime p plays the role of the “indivisible” Consider now the hypothesis: the number of prime numbers is finite. In other words, there exist a set of numbers, S, such that the entire number system can be generated from them through multiplication. Well, we know this is not true (the number of prime numbers is infinite). This is similar to the fact that any formal logic system is imcomplete: the analogy is provided by the prime numbers as axioms. An infinite number of “axioms” is needed to fill Z. Compressive processes are probably very basic mechanisms in nervous systems. As mentioned above, I would also like to argue that counting, which is at the root of mathematics, is just such type of compactification. Simple repetition is perhaps the most common “pattern” detected by the brain. Counting is just the next step after noticing repetition. And with the concept of repetition also comes the notion of multiplication and primality. More fundamentally, repetition is the core of algorithmics. A really interesting definition of mathematics which is very fitting in this paper is provided in [32]: mathematics [...] is simply the discipline par excellence that draws the conclusions logically implied by any given set of axioms or postulates. In other words, mathematics provides the necessary tools for model building. I think this is a very nice and direct definition for mathematics, and it clarifies the role of mathematics in the human brain. Since we hold in this paper that the brain is a model builder, a compression machine, mathematics provides the fundamental substrate to do that, because compression is the reduction of a large set of data or conclusions into a small program or set of axioms. Not all instances of information compression are so straightforward. Here is a more challenging case: consider the data sequence S=1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548. Nothing seems to repeat in a simple manner—but it does! The reader will no doubt realize this is the numerical sequence for π − 3. Euler found a nice expression for π which we can use to construct an algorithm to reproduce this sequence (although we will have to implement an algorithm for taking square roots, summing, etc.): √√√√6 ∞∑ To obtain π to a given digit of accuracy, we need to compute a finite number of calculations and sums. To go beyond, the process is repeated. Thus, the algorithm to describe the above sequence (or, more importantly, an arbitrary longer version of it) is rather short. Recently, a digit-extraction algorithm for π in base 16 has been discovered [2], 8n+ 1 8n+ 4 8n+ 5 8n+ 6 Using this algorithm (a repetitive process!) the authors have managed to show, so far, that π is random in base 2 (i.e., normal5). Compression programs like gzip will not compact much a large file describing6 π because they are not flexible enough pattern detectors. Their approach to simplicity (compression) is not sophisticated enough. Pi is characterized by the non-repetition of its digits, which appear randomly (as mentioned, π is believed to be a normal number, totally democratic in its distribution of digits)! This is illustrative of the difficulty in the general compression problem. The generation of such complex sequences out of simple rules is the subject of many studies. Chaitin would call them computable numbers (and he would remark that most numbers are in fact not computable [6]). Wolfram treats this subject 5http://crd.lbl.gov/∼dhbailey/dhbpapers/bcrandom.pdf 6For π digits, see ftp://uiarchive.cso.uiuc.edu/pub/etext/gutenberg/etext93/pimil10.txt. ftp://uiarchive.cso.uiuc.edu/pub/etext/gutenberg/etext93/pimil10.txt in depth in his book, “A New Kind of Science” [44] from the point of view of Celullar Autamata. Although Shannon Entropy was the first attempt to quantify the complexity in data, it represents only a timid first step in the direction of algorithmic complexity. For example, the Shannon Entropy for the digits of π is maximal and cannot capture its inherent simplicity. The order in the digits of π is crucial to its simplicity. Shannot Entropy does not consider that and provides only a statistical analysis of complexity. Although repetition is not obvious in the above sequence, repetition is at the heart of any compression algorithm. This is because such an algorithm will have a “FOR” loop: the algorithm will state that a certain number of operations needs to be repeated over and over. Thus, we can state that repetition is at the essence of compression. An “unpacker”, a system capable of running a program, is of course needed to realize or implement the repetition procedure. The brain aims to simplify (compress) and manipulate incoming information. “Understanding” gives us power, speed of response, awareness of change. Our algo- rithms usually never reach the Kolmogorov compressive limit. This is due to noise, the fact that we only have access to partial information about our environment and also to our own modeling limitations. In what we could call “lossy compression”, information is modeled by some program which achieves moderate compression and predictability. This is of course sufficient for most purposes. In addition, there are many aspects of the data streams that a brain may not care about, and this is another important aspect: how to distinguish between what we want to model and what we don’t care about. A useful image here is that of a monkey in the jungle, enjoying a tasty banana by her lone self. Imagine an environment full of complex sounds, the daily soundtrack in tarzan-land. All these sensory inputs reach the ears of our monkey, and are acquired and processed by her brain. The monkey is relaxed. Suddenly, the jungle grows quiet. The monkey detects that a rule has been broken (the rule of “pseudo- random sound”, perhaps) and looks up, alerted. This monkey’s brain may very well have saved her, alerting her on time about the possible presence of predators. Similarly, in the same way that we try to model physical phenomena, from the micro to the macro, we must learn to model and control our environment. This is also carried out by discovering rules and simplifiers. Riding a bicycle entails the same discovery processes. We must begin by attaching a set of rules to the bicycle concept (invariants, such as shape, 2 wheels), and then learn to control the dynamics of riding a bicycle: we must encode Newton’s laws in our reflexes. Human beings, for good or ill, seem to have mastered well beyond other species the art of modeling. It is precisely this ability that puts us on top of the food chain, and in control of our trembling planet. In some ways, it is perplexing how beyond we are from our fellow travelers on Earth and how fast we have distanced ourselves from our evolutionary companions. From the point of view of the planet (as in, e.g., the Gaia Hypothesis) and our current lifestyle, this is definetely not a good thing. 1.3. An example: compressing GPS signals Here is a somewhat technical example that illustrates the relationship between com- pression and modelling, one that has to do with with Global Navigation Satellite Sys- tem (GNSS) communications. GNSS emit rather weak signals in the form of pseu- dorandom codes. These signals can be detected on Earth through cross-correlation with locally generated copies with the same codes, although by the time they reach the ground the emitted signals are well below the thermal noise level. When such signals are captured, e.g., through 1-bit sampling of the down-coverted data and storage into a hard drive, the resulting files appear to be basically noise. This is only partly due to the pseudo-random nature of the GPS signal. By the time it reaches the ground, the resulting signal is the sum of the original signal and a much stronger noise component. After 1 bit sampling, the voltage time series appears to be a random sequence of ones and zeros. Yet, correlation of this information with a GPS code replica (a model), produces a non-zero result—a correlation spike appears when data and replica are aligned in time (a phenomenon which provides the basis for GPS signal detection, of course). This fact can provide the means to compress a GPS bit capture file. To see this, write S(t) = R(t) + E(t), where S(t) is the bit stream, R(t) is the model bit stream, and E(t) is the error stream. If R(t) is indeed a good model, E(t) will have more zeros than ones, and can therefore be compressed using simple techniques. So even an approximate model can be very helpful for compression. Finally, the goal may actually be useful lossy compression. That is, the user or cognitive system (in this case, a GPS receiver) may not care to keep track of the noise part. Then, essentially, S(t) will be compressed, in a lossy fashion, to a concept such as “PRN NN with a delay of X ms and a Doppler offset of Y Hz”. In general, it will be pretty hard to define what noise is, but a first attempt may be to say that noise is whatever information does not improve the chances of survival. 1.4. On Gödel’s theorem and Algorithmic Complexity Algorithmic complexity provides also the backbone to study mathematical knowl- edge. The concept of compression applies as well to mathematical theorems and anti-theorems (statements whose negation are theorems). Once we can prove some- thing, thus transforming a statement into a theorem, we assert we can derive it from first principles by applying simple rules. Then, we have “compressed” that statement, together with all its brothers and sisters. Gödel showed that some state- ments cannot be thus compressed: they cannot be reached from axioms and rules. This concept, and the complexity-compression twist, are very nicely described in Chaitin’s paper in the book Nature’s Imagination, “Randomness in arithmetic and the decline and fall of reductionism in pure mathematics” [6], a truly marvelous piece. At a simpler level, something like that happens with prime and composite num- bers. As we discussed above, a composite number can be compressed, in some sense, into the product of other primes. A prime number cannot: it cannot be reached through the use of smaller primes and the multiplication rule. A prime number is an island, itself a generator of new truths (new composites). As explained by Chaitin and discovered also by Turing, Gödel’s theorem can be stated in the context of the following fact with regards to a Formal Axiomatic System: Consistent + Complete → There Exists a decision procedure. I include here a short glossary to interpret this statement (see [6] for a clear exposition): • Formal axiomatic system: axioms plus rules to produce theorems. • Consistent: you cannot prove A and not-A. • Complete: given B, you can prove B or not-B. • Decision procedure: given B, an algorithm to test if B is, or is not a theorem. If system is Consistent and Complete, then there exists a decision procedure: just go through all the proofs and check that wheter B or not-B pop out. Since by definition B or not-B are theorems, sooner or later the proof will be found (it exists). • Godel: there exist uncomputable statements in any consistent formal ax- iomatic system, making it incomplete. • Turing: There exist uncomputable real numbers. Hence, there cannot exist an algorithm to tell you if a program will stop (the halting problem). Hence, the halting problem is unsolvable....a Godel statement: the unsolvability of the halting problem. Also, there is no decision procedure. Chaitin argues that mathematics should perhaps become more like physics: laws (axioms) are added as needed, because they may be Godel statements: “...physicists are used to the idea that when we start experimenting at a smaller scale, or with new phenomena, we may need new principles to understand and explain what’s going on. Now, in spite of incompleteness mathematicians don’t behave at all like physicists” (page 42). This may be true, but many physicists believe in the existence of a TOE (Theory of Everything) capable of generating, from a finite set of first principles, all phenomena. I think Godel’s proof of incompleteness may be a severe blow to the concept of a TOE. As such, the theory will be a mathematical theory (based on a formal axiomatic infrastructure), and it will probably be incomplete: there will be state- ments which cannot be handled by the theory, and which will need to be added as new axioms—just as it happens in the number theory of Godel. These may be, for instance, statements about the the masses of an infinite set of elementary particles, or the dimensions of space-time. 1.5. Turing machines in dynamical systems: towards abso- lute complexity The Turing machine computational paradigm is a very powerful and successful one. How does it relate to the concept of Entity, to the paradigm of a brain with sensors interfacing to the universe? And what about the dichotomy of computer, program and data? If we think of a real physical system as a computer, and dynamics (classical or quantum) as the mechanism for computation, where is the program, where is the data, where is the computer? In the terms of the Turing description for computation, can we identify the tape, the state machine and the transition rules? One cannot help but thing that these dichotomies are somewhat artificial. Physically, there are no obvious boundaries: there is only a dynamical system. Turing Self-Entity Box Brain Stateo Initial Brain State Staten Neuroactivations Tape Universe Transition Rules Neurodynamics Tape head read Sensors Tape head write Actuators Table 1: The Turing and Brain-Universe or Self-Entity paradigms compared. One of the key aspects is the tape-head control, which manages the information interfaces. It is interesting to note here that we can describe our brain, sensor-actuator, universe, Self-Entity paradigm purely in Turing terms. In a Turing machine we have a tape and a state machine which reads and writes on the tape. The machine has a tape head for this purpose, and it is controlled by the Turing state machine to move across, read and write on the tape. In terms of the concepts described in these notes, the tape head acts both as a sensor and as an actuator. The tape head, instructed by the state machine driving it, chooses which data to read (it moves across the tape) and where and how to write on it. In our language, the tape is the universe, of course. And the State Machine, with its transition rules, is the Brain or Self-Entity. The parallel is pretty clear (see 1.5.), yet there is something to clarify further. In our description, the brain is a modelling tool, exploring the universe and seeking to find simple rules, invariants, to compress the tape. Can we program these operations? The answer I believe is yes, we can use a minimization paradigm. That is, we would define a cost function which evaluates the compression power of the models the program comes up with. The program is to create and test models, and shoose the shortest ones. Of course, a GA type of algorithm can be used for this, it seems pretty natural. Of course, the field of machine learning is devoted to this problem, and progress slow. The problem is very difficult. Why did Turing machines arise in the universe dynamical system? Because they are the means to compression and therefore “survival”. The universe is a physical, complex dynamical system. Evolution and natural selection apply to the evolution of physical variables. As dynamics unfold, as time passes, invariant structures in the data are naturally selected: that which perdures, remains. If this picture is useful, it follows that Turing machines, or self-entities, are emergent features of the dynamics, attractors in the chaotic sense of the word. I always thought that evolution and natural selection are really mathematical laws, with a reach well beyond the biological. That which perdures, is. Here this statement refers only to information. Recently, authors have discussed such things as the computational power of the universe [24] or of simple oscillating systems [36] and diverse physical media [18]. I believe the tools developed in those papers can be used to do away with the dichotomy of computer-algorithm length. That is, perhaps there is an absolute def- inition of KC: the shortest description of a computer plus algorithm length. Funda- mental physics may provide us with the ultimate, absolute programming language, and an absolute definition of KC. At any rate, it is important to keep in mind that we cannot focus only on the length of programs: the hardware to run programs is also important, and has to be taken into account in the evolution/natural selection analysis of Kolmogorov Complexity. From the physical point of view, there is only one computer, the universe as a whole. As a computer, it uses physical law/dynamics as its transition rules, and its state as the Turing tape. It is also interesting to note that the “tape” is the computer itself. Although this sounds a bit strange, in reality it is no different than a usual computer. A real computer is a dynamical, physical system. The programmer uploads a program, thereby setting up the “Initial Conditions” and “Boundary Conditions” of the system. When the return key is hit, the computer, as a dynamical system, unfolds. If the program is a good one, the system will reach a “stationary point” (it will stop) with information of relevance to the programmer. The universe, as a computer, is capable of simulating other computers, like the one I am using to type these notes. As a computer, the universe is definitely a universal one! It is also noteworthy that the “computer embedding” structure is actually im- plemented technically. A PC is a Turing Machine in the physical sense, using the laws of physics. It is used to emulate or “embed” another Turing machine, at the machine language level. This Turing machine is used to embed yet another one, the operating system...and son on. We thus have layers and layers of computers...the universe, your PC, the OS, your program, finally. Perhaps a better way to look at things is to say that the universe, or any other dynamical system, provides the substrate or infrastructure for the emergence of Turing CA PC Brain Ocean Universe PDE Evolution Box Board Hardware Wetware atoms Q-fields paper medium Stateo lineo In. State In. Brain State In. State IC IC In Pop Staten linen Soft. State Neuroactivations Ocean State Q-State Var. Time slice Popn Tape line out Data Memory Universe Ocean State QF history Var. history All pop Trans. Rules CA rules Program Neurodyn. Navier-Stokes Q dynamics Equation+BCs GA rules Tape in lineo input data Sensor in ICs ICs ICs popo Tape out line out output data Body out state Final State Final State popn Table 2: Different computational paradigms and their relation to the Turing Ma- chine. The Turing machine, a Celullar Automaton (CA), a personal computer (PC), a brain, the ocean, the Universe, a partial differential equation (PDE), a Genetic Algorithm or Evolution. Turing machines. The universe itself is not computing anything at all. The universe is not a computer. Computers are emerging properties of the universe. But can computers not simulate the universe itself? 1.6. Using Rule 110 for GA based compression Here we discuss an approach to the generic compression problem—finding a program to effectively compress a dataset. In general, one could say that a computer is needed and some sort of programming language space to search the solution in. I consider here a Cellular Automaton (CA). One can think of a CA as a dynamic system described by an infinite set of stacked arrays in which we start from a given array with discrete values (e.g., 1 and 0) (the initial condition) and an iterative rule for dynamics. The CA is defined then by the rule that describes the generation of the next array. The rules can be many. If we focus on only nearest neighbor rules, and a binary valued system, there are only 256 rules 22·2·2. The reader is directed to [44] for more on CAs. What I would like to disccuss here is the idea that based on the fact that the CA using Rule 110 (see Wolfram NKS [44]) is a Universal Turing machine7. That is, give the right inputs, it will emulate any Turing machine (computer and program). So we have a computer (a CA running rule 110), and a rather simple description of the programming language space (the initial condition). Now, CA programming is rather amenable to a search based on Genetic Algo- rithm (GA) implementation. A genetic algorithm is a procedure emulating (simu- lating) natural selection to find solutions to a minimization problem. The solutions are born, reproduce (sexually) and die (natural selection). If the solution space can be described by a discrete multidimensional space, then it is easy to code the 7See http://plato.stanford.edu/entries/turing-machine/ for a definition of Turing machine. http://plato.stanford.edu/entries/turing-machine/ 0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 10 0 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100 Generation Best: 0.29 Mean: 0.32638 0 20 40 60 80 100 Number of variables (100) Current Best Individual Best fitness Mean fitness 10 20 30 40 50 60 70 80 90 100 Generation Best: 0.29 Mean: 0.32638 0 20 40 60 80 100 Number of variables (100) Current Best Individual Best fitness Mean fitness Figure 3: Test solution with a box fit to target (top middle). The solution DNA is composed of 100 variables. The first 8 define the CA rule, while the rest provide the initial conditions. Top left, with any CA rule, and on the right with a fix for rule 110. algorithm. The array describing the solution candidates is used as DNA in evolu- tion. Coding is the hardest part, because it should be done to best exploit the GA searching procedure. What I have implemented is a GA to evolve the right initial conditions (the program) to have rule 110 emulate a set of data. The algorithm can also search in a more general rule space if desired. I would like to use Taitin’s problem (discussed below) as a test case. So far, I can show basic results using rather simple patterns (see Figures 1.6. and 1.6.). The results are encouraging but also illustrative of the fact that searching in the space of programs is not trivial! In connection with this, it would also be interesting to see if a CA approach to a Pattern Lock Loop machine would be useful. 0 20 40 60 80 100 120 140 0 5 10 0 20 40 60 80 100 120 140 10 20 30 40 50 60 70 80 90 100 Generation Best: 0 Mean: 0.01612 0 50 100 150 Number of variables (150) Current Best Individual Best fitness Mean fitness 10 20 30 40 50 60 70 80 90 100 Generation Best: 0.31 Mean: 0.36318 0 50 100 150 Number of variables (150) Current Best Individual Best fitness Mean fitness Figure 4: Test solution with a line fit (time series) to target (top middle). The target is a repetition of the basic pattern [1 1 0 0 0 0 0 1 1 0 ]. The solution DNA is composed of 150 variables. The first 8 define the CA rule, while the rest provide the initial conditions. Top left, with any CA rule (112 is selected), and on the right with a fix for rule 110. 1.7. Kolmogorov Complexity and the Value of Scientific Theories There are two ways, in my opinion, of furthering scientific knowledge. One is by developing theories that can best match the observations one is trying to “explain”. This is the standard way that has been used to assess the value of a theory. For instance, General Relativity is a better theory than Newton’s because it matches in a better way the observed data and it can actually predict correctly new phenomena or otherwise poorly unexplained observations (e.g., the perihelion of Mercury, or the existence of black holes). However, I would like argue here in the context the of Kolmogorov Complexity that a theory that leads to equal predictions but using a “shorter program” is also an instance of scientific progress. If we can produce the same data stream with a simpler model, we have advanced our science. At this point, the reader should not be surprised by this statement! Thus, a theory that “pares down” concepts like “time”, “aether”, and “scale”, and leads to the same predictions as the older, more complex theories, represents advance. Usually, and perhaps surprisingly, such simpler theories will not only re- produce the old observed facts but also lead to new results and predictions (General Relativity is again a good example). This is just a way to rephrase Occam’s Razor8 One should not increase, beyond what is necessary, the number of entities required to explain anything. The better perfomance, or existence, of simpler theories could indicate that reality itself “runs” on simple theries. Searching for a simpler theory would then get us closer to the target, with better performance. How simple is really reality? What is the algorithmic complexity of reality? 8Ssee http://pespmc1.vub.ac.be/OCCAMRAZ.html for more details. http://pespmc1.vub.ac.be/OCCAMRAZ.html 2. THE BRAIN Part of the motivation for this paper is the realization that we must put the brain at center stage if we are going to advance in fundamental science (physics if you ask me). We do not really know what is “outside” our brains. All we know is what we can interpret and model using our sensors and what they tell us about the outside world. In fact, for all we know, there may not be an outside world at all. But we suspect there must be a unifying source somewhere, however, as the inputs provided by our different sensors reveal coherence. This is not a proof of anything, of course. Given enough time we could find any finite sequence we want in a randomly generated stream (as it is conjectured for the digits of numbers such as π. Come to think of it, this thought is not so distant from present concepts such as the Cosmic Landscape (see below for a discussion of Susskind’s Cosmic Landscape [41]). But I think most of us expect to find simple explanations at the end of the line. The brain receives input, an information multi-channel (temporal) stream. In order to gain an edge, it must try to implement a sufficiently fast predictive model of this information. The multi-channel aspect is rather important. Scientists trying to fool our senses in virtual reality environments (Presence researchers) know that synchronization, coherence, is a key aspect in a successful charade. The model in our brain is constantly evaluating its performance against reality. When there is a mismatch, the brain produces an Alert event. The model fails and this gives rise to an alert signal. The goal of learning is to produce as few alert events as possible. Perhaps this provides the ingredients for a suitable definition of pain. Pain may be equated to poor modeling and hence, painful surprises. In complexity terminology, the input stream has a given complexity which can be quantified. The brain develops a model that will, in general, fail to reach the compression limit (due to hardware limitations, for instance, or simply for algorith- mic search limitations). Model minus data will not be zero. The output “alert” stream carries the uncaptured complexity. 2.1. The closed Loop: self-modeling Part of the model of the universe that our brain seeks must take into account the model itself: the brain is not an input-only system, it produces output, and this affects the Universe (i.e., the input). There is something very mysterious about environment Information Human Machine From Information to Models Figure 5: The brain creates the model of reality through information interaction (in and out) with the universe “outside” using sensors and actuators. If we “hack” the inputs and outputs to the brain with an intelligent, pursposeful information stream, we can put this brain, i.e., fool it into believing, in an artificial environment. There is also the very interesting possibility of bypassing the senses and going directly from machine to brain in both directions. this self-coupling, and it may be directly related to consciousness or at least self- awareness (consciousness is probably a more primitive feature). In particular, our brains decide which input streams to select and focus on (e.g., by moving or looking in a different direction, by selecting and following a particular conversation in a a party, or by changing a tv channel9). Modern views of AI state that “embodiment” is a fundamental aspect of cognitive systems. This view is easily interpreted here by stating that the cognitive agent needs to be able to act in the information environ- ment and regulate the input information stream. That is, the agent can control, to some extent, the information interfaces (sensors), and actuators. We will see again this concept in a re-analysis of the Turing machine concept below. In our concept of the Universe we must include our immediate universe, our limbs and body. We also have to model our inertias, our latencies, our strengths and limitations. Walking is a very complex act! The propioceptive system is a key element in this self-monitoring loop, and it is probably tightly linked to self- awareness. In summary, we have to model both the Universe and our impact on the Universe 9If you still watch that thing. (something which may be the real meaning of “us”). What we do affects us, and Reality kicks back [10]. This may be the key to the self-notion (self-awareness vs. awareness, which is a more primitive notion). The interaction between self and universe can be conceptualized like the inter- action between two programs, one for the self, the other for the universe, where at least one seeks to model the other. Of course, this is just an approximation. In principle, there is only one program running: the universe. 2.2. Pain and Consciousness Why do we feel pain and pleasure? We can easily argue that this is done to ensure our survival, of course. Pain stops us from doing things that may harm us, or makes us move into more favorable situations (hunger, thirst, sex). We are rewarded with negative pain—pleasure. But more specifically, I would like to suggest that Pain and its negation (Pleasure) are also the fundamental phenomena that drive us to model the Universe more efficiently. Pleasure is the reward to a good, simple model. When we have the right models we improve our chances of getting where we want. And pain is the punishment for a bad model. A scientist feels pleasure when discovering a good theory, and pain all along the way. We feel pleasure when we “get” a joke, when we “get” a complex song. In fact, pleasure and pain are the sole drivers of action, and they seem to be truly fundamental for animal behavior. One can also think of pain and pleasure and emotions in general, as the “firmware” in the brain that defines the goal functions for cognition. As firmware, it is some- thing we cannot easily reprogram, and it resides deep in our brain. 2.3. Sleep and Memory Why do we need to sleep? For several reasons, most likely. Although many explana- tions have been put forward this is as of yet an unanswered question. Restoration, repair and learning have all been proposed [28, 40, 13]. In my view, the last case is the most convincing one, i.e., the simplest theory to explain the facts, or at least a very interesting possibility, and I will analyze it with the conceptual tools developed here. The idea I would like to propose is that during the day we gather more or less raw information and put it into a temporary buffer. This buffer is adapted and optimized for the amount of information we gather during our waking hours, using the time-off at night strategy we have evolved. During night, sensory inputs are shut down (a hallmark of sleep) and the accumulated data is processed: the apparent complexity of the ingested information is processed and reduced as much as possible, to translate it into rules, or short algorithms to be hardwired. The buffer is then mostly emptied. The discovered rules (they are no doubt approximate rules) are better kept. They support our long-term memories and long-term models. Thus, during the day we “exercise” our models (put them to work) and gather further information. During the night, we recess into our self-universe, shutting off sensors and actuators, and we indulge into model development. Of course, finding regularities (rules) in data economizes the process of memo- rization. As we discussed above, instead of memorizing the sequence for π (which is infinite), just store the algorithm, it is much more economic. Compressed mem- ories are somehow less fragile. But compression and model building are equivalent processes, and the result of a smaller file size is not the only advantadge. A model is a much more practical implementation of knowledge than raw data. Control sys- tems for things like, e.g., a body confronted with the task of snowboarding, require efficiency and speed. Relying on old visual and propioceptive raw data to decide the strategy for the next bump won’t do it. A model, a set of rules to decide the next move is a much more efficient approach. And implicit in such a model lies the body model and a good portion of Newton’s laws. I remember dreaming, as a young boy, the relived experience of going down a bumby slope after the first days of the season. Just as if my body was making an effort to get up to speed—learning in my sleep. I just read something on model building in a news feature and related article in Science (Nov 2006). In this case, the cognitive agents were robots programmed to modify models of their own bodies (and, extrapolating, their environment) from past experience and in this way improve locomotive performance [1, 5]. In particular, the cycle would include the design of specially directed experiments to select among competing models. This is achieved through “actuation-sensation” interaction with the environment, of course. The methodology is an in the scientific method, where data suggest models, which are then tested via dedicated experiments. Is this what we do when we sleep? Model building and simulated testing? Another interesting piece of information comes from the work reported by Mas- simini et al. [29], in which a link is made between the phenomenon of consciusness and brain connectivity. It is shown that in NREM sleep brain connectivity is de- creased as compared to a wakeful state. This indicates that the brain “decouples” during NREM sleep, and that consciussnes is linked to integratory processes in the thalamocortical system. If learning occurs during NREM sleep, can we argue that such decoupling is needed? Is the brain fine tuning subsystems, and then trying them out during REM sleep? Although this is a question we cannot answer at the moment, let me point out that some work I have been doing with other colleagues may provide useful tools to further investigate brain connectivity during sleep [34]. In this work we analyze the complex networks associated with brain electri- cal activity. Multichannel EEG measurements were first processed to obtain 3D voxel activations using the tomographic algorithm LORETA. Then, we computed the correlation of the current intensity activation between voxel pairs to produce a voxel cross-correlation coefficient matrix. Using several correlation thresholds, the cross-correlation matrix was transformed into a network connectivity matrix and analyzed. To study a specific example, we selected data from an earlier experiment focusing on the MMN brain wave. The resulting analysis highlighted significant differences between the spatial activations associated with Standard and Deviant tones, with interesting physiological implications. When compared to random data networks, physiological networks are more connected, with longer links and shorter path lengths. Furthermore, as compared to the Deviant case, Standard data net- works are more connected, with longer links and shorter path lengths–i.e., with a stronger “small worlds” character. The comparison between both networks shows that areas known to be activated in the MMN wave are connected. In particular, the analysis supports the idea that supra-temporal and inferior frontal data work together in the processing of the differences between sounds by highlighting an in- creased connectivity in the response to a novel sound. This work was a first step into an interesting direction, I think, because it is clear that the brain is a complex network and complex network theory has today much to offer [4]. A strange consequence of the hypothesis that sleep is used for model building based on daytime data acquisition as discussed above is that our brain buffers would be different if we lived in a planet with a 40 hour cycle, or if we were surrounded and driven by more complex phenomena. However, this idea is probably testable. The more complex a task you have to learn, the more you will have to sleep. To some extent, overdoing it, burning the candle, allows us to put more in our buffer, which is essential to discover rules about complex phenomena. And following this line of thought we can also ask, Why do we dream? Dreaming is perhaps related to the firing off of our memories in a random way, in the process of looking for algorithms. Stephen LaBerge defines the difference between awake and dream-like states of consciousness based on sensory input. Dreaming is like being awake without the constraints and realignment from sensory input. During waking hours, sensory input is constantly used to realign our models. This may seem to imply that our models are too unstable, but I believe it is important that this be so: we need to learn new things, and retain flexibility in our model searching. Dreaming may be the means to test our models in a safe way. Using stored data from past experiences, the dreamer tests his/her new algorithm for riding a bycicle, say, or for skiing. This provides a safe testbed. From this point of view, dreaming is not the most fundamental block necessary for learning, rather one of the last steps. Recent research on the importance of REM and other stages to procedural learning shows the relevance of REM is unclear, while the deeper sleep stages seem to play the key role. In its simplest form, memory is a tool to store raw information. We would therefore expect short term memory to represent data in a less processed manner. After this data is processed and simplified, it is encoded as long term memory. The prediction we make is therefore that short term memories have a higher apparent Kolmogorov complexity, while long term ones are more compressed. As as species, or a group of brains, humans underwent a revolution with the invention of speech. Speech allowed us to share the burden of memory and process- ing. Memories could be passed from generation to generation via word-of-mouth— clearly an advance, although certainly error-prone! The subsequent invention of writing allowed humans to store in more permanent form both data and models. The transition to modern times, with the provision of an enormous power to access universal data and models will certainly have a profound impact. Thanks to devel- opments such as the Internet and Google, it is unlikely that the old emphasis on blind memorization will survive for long, as we now have very fast and accessible exterior memory buffers. Emphasis will shift to data handling and processing. The true experts of the future will be masters at data-interfacing and model making, more than at storing facts. The universe outside our brains is better becoming our Turing tape, supporting our computation and making us better Cyborgs [9]. An alternative is science fiction’s memory chip, or a direct brain interface to an external database. These technologies, far fetched as they may now sound, are surely in the horizon. 2.4. Neuroscience as part of the program to understanding Science and the Universe The title of this section refers to my understanding that, although physics, tra- ditionally the most fundamental science (i.e, seeking to understand all aspects of objective reality), and neuroscience, a scientific endeavour focusing on the workings of brains and hoping to relate them to subjective phenomena, have been separate fields, the division is artificial and a better theory will have to treat them as part of the same endeavor, in a unified way. One aspect that needs less in terms of an explanation is rather obvious. Neuro- science has now entered the realm of quantitative science, thanks to the availability of sensors (e.g., multi-channel EEG, fMRI, direct chip to neuron interfaces, etc.) and computational resources powerful enough to attack the problems of data pro- cessing and model making. It is a new Era for neuroscience, and I expect great advances in the following decades. Similarly, and although physicists has been able to work quantitatively now for centuries, the advent of computers has opened new lines of work, allowing them to model phenomena using tools beyond analytically tractable differential/integral equations. As briefly discussed before, there are mechanisms in place in the brain that are well adapted to modeling patterns and detecting change. Some of the exciting exper- imental work carried out today (which is what sparked my interest in compression) involves studying the response of the brain to sound input sequences. It is possible today, with accurate timing to send some stimulus to the brain (visual, auditory, haptic) and then measure the response as seen in EEG. Typically, it is necessary to average the response over several hundreds or thousands of such events in order to filter out the noise (the brain seems to make a lot of noise to the untrained ear) and observe what is called an Event Related Response (ERP). In Mismatch Negativity experiments (MMN), which are a further specialization of ERPs, auditory patterns are generated and the response of the brain is measured using an EEG recording system (see [35, 17]). The experiments can then measure the change in response when the patterns are broken. The phenomenon of MMN basically illustrates the fact that the brain, at some very primitive level, is able to learn patterns. We can also measure how many iterations are needed before the pattern is assimilated, and we can try to understand the type of pattern that the brain’s mechanisms involved are best at “understanding”. In this context, the brain acts like a “Pattern Lock Loop” system. At Starlab we carry out research with ERP/MMN for different applications. An interesting (very) preliminary result of our work together with UBNL is that simpler patterns appear to be more easily learned. The way this is observed is by comparing the response of the brain to the perfect sequence. When there is a deviant in the sequence, the EEG records show a change with respect to the standard response. This illustrates the fact that the brain is acting like on-the-fly modelling tool. What makes this especially interesting is the fact that the level of response to the deviant appears to be related to the complexity of the sequence. In the first experiment three sequences were tested: 1: ABABAB ABABAB ABABAB .... 2: AABABA AABABA AABABA .... 3: BAABAB BAABAB BAABAB ... The reader may notice that the sequences have been ordered according to complexity (which one is the easiest to memorize?). Even a simple program for compression, like gzip, is more efficient at compression with the first, then the second, then the third sequences. The first sequence is efficiently described by the AB subunit and the space (common to all). To describe the second one, we need A twice and then BA twice plus the space. The last sequence requires also two subsequences, BA and AB, the first one repeated once, the second twice. Our results are rather preliminary and we have not investigated yet is the num- ber of iterations needed for the brain to learn and assimilate the pattern. It is worth mentioning here that present understanding seems to point out that simple brain mechanisms are responsible for learning these sound sequences. The subjects in the experiments are not paying attention to the sounds, they are told to read or undertake other high level activities. In fact, depending on the severity of their con- dition, patients in a coma are also capable of assimilating the patterns and detecting change. There appear to be rather low level “pattern lock loop” mechanisms in our brain. Are these the building blocks of reality? They certainly occur at the very first processing stages, and drastically reduce the data influx passed on to higher processing levels. How can we describe the algorithmic complexity of these sequences? In order to do that we will have to chose a programming language. We will use a rather simple one. It will contain the simbols A, B, S (space) and Rn() (repeat n-times, with 0-times meaning “forever”). For instance 1: ABABAB ABABAB ABABAB ....= R_0(R_3(AB)S) 2: AABABA AABABA AABABA ....= R_0(R_2(A)R_2(BA)S) 3: BAABAB BAABAB BAABAB ....= R_0(R_1(BA)R_2(AB)S) We can see that the length of the programs grows as expected. Is this how the human brain does this encoding? How can we test it? 2.5. Music and dripping faucets Why do we like music? This is a very interesting question. I think (and this has been said before) that there is a profound relationship between music and mathematics. As a species we seem to be quite generally and cross-culturally attached to the generation of organized sound (a working definition for music). We can frame this question in the present context of complexity and answer that we like musing because music is pattern, and we are addicted to pattern catching. This is the reason why difficult (good) music takes a few rounds of listening before we “get” it (read that as “model it”). Could MMN and music be closely related? Perhaps the same could be said of humor. When we smile and feel pleasure on getting a joke, we may be experience a litte Eureka moment in which the joke data stream is simplified by the joke meaning. Why do we hate the sound of a dripping faucet? Because the timing of the drops is chaotic, we can’t find a rule for it, yet we are teased by a myriad of possible patterns. This process of never ending pattern search becomes physically painful (recall that we hypothesized above that pain is associated with mismodelling). It is probably exhausting, and we could think of measuring this effort using a tool like fMRI (which is used to measure metabolic activity). In summary, when we listen to a piece of music, we are probably engaging a “tracking” program in our brain, with an associated model. Listening to a good song is an active process. In a sense, we are unwrapping a model in real time and testing against the song. It is a process inverse to model building. Perhaps there are a few parameters (tempo, transposition, instruments) that we need to adjust to sustain the tracker, as in a Kalman filter. In doing that, we are simulating reality (the song) and benchmarking it with the outside reality (a good book on this subject is “This is your brain on music”, by D. J. Levitin [22]). 2.6. Handling information Why is it said that men and women think differently ... or physicists and fashion designers...? Because depending on our universe of interest, we need to develop different information encoding and management systems for simplification: infor- mation handling processes adapted to our needs and our environment. This kind of observation is part of the philosophy of the old Starlab Media Impossible Consor- tium (Starlab NV, also known as DF1). The Media Impossible project targeted the development of technologies for automated meaning extraction of media. Something to allow asking questions such as “show me the portions of the game on tv with goals”, or “show me all the movies I have in which Scarlet Johansson is wearing a green tie”. As human beings we are concerned with specific aspects of our envi- ronment, and we develop rules to handle the part of the information environment we care about. We have concepts like “person” or “shirt”, and even “justice” or “cup”. These refer again to invariants, or approximate invariants in our universe. Depending on who or where we are, we have to refer to and work with different invariants. A recent article in Nature [43] discusses the relationship of the propioceptory system with our physical modelling capabilities. Intuitively, it makes sense that we have internalized physical models as a result of our interaction with the universe. I am certain I knew quite a bit of physics before my first course in 6th grade, just from my own body experimentation. I would like to add here the conjecture that women and men think differently, and in particular, have different affinity with physics and engineering, because they have different physical and social specializations which make them interact with the Universe differently. Their model of the universe is slightly different than men’s. 2.7. Oscillations and resonance Neuroscience research, and especialy indicates that the brain can be thought of as a ringing bell, and that oscillations and their phase relationships are tied information processing and influenced by stimulus presentation [27]. A bell will resonate, when the correct air pressure time-series pattern is coupled to it. A sine wave is a simple pattern. The pattern is actually the periodic pressure time series, that is, the coherence and regularity of the pressure time series with respect to the Master Clock. Can we conceive of a different computation/compression paradigm based on coupled oscillators (oscillating rings, such as those described by Rietman and Tilden [36, 37], or even mechanical oscillators and beyond [18])? Can we make a chaotic system capable of “tuning” to patterns? This would seem to be closer in spirit to the concept of computation using dynamical systems. Could we design such a model and simulate it? This is the goal of creating a “Pattern Lock Loop” machine. The oscillatory paradigm seems to be taking a more and more prominent role both in neuroscience and in robotics. The common link is non-linear dynamics. Attractors can provide a mechanism for dynamical based memory storage and pro- cessing. In effect, memories can be represented or encoded by particular attractors. Biologically, the brain complex can indeed be seen as a very large set of coupled dynamic components. Bio-inspired approaches to robotics seem to be yielding in- teresting results. 2.8. Presence Reality, according to this paper, is a model or construct in the brain of the beholder. As has been argued, among all the possible models that can account for our observa- tions, the simplest ones are ranked higher by the brain. The only limitation to our construction of reality is the capability of the brain to come up with good, simple models. Nevertheless, it should be clear that the enhancement or manipulation of sensorial input to our brain should therefore have a very powerful effect. Presence Science studies how the human brain constructs the model of reality and self using replacement/augmentation of sensorial data (VR and beyond). Presence belongs to a wider class studying how cognitive systems build models of their environment and interact with it. The field was originally inspired by the subjective experience of being “there” and how to achieve and modify that experience in virtual or augmented environments. In more general terms, then Presence is an emerging field focusing on under- standing the cognitive experience of being (somewhere, someone, sometime, etc.) and developing technologies to generate and augment it (being someone or some- thing, somewhere, sometime, without physically being there). By nature a deeply interdisciplinary field, Presence spans a wide range of sub- jects: from neuroscience and cognition to artificial intelligence, sensors and systems. Aided by new technologies such as virtual and augmented reality, AI, wearable dis- plays and high-end cinemas, Presence research aims at empowering us to achieve realistic feelings and experiences when immersed in a virtual environment. The feel- ing of being somewhere is the result of a complex interaction of many technological, nontechnological and personal factors. A fundamental understanding of these fac- tors will allow for construction of virtual and augmented environments with greater effectiveness. Presence research will lead to many new technologies and enable pow- erful applications: communications, learning, robotics, etc, that are more affordable and usable in the workplace, at home, in school and even on the move. Although one can detect elements of Presence in simple situations (reading a book, watching a good movie, holding an engaging telephone call), true Presence tar- gets full inmersion, and this requires sophisticated technologies for Virtual Reality— some of which have yet not been invented. Note here that Presence is not the same as Virtual Reality. The field of virtual reality focuses on technology development. Presence focuses on the cognitive aspects needed for ... presence, and it shall guide technology development. The following fundamental research pillars of Presence can be identified: Hu- man Cognition, Machine Cognition, and brain-machine interaction. All of these are essential aspects of Presence. And they have at their center or overlap the phe- nomenon, the subjective feeling of some reality with other mediated agents, human or machine. As one important working methodology, Presence uses the benchmark10 for the successful replacement/augmentation of sensory data with virtual gener- ated data with success defined by analyzing the response of the subject in phys- iological, behavioral and subjective terms in relation to a real situation (this is the concept of “ground truth”). But Presence can deliver much more than normal reality. As a field, Presence, in more than a way, is the science of existence. It can also be called the science of illusion. Research in this field aims to understand and measure our experience of being (somewhere, somehow, ...). In many ways, we have all encountered situations in which some means are used to alter our “presence”. Two examples will suffice. The first is what I would call the experience of “nightlife”. Why do people party more at night (as opposed to during the day)? Because at night the photon flux from the sun is greatly reduced, and therefore so is the natural sensorial input into our brains. At nightime it is easier to artificially manipulate our sensorial input. Anyone who has experienced a disco bar knows what I am talking about. A more primitive form of Presence is illustrated perhaps by the 3D vision draw- ings fashionable in the 90’s, where the sensorial input to both eyes can be made “coherent” by slight divergence of the eyes’ gazes, and thereby provide a simpler explanation for the brain. Coherence of sensorial input, understood as a simplifying agent, is an important elemente for Presence. 10Definition from EU FP6 Presenccia Integrated Project, coordinated by Mel Slater, UPC (2005). Another example is the “presence” one can feel when reading a good book. You can feel you are “there”. Here we have a sensorial replacement by sensorial input generated by our own brains (at some level). Imagination is perhaps the best known tool to experience presence. And, of course, dreams can be extremely vivid. In some ways, humanity has been striving for higher Presence: storytelling, books, theater, cinema, caves and now virtual reality... Reality is the ultimate technological challenge. Following the logic laid out in this essay, we can state that in order to echieve “more” Presence, simplicity in the inputs is a key aspect. We state this as an hypothesis: Given alternative models for the sensorial input, a brain will select the simplest one it can construct. For example, in the “fake” hand experiments using a rubber arm, virtual, or mixed reality setup as described in [20], the subject will tend to select the simplest expla- nation. In the experiment, the real forearm is hidden, and a fake one is displayed and stroked. While the fake arm is stroked, the real one is stimulated but hidden from view. In this case, the models the experimental subject observing the fake hand and experiencing real hand stimulation would consider could be • That is simply my hand being stroked • That is my hand being stroked by a brush as seen through a TV • That is the image of my hand as seen through a TV, but stroked by a real brush • There is a complex set-up in place to fool me into believing that that is my The complexity of the “explanation” of the experience increases from top to bottom. The rubber hand experiment can be fitted by the first model. Of course, the better rendition of a hand we can provide, the less noise the subject will need to deal with to accept the illusion. Another example is provided by the “Pinocchio” illusion. In this set up, a blind- folded subject is made to stroke a third party nose while his/hers is simultaneously stroked. The coherence of inputs (haptic inputs through nose, hand and propiocep- tion) supports the “I have a long nose” theory. A higher level explanation involving the cortex (“there is a set-up to fool me”) is more complex and disfavored: keep explanations simple. To summarize, we conjecture here that the feeling of Presence, as measured by subjective or objective ways, is increased if the induced sensorial input (the input data stream), has a low complexity, i.e., it can be modelled in a simple manner by the subject’s brain. Coherence in the inputs, in this sense of there being a simplifying model, is an important element to enhance Presence. Bayesian expectations are also an important aspect: explanations with a better match with past experiences are inherently simpler. 3. LIFE What is Life? Who invented that? I would like to argue that Life can be analyzed in information theoretic terms as well: A living being, or Entity or agent, can be defined to be a replicating program that successfully encodes and runs a (partial) model of reality, thus increasing its chances of survival as a replicating program. Thus, an Entity in this discussion is a perduring information conformation. For example, a prion, in an information or algorithmic sense, is a stable information conformation (representing all prion instances). There is only one “prion algorithm”, and this is the perduring Entity. The terms “agent” or “entity” are interchangeable here. The first is a more com- mon term in the AI community. The second highlights more the implicit subjectivity of being. If we imagine a Celullar Automaton model of the universe, a perduring informa- tion entity is a subset of cells that more or less perdure under time evolution—e.g., we can imagine a set of cells which in time moves around the board (as in the Game of Life). Here we face again the boundary problem: there is no natural bondary line between a program, data and computer in the physical universe. In the same way, there is no obvious way to draw the line between “me” and the Universe, and the concept of Entity is already fraught with ambiguity. In the same way, the Turing machine paradigm is lacking. It appears to require an artificial boundary, not unlike the observer-universe boundary in Quantum Mechanics, which has created so many unresolved conceptual problems. Here we encounter a recurring problem. The concept of compression relies on the concept of “data stream”. Like water in a river, information flows from the universe to the entity. Time is needed in this discourse—apparently. Time is needed for to talk about Computation. Without the concept of Time, all we are left with is the Program. But note that the Program contains already everything, the Computation is intrinsic to it. In this sense, “Time” is in the sequence of digits of π, each digit a tick of the clock. The algorithm freezes Time. The dichotomy of Program and Computer is also associated to Time—the com- puter provides for time evolution, it is what in QM is called a Propagator—so this reflection may provide the way to do away with it. Part of the code that makes up a spider includes the rules to build a web. A spider web, in this sense, is the output of a smart program that encodes part of reality (the reality of flies, wind, rain, hunger, etc.). A successful DNA string encodes part of reality. The input of this DNA code is its immediate chemical environment. Its output is the proteins and other RNA strings it will make, given the input. A successful DNA molecule can also control its environment to a healthy extent. In order to do that it must encode a good model of how this environment is expected to behave. In fact, we should really think of DNA-based life in the context of a theory of interacting Turing machines—at least as a first step. Another way of seeing this is to realize that by looking into the DNA of a living organism we can gain knowledge about the environment in which it is meant to populate. It follows that the mutual information of the code and its environment must then be non-zero. We can find similar ideas in [10]: So living processes and virtual-reality renderings are, superficial differ- ences aside, the same sort of process. Both involve the physical em- bodying of general theories about and environment. In both cases theses theories are used to realize that environment to control, interactively, not just its instantaneous appearance but also its detailed response to gen- eral stimuli. Genes embody knowledge about their niches. Everything of fundamental significance about the phenomenon of Life depends on this property, and not on replication per se.....Life is the physical embodiment of knowledge. The Brain is therefore a natural evolutionary step in Life: to build a large organism and complex organism it is more convenient to pass from few to many sensor, to getter a better “view” of reality. The brain has perhaps evolved to process massive input from sensors. Natural selection drives the universe towards the establishement of perduring complex entities. Why? Because to perdure, information about the environment must be encoded in an efficient, executable form in the perduring entity—and the environment is complex. Anything that lasts must “know” about its environment and be fast in its knowledge11. 11Here is an interesting idea for a virtual experiment: run Tierra with a complex environment background, to demonstrate the evolution of complex programs. This is a generalization of the usual genetic/reproductive description of natural selection. Thus, Natural selection is the reason behind our Kolmogorov minimization capacities. Only those entities that can model and thus have the opportunity to control their environment the best will survive. This is entirely in line with the definition of Life given above. However, Natural Selection and Evolution can be based on a biological information handling (such as DNA and genes), or on other concepts (e.g., Dawkins’ memes). The key concept is that of a program—regardless of its physical instantiation of the memory and computation system. The high level activity we call “Physics” is a clear example of our brain’s never ending goal of simplifying the universe (of sensory inputs). A simple example are Maxwell’s equations, which provide a good model for electromagnetic phenomena (and associated data streams). In four lines (and the machinery behind) we can com- press radically the information associated to electromagnetic pheonomena. Physical theries achieve their maturity when the can account for a wide range of phenomena with a few equations and axioms. A more fundamental illustration of compression comes from the concept of in- variant. Take the concept of “mass” or “charge”. What is mass? Mass, a physicist will tell you, is simply an invariant. Mass is a rule that says “In your data, if you combine the information from this and that in such a way, the result will not change over time or space. This quantity can be used to predict future dynamics.” In special relativity, (invariant or rest) mass is the invariant associated to elemetary particles by (mc2) = E2 − (cp)2 (where c is the speed of light). That is, it em- bodies a rule that tells us how to combine energy and momentum measurements is a particular way, to get always the same number, no matter what the particle is doing or in which frame of reference we are measuring from. The concept of mass, at fundamental level, is a rule, a model for what we observe. The same applies to charge, spin, energy, particle, etc., and even time. None of these really exist in the classical sense of the word. They are just elements in our models of what exists. Julian Barbour states in his recent book, “The End of Time” What we call time—in classical physics at least—is simply a complex of rules that govern change. According to this point of view, Time is definitely not a fundamental concept, just another rule, another gear in our model. So Time should not be a fundamental block in our Theory of Everything! The same can be said of any other physics rule: mass, space, spin, charge, ..., none of them are fundamental reality blocks. The substrate of Reality is surely not “material”. Materialism is a red herring. Rules, invariances: there is simplicity hiding in the apparent chaos. Conserva- tion laws are a special type of invariance laws: they refer to invariance under time translation. They are part of the Program in plain language. In classical physics, once the equations of motion are solved, the entire history can be succintly sum- marized by defining and giving a value to the invariants. This can be achieved, for example, by a clever change of coordinates that make the Hamiltonian zero. There is another interesting connection between the Kolmogorov limiter and time. Once you have the limit program, you have frozen Time. Time is related to the data stream, but once the Program has been found, time is just a crutch that falls off. If the program is not the limit program, then you need less data stream. In a way, you have less need for time. Once you have an invariant, Time disappears. This is an interesting connection. From this point of view, the Brain seeks to do away with Time as well. Physics is based on “dynamical laws”. Time is part of the “unfolding”, or decompression of the universe model. If someday we do find the Theory Of Everything, it will read something like Maxwell’s equations, something of the form12 dF = 0. The goal of finding a Theory Of Everything is to find the ultimate compression algorithm. We will then have the algorithm to generate the universe in a faithful way. If we do find a final model, we will still need a big computer to unfold it into predictions. In, fact, there may not a faster way than to ”run a universe” to see what phenomena the model contains. But what is the Kolmogorov Complexity of the Universe? And where is the boundary across programs? Where does the program end and the computer begin? Such distinctions are clearly artificial. From a certain point of view, Mathematics can be see as providing the computer language to write the programs of physics. Of course, if the algorithm can itself be simplified, we will not have finished yet. The form of the final theory, if we find it, may be different than expected. 12Feynman points out in his Lectures that one can vacuously write all the equations of physics in an apparently simple form: U = 0. This is not what we mean here! Inventing notation is not a valid way for compression. Susskind, in his recent book titled ”The Cosmic Landscape” [41] addresses the is- sue of simplicity of Physics Laws in a direct matter. Pointing out the extreme fine tuning of physical constants and particle types for the benefit of Life (notably, the cosmological constant) he asks if we can expect to find a simpler physical model to encompass all, or if there isn’t a real explanation, if a fine tuned theory to explain all the miracles does not exist. The second option would be compatible with the existence of a multiverse (megaverse as Susskind calls it) that would allow for all possible combinations of physical constants (and theories) in a multidimensional cosmic landscape. In this framework, we observe the precise and miraculous combi- nations of constant values that we do because we, as human beings we are part of that solution, of that point in the the cosmic landscape. This is the modern form of the Anthropic principle. That is, there are many other universes in the multiverse, but in most of those there are no human beings (or organic matter, or even stars) there to observe it. It was once hoped that String Theory would impose a complete set of constraints on all the constants and particle types, but this is not the case. There is a landscape of solutions. This landscape gives scientific credibility to the anthropic principle. This principle is in fact very much in line with the ideas in this paper, where the brain is at center stage. Anyhow, I feel it is rather premature to jump to conclusions on the Landscape and the Anthropic principle. After all, string theory is not the widely accepted way forward (although this may now be changing). Nevertheless, the landscape possibility is rather interesting. The question then is, what law defines the landscape (in this case some final form of String theory). The goal of simplicity in explanation is certainly still a valid guiding theme. These new developments are in no way a departure from traditional scientific work. Here I would just like to point out something I concluded during my thesis work [31]. That is, that gauge theories are just a mathematical constructs to be able to work in topologically non-trivial phase spaces. This concept seems to mirror nicely the point of view of Barbour on time. In essence, time is gauge. There is something of a mistery here: to simplify things we must enlarge first the space of discourse! This sounds like a shift of complexity from the program to the programming language (which now needs to include, e.g., ghosts in the BRST formalism). At any rate, the point is that by adding dimensions or degrees of freeedom to a problem the description turns out to be simpler (in the algorithmic sense). In reality, any calculus student knows this from the use of Lagrange multipliers to solve a constrained, minimization problem. Adding the Lagrange multiplier dimension (one per constraint) make the problem easy to solve, as opposed to first imposing the constrain and then solving the minimization problem. I think there is something deep here that appears in fields as diverse such as constrained minimization theory, Gauge theory [33] and the theory of Support Vector Machines [8]. In the latter case, the problem of classification using hyperplanes in some dimension is solved by a map into a higher dimensional case, where the it is possible to separate cleanly the two desired regions using a dividing (linear) plane—see Figure 6. Embedding and non-linearity are both needed because the topology of an easy-to-classify problem is trivial, while in the starting feature space topology of the class regions will be complex. 3.1. The Theory of Interacting Turing Machines, Aging and Cancer Consider a fertilized egg, the fusion of sperm and ovulo. This is a program, encoded in the DNA sequence. This program is immersed in an environment, the chemical environment of input and outputs. Part of the program includes a statement that says, “when the environment is such and such, divide”. We can then think of this as a Turing machine, interacting through the environment. An organism is a society of cells, so we can say that at the end an organism is made up of a very large number of Turing machines. They form and organism because the communicate efficiently and because they are somehow “aligned”, acting coherently to help the organism survive. As time goes by, small changes occur in each Turing machine, and these changes are not all the same. This degrades the coherence of the organism, and the organism ultimately dies. This is perhaps why we reproduce starting all over again, from a single, coherent program, the single fertilized egg. In a recent paper it is argued that evolving from a single fertilised egg makes evolvability more likely [43]. The issue of aging is also related to cancer. At some point it appears that some cells in our body rebel against the system and become autonomous, rebellius, hijacking organism resources until it ultimately fails. This is a rather mysterious event. It would seem, from a natural point of view, that the tumor and its colonies become in many ways an independent organism. That is, a tumor is a rather complex machine, capable of hijacking resources, evade safety mechanisms, etc. Such an event would not appear to be an accident—rather the opposite, there seems to be purpose here. However, if the tumor is a new organism, a parasite of the host, how does it reproduce, how does it enter the natural selection game? It is interesting to note that recent studies link cancer and stem cells—cells not unlike the original fertilized egg (see Scientific American July 2006). 1d space 2d space Class 1 Class 1 Class 2 Class 1 Class 2 Figure 6: How to solve a classification problem by increasing the number of dimen- sions and using a non-linear map. In the 1d space case there is no “plane” (point) which can divide the space into two sections with the correct representation. In the 2d case, after a non-linear embedding, this is possible—a hyperplane exists that separates the two classes. Alternatively, cancer is a fast self-destruct mechanism to expedite demise, as we discuss next. 3.2. Natural selection as Computation Why do organisms age and die? An idea I would like to consider is that this is because this is the solution found by Nature (in our local universe) to make perduring entities. Here we could distinguish between entities and meta-entities. Entities are what I called before living organisms. Meta-entities would be the pattern defining the Entity. For instance, a prion would be considered an Entity, and there can be many instances of this entity. There is only one meta-Entity, though, the pattern for a prion. That is, having organisms die allows for more room for reproduction, mutation and genetic algorithmic improvement. Clearly, in a finite world with finite resources, without aging or death, there is no space for evolution—there is no room for repro- duction once the resource capacity of the habitat is exhausted. And if we are in competition with other evolving organisms, that is not a good strategy. So, we must die and reproduce. And the perduring entity is not the individual, but the individ- ual “stream”. Here I note that environmental stress is a cause for cancer—which is a fast route to demise. Perhaps we accelerate our “turnover” when the environment sends us agressive signals. Evolution here is to be thought of as a form of computation, the means to obtain perduring entities. Of course, evolution is only needed if “perdurance” is under stress. If we have found the solution to perdure without biological evolution (or, at least, natural bio- evolution), aging and mortality are no longer a must. I think we will some day stop and eliminate them in our current “individuality” culture. The stream will stop, the pattern will remain. So, this reasoning fits well with a picture in which: • The Universe is computation. • Entities are programs (in an abstract sense, that is, two copies of a running program are effectively the same program). • Programs evolve. This is also a form of computation (a Genetic Algorithm). • Programs that perdure, perdure and are around. Note that “perdurance” can result from successful reproduction (bacteria) or high resilience (diamond). • A program will perdure if it has captured the essence of its environment. • In order to perdure, sometimes it suffices for a program to evolve slowly. Other times, when the environment demands it, it better evolve faster—competition is tight. • Computation cycles and their duration, at some level, are defined by the die- mate/mutate-reproduce cycle (GA). • Computation (reproduction) consumes resources, and it should be avoided if possible. • Thus, there is an optimal “age” for every program which depends on the environment: a hostile environment to the “program”, invites a faster clock (shorter lifetime, perhaps through cancer or other self-destruct mechanisms), a benign environment the opposite. Turtles and elephants have long lifespans. Bacteria the opposite. It is worth noting that recent research (e.g., see New Scientist, April 19th 2003, p 27) indicates that organisms have a “lifespan knob”. That is, there are molecular knobs that can alter lifespan. These ideas suggest a virtual silicon experiment: let the lifespan itself be a genetic trait that can be altered, and see how a species adapts to different environments (with limited resources penalizing fast reproduction). In terms of a GA, we are saying that the cost function is changing with time (perhaps suddenly). As a result, mutants with shorter lifespans will adapt faster and take over. Finally, note that the title of this section hints at the fact that there are many other forms of computation. Even in evolutionary computation, the use of “repro- ductive hardware” is not strictly necessary. Memes are another example of programs that can evolve and perdure. The use of DNA for information technology is probably just one approach among many possibilities. 3.3. Maximum Entropy Production Several papers have appeared recently that point out to a very interesting link be- tween complexity and non-equilibrium thermodynamics. It has been shown that non-equilibrium systems such as the Earth (driven by the Sun) settle into a state of maximal entropy production given the available limits (constraints). In statistic mechanics terms, such states have the highest probability. There could be an in- teresting link between MEP and KC in the context of Life. If systems such as the Earth-Sun seek MEP, then Life may search for powerful analysis tools of the hyper entropic data streams. The universe, our universe is a complex place. 4. PHYSICS 4.1. Time and Kolmogorov Complexity What is Time? Julian Barbour calls it The Great Simplifier, in reference to Ephemeris Time. I would like to argue that our brains have found Time precisely because it greatly simplifies our understanding of the universe. We find, in Barbour’s book, an interesting quote of Mach that says it all: ... time is an abstraction, at which we arrive by means of the changes of things. In Barbour’s book we find a very clear exposition of the so-called Tait’s problem. The problem is as follows (the reader is invited to [3] for details). Suppose we gen- erate on a computer a set of numbers (triads), representing the separation between 3 particles in a free motion, non-interacting: ...... ..... ...... 7.07107 10.6301 8.06226 6.08276 9.21954 7.07107 5.09902 7.81025 6.08276 4.12311 6.40312 5.09902 3.16228 5.00000 4.12311 2.23607 3.60555 3.16228 1.41421 2.23607 2.23607 1.00000 1.00000 1.41421 1.41421 1.00000 1.00000 2.23607 2.23607 1.41421 3.16228 3.60555 2.23607 4.12311 5.00000 3.16228 5.09902 6.40312 4.12311 6.08276 7.81025 5.09902 7.07107 9.21954 6.08276 8.06226 10.6301 7.07107 9.05539 12.0416 8.06226 10.0499 13.4536 9.05539 ..... ..... .... Let us now ask the question, “What is the shortest program capable of generating these numbers?”. These numbers have been generated by the following scheme. Figure 7: Three free particles make a simple universe. First, the 3 particles are put in 3-space, like we usually do: ~P1 = (1, 1, 1) + τ(1, 1, 1) (1) ~P2 = (0, 1, 0) + τ(0, 1, 1) (2) ~P3 = (1, 1, 0) + τ(1, 0, 1). (3) Then, the distances between the particles are generated, r12 = (~P1 − ~P2) · (~P1 − ~P2) (4) r23 = (~P2 − ~P3) · (~P2 − ~P3) (5) r13 = (~P1 − ~P3) · (~P1 − ~P3) (6) The parameter labeling “time”, τ , is just that, a parameter. Could we write a program able to infer these equations (in the form of an al- gorithm, or in a neural network) from the data? This is what our collective and historical brain has been able to do. r12 coordinater13 coordinate Figure 8: The Platonia view of the 3 particle universe’: only the separation between the particles is relevant. Time arises as a Kolmogorov minimizing construct to simply describe the data. 4.2. Time capsules Time capsules, as described by Julian Barbour, are special states in Platonia in which a “coherent history” is described. They are in a sense very structured and “consistent”. Barbour uses the example of geological records which provide consis- tent accounts of “historical developments”. I would like to use the term of mutual- information to refer to consistency: consistency can be quantified using the mutual information concept. If you take a data stream and it can be highly compressed (i.e., it is simple), they it must follow that subsets of the data have a high mutual information content. In another way we can see that the concept of Time is redundant. Time is needed to have the concept of “Computation”. Computation is intrisically a dynamic pro- cess. But as it should be clear that computation does not contribute anything to the information of the initial state: everything is already in the program plus data, which is available at time zero (for those familiar with the concept, a canonical transformation exists which makes the Hamiltonian zero). Both Time and Compu- tation (hence computers) are probably just auxiliary concepts (i.e., gauge concepts in the language of gauge theory). 4.3. Kolmogorov Complexity and the Wheeler-DeWitt equa- Can we show that the states selected by the Wheeler-DeWitt equation highlight configurations with a low Kolmogorov Complexity? What I would like to suggest is that Time Capsules represents minima in the Kolmogorov Complexity of the universe configuration. From this point of view, the quantum Wheeler DeWitt equation would be interpreted as an information-theoretic equation: it selects states which minimize the Kolmogorov Complexity of the universe. For example, consider the fact that in the Time Capsule you find yourself now in, mechanical conservation laws are an important holding rule (if this is not the case, you are in a different type of universe!). In the records you will find around you things like mass, the amount of water in a glass, energy and such, are all (at least approximately) conserved. The fact that this happens seems to imply our brains like to exist (or can only exist) in intrinsically simple universes. 4.4. The arrow of time: from simple to complex Why do we remember the past, yet have a very limited picture of the future? This is a deep and difficult question. I would like to argue that we do have a certain memory of the future (we can predict the next 5 seconds reasonably well, at least in our immediate environment). We can predict that mass will be conserved, that the outside temperature will not change too much..that we will take a plane in the afternoon.... If we follow Barbour’s thesis, in which the past is just a structure in the brain (now!), then why do we seem to do a better job in one temporal direction than in the other? Is this a problem related to the Kolmogorov Complexity of the Past and of the Future? It is certainly not a property of the data itself. It is fascinating to think of this problem when only Nows exist! Another point: why do we only have measurements of the past? Note that the rest is pure data extrapolation (IC, BVP, Newtonian mechanics, Analytic Con- tinuation...). Is it really true? Do we not have imperfect models of the future (extrapolation all the same)? 4.5. What is computation in Barbour’s Platonia? The universe is information. In the platonic terms of Barbour, I think it suffices to state that the universe is just a number: the number specifies the state of the system. There is no time, nor space. Only information. This picture is devoid of space and time, matter, energy.... These are derived emerging concepts. There are patterns in the number, and these can be compressed through the use of these concepts. It would follow that these patterns are compressed by subportions of the uni- verse...our brains! 4.6. Data, program, computer One of the concepts that do not sustain a thorough study is the dichotomy of computation, in its traditional terms. A computer is an entity different from the environment: there is the input, the computer, and the output. Such an approach is not useful if one wishes to think of the Universe as a computer. It is possible to define portions of the Universe as computers, but, as usual in cosmology, the extension of the concept to the whole Universe is really troublesome. This situation happens in Quantum Gravity, for instance. When dealing with the whole, a Machian approach is needed. Consider a close system filled with bouncing balls (the billiard universe). The system is in some way computing its own future constantly. But where is the Turing machine, where is the tape? Note that this “computer” can be described in a timeless way by the constants of the motion coordinates (or, more directly, by the Hamiltonian and the initial conditions). A subset of billiard balls can be selected as the computer, the rest as the tape, for instance. Both the timeless and the usual time description are in this picture. 4.7. Working with partial information: Statistical and Quan- tum Mechanics Human brains have access to limited information about the universe. Our sensor systems tap only to data combinations using a limited number of inputs, etc. This is natural and efficient—why perceive everything? The implications of this simple fact are rather interesting. Consider Statistical Mechanics (SM). This is a beautiful subset of physics which deals with what knowledge we can extract from physical so-called macrostates. Now, strictly speaking, a macrostate is not a state at all! When we say that a system is in a macrostate we really mean that we have some limited information about the microstate the system is in, and that this limited information defines a region of phase space in which the microstate lies. A macrostate is really a set of microstates. In fact, the first clue to the crucial role of information in physics, in my opin- ion, arises from Statistical Mechanics (SM for short). The key concept in SM is microstate and counting. Counting is made possible, especially in the quantum ver- sion of SM, because the number of states is countable. Recall that a set is countable if it can be put into a 1-1 relationship with the integers. Being able to count enables the quatification of the number of states in a macrostate. Another important clue, that in my view detaches physics from classical theo- ries of reality comes from quantum mechanics and the concept of the wavefunction. Without going into much detail, I just want to emphasize that in quantum mechan- ics we are forced to talk about measurements, and reality is described by rules to predict the outcome of experiments. One can and probably should work on quan- tum mechanics without paying attention to the existence of something ”real”. The ”essence” of reality is not needed to do quantum mechanics. Special Relativity began in a similar way, by questioning things we take for granted in ”reality”. Con- cepts such as space-time or simultaneity, which we take for granted, were questioned by Einstein with great profit. He was challenging, in a way, what we take to be the ”essence” of reality. To the question of what is information, and its relation to SM, I propose the following answer. Information can be thought of as a set of rules to describe, fully or partially, a microstate. In mathematical terms, information is associated to a set of constraints. In algorithmic or logic terms, these are described by a set of rules. A microstate is fully specified by a complete set of rules. A macrostate is specified by a partial set of rules. The Entropy of a macrostate is the number of bits to specify the missing set of rules to fix the macrostate into a microstate. This is an algorithmic definition. In this way we can talk about the Entropy of a microstate. Let L be the length of the program required to fully specify a microstate S. The entropy of a macrostate is E = L−M , where M is the number of bits used to describe the macrostate. Here we assume that we are working at the KC limit. In normal statistical mechanics the ”programming language” is simple. It is a list specifying the position and momenta of each particle, or otherwise specifying the state. In QM things become more complicated due to the fact that all particles are undistinguishable. In such scenarios, the lenght of the list is equivalent to the more familiar concept in which the entropy of a macrostate is defined by the ”number of mcro-states” it is really referring to. The Entropy of a macrostate is the amount of missing information needed to fix it into a micro-state. According a Noether’s theorem, invariances in the action lead to conservation laws. For example, time invariance of the action leads to conservation of energy, and space displacement invariance leads to conservation of momentum. Following the approach explained here, we could say that time and space are concepts derived from the need of human brains to seek invariances. We can talk here about a more information oriented definition of a self-entity: as self-entity is a semi-isolated information system using a controllable information membrane. It is the ability to control information contamination from the exterior that really makes the self-entity. As self-entities we focus (and this is probably not a coincidence) on variables such as temperature, which are rather immune to this lack of knowledge: their probability distribution is very peaked. Well, we deal with averages of things, and the law of large numbers comes to rescue. We can more easily predict. However, note that when we say that the temperature of the system is T (canoni- cal ensemble), or that the total energy of the system is E (microcanonical ensemble) we are really defining a constraint for the state of the system—a constraint in a huge phase space. Similarly, the entropy of a macrostate is a measure of our lack of information, about the size of the phase space in which the microstate can be given the information we have. If a macrostate is really a set of microstates, entropy is a measure of this set: it is just a count of how many microstates are in the set. Wouldn’t it be interesting to work with Algorithmic Complexity instead of En- tropy? That is, instead of counting how many microstates there are, we should pro- vide the length of the shortest program providing a description of the microstates. In Quantum Mechanics (QM), it is now being forcibly argued that the state is not a physical state at all: it is a representation of the the information our brain has about the system. This is very similar to what was just discussed on statistical mechanics. Furthermore, Quantum mechanics carefully describes what our sensors can and cannot tell us about the universe—think of the Heisenberg principle. As described in Section 1.1., Quantum mechanics is first and foremost a theory describing what information an observer can access about the universe. So here is an interesting common aspect to Quantum and classical Statiscal Mechanics. In both Statistical and Quantum mechanics we work with something called a state which is not the state at all (the classical microstate): call it the quantum state or the classical macrostate. In both cases it appears that this state represents our knowledge of the “real” state. As little detour, I would like to go back for a moment to Cellular Automata (CAs). As described above in Section 1.6., there is one rule for elementary binary CAs which is a universal computer and therefore irreducible—rule 110 discussed above. This means that in order to model what will happen in the future, there is no other approach than brute force computation of every single step. Such data streams are therefor incompressible, and their kolmogorov complexity is basically the length of the data stream. In [21] it is shown that it is possible to “coarse grain” such CAs and make them compressible. This means to work with averaged out effective forms of the CA. Something like what we do everyday, when we work with incomplete information. In other words: even though the original data stream is incompressible (irreducible), if we accept to work with some effective data, we can compress it. I think there is a clear connection here to the discussion above, in which we have highlighted the coarse graining aspects of two fundamental theories. Working with macrostates may allow for the development of models. This may be sufficient for, e.g., survival (unless the essence is in the details). Another interesting point is that in Statistical mechanics you have as the most primitive concept the microcanonical ensemble. The microcanonical ensemble is the set of microstates defined by fixing an overall total system energy. It is a rather natural concept, because with time translation invariance in physics laws you have conservation of energy. We could think of it as the statistical mechanics framework of the universe. However, if you are interested about the physics of a subsystem which by itself may not conserve energy (because of interaction with the rest of the system), you work with the canonical ensemble: this is done by averaging out over all states outside the subsystem of interest (“tracing out”). Out of this very inter- esting process, the canonical ensemble is born and the canonical partition function is defined. This looks a lot like a the Hamiltonian propagator in QM, except that time is changed by a complex quantity, 1/kβ (see the lucid discussion in Feynman and Hibbs [15]). The connection between statistical mechanics and classical dynam- ics (the Hamiltonian) has to do, no doubt, with the ergodic hypothesis. In SM, the Hamiltonian is our “chaperon” or escort of phase space: it takes us, eventually, everywhere where the energy in constant. Thus, we can do time averages for con- stant energy phase space averages—which is what we want in the microcanonical ensemble. All accessible microstates are equally probable over long period of time, or, equivalently, time averaging and averaging over the statistical ensemble are the same. In the SM of a free gas, for instance, the Hamiltonian is the sum of the Hamil- tonians for each particle. A related fact: Planck’s constant has the units of action. Action units are volume units in phase space. Both in QM and SM, to compute the Feynman path integral or the statistical partition function, the key operation is a sum over volumes in a large phase space. In SM the space is large because you have many particles. In the QMs of a single free particle you sum over all possible paths in time. A phase space path integral is really a sum over a super-phase space though (think about the ergodic hypothesis!). In the QM phase space path integral, a single particle moving in time emulates, mathematically speaking, a statistical ensemble of free particles. The resulting expression for the propagator is symbolized by DxDp e pdx−Hdt DxDpe dt(pẋ−H) , (7) where the measure here means DxDp ≡ dxidpi . (8) The QM measure for the integral is that of a phase space in which there are N particles. The propagator is a weighted sum over this (super) phase space. No wonder there is a strong (mathematical) connection between the QM and SM—as also discussed in Feynman and Hibbs [15]. An important difference is the imaginary term in QM. This means, somehow, that instead of having on probability you have two numbers. And the, of course, there are all the aspects of interpretation of the amplitude. Time is in this picture from the beginning, however, because in the microcanon- ical ensemble you define a total energy of the universe, and energy and time are strongly related (canonical conjugates). By the way, this “canonical” process of tracing degrees of freedom out is also in discussions by Barbour on how inertia arises from the distribution of matter in the universe. The total universe is postulated to have fixed (zero) angular momentum, but local subsystems do not need to conserve momentum. Inertia can be seen to arise from this “effective” theory. 5. CLOSING The underlying theme of this set of notes on the theme of Kolmogorov Complexity has been the relationship between Brain and Reality, or Neuroscience, Information theory and physics. I have argued that we cannot really progress in our understand- ing of Reality without understanding the Reality Machine, the Brain. I have also argued that compression is the main task of the Brain, and that from compression the picture of Reality arises. The discussed conceptual framework can be summarized as follows: • The brain receives all its information flows from the senses, and constructs models for the incoming data. All we can assert for certain is that Information exists. The brain is really a Turing machine, with the tape head role being played by active sensors and actuators. • The universe is a dynamical system providing the infrastructure for the evo- lutive emergence of Turing machines. • These models generate the concepts we associate with reality. They are reality as far as the brain is concerned. • It is a key aspect that we have access only to a limited set of information, that we cannot really identify the Universe microstate. This is due to our lim- ited sensing powers, and this in turn is driven by evolution. This limitation is no doubt closely related to the information regulation rules in Quantum Mechanics and Statistical Mechanics which we have “discovered” and their focus on working with partial information. As organisms, we have developed sensors and algorithms focusing on limited information availability, e.g., tar- geting quantities which are stable (“the temperature of a system”) and useful to us (“the position of the particle”). • The driving force in the brain and in model building is compression. Thereby arise concepts such as space and time, “the Great Simplifier”. If a model and its concepts can compress the data, then we assign a level of reality to them. Useful concepts and models are real. • Evolution and Natural selection play an important role. Models for the uni- verse evolve. Our brains themselves have evolved as compressors. Compression is power. DNA itself contains compressed representations of “reality”. • In a more speculative and platonic vein, the universe can be conceptualized as a number. This number can be thought of as representing the dynamical data stream. In this number (perhaps π!) there are patterns, and these interact. Some of these are Turing machines. Somehow, some portions of the Universe contain information on the rest. Our brains do that, DNA does that. As argued by Barbour, this is the mechanism through which the concept of time arises. The Nows contain information about the Pasts. Here I will try to list a series of more concrete questions to address as part of this research program: 1. How can we in practice differentiate and quantify complexity? Can we differ- entiate the DNA sequence from a live organism from a random sequence? 2. How much can we compress the DNA sequence for “Drosophila”? What is its Kolmogorov Complexity? Does this correlate in some sense with the complexiy of its “effective universe of interaction”? 3. Can we discover the encoding mechanisms of the brain by studying learning processes as described above? Does the brain encode in a specific way, opti- mized for our environment? How does the Pattern Lock Loop machine work? Can we replicate it in a simulator or electronic hardware? 4. What experiments can we design to test the role of KC and “pattern reso- nance” in brain function? 5. Is there any fundamental difference between Reality and Virtual Reality? The area of Presence is precisly addressing this question by defining experimental benchmarks of what type of information baths feel real. 6. What is the Kolmogorov Complexity of the Universe (this is a difficult one, but physicists are working on it)? 7. What is the Kolmogorov Complexity of a dream universe? 8. Can we measure the coherence (which can also be defined in terms of the Kolmogorov Complexity of the entire set of programs in an organism) of an organism and correlate it with age? 9. Can we show that the states selected by the Wheeler-DeWitt equation high- light configurations with a low Kolmogorov Complexity? In a sense they do, because they satisfy that equation. What makes that equation special in the complexity context? 10. Can we devise pattern lock loop programs? A possible approach may be provided by CAs (rule 110) and GAs. This is an interesting, practical area of work with “real” potential applications. Acknowledgments I wish to thank Ed Rietman, Walter van de Velde, Carles Grau, Josep Marco, Julian Barbour, Bert Kappen, Isabel Bouhom and Rodrigo Quiroga for useful comments during the writing of this essay. This research has been funded by Starlab. References [1] Christoph Adami, What do robots dream of?, Science, Vol 314, Nov 17 2006, 1093–1095 [2] David Bailey, Peter Borwein1and Simon Plouffe, On the rapid computation of various polylogarithmic constants, Mathematics of Computation, vol. 66, no. 218 (April 1997), pg. 903–913 [3] Barbour, J., “The End of Time”, Oxford U. Press, 1999 [4] Albert, R., Barabasi, A-L, Statistical mechanics of complex networks, Rev. Mod. Phys., vol 74, pp 47–97, 2002. [5] J. Bongard, V. Zykov, H. Lipson, Resilient machines through continuous self- modeling, Science, Vol 314, Nov 17 2006, 1118 [6] Chaitin, G. J., Randomness in arithmetic and the decline and fall of reduc- tionism in pure mathematics”, paper in Nature’s imagination, John Cornwell, Ed., Oxford U. Press, 1995, p. 27–44. See also http://arxiv.org/abs/chao- dyn/9304002. [7] Cover, T., Thomas, J.A., Elements of Information Theory, John Wiley & Sons, [8] Cristianini, N., Shawe-Taylor, J., An Introduction to Support Vector Machines and Other Kernel-based Learning Methods, Cambridge University Press, 2000 http://arxiv.org/abs/chao-dyn/9304002 http://arxiv.org/abs/chao-dyn/9304002 [9] Clark, A., Natural-Born Cyborgs: Minds, Technologies, and the Future of Hu- man Intelligence, Oxford U. Press, 2003. [10] Deutsch, D., “The Fabric of Reality”, Penguin Books, 1997 [11] Dirac, P.A.M., Lectures in Quantum Mechanics, Belfer Graduate School of Sciences (Yeshiva U. NY, 1964) Dirac, P.A.M., Can J Math 2, p 129, 1950 [12] Halliwell, J J, “The Wheeler-DeWitt Equation and the Path Integral in Min- isuperspace and Quantum Cosmology”, in Conceptual Problems of Quantum Gravity, A Ashtekar, J Stachel eds., Einstein Studies v.2, 1991 [13] Huber, R., Ghilardi, M.F., Massimini, M., Tononi, G., Local sleep and learning, Nature, Vol 430, July 1st 2004, p. 78–81 [14] Feynman, R.P., Feynman Lectures on Computation, Perseus Publishing, 1996. [15] Feynman, R.P., Albert R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). [16] Gammerman, G., Vovk, V., Kolmogorov Complexity: Sources, Theory and Applications, The Computer Journal, p. 252, Vol. 42, No. 4, 1999 [17] Grau C, Escera C, Yago E, Polo MD. Mismatch negativity and auditory sensory memory evaluation: a new faster paradigm. NeuroReport 1998;9:2451–2456. [18] Harding, S.L., Miller, J.F., Rietman, E.A, Evolution in materio: exm- ploiting the physics of materials for computation, http://arxiv.org/abs/cond- mat/0611462 [19] Hawkins, J., On Intelligence, Times Books, 2004. [20] IJsselsteijn, W, Kort, Y., Haans, A., Is this my hand before me? The rubber hand illusion in reality, virtual reality and mixed reality, Proceedings of the 8th international workshop on Presence, Presence 2005. [21] Israeli, N., Goldenfeld, N., Computational Irreducibility and the predictability of complex physical systems, Phys. Rev. Lett. Vol 92 number 7, 20 Feb 2004. [22] Levitin, D.J., This is your brain on music, Dutton 2006 http://arxiv.org/abs/cond-mat/0611462 http://arxiv.org/abs/cond-mat/0611462 [23] , Li, M., Vitanyi, P., An Introduction to Kolmogorov Complexity and Its Ap- plications, Springer Verlag (1997) [24] Seth Lloyd, The computational capacity of the universe, Phys.Rev.Lett. 88 (2002), http://arxiv.org/abs/quant-ph/0110141. [25] Seth Lloyd, The Computational Universe, 2006 [26] Mach, E., The Science of Mechanics: A Critical And Historical Account of its Development, Open Court, LaSalle, IL, 1960 reprint of original in German, 1883. [27] Makeig, S. et al. Dynamic brain sources of visual evoked responses. Science 295, 690-694 (2002) [28] Maquet, P., Laureys, S., Peigneux, P., Fuchs, S., Petiau, C., Phillips, C., Aerts, J., Del Fiore, G., Degueldre, C., Meulemans, T., Luxen, A., Franck, G., Van Der Linden, M., Smith, C. and Cleeremans, A. Experience-dependent changes in cerebral activation during human REM sleep, Nature Neuroscience, 3:831– 836, 2000. [29] Massimini, M., Ferrarelli, F., Huber, R., Esser, S.K., Singh, H., Tononi, G., Breakdown of Cortical Effective Connectivity During Sleep, Science, Vol 309, Setp 2005, p. 2228–2232. [30] Nelson, L., While you where sleeping, Nature vol 430, p. 964, 26 August 2004, and references therein. [31] Ruffini, Giulio, “Quantization of Simple Parametrized Systems”, Dissertation, UC Davis, 1995. [32] Nagel, E., Newman, J.R. Newman, Gödel’s Proof, Revised edition with fore- word by D. R. Hofstadter, NY Press 2001. [33] Ruffini, G., Marco, J., Grau, C., Spherical Harmonics Interpolation, Computa- tion of Laplacians and Gauge Theory, arXiv:physics/0206007, 2002. [34] Ruffini, G., C. Ray, J. Marco, L. Fuentemilla, C. Grau, Complex networks in brain electrical activity, arxiv.org physics/0511137. [35] Näätänen R, Gaillard AWK, Mäntysalo S. Early selective-attention on evoked potential reinterpreted. Acta Psychol Amst 1978;42:313–329. http://arxiv.org/abs/quant-ph/0110141 http://arxiv.org/abs/physics/0206007 http://arxiv.org/abs/physics/0511137 [36] Rietman, E.A., Tilden, M.W., Askenazi, M.: Analog computation with rings of quasiperiodic oscillators: the microdynamics of cognition in living machines. Robotics and Autonomous Systems 45 (2003) 24926 [37] Rietman, E.A.,Hillis, R.W., Neural computation with rings of quasiperiodic oscillators, PSI-SR-1278 [38] Smetacek, V., Mechsner, F., Making Sense (Propioception: is the sensory sys- tem that supports body posture and movement also the root of our understand- ing of physical laws?), Nature, vol 432, November 4th 2004, p.21. [39] Solomonoff, R., Two Kinds of Probabilistic Induction, The Computer Journal, p. 256, Vol. 42, No. 4, 1999 [40] Sejnowski TJ, Destexhe A., Brain Res. 2000 Dec 15;886(1-2):208-223. [41] Susskind, L., The Cosmic Landscape, Little, Brown and Company, 2005. [42] Henneaux, M., Teitelboim, C., Quantization of Gauge Systems, Princeton Uni- versity Press, 1992 [43] Wolpert, L., et al., “Evolution and the egg”, Nature vol 420, p. 745, December 2002. [44] Wolfram, S., A New Kind of Science, Wolfram Media 2002. INFORMATION Information and Sensors Kolmogorov Complexity, Science and Evolution An example: compressing GPS signals On Gödel's theorem and Algorithmic Complexity Turing machines in dynamical systems: towards absolute complexity Using Rule 110 for GA based compression Kolmogorov Complexity and the Value of Scientific Theories THE BRAIN The closed Loop: self-modeling Pain and Consciousness Sleep and Memory Neuroscience as part of the program to understanding Science and the Universe Music and dripping faucets Handling information Oscillations and resonance Presence LIFE The Theory of Interacting Turing Machines, Aging and Cancer Natural selection as Computation Maximum Entropy Production PHYSICS Time and Kolmogorov Complexity Time capsules Kolmogorov Complexity and the Wheeler-DeWitt equation The arrow of time: from simple to complex What is computation in Barbour's Platonia? Data, program, computer Working with partial information: Statistical and Quantum Mechanics CLOSING

0704.1148

Aspects of stochastic resonance in reaction-diffusion systems: The nonequilibrium-potential approach

EPJ manuscript No. (will be inserted by the editor) Aspects of stochastic resonance in reaction–diffusion systems: The nonequilibrium-potential approach Horacio S. Wio1,a and Roberto R. Deza2 1 Instituto de F́ısica de Cantabria, Universidad de Cantabria and CSIC E-39005 Santander, Spain 2 Departamento de F́ısica, FCEyN, Universidad Nacional de Mar del Plata Deán Funes 3350, 7600 Mar del Plata, Argentina Abstract. We analyze several aspects of the phenomenon of stochastic resonance in reaction–diffusion systems, exploiting the nonequilibrium potential’s frame- work. The generalization of this formalism (sketched in the appendix) to extended systems is first carried out in the context of a simplified scalar model, for which sta- tionary patterns can be found analytically. We first show how system-size stochas- tic resonance arises naturally in this framework, and then how the phenomenon of array-enhanced stochastic resonance can be further enhanced by letting the diffusion coefficient depend on the field. A yet less trivial generalization is ex- emplified by a stylized version of the FitzHugh–Nagumo system, a paradigm of the activator–inhibitor class. After discussing for this system the second aspect enumerated above, we derive from it—through an adiabatic-like elimination of the inhibitor field—an effective scalar model that includes a nonlocal contribu- tion. Studying the role played by the range of the nonlocal kernel and its effect on stochastic resonance, we find an optimal range that maximizes the system’s response. 1 Introduction Stochastic resonance (SR) is nowadays a paradigm of the constructive effects of fluctuations on nonlinear systems [1,2]. Sketchily, the phenomenon occurs whenever the Kramers’ rate for the transition between attractors matches the typical frequency of a signal which is incapable by itself to trigger that transition (i.e. it is subthreshold). Whereas several measures of SR can be defined [the signal-to-noise ratio (SNR) and the spectral amplification factor (SAF) being the main ones], theoretical analysis is usually carried on in terms of the two-state approximation [1]. Since its discovery a quarter of century ago—and besides exploring related phenomena like e.g., coherence resonance [3]—interest has gradually shifted towards increasingly complex systems, networks and nonlinear media being the main directions. Instances of this trend are the experiments carried out to explore the role of SR in sensory and other biological functions [4], and experiments in chemical systems [5]. Our concern throughout this review will be with nonlinear media that can be described as reaction–diffusion (RD) systems, namely those that can be thought of as a collection of diffusively coupled nonlinear units. The possibility of enhancing the system’s response through the coupling of those units [6,7,8,9,10,11,12] has been among the issues explored during the last decade, together with the “naturalness” problem (how does nature manage to make the a e-mail: [email protected] 2 Will be inserted by the editor system’s response less dependent on a fine tuning of the noise intensity) or that of searching for different ways to control the phenomenon [13,14]. In dissipative dynamical systems, the very notion of Lyapunov’s function is as useful as that of attractor itself. Even when often it cannot be explicitly computed (because integrability conditions are not readily met), it allows for picturesque reasoning in terms of “energy land- scapes” or “attraction basins”. When those dynamical systems are submitted to forces that can be modeled (à la Langevin) as stochastic, a new meaning—statistical in nature, akin to the notion of “free energy”—is added to the picture (in fact, it is worth mentioning that the first function known to have the Lyapunov property was Boltzmann’s H-function). Moreover, even for vanishing noise intensity, the very existence of stochastic terms (a “transport matrix”) can render the system well conditioned regarding integrability conditions. That was the rationale behind the definition of nonequilibrium potential (NEP) [16,17,18,19], two approaches to which are described in the Appendix. Such NEP is a special Lyapunov’s function of the associated deterministic system, which for nonequilibrium systems plays a role similar to that played by a thermodynamic potential in equilibrium thermodynamics [16]. It is closely related to the sta- tionary solution of the system’s Fokker–Planck equation, and characterizes the global properties of the dynamics: attractors, linear and relative stability of these attractors, height of the barri- ers separating attraction basins. In addition, it allows to evaluate the transition rates among the different attractors [16,17,18,19,20]. Regarding the problem of SR in extended systems, it was shown that the knowledge of the NEP allows to obtain a rather complete picture of the behavior of the output signal-to-noise ratio (SNR). The novelty with nonequilibrium extended systems is that even pointlike attractors in the medium’s infinite-dimensional phase-space can be nontrivial field configurations (real-space patterns). In a series of recent papers we have studied the SR phenomenon for the transitions between two different patterns [8,9,10,11,12,15], exploiting the concept of nonequilibrium potential. In this review we discuss some recent results concerning different aspects of SR in RD systems. In Sec. 2 we discuss the phenomenon of system-size stochastic resonance (SSSR), and show how can it be analyzed and understood within a NEP framework [21]. In Sec. 3, after reviewing a recent study on the enhancement of the SNR found for a scalar system with density-dependent diffusivity [12], we discuss its extension [22] to an array of FitzHugh–Nagumo units [23]. In Sec. 4—through an adiabatic-like elimination of the inhibitor field in an activator–inhibitor system—an effective scalar system with a nonlocal term is derived, and the role of the local and nonlocal interactions on the SR response studied. The main conclusions are finally summarized in Sec. 5. 2 System-size stochastic resonance Recent studies on biological models of the Hodgkin–Huxley type [24,25] have shown that ion concentrations along cell membranes display intrinsic SR-like phenomena as the number of ion channels is varied. A related result [26] shows that even in the absence of external forcing, the regularity of the collective firing of a set of coupled excitable FitzHugh–Nagumo units is optimal for a given number of elements. From a physics point of view, the same phenomenon—called system-size stochastic resonance (SSSR)—has also been found in an Ising model as well as in a set of globally coupled units described by a φ4 theory [27]. It has been even shown to arise in opinion formation models [28]. Since the SSSR phenomenon is peculiar to extended systems, there is an obvious interest in describing it within a NEP framework, that offers a very general framework for the study of the dependence of SR and related phenomena on any of the system’s parameters. Here we discuss in some detail a one-component (“scalar”) RD system [21], and briefly refer to other cases analyzed in [27] and [29]. 2.1 Review of the scalar model For one-variable dynamical systems, the Lyapunov’s function can always be found by quadra- ture. This property can be readily translated to scalar RD systems: the Lyapunov’s functional Will be inserted by the editor 3 F [φ] fulfills the “potential” condition ∂tφ(y, t) = −δF [φ]/δφ(y, t), where δ/δφ(y, t) indicates a functional derivative. This is also the NEP for a scalar transport matrix (i.e. a multiple of the unit matrix in the medium’s infinite-dimensional phase space). The specific model we shall focus on here has a piecewise linear reaction term, that mimics general bistable RD models [23], e.g. those with a cubic-like reaction term. In the following, we shall exploit some of the results on the influence of general (partially reflective or albedo) boundary conditions found in [30], as well as previous studies of the NEP [17] and of SR [8,9,11,15]. The particular dimensionless form of the deterministic model we start with is [30,8,9] − φ+ φhΘ(φ− φc), (1) where Θ(z) is the Heaviside step function, φc is the value at which the piecewise-linear “reaction term” has the jump, and D is a phenomenological “diffusion coefficient”, not necessarily related to γ in Sec. 2.2 [in Graham’s approach, Eq. (15), D/(∆y)2 would be the matrix elements Qνν±1 in a discretization of the Laplacian]. All the effects of the parameters that keep the system away from equilibrium (such as the electric current in electrothermal devices like the ballast resistor [23,30], or some external reactant concentration in chemical models) are now included in φc. For the system to display a bistable behavior, it must be 0 < φc < φh. We consider here the class of static structures φ(y) studied in [30]. They are even solutions to the stationary (∂tφ = 0) version of Eq. (1) in the bounded domain [−yL, yL], with equal albedo boundary conditions (b.c.) at both ends ∂φ(y, t) y=±yL = ∓k φ(±yL, t). k > 0 is called the albedo parameter: the limit k → 0 yieds Neumann’s b.c. ∂yφ(y, t)|y=±yL = 0 and the k →∞ one, Dirichlet’s b.c. φ(±yL, t) = 0. The explicit form of these static patterns is φ(y) = φh  sinh(yc)ρ′ k, yL+y√ k, yL√ −yL ≤ y ≤ −yc, 1− cosh(y)ρ yL−y±c√ k, yL√ −yc ≤ y ≤ yc, sinh(yc)ρ′ k, yL−y√ k, yL√ yc ≤ y ≤ yL, where ρ(k, ζ) = sinh(ζ) + k cosh(ζ) and ρ′(k, ζ) = ∂ρ/∂ζ. The coordinate values y±c at which φ(yc) = φc are y±c = yL − ln Z2 + 1− k2 1 + k with Z = . (3) Each real solution y±c < yL to Eq. (3) represents a structure with a central “activated” zone (φ > φc) and two lateral “resting” regions (φ < φc). Figure 5 in [30] displays the relation yc/yL vs k, for several values of φc/φh. Typical shapes of the arising patterns are shown in Fig. 1. Through a linear stability analysis it has been shown [30] that the structure with the smallest “excited” region [that is with yc = y+c , denoted by φu(y)] is unstable, whereas the other one [with yc = y c , denoted by φ1(y)] is linearly stable. The trivial homogeneous solution φ0(y) = 0 exists and is linearly stable for any parameter set. These two linearly stable solutions (φ0 and φ1) are the only stable static structures under albedo b.c. We will concentrate on the region of values of φc/φh, yL and k where φ1 exists. For the finite system with albedo b.c., the NEP is a functional of φ and a function of k, yL and φc/φh. It has the expression [17] F([φ], φc/φh, k, yL) =∫ yL ∫ φ(y,t) [−φ′ + φhΘ(φ′ − φc)] dφ′ + ∂φ(y, t) φ2(y, t) 4 Will be inserted by the editor Fig. 1. Inhomogeneous static solutions to Eq. (1) for yL = D = φh = 1.0 and φc = 0.193 (highlighted by the dashed horizontal line). Dash-dotted upper curve: φ1(y) for k = 1.0 (in this case, yc ≡ y−c > yL). Lower curves: φ1(y) (solid line, for which +y c is highlighted) and φu(y) (dotted line, for which −y+c is highlighted) for k = 7.0. Fig. 2. NEP F([φ], k, yL) evaluated at the inhomogeneous stationary solutions φ1(y) (lower branch) and φu(y) (upper branch), as a function of: (a) system’s size yL, with k = 3.0; (b) albedo parameter k, with yL = 1.2. The remaining parameters are D = 1.0, φc/φh = 0.193. The NEP for the homogeneous stationary solution φ0(y) coincides with the horizontal axis. When the NEP is evaluated at the inhomogeneous static solutions of Eq. (1) [Eqs. (2) and (3)] it takes the explicit form [8,17] Fu,1(φc/φh, k, yL) = F([φu,1], φc/φh, k, yL) (4) = φ2h −y±c (1− 2φcφh + sinh y±c / ) ρ(k, (yL − y±c )/√D) k, yL/ while at the trivial solution φ0 ≡ 0 it is F([φ0], yL) = F0 = 0. Figure 2a depicts the nonequilibrium potential F([φ], yL) as a function of the system’s size yL, keeping the albedo parameter k and the ratio φc/φh fixed. The curves correspond to the NEP Fu,1(yL), whereas F0 coincides with the x–axis. Our focus is the “bistable zone” yL ' 0.72, where φ1(y) exists. The unstable structure φu(y) is a saddle point for F [φ] (in the medium’s infinite-dimensional phase space), so its NEP Fu(yL) > 0 (upper branch in Fig. 2a). On the other hand, both φ1(y) (lower branch) and φ0 (x–axis) are local minima of the NEP. One immediately notices that ∆F1(yL) ≡ Fu(yL)−F1(yL) is an (almost linearly) increasing function of yL (this has a profound implication for SSSR, as we shall see below). Equation (3) has real solutions only for yL ' 0.72. This corresponds to a supercritical saddle–node bifurcation, at which both inhomogeneous structures pop up. Now, the most important feature in Fig. 2a is that F1(yL) vanishes at a certain system’s size y∗L (≈ 1.0 for the given values of k and φc/φh). At that point, the stable inhomogeneous structure φ1(y) and the trivial solution φ0(y) exchange their relative stabilities. Will be inserted by the editor 5 For completeness, in Fig. 2b we plot F([φ], k) for the same value of φc/φh as in Fig. 2a. For the chosen value of yL it is always F1(k) < 0, and correspondingly there is no “stability exchange” as a function of k. Also, the initially large ∆F1(k) decreases with k as Fu(k) and F1(k) tend (for k →∞) to the values corresponding to Dirichlet’s b.c. [17]. 2.2 Results for SSSR By including an additive spatiotemporal noise source ξ(y, t) [15,33], Eq. (1) becomes a stochastic partial differential equation for the random field φ(y, t). The simplest assumptions about ξ(y, t) are that it is Gaussian, with zero mean and a correlation function given by 〈ξ(y, t)ξ(y′, t′)〉 = 2γ δ(t− t′)δ(y − y′), where γ denotes the noise strength. As discussed in [8,9,11,15], known results for activation processes in multidimensional sys- tems [31] allow us to estimate the activation rate using the following Kramers’-like expression for the mean first-passage time for the transitions between attractors 〈τi〉 = τ0 exp ∆F i(yL) where ∆F i(yL) = Fu(yL) − F i(yL), i = 0, 1. The prefactor τ0 is usually determined by the curvature of F [φ] at its extrema. On one hand, it is typically several orders of magnitude smaller than the average time 〈τ〉, while on the other it does not change significatively when varying the system’s parameters around the “bistable point” y∗L, where F([φ0], y L) = F([φ1], y L). Hence we may simplify the analysis by assuming here that τ0 is constant, and scale it out of our results. The behavior of 〈τ〉 as a function of k and φc/φh has been shown in [8,9,17]. As done in [8], we now assume that the system is (adiabatically) subject to an external harmonic variation of the parameter φc: φc(t) = φ0c + δφc cos(ωt) [9,15], and exploit the “two- state approximation” [1] as in [9,11,15]. Such approximation reduces the whole dynamics on the bistable potential landscape to one where the transitions occur only between the states associated to the bottom of each well, hence the only dynamical contents resides in the transition rates. Up to first order in the amplitude δφc (assumed to be small in order that the periodic input be sub-threshold) the transition rates Wi adopt the form µi ∓ αi cos(ωt) , (5) where (at constant φh) µi ≈ exp[−∆F i(φ0c , yL)/γ] and αi ≈ µi (∂∆F i/∂φc|φ0c ), i = 0, 1. The quantity inside parentheses can be obtained analytically using Eq. (4). These results allow to calculate the autocorrelation function, the power spectrum density and finally the SNR, that we indicate by R. The detailed calculation can be found in the appendix of [11]. Up to the relevant order (the second) in the signal amplitude δφc, we obtain 4µ0µ1 (α0µ1 + α1µ0)2 µ0 + µ1 µ0 + µ1 Φ, (6) where we have used the form of the αi to reduce the expression, and defined Φ = [2φh yc(yL)]2. Figure 3 (left) is a plot of R as a function of the noise intensity γ for a fixed system’s length yL, displaying the typical maximum that has become the fingerprint of the SR phenomenon. In Fig. 3 (right), the roles of γ and yL are exchanged (R is plotted as a function of yL for fixed γ). Such a response is the expected one for a system exhibiting SSSR. In both cases, the values of k and φ0c/φh are kept fixed. Within the NEP context and in this kind of systems, the phenomenon arises due to the breakdown of the NEP’s symmetry. This means that (as shown in Fig. 2) when varying yL, both attractors can exchange their relative stability. For yL = y∗L ≈ 1 both stable structures— the inhomogeneous one φ1(y) and the trivial one φ0—have the same value for the NEP. For yL < y L, φ1(y) becomes less stable than φ0 so transitions from φ1(y) to φ0 are more frequent 6 Will be inserted by the editor Fig. 3. Left: SNR vs the noise intensity γ, with system’s size yL = 1.1. Right: SNR vs yL, for γ = 0.1. The remaining parameters are k = 3.0, D = 1.0, and φ0c/φh = 0.193. (the barrier is lower) than in the reverse direction, thus reducing the system’s response. When yL ∼ 0.72, φ1(y) and φu(y) coalesce and disappear, and the response is strictly zero (within the linear response scheme implicit in the two-state approximation). When yL > y∗L, φ1(y) becomes more stable than φ0, making now transitions from φ0 to φ1(y) more frequent than in the reverse direction, and reducing again the system’s response. Clearly, the system’s response has a maximum when both attractors have the same stability (yL = y∗L), and decays when departing from that situation. Hence, for this system and within this framework, SSSR arises as a particular case of the more general discussion done in [11]. It should not come as a surprise to find an analogy with the mechanism of double stochastic coherence described in [32], where the NEP’s symmetry is induced by (an additional, multiplicative) noise. By comparing figures 2a and 3 it becomes apparent that the value of yL at which the SNR has its maximum differs slightly from y∗L (where the crossing between F 1 and F0 takes place). The origin of this discrepancy is the following: whereas on qualitative grounds we have argued that the maximum of the SNR should be related to the potential being symmetric (both wells having the same “energy”) [11], the exact condition is that the transition rates between both wells be equal. In general, due to small differences between the curvatures at the bottom of each well, those rates become equal for values of yL slightly different from the one at the symmetric case. Although by adopting here a constant value of τ0 we have assumed equal curvatures, there is still a difference in the values of the αi, since the ∂∆F i/∂φc|φ0c , i = 0, 1 are slightly different (a fact reflected in the dependence of Φ on yL). Additional light can be shed on the phenomenon when viewed from a different angle. Figure 4 is a plot of R as a function of k at fixed values of γ, yL and φ0c/φh. It exhibits a broad resonance since for k not too large (indicating a high reflectiveness at the boundary or a reduced exchange with the environment) R increases with k, whereas it slightly decreases for larger k values (where the system’s boundaries become absorbent). An explanation of this behavior in terms of the NEP has been given in [29]: as already observed, the NEP’s symmetry is broken for this value of yL; moreover, whereas the lower branch in Fig. 2b goes rapidly towards the value corresponding to Dirichlet’s b.c. the upper branch keeps increasing, thus degrading the SNR. In any case, the fact that the resonance is broad indicates the robustness of the system’s response with regard to k, a parameter that (together with γ) encodes the coupling with the environment. We stress the fact that the NEP framework put forward in this review allows to study SSSR between whole patterns. The explanation offered in [27] to the phenomenon resorted to a collective variable X ≈ (1/N) j=1 xj , and to the fact that the noise in the effective stochastic differential equation for X scaled with size. In [29] it was shown that all the cases discussed in [27] can be put within the same NEP framework than the above studied scalar model. In fact, the aforementioned almost linear increasing dependence of ∆F1(yL) on yL can be interpreted as a noise scaling with size. There are however situations where the NEP’s symmetry is retained as the system’s size is varied. We may then speak of a genuinely noise-scaling SSSR, in contrast to the cases that could be called NEP symmetry breaking SSSR [29]. Will be inserted by the editor 7 Fig. 4. SNR vs k for yL = 1.2, γ = 0.1 and φ c/φh = 0.193. 3 Case of selective coupling In this section we analyze SR in two extended systems with density-dependent diffusive-like cou- pling: an extension of the scalar RD model considered in Sec. 2 [12], and an array of FitzHugh– Nagumo [23] units. 3.1 Scalar model Here we extend the one-component RD model discussed in Sec. 2 by letting the diffusive pa- rameter D in Eq. (1) depend on the field φ(x, t). As a matter of fact, since in the ballast resistor [23,30] the thermal conductivity is a function of the energy density, the resulting equation for the temperature field includes a temperature-dependent diffusion coefficient in a natural way. The form of the governing equation is now ∂tφ(x, t) = ∂x [D(φ)∂xφ] + f(φ) + ξ(x, t), (7) with ξ(x, t) and f(φ) as in Sec. 2. As it was done for the reaction term, a simple choice (that retains however the qualitative features of the system) is to consider the following dependence of the diffusion term on the field variable D(φ) = D0[1 + hΘ(φ− φc)]. For simplicity, here we choose the same threshold φc for the reaction term and the diffusion coefficient. We assume the system to lie in a bounded domain [−L,L], with Dirichlet b.c. at both ends: φ(±L, t) = 0. The form of the patterns is analogous to what has been obtained in Sec. 2, the only difference being that in the present case dφ/dx|xc is discontinuous and the area of the “activated” central zone depends on h. As before, the indicated patterns are extrema of the NEP: the unstable pattern φu(x) is a saddle-point of this functional, separating the attractors φ0(x) and φs(x). For the case of a field-dependent diffusion coefficient D(φ(x, t)) as described by Eq. (7), the NEP reads [12] F [φ] = D(φ′)f(φ′) dφ′ + Given that ∂tφ = −[1/D(φ)](δF/δφ), one finds dF/dt = − (δF/δφ)2 dx ≤ 0, thus warranting the Lyapunov’s functional property. Whereas in Sec. 2 we kept φ0c constant and varied yL, we now vary instead φ c at constant L. Similarly as before, both linearly stable states have the same value of the NEP (i.e., they are equally stable) at some value φ∗c of the threshold. The way F [φ] depends on φ0c resembles the dependence on yL shown in Sec. 2, but now ∆Fs is an (almost linearly) decreasing function 8 Will be inserted by the editor 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 γ (arb. units) -0.2 -0.1 0.0 0.1 0.2 h (arb.units) (arb. units) 0.90 0.95 1.00 1.05 1.10 Fig. 5. Left: SNR vs the noise intensity γ, for D0 = 1.0 and h = 0.0 (full line), −0.25 (dashed line) and 0.25 (dotted line). Right: the maximum Rmax of the SNR curve as a function of the selectiveness h of the coupling, for D0 = 0.9 (dashed line), 1.0 (full line) and 1.1 (dotted line). The arrows a and b indicate the response gain due respectively to a homogeneous increase of the coupling and to a selective one. The larger gain in the second case is apparent. The inset shows the dependence of Rmax on D0 for h = −0.25 (lower line), 0.0 and 0.25 (upper line). The remaining parameters are L = 1.0, δφc = 0.01 and Ω = 0.01. of φ0c and both inhomogeneous structures coalesce and disappear through a subcritical saddle– node bifurcation. As in the previous case, we analyze only the neighborhood of φ∗c . We shall moreover consider only the neighborhood of h = 0, where the main trends of the effect can be captured. Figure 5 (left) depicts the dependence of R on the noise intensity γ for three values of h, each curve displaying the typical SR maximum. Figure 5 (right) is a plot of the value Rmax of these maxima as a function of h. The dramatic increase of Rmax (several dB for a small positive variation of h) is apparent, and shows the strong effect that the selective coupling (or field-dependent diffusivity) has on the system’s response. It must be noted that the only two approximations made in order to derive our results— namely the Kramers-like expression and the two-level approximation used for the evaluation of the correlation function [12]—break down for large positive values of h because for increasing selectivity the curves of F [φ] vs φ0c shift towards the left [12], which in turn means that the barrier separating the attractors at φ∗c tends to zero. This effect is basically the same as the one discussed in Refs. [9,11] in connection with global diffusivity D0. It is also worth noting that except for the two aforementioned approximations, all the previous results (e.g. the profiles of the stationary patterns and the corresponding values of the nonequilibrium potential) are analytically exact. 3.2 FitzHugh–Nagumo model Here we study an array of FitzHugh–Nagumo [23] units, with a density-dependent (diffusive- like) coupling. The NEP for this system was found within the excitable regime and for particular values of the coupling strength [19]. In the general case, however, the form of the NEP has not been found yet. Hence, we have resorted to a study based on numerical simulations, analyzing the influence of different parameters on the system’s response. Nevertheless, the idea of the existence of such a NEP has always underlied this study. The results show that the enhancement of the SNR found for the scalar system [12] is robust, and that the indicated non-homogeneous coupling could clearly contribute to enhance the SR phenomenon in more general situations. We consider a simplified version of the FitzHugh–Nagumo model [12,17,23], which has been useful for gaining qualitative insight into the excitable and oscillatory dynamics in neural and chemical systems. It consist of two variables: – a (fast) activator field u, that in the case of neural systems represents the voltage variable, while in chemical systems represents the concentration of an autocatalytic species. Will be inserted by the editor 9 – an inhibitor field v, associated (within a neural context) to the concentration of potassium ions in the medium, and that in a general chemical reaction inhibits the generation of the u species. Instead of considering the usual cubic-like nonlinear form, we use a piecewise linear version ∂u(x, t) Du(u) + f(u)− v + ξ(x, t) (8) ∂v(x, t) Dv(v) + β u− α v, (9) where f(u) = −u + Θ(u − φc), and ξ(x, t) is a δ–correlated white Gaussian noise, as before. γ indicates the noise intensity and φc is the “discontinuity” point, at which the piecewise linearized function f(u) presents a jump. The parameter � = τu/τv � 1 indicates the timescale ratio between the (fast) activator and the inhibitor. Together with α and β, it is chosen to correspond to the excitable regime. We consider Dirichlet b.c. at x = ±L. Although the results are qualitatively the same as those that could appear considering the usual FitzHugh–Nagumo equations, this simplified version allows us to compare directly with the previous analytical results for this system [12,15]. As in [12], we assume that the diffusion coefficient Du(u) is not constant, but depends on the field u according to Du(u) = D0u [1 + hΘ(u− φc)]. This form implies that the value of Du(u) depends “selectively” on whether the field u fulfills u > φc or u < φc. D0u is the value of the diffusion constant without such “selective” term, and h indicates the size of the difference between the diffusion constants in both regions [if h = 0 then Du(u) = D0u constant]. Dv(v) is the diffusion for the inhibitor v, that we assume to be homogeneously constant. This system is known to exhibit two stable stationary patterns. One of them is u(x) = 0, v(x) = 0, while the other is one with nonzero values. Further, we consider, as before, that an external, periodic, signal enters into the system through the value of the threshold φc, φc(t) = φ0c + δφ cos(ωt), where ω is the signal frequency, and δφ its intensity. All the results were obtained through numerical simulations of the system. The continuous version of the system indicated by Eqs. (8), (9) was transformed into a second-order spatially discrete one u̇i = Dui∆ui + (Dui+1 −Dui−1)(ui+1 + ui−1) + f(ui)− vi + ξi(t), v̇i = Dv∆vi + β ui − α vi, with ∆φi ≡ (φi−1 + φi+1 − 2φi). The extensive numerical simulations performed for a set of equations were done exploiting Heun’s algorithm [33]. In this spatially-extended system there are different ways of measuring the overall system’s response to the external signal. In particular, we show the evaluated output SNR in two different ways (the units being given in dB): – SNR for the middle element of the chain evaluated over the dynamical evolution of uN/2, that we call SNR2 (however, having Dirichlet b.c., the local response depends on the distance to the boundaries). – In order to measure the overall response of the system to the external signal, we computed the SNR as follows: We digitized the system’s dynamics to a dichotomic process s(t): At time t the system has an associated value of s(t) = 1 (0) if the Hilbert distance to pattern 1 (0) is lower than to the other pattern. Stated in mathematical terms, we computed the distance D2[·, ·] defined by D2[f, g] = dx [f(x)− g(x)]2 in L2([−L,L]), the Hilbert space of the real-valued functions in that interval. At time t, a digitized process is computed by means of s(t) = 1 if D2 [Pu1 (x), u(x, t)] < D2 [Pu0 (x), u(x, t)] 0 if D2 [Pu1 (x), u(x, t)] ≥ D2 [Pu0 (x), u(x, t)] 10 Will be inserted by the editor Model Numerical α β � φ0c δφ ω Dv ∆t N 0.3 0.4 0.03 0.52 0.4 5π/8 1.0 10−3 51 Table 1. Fixed parameters for the FitzHugh–Nagumo model with Du(u) Fig. 6. SNR2 and SNRp vs the noise intensity γ, for D u = 0.3 and h = −2.0 (+), −1.0 (4), 0.0 (�), 1.0 (�), and 2.0 (©). Both measures reveal a systematic enhancement of the SNR as h increases. -2 0 2 4 6 8 10 -2 0 2 4 6 8 10 Fig. 7. SNR2 and SNRp vs the selectiveness of coupling h, for D u = 0.3 and γ = 0.032 (©), 0.32 (�), 0.6 (�), 1.2 (4), and 3.2 (×). We call this “global-like” measure SNRp. The parameters kept fixed have been summarized in Table 1. The simulation was repeated 250 times for each parameter set, and the SNR was computed by recourse of the average power spectral density. Figure 6 depicts the results for the different SNR measures we have previously defined, as functions of the noise intensity γ. For both measures it is apparent that there is an enhancement of the response for h > 0, when compared with the h = 0 case, while for h < 0 the response is smaller. In Fig. 7 we show again the response’s measures, but now as functions of h. We have plotted the maximum of each SNR curve for D0u = 0.3, and γ = 0.01, 0.1, and 0.3. It is clear that there exists an optimal value of γ for which the response is largest. The rapid fall in the response for h < 0 is also apparent. In Fig. 8 we show the dependance of SNR on h, for different values of the diffusion which depends on the activator density D0u. It is apparent that the response becomes larger when the value of D0u is larger. However, as was discussed in [8,15], it is clear that for still larger values of D0u, the symmetry of the underlying potential (that is the relative stability between the attractors) is broken and the response finally falls down. The previous figures clearly show that the response to the external signal grows with the “selectiveness” of the coupling, showing the robustness of the phenomenon presented in [12,15]. Will be inserted by the editor 11 -2 0 2 4 6 8 10 -2 0 2 4 6 8 10 Fig. 8. SNR2 and SNRp vs h for γ = 0.032 and D u = 0.0 (©), 0.1 (�), and 0.3 (�). 4 Nonlocal Interaction Let us consider again a system like the one described by Eqs. (8), (9), but now assume that Du and Dv are constant. In Ref. [18] it was assumed that the inhibitor-like field has a diffusive transport behavior, and is fast enough that can be adiabatically eliminated, thus yielding an effective scalar RD equation with a nonlocal term, characterized by a diffusive kernel G(x, x′). After briefly reviewing the derivation of the NEP for that situation, we shall assume in this section that the transport mechanism of the adiabatically eliminated inhibitor-like field is of nondiffusive character, thus yielding a kernel H(x, x′) that is more localized in space, and with a controllable interaction range. Following Ref. [18], let the system be defined by ∂tu(x, t) = Du∂ xu(x, t)− u(x, t) +Θ[u(x, t)− a]− v(x, t), �−1 ∂tv(x, t) = Dv∂ xv(x, t) + βu(x, t)− αv(x, t), (10) where � was defined after Eqs. (8), (9), and let it be confined to the domain [−L,L], with Dirichlet b.c. at both ends: u(±L, t) = v(±L, t) = 0. Contrarily to the standard hypothesis, we now assume that the inhibitor is much faster than the activator (i.e. τv � τu). In the limit �→∞, we can rewrite Eq. (10) as ∂tu(x, t) = Du∂ xu(x, t)− u(x, t) +Θ[u(x, t)− a]− v(x, t), 0 = Dv∂ xv(x, t) + βu(x, t)− αv(x, t). In the last pair of equations we can eliminate the inhibitor (which is now slaved to the activator) by solving the second equation using the Green’s function method [−Dv∂2x + α]G(x, x ′) = δ(x− x′), v(x) = β dx′G(x, x′)u(x′), where the Green’s function G(x, x′) is given by G(x, x′) = [sinh k(L− x′)/ sinh 2kL] sinh k(L+ x) x < x′, [sinh k(L+ x′)/ sinh 2kL] sinh k(L− x) x > x′, with k = (α/Dv)1/2. This slaving procedure reduces our system to a nonlocal equation for the activator only, that has the form ∂u(x, t) ∂2u(x, t) + f(u)− β G(x, x′)u(x′) dx′. (11) 12 Will be inserted by the editor From this equation, and taking into account the symmetry of the Green’s function G(x, x′), we can obtain the Lyapunov functional for this system, which has the form F [u] = f(w) dw + dx′G(x, x′)u(x′)u(x) . (12) This spatial nonlocal term in the NEP takes into account the repulsion between activated zones. When two activated zones come near each other, the exponential tails of the inhibitor concentration overlap, increasing its concentration between both activated zones and creating an effective repulsion between them. Hence the Green’s function plays the role of an exponential screening between the activated zones. In Ref. [9] the knowledge of such NEP was exploited to study SR on the system indicated by Eq. (11). The starting point of our present analysis will be the effective, nonlocal and stochastic RD equation for the real (activator-like) field φ(x, t), analogous to Eq. (11), defined in the one dimensional domain x ∈ [−L,L] by + f(φ)− β H(x, x′)φ(x′) dx′ + ξ(x, t), where the diffusivityD is constant, and we assume a cubic nonlinear term f(φ) = φ (φ−b) (2−φ). Here ξ(x, t) is an additive Gaussian white noise, as in the previous cases. As before, the system is subject to Dirichlet b.c. φ(±L, t) = 0. Similarly to Eq. (11), this system could be written in a variational form, with the func- tional F [φ] given by Eq. (12). As anticipated, here we consider a nondiffusive kernel, with a controllable interaction range. In order to keep our analysis simple we propose the following H(x, x′) = 1/2; |x− x′| ≤ l 0; |x− x′| > l, (13) which allows the analysis by just varying the interaction range 2l. The new effective RD equation contains local and nonlocal couplings (corresponding to the diffusive and the nonlocal contribution, respectively). The last one contains the nonlocal kernel given by Eq. (13), with a variable range 2l, that corresponds to the interaction of the field at points x′ ∈ [x − l, x + l]. However, such points will contribute if and only if they are inside the domain [−L,L]. We are now in position to study the role played by the nonlocal kernel (particularly by its range 2l) on the SR phenomenon. As before, the SR between stationary solutions was investigated in terms of the two-state approach (all the details about the procedure and the evaluation of the SNR can be found in Refs. [11,15]). As usual, we subject our system to a weak external signal b = b0 + A(t) = b0 +∆b cos(ωst), rocking the NEP [15]. In order to have a subthreshold signal, the amplitude ∆b should satisfy ∆b� b. We have chosen b0 as the value of b at which F [φs] = F [φ = 0] = 0 when β = 0. Up to first order in the small amplitude ∆b, the transition rates Wi and the functions αi have the form indicated in Eqs. (5), (6). But now Φ, that depends on the inhomogeneous attractor φs, has the form After fixing the length of the system L and the kernel range 2l we can use the above indicated expressions to find the SNR, that shows the usual bell-shaped form of stochastic resonance as a function of the noise intensity. In Fig. 9 we show the dependence of the SNR on the kernel range 2l for a fixed value of 2L. There is a nonmonotonic behavior in the system’s response against variation of 2l, that can be explained by the following facts: – On one hand, the transition rates are decreasing functions of the range 2l for fixed 2L. Therefore, the ratio µ1µ2/(µ1 +µ2) in the expression for the SNR also reflects this behavior. Will be inserted by the editor 13 – The other factor in this expression has a maximum for a kernel range that corresponds to “first neighbor” sites, namely x = x′ ± l. Fig. 9. SNR as a function of 2l for 2L fixed. The parameters are D = 0.6, 2L = 6.25, β = 0.02 and b0 = 0.719123. The maximum in the system’s response as a function of the kernel range is due to the interplay between these two factors. From this analysis of the comparative weights of the local (diffusive) and nonlocal terms contributing to the SR response, it is apparent that the range of such nonlocal kernel has an optimum value yielding a maximum for the SNR [36]. 5 Conclusions We have discussed three different aspects of the phenomenon of stochastic resonance in reaction– diffusion systems, within the nonequilibrium potential’s framework. In first place we have dis- cussed system-size SR in a scalar model. Even though we have not shown the details here, it has been also possible to also study other cases [29]. In particular, a model of globally coupled nonlinear oscillators discussed in [27], showing that it can also be described within the NEP framework, with SSSR arising through an “effective” scaling of the noise intensity with the system’s size. In second place we presented a study of SR in systems with a density-dependent (diffusive- like) coupling. We initially discuss the case of a scalar system [12], and afterwards extent the analysis to an array of FitzHugh–Nagumo units, with a field-dependent activator diffusion [22]. For the second system, when both diffusions are constant (that is: Du > 0 and Dv = 0), has a known form of the NEP [15]. However, in the general case we have not been able to find the form of the NEP (but the idea of such a NEP is always underlying our analysis) and have to resort to an analysis based on numerical simulations. The result shows that the system’s response is enhanced due to the particular form of the non-homogeneous coupling. From such results, we can conclude that the phenomenon of enhancement of the SNR, due to a selectivity in the coupling, initially found for a scalar system [12] is robust, and that the indicated non- homogeneous coupling could clearly contribute to enhance the SR phenomenon in very general systems. Finally, we analyzed an activator-like field including a nonlocal contribution that arise through an effective adiabatic elimination of an auxiliary (inhibitor-like) field. By exploiting the knowledge of the nonequilibrium potential in such a case, we have analyzed the dependence of the SNR on the nonlocal interaction kernel range, founding that there is an optimal value of the kernel range, yielding a maximum in the system’s response, corresponding to a very localized interaction. The indicated results clearly show that the “nonequilibrium potential” (even if not known in detail [10]) offers a very useful framework to analyze a wide spectrum of characteristics associated to SR in spatially extended or coupled systems. For instance, within this framework, 14 Will be inserted by the editor the phenomenon of SSSR looks—as other aspects of SR in extended systems [11]—as a natural consequence of a breaking of the symmetry of the NEP [27]. Appendix: Brief review of the nonequilibrium potential scheme Loosely speaking, the notion of NEP is an extension to nonequilibrium situations of that of equilibrium thermodynamic potential. In order to introduce it, we consider a general system of nonlinear stochastic equations (admitting the possibility of multiplicative noises) q̇ν = Kν(q) + gνi (q) ξi(t), ν = 1, . . . , n; (14) where repeated indices are summed over. Equation (14) is stated in the sense of Itô. The {ξi(t)}, i = 1, . . . ,m ≤ n are mutually independent sources of Gaussian white noise with typical strength Graham’s approach The Fokker–Planck equation corresponding to Eq. (14) takes the form Kν(q)P + ∂qν ∂qµ Qνµ(q)P (15) where P (q, t; γ) is the probability density of observing q = (q1, . . . , qn) at time t for noise intensity γ, and Qνµ(q) = gνi (q) g i (q) is the matrix of transport coefficients of the system, which is symmetric and non-negative. In the long time limit (t→∞), the solution of Eq. (15) tends to the stationary distribution Pst(q). According to [16], the NEP Φ(q) associated to Eq. (15) is defined by Φ(q) = − lim γ lnPst(q, γ). (16) In other words Pst(q) d nq = Z(q) exp +O(γ) where Φ(q) is the NEP of the system and Z(q) is defined as the limit lnZ(q) = lim lnPst(q, γ) + Here dΩq = dnq/ G(q) is the invariant volume element in the q-space and G(q) is the deter- minant of the contravariant metric tensor (for the Euclidean metric it is G = 1). It was shown [16] that Φ(q) is the solution of a Hamilton–Jacobi-like equation (HJE) Kν(q) Qνµ(q) and Z(q) is the solution of a linear first-order partial differential equation depending on Φ(q) (not shown here). Equation (16) and the normalization condition ensure that Φ is bounded from below. Fur- thermore, from the separation of the streaming velocity of the probability flow in the steady state into conservative and dissipative parts, it follows that dΦ(q) = Kν(q) ∂Φ(q) Qνµ(q) i.e. Φ is a LF for the dynamics of the system when fluctuations are neglected. Under the deterministic dynamics, q̇ν = Kν(q), Φ decreases monotonically and takes a minimum value on attractors. In particular, Φ must be constant on all extended attractors (such as limit cycles or strange attractors) [16]. Will be inserted by the editor 15 Ao’s approach An alternative way to look into this problem is due to Ao [37]. Let us refer again to the system in Eq. (14). Following [37], we introduce now the auxiliary matrix H−1(q) = S(q) + A(q), with S(q) a symmetric matrix while A(a) is an antisymmetric one. H−1(q) is now used to rewrite the initial system as H−1(q) q̇ = H−1(q) K(q, t) + H−1(q) ξ(q, t) = −∇Φ(q, t) + η(q, t), where −∇Φ(q, t) = H−1(q) K(q, t), η(q, t) = H−1(q) ξ(q, t) and H−1(q) q̇ = −∇Φ(q, t) + η(q, t). The new stochastic variables, η(q, t), fulfill 〈η(q, t)ηT (q, t′)〉 = 2 S(q) δ(t− t′) = 2 G−1(q)Q(q)[G−1(q)]T δ(t− t′), that imposes a condition on the arbitrary definition of S(q) as we have [S(q) + A(q)] Q(q) [S(q)−A(q)] = S(q). As shown by Ao, the last equation also implies H(q) + HT (q) = 2 Q(q). We also have H−1(q) K(q, t) = ∇× [−∇Φ(q, t)] = 0. From the previous equations, we have in principle all the needed conditions to determine H−1(q), and to obtain from it the potential Φ(q, t). The interesting feature of this approach is that it resorts neither to Pst(q) nor to the small-noise limit, thus being applicable in principle to more general situations. Acknowledgements The authors acknowledge the collaboration of B. von Haeften, S. Bouzat, C. J. Tessone, G. G. Izús, M. Kuperman, S. Mangioni, A. Sánchez, F. Castelpoggi, in different aspects and/or stages of this work. HSW thanks the European Commission for the award of a Marie Curie Chair at the Universidad de Cantabria, Spain. References 1. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998). 2. T. Wellens, V. Shatokin, and A. Buchleitner, Rep. Prog. Phys. 67, 45 (2004). 3. A. S. Pikovsky and J. Kurths, Phys. Rev. Lett. 78, 775 (1997). 4. J. K. Douglas et al., Nature 365, 337 (1993); J. J. Collins et al., Nature 376, 236 (1995); S. M. Bezrukov and I. Vodyanoy, Nature 378, 362 (1995). 5. A. Guderian, G. Dechert, K. Zeyer, and F. Schneider, J. Phys. Chem. 100, 4437 (1996); A. Förster, M. Merget, and F. Schneider, J. Phys. Chem. 100, 4442 (1996); W. Hohmann, J. Müller, and F. W. Schneider; J. Phys. Chem. 100, 5388 (1996). 6. J. F. Lindner et al., Phys. Rev. E 53, 2081 (1996). 7. A. R. Bulsara and G. Schmera, Phys. Rev. E 47, 3734 (1993); P. Jung, U. Behn, E. Pantazelou, and F. Moss, Phys. Rev. A 46, R1709 (1992); P. Jung and G. Mayer-Kress, Phys. Rev. Lett. 74, 2130 (1995); J. F. Lindner et al., Phys. Rev. Lett. 75, 3 (1995); F. Marchesoni, L. Gammaitoni, and A. R. Bulsara, Phys. Rev. Lett. 76, 2609 (1996). 8. H. S. Wio, Phys. Rev. E 54, R3075 (1996); H. S. Wio and F. Castelpoggi, Unsolved Problems of Noise, Proc. Conf. UPoN’96, C. R. Doering, L. B. Kiss, and M. Schlesinger, Eds. (World Scientific, Singapore, 1997), pg. 229; F. Castelpoggi and H. S. Wio, Europhys. Lett. 38, 91 (1997). 9. F. Castelpoggi and H. S. Wio, Phys. Rev. E 57, 5112 (1998). 10. H. S. Wio et al., Physica A 257, 275 (1998); M. Kuperman, H. S. Wio, G. Izús, and R. Deza, Phys. Rev. E 57, 5122 (1998). 11. S. Bouzat and H. S. Wio, Phys. Rev. E 59, 5142 (1999). 16 Will be inserted by the editor 12. B. von Haeften, R. Deza, and H. S. Wio, Phys. Rev. Lett. 84, 404 (2000). 13. C. J. Tessone, H. S. Wio, and P. Hänggi, Phys. Rev. E 62, 4623 (2000). 14. M. A. Fuentes, R. Toral, and H. S. Wio, Physica A 295, 114 (2001). 15. H. S. Wio, S. Bouzat and B. von Haeften, in Proc. 21st IUPAP Int. Conf. on Statistical Physics, STATPHYS21, A. Robledo and M. Barbosa (Eds.), Physica A 306C, 140-156 (2002). 16. R. Graham, in Instabilities and Nonequilibrium Structures, Eds. E. Tirapegui and D. Villaroel (D. Reidel, Dordrecht, 1987); R. Graham and T. Tél, Phys. Rev. A 42, 4661 (1990); R. Graham and T. Tél, in Instabilities and Nonequilibrium Structures III, E. Tirapegui and W. Zeller, eds. (Kluwert, 1991); O. Descalzi and R. Graham, Phys. Lett. A 170, 84 (1992); O. Descalzi and R. Graham, Z. Phys. B 93, 509 (1994); H. S. Wio, in 4th Granada Seminar in Computational Physics, Eds. P. Garrido and J. Marro (Springer-Verlag, Berlin, 1997), pg.135. 17. G. Izús et al., Phys. Rev. E 52, 129 (1995); G. Izús et al., Int. J. Mod. Phys. B 10, 1273 (1996). 18. G. Drazer and H. S. Wio, Physica A 240, 571 (1997). 19. D. H. Zanette, H. S. Wio and R. Deza, Phys. Rev. E 53, 353 (1996); F. Castelpoggi, H. S. Wio and D. H. Zanette, Int. J. Mod. Phys. B 11, 1717 (1997); G. Izús, R. Deza and H. S. Wio, Phys. Rev. E 58, 93 (1998); S. Bouzat and H. S. Wio, Phys. Lett. A 247, 297 (1998); G. G. Izús, R. R. Deza and H. S. Wio, Comp. Phys. Comm. 121-122, 406 (1999). 20. B. von Haeften et al., Phys. Rev. E 69, 021107 (2004); B. von Haeften et al., in Noise in Complex Systems and Stochastic Dynamics, Z. Gingl, J. M. Sancho, L. Schimansky-Geier, and J. Kerstez (Eds.), Proc. SPIE 5471, 258-265 (2004). 21. H. S. Wio, in Noise and Fluctuations, Proc. 18th Int. Conf. on Noise and Fluctuations-ICNF2005, Eds. T. Gonzalez, J. Mateos, D. Pardo, AIP 780, 55 (2005). 22. C. J. Tessone and H. S. Wio, Physica A 374, 46 (2006). 23. H. S. Wio, An Introduction to Stochastic Processes and Nonequilibrium Statistical Physics (World Scientific, Singapore, 1994); A. S. Mikhailov, Foundations of Synergetics I (Springer-Verlag, Berlin, 1990). 24. G. Schmid, I. Goychuk, and P. Hänggi, Europhys. Lett. 56, 22 (2001); ibid., Phys. Biol. 1, 61 (2004). 25. P. Jung and J. W. Shuai, Europhys. Lett. 56, 29 (2001); J. W. Shuai and P. Jung, Phys. Rev. Lett. 88, 068102 (2003). 26. R. Toral, C. Mirasso, and J. Gunton, Europhys. Lett. 61, 162 (2003). 27. A. Pikovsky, A. Zaikin, and M. A. de la Casa, Phys. Rev. Lett. 88, 050601 (2002). 28. C. J. Tessone and R. Toral, Physica A 351, 106-116 (2005). 29. B. von Haeften, G. G. Izús, and H. S. Wio, Phys. Rev. E 72, 021101 (2005). 30. C. L. Schat and H. S. Wio, Physica A 180, 295 (1992). 31. P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990). 32. A. Zaikin et al., Phys. Rev. Lett. 90, 030601 (2003). 33. J. Garćıa-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems (Springer-Verlag, New York, 1999). 34. It is worthwhile noting that when the parameter h is large enough, under some circumstances the coupling term might become negative, in what is known as “inhibitory coupling” [35]. This is a very interesting kind of coupling that has attracted much attention, both in neural and chemical context, that we will not discuss here. 35. P. Dayan and L. F. Abbott, Theoretical neuroscience: Computational and mathematical modeling of neural systems (MIT Press, Cambridge, 2001). 36. B. von Haeften and H. S. Wio, Physica A 376, 199 (2007). 37. P. Ao, cond-mat/0302081 (2003); P. Ao, J. Phys. A 37, L25-L30 (2004). http://arxiv.org/abs/cond-mat/0302081 Introduction System-size stochastic resonance Case of selective coupling Nonlocal Interaction Conclusions

0704.1149

Distortion of Gravitational-Wave Packets Due to their Self-Gravity

Lumdiff_self_k05_tau.tex Distortion of Gravitational-Wave Packets Due to their Self-Gravity Bence Kocsis1, 2, ∗ and Abraham Loeb1, † 1Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 2Institute of Physics, Eötvös University, Pázmány P. s. 1/A, 1117 Budapest, Hungary (Dated: October 25, 2018) When a source emits a gravity-wave (GW) pulse over a short period of time, the leading edge of the GW signal is redshifted more than the inner boundary of the pulse. The GW pulse is distorted by the gravitational effect of the self-energy residing in between these shells. We illustrate this distortion for GW pulses from the final plunge of BH binaries, leading to the evolution of the GW profile as a function of the radial distance from the source. The distortion depends on the total GW energy released ǫ and the duration of the emission τ , scaled by the total binary mass M . The effect should be relevant in finite box simulations where the waveforms are extracted within a radius of 2M . For characteristic emission parameters at the final plunge between binary BHs of arbitrary spins, this effect could distort the simulated GW templates for LIGO and LISA by a fraction of 10−3. Accounting for the wave distortion would significantly decrease the waveform extraction errors in numerical simulations. PACS numbers: I. INTRODUCTION The observation of gravitational waves (GWs) is ex- pected to open a new window on the universe within the following decade. First generation GW detectors (In- LIGO [76], VIRGO [77], TAMA [78], GEO [79]) are al- ready operating at or close to their design sensitivity lev- els and the development of the next advanced-sensitivity GW detectors (Advanced LIGO [80], Advanced Virgo [81], LCGT [82]) and the space-detector LISA [83] are well underway. It is now increasingly important to fully understand the precise characteristics of the GW wave- forms that we expect to observe. The most luminous GW sources are expected to be associated with mergers of BH binaries. The physical understanding of these sources has greatly improved by recent breakthroughs in numerical relativity [1, 2, 3]. It is now finally possible to simulate the merger of a BH bi- nary, from the initial circular inspiral, through the plunge to a common surrounding horizon, to the final ringdown, as the remnant settles down to a quiescent stationary Kerr-BH. It is widely believed now that existing simu- lations are sufficiently precise to allow targeted searches for these waveforms in real data [4]. In fact, it has been recently shown that the errors are not even limited by the numerical precision of the simulation (∼ 10−5), but the GW extraction method itself entails a much larger uncertainty (∼ 10−3) [5]. In this paper, we demonstrate that the self-gravity during the propagation of gravita- tional radiation in the zone of wave extraction of numer- ical simuations leads to the distortion of the waves, cor- responding to similar magnitude modifications in typical cases. ∗Electronic address: [email protected] †Electronic address: [email protected] A. Description of the effect Let us imagine a compact spherically-symmetric con- figuration of matter (representing the remnant) and a rapidly expanding sphere of massless particles (represent- ing the radiation) carrying away some of the initial mass of the system (Fig. 1). First, let us assume Newtonian gravity and spherical symmetry. In this case, the var- ious shells are pulled back only by the gravity of the mass interior as if it was concentrated to a point mass at the center of the sphere, and the effect of the outer enclosing shells exactly cancels out. Thus, the particles on the outermost shell are always attracted by the total mass, including the mass of the radiation, but the inner- most shells experience only the gravity of the remnant. Therefore, the gravity of the radiation implies that the innermost shells of radiation will be continuously catch- ing up to the outer boundary during their journey from the source to the observer. Do we expect an analogous effect to exist also for grav- itational waves in full general relativity? First, let us consider conventional (i.e. non-gravitational) radiation. In analogy to the Newtonian gravitational pull, relativis- tic test particles are slowed down by gravity: the null- geodesics in a gravitational field experience the so-called Shapiro delay [6], decreasing the radial coordinate ve- locity with increasing gravity. Furthermore, according to Birkhoff’s theorem, the spacetime outside a spheri- cally symmetric distribution of energy is equivalent to the spacetime of a point-mass placed at the center of the sphere, the Schwarzschild metric, and the spacetime inside a cavity is the free-space Minkowski spacetime. More generally, the spherically symmetric expansion of collisionless radiation is a known simple exact solution of the Einstein equations, the Vaidya metric [7, 8]. This solution has exactly the same characteristics as the New- tonian example, whereby various shells react only to the mass interior to them, i.e. they move on world lines ne- http://arxiv.org/abs/0704.1149v4 mailto:[email protected] mailto:[email protected] Mf ∆τ ′ FIG. 1: A sketch of the effect under consideration. The co- alescence of two BHs in a binary of total initial mass M0 results in the emission of a burst of gravitational radiation which carries away a non-negligible ǫ fraction of M0. The remnant BH mass is Mf . The proper temporal width of the wave-packet for a hypothetical observer fixed at a radial dis- tance r is ∆τ . As the packet propagates outwards, it (1) expands due to gravitational redshift of the initial mass M0 (solid lines), (2) contracts due to the mean self-gravity of the radiation (dotted line, ∆τ ′), and also (3) distorts its profile due to the self-gravity of the radiation (not shown). As a re- sult the inner shells begin to catch up, and the proper time separation in excess of gravitational redshift from the front of the burst decreases with distance. Consequently, the net luminosity of the radiation burst changes with distance. glecting the exterior shells and the effect of the interior shells is the same as if they were concentrated to a point mass at the center. Next, let us turn to the case of gravitational radiation. The effect of the self-energy of gravitational radiation can be accounted for in the first nonlinear-order approxima- tion of the Einstein field equations by attaching terms of order h2 to the stress-energy, Tij , considering these terms as sources in addition to the regular radiation fields [9, 10, 11, 12, 13, 14, 15]. Here hij is the wave amplitude which is the correction to the background metric. If the wavelength of the GW wave-packet is much smaller than the size of the wave envelope, the evolution of the wave- packet is determined by the WKB cycle-averaged effec- tive stress-energy tensor [10, 11], independent of the spe- cific wave-characteristics of the radiation. In this regime, we may treat the GW packet as an ensemble of relativis- tic particles for which our previous arguments apply. In conclusion, we anticipate that (i) the wave-envelope will continuously expand due to the redshift of the initial mass of the binary, (ii) it will contract due to the self-gravity of the radia- tion, and (iii) in analogy to electrodynamics, we expect that the distortion of the wave envelope would lead to a continuous adiabatic modification in the GW fre- quency. The purpose of this paper, is to quantify these expec- tations for typical BH merger waveforms using simple models and to demonstrate that this effect should be accounted for in relation to numerical simulations and observed merger waveforms. B. Related literature To our knowledge the effect of self-gravitational dis- tortion of GWs had not been explicitly recognized previ- ously. We elaborate on the relation of the self-distortion effect to numerical general relativity, analytical investiga- tions like the multipolar post-Minkowskian (MPM) and post-Newtonian (PN) theory, and the studies of the scat- tering of gravitational radiation in curved spacetimes. The self-gravitational distortion of GWs is a relatively small effect on short scales currently accessible to numer- ical simulations. Current state-of-the-art simulations of binary BH mergers are restricted to the central strong- gravity domain near the black holes, and extract gravity waves from the boundary of this domain. The standard method of extracting and extrapolating the waveforms to larger distances, is based on the Regge–Wheeler–Zerilli- Moncrief perturbation formalism [16, 17, 18]. This is a linear-order representation of the Einstein field equations and so it neglects self-energy effects of order h2. Cumula- tive nonlinear effects like the self-distortion effect should lead to systematical deformations of the linear waveform extracted at different radii, which can in principle be dis- covered by a rigorous convergence test. In fact, nearly all papers on simulated merger GWs study the conver- gence behaviour in some detail. However, due to com- putational limitations, the extraction radius is currently restricted to r ∼ < 50M , and the extraction has been pre- formed on only a few, typically 3–4 different radii with the extrapolation done empirically based on these radii [1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27]. Recently, Pazos et al. [5] reported a systematic effect of order 10−3 (for extraction radii r ≤ 80M), which is much larger than the numerical precision of the simulation ∼ 10−5. This is roughly the same order of magnitude systematic effect that we expect for the GW distortion for the given radii (see below). Note however, that Berti et al. [25] showed that the convergence behavior is also largely sensitive to simulation assumptions. On small scales probed by these simulations, other near-field nonlinearities might also be of equal importance. A precise treatment of the waveforms at large radii is possible by analytical methods, such as the MPM- expansion introduced by Thorne [14] (see also [13]) and developed further by Blanchet & Damour [28, 29, 30]. In Ref. [14], Thorne introduced the concept of “local-wave zone,” which is outside the dynamical zone of wave gener- ation, but where nonlinear effects are still important. In this region the propagation of gravity waves is expressed in terms of an expansion in powers of hij as an infinite sum of multipole contributions with rapidly decreasing amplitude, which needs to be matched to the dynamical gravitational field generated by the source. The GWs in the local wave zone are given formally by the MPM expansion, whose terms correspond to different powers of the gravitational coupling constant G. In the PN ap- proach, the dynamical wave generation is calculated an- alytically in an infinite series in the inverse speed of light c−2. Matching the PN and MPM expansions in the local wave zone is a successful method for the calculation for steady source GWs produced by relatively slow motions, like the inspiral phase of BH mergers where the distance between the BHs is large enough to allow an adiabatic quasicircular orbit at r > 6M . To date, the PN wave- forms for circular binary inspirals are available to 3.5PN order for general mass ratios [31] (which is the highest order that is expected to have a measurable contribution for circular inspirals by a LISA-type detector [32]) and 5.5PN order for extreme mass ratios and no BH spins [33]. To our best knownledge, the self-gravitational distortion effect has not been identified in these works. However, in this paper we show that the radius for a fixed lumi- nosity shift or a fixed frequency shift is linearly sensitive to the energy density of the radiation. The luminosity at the inspiral phase of binaries is less than 1% of the lu- minosity at the final plunge [26]. Therefore, even if it is negligible for inspirals, the self-gravitational modulation of GWs could be significant for the final plunge. We expect the self-gravitational distortion of GWs to be consistent with the PN/MPM expansion, and to have corresponding PN tail counterparts [34]. The tails of GWs are caused by the scattering of linear waves on the spacetime curvature generated by the total mass- energy of the source [35], which is related to the Shapiro time-delay [6] of the radiation crawling out of the back- ground gravity of the source. At 2.5PN order beyond the Newtonian quadrupole formula, GW tails scatter off the monopole field of the remnant [36, 37, 38]. Furthermore, above 3PN order, the tails of the tails are produced by curvature scattering of the tails of the waves themselves, associated with the cubic nonlinear interaction between two mass monopole moments and the mass quadrupole of the source [39]. In this paper we show that the modifica- tion caused by the spherical self-gravitational distortion of GWs is to lowest order proportional to the original waveform (i.e. without this effect) times the energy den- sity of the waveform. Since the energy density is pro- portional to the square of the amplitude, this possibly implies to lowest order a monopole-quadrupole2 type in- teraction counterpart. The self-gravitational distortion effect is also related to the “memory effects” (or “hereditary effects”) of gravi- tational radiation, since it is a cumulative effect that de- pends on the full past history of the radiation, as opposed to regular PN terms which only depend on the instanta- neous retarded fields. Other known hereditary GW ef- fects are the tails of GWs [30, 40], the Christodoulou effect [41, 42], and the GW recoil kick [43]. Finally, nonlinear effects were also examined for the in- teraction of plane GWs on a free space (i.e. Minkowski) background [44, 45, 46, 47, 48, 49]. The nonlinear terms in the scattering problem are found to exactly cancel up to fourth order, leaving no self-phase-modulation effect for GWs in vacuum to this order. However, the geome- try of this case is very different from the one discussed in this paper where a curved background is initially present. It is also unclear whether there are self-phase modulation effects at higher nonlinear orders. Fortunately, our ap- proach does not face such convergence issues, since we adopt the exact (i.e. non-perturbative) solution of the Einstein equations in the spherical WKB approximation of an expanding radiation shell. The present paper aims to quantify the self- gravitational distortion effect for recently compiled merger waveforms. In § II, we list the main properties of the waveforms relevant for our study. In § III we present our analysis in spherical symmetry, and derive the results for the self-gravitational distortion of signal duration and the luminosity profile. In § IV we summarize the main conclusions, and then discuss their implications. Finally, we discuss the validity of the spherical approximation and consider the possible effect of the anisotropy in the Appendix. We use units with G = c = 1 and a metric signature of (−1, 1, 1, 1). II. MERGER WAVEFORMS To illustrate our effect, we adopt a simplified treatment for the merger GW waveforms. Table I lists the total ra- diated mass ∆mtot relative to the initial total mass M0 for various encounters between compact objects found in the literature. While all of these encounters would have a nonnegligible GW self-gravitational effect, the inspiral– merger–ringdown events are expected to have the most prominent event rates for interferometric GW detectors. Inspiral detection rate estimates are between 0.3−3 yr−1 and several per day for NS/BH mergers for inLIGO and adLIGO, respectively [50]; 3 and 100 yr−1 for stellar BH/BH inspirals in globular clusters [51] and in galac- tic nuclei [52] with adLIGO, respectively; 1–100 yr−1 and 30 yr−1 for supermassive (SMBH) and intermediate mass BH (IMBH) [53, 54, 55] and SMBH/SMBH inspirals [56, 57] with LISA, respectively. The dynamic time of the encounter is proportional to the total mass; consequently we do not consider extreme mass ratio inspiral–mergers (EMRI), as the GW luminosity of these sources is much smaller. Although high velocity stellar BH encounters can be very bright in GWs, these events are expected to be rare, less than 1 yr−1 for adLIGO or LISA [58]. In this analysis, we focus on equal mass inspiral–merger– ringdown waveforms. Generally, the waveforms can be expanded in multi- TABLE I: Total radiated mass relative to the initial total mass, ǫ, for bright GW encounters. For a detailed comparison of BH inspiral computations see Ref. [59]. Objects Encountera Spins orien- Refs. ǫ tationb [%] NS–BH head-on (0.0,0.0) [60] 0.01 BH–BH head-on (0.0,0.0) [61] 0.05 BH–BH head-on (0.1,0.1) [61] 0.06 NS–BH orbiting (tidalc,0) [62, 63] 0.1 BH–BH head-on (0.2,0.2) [61] 0.12 BH–BH head-on (0.0,0.0) [64] 0.13 BH–BH whirl (0.0,0.0) [24] 0.5–3 BH–BH grazing (0.9,0.7) − ⊥ − [65] 0.9 BH–BH grazing (0.9,0.7) + ⊥ + [65] 1.0 BH–BH grazing (0.0,0.0) [65] 1.2 BH–BH inspiral (0.2,0.2) + ‖ + [20] 1.8 BH–BH inspiral (0.1,0.1) − ‖ − [20] 2.0 BH–BH inspiral (0.2,0.2) − ‖ − [20] 2.0 BH–BH inspiral (0.8,0.8) − ‖ − [22] 2.2 BH–BH inspiral (0.1,0.1) + ‖ + [20] 2.4 BH–BH inspiral (0.0,0.0) [20] 2.5 BH–BH inspiral (0.0,0.0) [3, 21] 3.2 BH–BH inspiral (0.0,0.0) [19, 22] 3.5 BH–BH inspiral (0.0,0.0) [25, 27] 3.7d BH–BH inspiral (0.1,0.1) + ‖ + [26] 5.2 BH–BH inspiral (0.8,0.8) generale [23] 5–6 BH–BH inspiral (0.8,0.8) + ‖ + [22] 6.7 BH–BH r.whirl (0.0,0.0) [24] 15f BH–BH r.head-on (0.0,0.0) [66] 16 BH–BH r.head-on (1.0,0.0) ⊥ [67] 17 η BH–BH whirl (0.0,0.0) [24] 24h BH–BH r.whirl (0.0,0.0) [24] 100h aHere “head-on” stands for a direct collision with v ‖ r initially, “orbiting” stands for the tidal stripping of a NS by a BH in close orbit, “grazing” stands for inspiral collisions in which the initial separation is within the final orbit in a merger, “inspiral” is the complete inspiral–merger–ringdown event, “whirl” stands for parti- cles approaching from infinity with some impact parameter leading to a quasi-circular whirl-type orbit before merger, “r.whirl” and “r.head-on” corresponds to relativistic initial velocities v ≈ 1 at r ≫ M . bHere we give the sign of S1 · J , the relationship between S1 and S2, and the sign of S2 ·J , where J is the orbital angular momentum. In case of a head-on collision with spins, we give the initial direction of the spin relative to the separation vector. cNS tidally locked dRef [25] provides the results for different mass ratios, m1/m2 = 1–4, and found that ∆m ∝ η2, where η = m1m2/(m1 + m2) e8 different choices of spin orientations fIn case the impact parameter is small enough to end up in a merger. gCalculated for m1 ≪ m2. (See η above at hIn case the impact parameters are fine tuned for the binary to approach the unstable circular orbit. poles [14] hµν = (1 + z)Anµν eiφn (1) where φ is the high frequency GW phase which is related to the instantaneous frequency through f = dφ/dt; Aµν is a slowly varying envelope describing how the instanta- neous amplitude changes over the waveform, the index n labels the various polarizations and multipoles, and dL is the luminosity distance, and z is the redshift. For binary inspiral merger waveforms, the (l = 2,m = 2) multipole (i.e. quadrupole) dominates the waveform [25, 26, 27]. For the sake of simplicity, we restrict our attention to a single monochromatic wave. During a single GW cycle, corresponding to a time interval f−1, the envelope Aµν can be regarded as con- stant. In the WKB approximation, the energy carried by the radiation can be calculated as a cycle-averaged quan- tity. The effective stress-energy tensor of the radiation is [11, 12] T µν = kµkν , (2) where A2 = AµνAµν is the squared effective amplitude and kµ = φ,µ is the wave number. Each infinitesimal volume of the wave can be at- tributed the mass-energy that it carries according to Eq. (2). In the spherically symmetric approximation, the to- tal luminosity (or more precisely the graviton number) is = 4πr2T tr = ktkr (3) With this equation it is possible to obtain the waveform [A(t), f(t)] within radius r from the luminosity function L(t). Based on the waveforms derived by numerical simula- tions (such as Fig. 25 in Ref. [26]), we adopt the following simple fit to the effective luminosity L(t) = L0 × [(t1 − t)/t1] −1.5 if − t0 < t < 0 1 if 0 < t < t1 exp[−(t− t1)/t2] if t > t1 where the intervals t < 0, 0 < t < t1 and t1 > t corre- spond to the late inspiral/final orbits, the peak luminos- ity at the plunge, and the ringdown phases, respectively, −t0 sets the initial time of the calculated profile, t1 rep- resents the characteristic timescale of the most intensive part of the radiation, t2 sets the ringdown decay rate, and L0 is the normalization luminosity. Numerical simula- tions show [26] that the characteristic frequency of the ra- diation rapidly increases after the inspiral and saturates at ω = 2πf ∼ M−10 /2 at t ∼ 0, where M0 is the initial mass of the source. The characteristic number of wave cycles during the brightest phase is N = ∆t/f−1 ∼ 3 over a time ∆t = 12πM0. Note that we assume that the WKB method is applicable to the waveform, imply- ing that L(t) does not change greatly over a single cycle. This condition is just marginally satisfied for these wave- forms. For a simple analysis, we assume that the total lu- minosity crossing a sphere at infinity L∞(t) is given by Eq. (4) with the following parameters: (t0, t1, t2) = (100, 10, 5)M0, and use ∆t = 12πM0 in reference to av- erage quantities below. Under the WKB approximation the luminosity (4) evolves independently of the carrier frequency f ∼ 1/(4πM0). The L0 normalization is set using the total radiated mass of the system as a frac- tion of M0 by ǫ = ∆mtot/M0 for the inspiral-mergers (3) listed in Table I. For pedagogical comparison purposes we distinguish the inspiral events only based on the nor- malization L0, and do not consider the variations in the shape of the waveform (e.g. ∆t). III. QUANTITATIVE ESTIMATES IN SPHERICAL SYMMETRY To give a first quantitative estimate of the magnitude of the wave-packet distortion due to self-energy, we start by computing the propagation of unpolarized radiation packets in the spherically symmetric Vaidya spacetime. The possible effect of anisotropies is discussed in the Ap- pendix. The Vaidya metric [7] in radiation coordinates (u, r, θ, φ) is ds2 = − 2m(u) du2 − 2drdu + r2dΩ, (5) where dΩ = dθ2 + sin2 θdφ2. This metric is an exact (i.e. nonperturbative) solution of Einstein’s equations in spherical symmetry in the eikonal approximation to a ra- dial flow of unpolarized radiation. Here u is the retarded time parameter which is constant along the world lines of radially outgoing radiation, and m(u) describes the mass function interior to u. Outside the radiation (i.e. where m(u) is constant in the space-time), the Vaidya metric (5) is the Schwarzschild solution in Eddington- Finkelstein coordinates [12]. For our approximate wave- forms decribed in § II, m(u) is M0 constant outside, it is quickly changing within a short range 0 ≤ u ≤ ∆utot across, and it is Mf = M0 −∆mtot inside the radiation shell. Here, ∆mtot and ∆utot are set by the simulated waveforms § II. For the metric given by Eq. (5) it is straightforward to derive the convergence of null-geodesics describing the world lines of radiation shells using Raychaudhuri’s equa- tion or the equation of geodesic deviation [68]. In either way, we find that radially outgoing shells of radiation sim- ply follow the world lines u(r) = constant, implying that the ∆u coordinate difference between the shells does not change during the propagation. The physical contrac- tion of radiation shells can be examined using the proper time measure between shells and the observed luminosity profile. A. Proper time duration First, we estimate the proper-time duration of the GW signal along the world-line of a hypothetical observer crossing the radiation shell. For simplicity, we restrict to observers at a fixed spacial coordinate (r, θ, φ). Then we have dr ≡ dθ ≡ dΩ ≡ 0 and Eq. (5) gives dτ2 = −ds2 = 2m(u) du2 (6) leading to dτ = 1− 2m(u)/rdu. Therefore, du can be interpreted as the infinitesimal proper time difference between two radiation shells at fixed radius approaching infinity. Thus, we adopt the notation du ≡ dτ∞, and similarly ∆u ≡ ∆τ∞ for integrated quantities. Finally, let us define mi = m(ui) for i ∈ {1, 2} and ∆m = |m2 − m1|. We set (m1,m2,∆mtot) = (M0,Mf , ǫM0) when referring to the total GW signal duration. Integrating between two arbitrary shells of radiation u2 and u1 gives 2m(u) du. (7) After expanding m(u) in a Taylor-series, the integrand becomes 2u〈m′〉 u2〈m′′〉 + . . . . (8) Substituting in Eq. (7), and using 〈m′〉 = −∆m/∆τ∞ and 〈m′′〉 = 0 to first order, we get ∆τ(m1,m2, r) = 3|m2 −m1| . (9) Setting m1 = M0 and expanding in terms of ∆m, we get r − 2M0 +O(∆m2) The leading order term can be indentified as the gravi- tational redshift for constant mass M0, the second term describes the correction due to the radiation mass. If we expand also in terms of powers of 1/r, the relative change in the proper time duration of the signal becomes ∆τ −∆τ∞ +O(r−2,∆m2). (11) Equations (9–11) describe the self-gravitational distor- tion between radiation shells in terms of proper time along world lines of r = const. One can notice that to leading order this is simply the gravitational redshift for the average enclosed mass between the shells. However, since ∆m changes along the wave packet non-trivally for a fixed radius as a function of time, the modification of the profile is generally not self-similar, leading to the dis- tortion of the luminosity profile as a function of radius. To make this point clearer, we correct for the average distortion of the signal and calculate the residual distor- tion of the signal. Let us define ∆τ ′ = (1 + z)∆τ −∆τ∞ (12) where (1+z) is a time-independent constant representing the “average gravitational redshift” at a given r, given by 1 + z = , (13) 〈m〉 ≡ (m1 +m2)/2 is the average mass, and to leading order . (14) Now let us take an arbitrary radiation shell enclos- ing mass ∆m relative to the shell enclosing mass M0 − 0.5∆mtot, i.e. we set m1 = M0 − 0.5∆mtot and m2 = m1−∆m in eq. (9). After correcting for the average grav- itational redshift using Eqs. (12) and (13), the residual relative distortion to leading order is +O(r−2,∆m2). (15) Figure 2 shows the residual distortion using the exact formula (Eq. (12), thick lines) and the leading order con- tribution (given by Eq. (15), dotted lines). At the typ- ical radius used by numerical simulations for waveform extraction, r = 50M⊙, the primary bulk waveform distor- tion changes the signal duration by M0/r ∼ 2%, and the secondary relative waveform distorsion between the front and the back of the signal is ǫM0/(2r) ∼ 7 × 10 −4 for a typical BH inspiral–merger with high spins ǫ = 7% (see § II). The figure also shows that the higher order effects beyond 1/r lead to an uncertainty of order 10−3–10−4 for r = (30–50)M0. B. Luminosity Profile Since Eqs. (9) and (11) are applicable to two arbitrary shells of radiation, we can use them to compute the evo- lution of an arbitrary initial radiation profile, whereas the luminosity is simply L = ∆m/∆τ , the total mass-energy crossing a sphere at radius r within proper time ∆τ . The profile at infinity is given by m(u) in radiation coordinates, or ∆τ∞(m), the proper time a shell enclos- ing mass m arrives at r = R where R → ∞, relative to the outermost shell of radiation [84]. Here ∆τ∞(m) can 10 100 1000 FIG. 2: The residual self-gravitational distortion of shells af- ter correcting for the bulk gravitational redshift. The thick curves show the change in the proper time duration of the signal at radial distance r from the source between shells en- closing 7% (top) or 3% (bottom) of the total mass, dotted lines correspond to the leading order term given by Eq. (15). The vertical lines highlight the typical radii used in numerical simulations for GW extraction. be any monotonically decreasing function, for which the luminosity at R in Eq. (3) is L∞(τ) = − d∆τ∞(m) , (16) where the minus sign originates from our definition of m: the shell labeled by the largest value of m arrives the ear- liest. The luminosity profile can also be obtained as the function of time, L∞(τ), using the relationship ∆τ(m). Conversely, for given L∞(τ), we can compute ∆τ∞(m) using Eq. (16). The luminosity at some other distance r can be obtained similarly if given ∆τ(m, r), the arrival time of mass m relative to the outermost shell at distance r. This function is given by Eq. (9), substituting the waveform ∆τ∞(m) for ∆τ∞, and (m1,m2) = (M0,m). The luminosity profile at r is then Lr(m) =− ∂∆τ(m, r) )−1/2 L∞(m) Therefore, the modification of the profile in Bondi ra- diative coordinates (m, r) the profile is distorted self- similarly. However, in terms of the observer proper time variable, ∆τ(m) = Lr(m) −1dm, Lr(τ), the modifica- tion to the profile will not be self-similar: Lr(τ) = ′)dτ ′ )−1/2 ′)dτ ′ Equation (18) relates the luminosity profile as a function of proper time at radius r, Lr(τ), to the profile at infinity, L∞(τ). Comparing Eqs. (17) and (18) shows the advan- tage of Bondi type radiative coordiantes as opposed to proper time. -100 -80 -60 -40 -20 0 20 40 r = 1000Mtot r = 12.5Mtot -100 -80 -60 -40 -20 0 20 40 r = 12.5Mtot r = 30Mtot r = 80Mtot FIG. 3: Our fits to the GW luminosity profile for binary BH inspiral merger simulations as a function of observer proper time τ and the evolution of the profile at various distances, r. The profile parameters are given in § II and ǫ = 7%. Top: The absolute profile (in units of c5/G) is shown for two extremes, a nearby distance (r = 12.5M0) and far-away distance (r = 103M0). Bottom: The difference between the GW luminosity profiles at infinity (i.e. r = 103M0) and three cases of smaller r, in units of peak luminosity at infinity. The peaks of the profiles are set to τ = 0. The main effect responsible for the differences seen in this figure is the bulk gravitational redshift. Figure 3 plots Lr(τ) for our fit to the luminosity profile at infinity of merging binary BHs L∞(τ) with ǫ = 7% (see § II). The top panel shows the absolute profile while the bottom panel shows the difference between the profile at some radius r and the profile at infinity, such that the peak of the profiles are at τ = 0. The bottom panel is useful to visualize the characteristic evolution of the profile. -100 -80 -60 -40 -20 0 20 40 r = 12.5Mtot r = 30Mtot r = 80Mtot -100 -80 -60 -40 -20 0 20 40 r = 12.5Mtot r = 30Mtot r = 80Mtot FIG. 4: The residual self-gravitational distortion to the lumi- nosity profiles after accounting for the avarage gravitational redshift, z = M0/r (top) or (M0 − 0.5∆mtot)/r (bottom), re- spectively. Other parameters are the same as in Fig. 3. In § III A, we have identified the two main effects re- sponsible for the convergence rate of the waveform to be the gravitational redshift corresponding to the aver- age mass and the self-gravitational effect. Indeed, the differences visible in Figure 3 are primarily due to the former. However, correcting for only the average gravi- tational redshift at each radius r leaves a nonnegligible systematic error with respect to the true signal. To see this, we substitute ∆τ∞/(1 + z) given by Eq. (13) into Eqs. (16) and (17), and refer to the corresponding lumi- nosity as the average gravitational redshifted luminosity profile, Lzr(τ). After subtracting from the true profile for each τ , the residual luminosity distortion is L′r(τ) = Lr(τ) − L r(τ), (19) where we set the reference time again to τ = 0 for the peak of the luminosity profiles. Figure 4 shows the residual self-gravitational distortion L′r(τ) for various radii in units of the peak luminosity at infinity, L∞,max. Naturally, the definition of the “av- erage gravitational redshift,” z, used for defining L′r(τ) makes a difference in the result. The top panel uses only the initial binary mass 〈m〉 = M0/r in Eq. (14) totally neglecting the gravity of the radiation shell, while the bottom panel has 〈m〉 = (M0 − 0.5∆mtot)/r, i.e. the redshift is chosen to account also for the average gravity of the radiation shell. In the later case we find a much quicker convergence for the waveform peak at increasing radii, but the former choice is more suitable for the early parts of the waveform corresponding to the late inspiral waveform. A comparison of Fig. 3 and 4 shows that the self-gravitational distortion is roughly an order of magni- tude smaller than the effect of the average gravitational redshift. C. Self-Gravitational Coordinate Effects In the previous sections we have derived the time du- ration and the luminosity profile of the radiation shell as it propagates radially outward from the source. We have assumed that the GW profile at each fixed arial radius r is parameterized by the proper time τ of a hypothetical observer fixed at that radius, in particular the luminosity Lr(τ) was the total mass-energy crossing a sphere at ra- dius r within infinitesimal proper time dτ . Therefore, the adopted time-coordinate τ corresponds to a synchronous gauge at each radius. Since physical observables depend precisely on proper measures, these coordinates allow a simple interpretation of the convergence characteristics of the GW profile at large radii. Other choices of coordinates would have introduced additional artificial distortion effects making the conver- gence characteristics of the waveforms much different. Consider for instance the “natural” coordinate system (t, r, θ, φ) in the spherically symmetric case that is chosen to be Schwarzschild both before and after the GW burst has arrived with masses M0 and Mf = M0 −∆mtot, re- spectively, and which changes smoothly in between these regions. An example of such a coordinate system can be derived from the Vaidya metric Eq. (5) with the implicit transformation t ≡ u + r + 2m(u) ln[r − 2m(u)] where m(u) describes the mass function interior to u (which is constant along the outgoing radiation world lines). In- deed, everywhere in the spacetime where dm/du = 0, these coordinates yield a Schwarzschild metric, and the Vaidya metric in radiation coordinates (5) is then sim- ply the Schwarzschild solution in Eddington-Finkelstein coordinates [12] in these regions. This map covers all relevant parts of the spacetime including the GW zone. The world-lines of radiation shells can be shown to follow , (20) where the second term is called the Shapiro-time delay [6] for a particle crawling out of the gravitational poten- tial of mass mu interior to it. After integration, we find that to first order the temporal separation of two radi- ation shells enclosing mass ∆m at radius r evolves to leading order as ∆t(r) = ∆t(R0)− 2∆m ln(r/R0), where R0 is an arbitrary initial radius. In these coordinates, the signal duration contracts uniformly in exponential distance intervals. Even though the metric is asymptoti- cally Minkowski (where t approaches τ for r ≫ M0), the resulting profile evolution is fundamentally different from ∆τ(r) given by Eq. (11)! The appearance of the logarithmic radial dependence of the waveform was first realized by Fock [69]. This ef- fect is specific to the harmonic coordinates and can be avoided if changing to Bondi type radiative coordinates [70, 71]. Blanchet & Schäfer [38] have shown that a sim- ilar logarithmic dependence of the GW tail leads to a tail-induced amplitude and phase shift (typically of order 10−7) for stationary sources. In contrast, the logarith- mic radial dependence of the wave contraction for merger waveforms can be significant for GW merger simulations. Between r = (20–40)M0, the waveform contracts in ∆t by a fraction of 5×10−3, which is just of the order of the current wave extraction precision [5, 26, 27]. This appar- ent logarithmic contraction effects can be avoided if one changes to the proper time variable as we have done in the previous sections. The remaining ∆τ(r) evolution is however a physical effect. Both the logarithmic ∆t(r) contraction and the phys- ical ∆τ(r) evolution (in particular the contribution de- noted by ∆τ ′(r) above) are consequences of higher order radiation effects in the Einstein equations beyond the scope of first order methods such as the Regge–Wheeler– Zerilli-Moncrief perturbation method used for extrapo- lating the numerical waveforms to infinity. Therefore these effects cause the extrapolated waveforms to be different when extracting GWs from numerical simula- tions at various radii by standard methods using no self- gravitational interaction. For a related recent analysis see Ref. [72]. IV. DISCUSSION A. Summary We considered the self-gravitational effect of gravi- tational radiation on the propagation of GWs from a compact source. We adopted simple approximations for the geometry of the radiation, by considering spherical symmetry on scales comparable to the radial width of the radiation packet. This approximation appears ade- quate for the quadrupolar (l = 2,m = 2) radiation pat- tern around binary BH sources in numerical simulations [25, 26, 27]. Nevertheless, we use the Appendix to ex- amine the maximal effects of anisotropy in the opposite (exaggerated) extreme, when the outgoing radiation is concentrated into a compact region. We find that irre- spective of the level of anisotropy, the gravitational radi- ation is distorted under the influence of its own gravity as it propagates. Contrary to the standard gravitational redshift, which is a uniform shift of the waveform, the self-gravitational effect depends on the intensity and is predominant only for the most intensive bursts of radi- ation causing a non-uniform distortion of the waveform. The self-gravitational distortion depends on distance to leading order as ∆m/r, and is therefore relevant on scales rsg/M0 ∼ < ǫ/δ where ǫ is the radiation efficiency and δ is the desired calculation accuracy. For BH binary merg- ers simulations ǫ ∼ 7% and δ ∼ 10−5 implying that rsg ∼ < 7 × 103M0. If the GWs are extracted within this region, the self-gravitational distortion should be taken into account. B. Testing the Effect with Numerical Simulations Numerical simulations based on the full set of Einstein equations for binary BH inspirals have not yet reported evidence for the waveform distortion effect considered here although they have shown that the waveforms do not converge within a fractional accuracy of δ ∼ 10−3 [5, 26, 27]. This is because the simulations are restricted to a limited volume, typically of radii ∼ (80–850)M , while the extracted waveforms are typically compared be- tween r ∼ 20–50M . For a radiation mass ∆m/M0 ∼ 7%, the primary effect is a shift of the waveform due to a logarithmic Shapiro time delay of the remnant, a uni- form gravitational redshift, and the self-gravitational ef- fect. We have shown that the logarithmic Shapiro de- lay does not show up if using proper measures to de- scribe the waveform, and the uniform gravitational red- shift is accounted for in the linear wave propagation mod- els. However, the residual self-gravitational effect in the GW luminosity has a characteristic profile that has to be subtracted when extrapolating the extracted waveform. The peak of the effective luminosity distortion reaches 2× 10−3 and 5× 10−4 at r = 30 and 50M0, respectively. Present-day numerical relativity simulations should al- ready be capable of directly measuring the relevance of our effect by artificially amplifying the gravitational radi- ation found at the extraction radius r ∼ 20M , and start- ing the simulation with these amplified initial conditions. For example, for a total GW energy ∆m/M0 = 30%, the effective luminosity distortion between r = 20M between 20–50M is several percent, which is well within simula- tion and extraction errors. For consistency, the simu- lation should confirm that the total energy content of the radiation does not change. We also expect the initial ringdown frequency (corresponding to the most energetic shell) to be smaller than the final ringdown frequency. In order to avoid errors caused by the self-gravitational distortion effect up to the desired numerical precision δ ∼ 10−5, the waveform extraction radius should be chosen to be rsg ∼ > 7 × 103M0. Alternatively, if the waveforms are extracted at smaller radii, the waveforms should be converted to Bondi type radiative coordinates and then extrapolated with the scaling 1/r. C. Observational Implications The self-gravitational waveform distortion is important for future observations of BH binary mergers. 1. The waveform distortion is expected to be resolv- able for the LISA instrument with respect to sim- ulated waveforms for total BH masses of (104– 109)M⊙. The total signal to noise ratio of merger waveforms is 104 for LISA observing zc ∼ 1 [4]. The distortion effect modifies the waveform ampli- tude and frequency by∼ (10−3–10−4) for numerical waveforms extracted between r = (30–80)M0. 2. The distortion involves a systematic modification of the waveform which needs to be accounted for in order to interpret observed merger waveforms and improve the estimation uncertainty of physical parameters beyond the uncertainty of the preced- ing inspiral signal. The signal to noise ratio of the final BH merger waveform is an order of magni- tude larger than for the inspiral, implying that the merger waveform has a potential to greatly reduce parameter estimation errors. Note that the relative accuracy using only the inspiral signal with LISA is expected to be 10−3–10−5 [73, 74] for estimating the component masses, which is smaller than the distortion effect. 3. This effect is different from the uncertainties caused by gravitational lensing [75], in that it is only an issue concerning the convergence properties of nu- merical simulations. Lensing causes an error of several percent on the inferred luminosity distance (due to the unresolved matter along the line of sight), and lensing errors increase with the source distance. In contrast, the self-gravitational effect is of order 0.1-0.01 percent for numerical simulations if the waveforms are extracted at 30M–80M and dies off quickly as 1/r. The wave distortion effect is of order 10−20 relative to the waveform ampli- tude for typical astrophysical scales. Therefore, the wave-contraction effect does not provide any addi- tional physical parameters for observations. Acknowledgments We thank George Rybicky, Irwin Shapiro and Kip Thorne for enlightening discussions. BK acknowledges support from a Smithsonian Astrophysical Observatory Predoctoral Fellowship and from NKTH Öveges József Fellowship. APPENDIX A: ANISOTROPY Our analysis considered only perfectly spherically- symmetric configurations. The approach was motivated by the quasi-spherical radiation patterns found around binary BH sources in numerical simulations [25, 26, 27]. In this Appendix, we would like to examine the sensitiv- ity of our basic results to deviations from sphericity. To gauge whether there is any such sensitivity, we analyse the most extreme case in which the outgoing radiation is concentrated into a highly compact region. But first let us define more precisely what we assumed so far. The derivation presented in § III requires that the radiation field is “initially locally spherically symmetric”, so that it is initially described by the Schwarzschild met- ric locally within some narrow solid angle ∆θ ∼ < ∆r/r be- fore the GW arrives, where ∆r ∼ c∆t is the radial width of the wave-packet along its propagation direction. But since the radiation propagating through this solid angle is in no causal contact with the radiation field expanding towards other directions, it cannot distinguish the ac- tual spacetime from a spherically symmetric one. Note that the outermost shells of radiation expanding along different directions are always causally disconnected by definition, and the interior shells of radiation can only be affected by the outer shells within ∆θ. Since we examine the distortion effect on large distances compared to the width of the burst r ≫ ∆r, spherical symmetry must only be required within a very narrow angle. This simple set of considerations implies that if high-order multipoles have a vanishing contribution at large distances, the re- sults derived in § III are applicable very generally for short bursts of radiation, ∆r ≪ r. Indeed, numerical simulations confirmed that the dominant contribution to the wave amplitude is given by the (l = 2,m = 2) multi- pole and higher order terms are suppressed by more than a factor of magnitude (see references in § II). In the re- mainder of this Appendix we demonstrate the validity of this simple conclusion through explicit calculations. We consider three variations on a toy model to estimate the effect of anisotropy. We start with the simplest model and refine this model by adding more details and com- plexity in the successive models. In each case, we discuss general implications for the model under consideration. In all models we consider the extreme opposite regime to spherical symmetry, namely that the radiation is maxi- mally clumped into two outgoing BHs L (leading) and T (trailing) of masses mL and mT , representing the leading and trailing edges of the radiation, respectively. We as- sume that L and T are moving in the same direction on light-like world lines, so that T lies in the causal past of L, but L is outside the causal past of T throughout their propagation. We assume that there is also a remnant Schwarzschild BH R with mass mR. The instantaneous radial position coordinate of R, T , and L at time t are 0, rT (t), and rL(t). We are interested in obtaining the world lines of BHs T and L to see how the coordinate separation ∆r(t) = rL(t) − rT (t) decreases with time as compared to the spherically symmetric result. In our first model we neglect the remnant R (setting mR = 0), assume that L moves with constant velocity vL in free space, and calculate the trajectory of T in the spacetime created by L. Subsequently, we will generalize L to move on a more general world line with a slowly changing ve- locity vL(t). Finally, we can turn on the remnant R in addition to L, and include the retardation effect when calculating the relative motion of T . We note that the spacetime of BHs moving at the speed of light have been calculated previously in Ref. [46], which found that BHs moving in the same direction do not interact. However, Ref. [46] assumed that the BHs move in free space and consequently adopted v = 1 for their velocity. In contrast, the BHs T and L travel on null-geodesics in the perturbed spacetime which is ini- tially the Schwarzschild spacetime. This difference gives rise to a non-trivial interaction between the BHs T and We compare our results to the spherical case, using the (t, r) coordinate system defined by Scwarzschild coordi- nates before and after the radiation shells as described in § III C. 1. No remnant mR = 0, constant vL velocity We start by assuming that L is a BH with constant velocity vL < 1 in free space, and wish to calculate the world line of T in this background. Here, we assume that no remnant is present, and that L and T move along the same spatial direction, which we denote by x. Thus it is sufficient to restrict our attention to the two dimensions (t, x) of the spacetime. Let us start by deriving the metric. In the coordi- nate system (t′, x′) comoving with L, the metric is the Schwarzschild metric ds2 = −(1−φ′)dt′2+(1−φ′)−1dx′2, where φ′ = 2m′L/|x ′|. Here x′ ≡ 0 corresponds to the BH L for all t′, and m′L = mL/γ is the rest mass of L, where mL is the energy carried by L in the original (t, x) co- ordinates and γ = 1/ 1− v2L is the Lorentz factor. To derive the metric in the (t, x) coordinate system, we ap- ply the diffeomorphism (t, x) = γ(t′+vLx ′, vLt ′+x′), i.e. a global Lorentz transformation, ds2 = − (1 − φ′)2 − v2L (1− v2L)(1 − φ dt2 + 1− v2L(1− φ (1 − v2L)(1− φ ′(2− φ′) (1− v2L)(1 − φ dtdx (A1) which can be rearranged as ds2 = [vL + (1− φ ′)]dt− [1 + vL(1− φ ′)]dx (1− v2L)(1 − φ [vL − (1 − φ ′)]dt− [1− vL(1 − φ ′)]dx (1− v2L)(1 − φ .(A2) Here φ′ is to be expressed as the function of the new coordinates (t, x), i.e. φ′ = γ−2φ where φ = 2mL/|∆x|, ∆x = xL−x, and xL = vLt is the instantaneous position of the singularity. The xT (t) null-geodesics describing the world line of T can be obtained by setting ds2 = 0. Equation (A2) shows that there are two solutions [vL + (1− γ −2φ)]dt − [1 + vL(1− γ −2φ)]dxT = 0, [vL − (1− γ −2φ)]dt − [1− vL(1− γ −2φ)]dxT = 0. These differential equations can also be obtained more simply by finding the null-geodesics in the comoving co- ordinates (t′, x′) first, and changing to the (t, x) coordi- nates only in the resulting equation. The null geodesics in the comoving coordinates are simply dx′T /dt ′ = ±|1−φ′| (see Eq. 20), and the Lorentz boost coordinate trans- formation of this differential equation leads instantly to (A3-A4). Therefore, the two solutions (A3-A4) describe the null geodesics approaching or receding the moving BH, respectively. We would like to find the solution for the T test particle approaching the source L from behind, namely Eq. (A3) for an initial condition xT < xL. This first-order dif- ferential equation can be solved analytically by a linear substitution. The coordinate velocity vT = dxT (t)/dt monotonously decreases from 1 to vL as the event horizon at xLhor(t) = vLt− 2γ −2mL is approached. In particular if vL = 1, i.e. the source L has the speed of light in the free-space background, then the trailing test particle T will not be delayed at all, vT (t) = 1 for all t. However, if vL ≪ 1 then T is considerably affected by the Shapiro delay near the horizon of L. In concluding the description of this model, let us sum- marize how the clumpy case compares to the spherically symmetric case of expanding radiation shells. First re- call that in the spherically symmetric case, L has no ef- fect on T throughout the dynamics regardless of vL or ∆r. In the clumpy case, the gravity of L delays the mo- tion of T . The magnitude of this delay is significant only if both of two conditions are violated: (a) vL ≈ 1 and (b) ∆x = xL − xT ≫ 2γ −2mL. What are the “typi- cal numbers” for these quantities? Eq. (20) implies that vL = 1 − 2M0/x (which is also true in the clumpy case, see § A3), implying that γ−2 ∼ 4M0/r, and for binary mergers ∆x ∼ 12πM0, mL < M0 − Mf ∼ < 0.06M0, we find that the two cases are equivalent to (a) x ≫ 2M0 and (b) x ≫ 0.2∆m ∼ > 0.01M0. Quite clearly, these conditions will not be violated for distances outside the dynamical regime of strong gravity e.g. x ∼ > R0 = 30M0. Thus, we expect only very minor modifications relative to the spherically symmetric case, even in the most clumpy case. For a quantitative estimate we need to integrate these modifications over the relevant distances which we describe next. 2. No remnant, slowly changing vL(t) Next we consider a slowly changing source velocity vL(t) for the BH L, continue to neglect a remnant R, and calculate the motion of T in this spacetime. Since L is assumed to move on a light-like world-line, it is not ef- fected by R and only responds to the background created prior to the production of the bursts. Thus we assume L moves on the null-geodesic of the background as de- scribed by (20) with vL = 1− 2mT /xL. If vL is slowly changing, we can consider vL to be con- stant during short time intervals with infinitesimal jumps on their boundaries. We can then find the correspond- ing world line segments of T by solving the differential equation (A2) and matching the boundary conditions of the successive segments by requiring continuity. In the limit that the length of the constant time intervals ap- proaches zero, the world line xT (t) at every instant is given by Eq. (A2) with vL now denoting the instanta- neous velocity, and φ′ referring to the instantaneous value of the potential: φ′ = γ−2φ, where φ = 2mL/|∆x| with ∆x = xL − x and xL = vL(t)dt. Thus, vL + 1− γ 1 + vL(1− γ−2φ) 2mL(1 − vL) ∆x− 2mLvL(1− vL) Substituting vL = 1−2mT /xL, the distance between the clumps of radiation satisfies 4mLmT xL∆x− 4mLmT 1− 2mTx Note that xL(t) can be used to express the width of the packet ∆x as a function of xL. To simplify the result, we use qL = mL/mT , set the units to the Schwarzschild radius 2mT = 1, and express (A6) in terms of the loga- rithmic distance variable, y = ln(xL − 1). Then = −1 + xL∆x− qL 1− x−1L ) (A7) which to first order in 1/xL becomes = −1 + . (A8) Equation (A8) shows that to leading order, the wave- packet packet contracts linearly in terms of the logarith- mic distance variable x. In the spherically-symmetric case, the inner shell T is not influenced by L and so dxT /dt = 1 instead of Eq. (A5), leading to d∆x/dy = −1. Therefore the distortion of wave-packets in the max- imally clumpy case is the same as in the spherically sym- metric case to leading order. The difference arises in the next order given by the second term in (A8) describing how the gravity of L Shapiro-delays the motion of T . This is typically of order (4 × 6%)/(12π) × x−1L ∼ 10 for xL ∼ 30M0 and gets exponentially smaller for expo- nentially larger distances. 3. Remnant included, changing vL The previous model assumed no remnant (i.e. mR = 0), and postulated that T propagated in the spacetime of L by assuming that the spacetime at T was the spacetime of L at the same instant. Here we consider a nonzero mR and account for the retardation of the effect of L as percieved by T . We assume a slowly changing velocity and that the gravitational perturbations are sufficiently small to allow simple superposition to leading order. We follow a simplified approach with the following essential assumptions: 1. The initial condition is a spherically symmetric Schwarzschild spacetime centered at x = 0. 2. Lmoves on a null-geodesic xL(t) of the initial back- ground metric of R and T , i.e. in a Schwarzschild metric centered at x = 0 for all t and for a mass mR +mT . The world-line of L follows (20) accord- ingly. 3. T moves on a null-geodesic xT (t) of the background metric of R and L, which we assume to be a sim- ple superposition gij = ηij + δg ij + δg L,ret ij . Here δgNij = g ij − ηij for a given metric, g ij , where ηij is the Minkowski metric, gRij is the metric of the remnant i.e. the Schwarzschild metric centered at x = 0 for all t with mass mR. The metric g ij is the stationary boosted Schwarzschild metric (A1–A2) with mass mL, velocity vL, centered at xL. The label ret stands for retardation, which we describe next separately. 4. We account for retardation by setting ∆tret = t − tret to be the light-travel time from L to T . For this we compute the inward propagating null-geodesics from L to T [i.e. between positions (tret, rL(tret)) and (t, rT (t))], based on the initial backgroundmet- ric of R and T (i.e. neglecting the gravity of L). To find the retardation time, we note that the inward propagating null geodesics satisfies dx/dt = −[1−2(mR+ mT )/x]. Since this is exactly the time-reversed world line of L, we get xT (t) = xL(t − 2∆tret). Integrating dt/dx for the world line of L between xL(t− 2∆tret) and xL(t), 2∆tret = ∫ xL(t) xT (t) xL − 2mR − 2mT dxL, (A9) from which ∆tret = xL − xT + (mR +mT ) ln xL − 2mR − 2mT xT − 2mR − 2mT (A10) The distance where T percieves L is xretL (t) = xL(t − ∆tret) and the separation is ∆xret ≡ x L − xT . Substi- tuting xT as xT = xL −∆x and xT = x L −∆xret into (A10), we can find ∆xret for given ∆x and xL. To first nonvanishing order in 1/xL, ∆xret = (mR +mT )∆x . (A11) The leading order term corresponds to the propagation at the speed of light in free space, vL = vT = 1. Note that the correction is proportional to x−2L , which is extremely small for the physical cases beyond the strong field zone. Finally, we define the retarded position and velocity xretL = xL − ∆xret, and v L = 1 − 2(mR + mT )/x which can be written in terms of xL and ∆x using Eq. (A11). The metric contribution δg L,ret ij of L at T at time t, is the boosted Schwarzschild metric (A1,A2) with instan- taneous velocity vretL , a singularity at x L , and distance ∆xret. Now we can redo the derivation presented in § A2 to find the motion of T , using the modified spacetime gij given above. Again we find two solutions for ds2 = 0 rep- resenting the ingoing and outgoing radiation. Expanding the outgoing solution in a series in x−1L , we find clumpy qR∆x+ 2qL(1 + q (A12) where qi = mi/(mR+mT ) for i ∈ {R, T, L} and distance units are chosen to be the Schwarzschild radius 2(mR + mT ) ≡ 1. In order to get the instantaneous shell width ∆x as a function of logarithmic distance y, we can redo the manipulations of (A6–A8) for the result (A12). To first order in 1/xL, clumpy = −qT + qR∆x+ 2qL(1 + q (A13) In the limit of no remnant qR = 0, we almost recover the solution derived previously in Eq. (A8). There is a factor 2 difference, which is the direct consequence of the retardation of the percieved distance ∆x, which had been neglected in Eq. (A8). Equation (A13) should be contrasted to the expansion of two spherically symmetric shells mT and mL in the presence of a remnant mR. We may expand the corre- sponding spherical solution of § III C in a series in 1/xL to first order: spherical = −qT + . (A14) The first two terms in Eqs. (A13) and (A14) are iden- tical. The correction describing the “Shapiro delay” of contraction in the clumpy case due to the gravity of L is 2qL(1 + q R)/(xL∆x). Substituting typical physical val- ues qR = 97%, qT = 3%, qR = 3%, and ∆x0 ∼ 6π ∼ xL0 in units of Schwarzschild radii, we see that the correc- tion is of order 10−4 initially, and becomes exponentially smaller at exponentially larger distances. In summary, even in the most extreme case of clumpiness, the radia- tion packet propagates to very high precision according to the spherically symmetric description. [1] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), arXiv:gr-qc/0507014. [2] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlo- chower, Phys. Rev. Lett. 96, 111101 (2006), arXiv:gr- qc/0511048. [3] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96, 111102 (2006), arXiv:gr-qc/0511103. [4] J. G. Baker, S. T. McWilliams, J. R. van Meter, J. Cen- trella, D.-I. Choi, B. J. Kelly, and M. Koppitz, ArXiv General Relativity and Quantum Cosmology e-prints (2006), gr-qc/0612117. [5] E. Pazos, E. N. Dorband, A. Nagar, C. Palenzuela, E. Schnetter, and M. Tiglio, Classical and Quantum Gravity 24, 341 (2007), arXiv:gr-qc/0612149. [6] I. I. Shapiro, Phys. Rev. Lett. 13, 789 (1964). [7] P. C. Vaidya, Physical Review 83, 10 (1951). [8] P. C. Vaidya and K. B. Shah, Progress of Theoretical Physics 24, 111 (1960). [9] D. R. Brill and J. B. Hartle, Physical Review 135, 271 (1964). [10] R. A. Isaacson, Physical Review 166, 1263 (1968). [11] R. A. Isaacson, Physical Review 166, 1272 (1968). [12] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Grav- itation (San Francisco: W.H. Freeman and Co., 1973, 1973). [13] K. S. Thorne and S. J. Kovacs, ApJ 200, 245 (1975). [14] K. S. Thorne, Reviews of Modern Physics 52, 299 (1980). [15] M. Efroimsky, Phys. Rev. D 49, 6512 (1994). [16] T. Regge and J. A. Wheeler, Physical Review 108, 1063 (1957). [17] F. J. Zerilli, Phys. Rev. D 2, 2141 (1970). [18] V. Moncrief, Annals of Physics 88, 323 (1974). [19] B. Brügmann, J. A. González, M. Hannam, S. Husa, U. Sperhake, and W. Tichy, ArXiv General Relativity and Quantum Cosmology e-prints (2006), gr-qc/0610128. [20] J. Baker, M. Campanelli, C. O. Lousto, and R. Taka- hashi, Phys. Rev. D 69, 027505 (2004), arXiv:astro- ph/0305287. [21] M. Campanelli, C. O. Lousto, and Y. Zlochower, Phys. Rev. D 73, 061501(R) (2006), arXiv:gr- qc/0601091. [22] M. Campanelli, C. O. Lousto, and Y. Zlochower, Phys. Rev. D 74, 041501(R) (2006), arXiv:gr- qc/0604012. [23] W. Tichy and P. Marronetti, ArXiv General Relativity and Quantum Cosmology e-prints (2007), gr-qc/0703075. [24] F. Pretorius and D. Khurana, Classical and Quantum Gravity 24, 83 (2007), arXiv:gr-qc/0702084. [25] E. Berti, V. Cardoso, J. A. Gonzalez, U. Sperhake, M. Hannam, S. Husa, and B. Bruegmann, ArXiv Gen- eral Relativity and Quantum Cosmology e-prints (2007), gr-qc/0703053. [26] A. Buonanno, G. B. Cook, and F. Pretorius, Phys. Rev. D 75, 124018 (2007), arXiv:gr-qc/0610122. [27] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. D 73, 104002 (2006), arXiv:gr- qc/0602026. [28] L. Blanchet and T. Damour, Physics Letters A 104, 82 (1984). [29] L. Blanchet and T. Damour, Royal Society of London Philosophical Transactions Series A 320, 379 (1986). [30] L. Blanchet and T. Damour, Phys. Rev. D 46, 4304 (1992). [31] L. Blanchet, Living Reviews in Relativity 9, 4 (2006). [32] K. G. Arun, Phys. Rev. D 74, 024025 (2006), arXiv:gr- qc/0605021. [33] T. Tanaka, H. Tagoshi, and M. Sasaki, Progress of The- oretical Physics 96, 1087 (1996), arXiv:gr-qc/9701050. [34] L. Blanchet, Classical and Quantum Gravity 15, 1971 (1998), arXiv:gr-qc/9801101. [35] A. G. Wiseman and C. M. Will, Phys. Rev. D 44, R2945 (1991). [36] E. Poisson, Phys. Rev. D 47, 1497 (1993). [37] A. G. Wiseman, Phys. Rev. D 48, 4757 (1993). [38] L. Blanchet and G. Schäfer, Classical and Quantum Gravity 10, 2699 (1993). [39] L. Blanchet, Classical and Quantum Gravity 15, 89 (1998), arXiv:gr-qc/9710037. [40] L. Blanchet, Classical and Quantum Gravity 15, 113 (1998), arXiv:gr-qc/9710038. [41] D. Christodoulou, Phys. Rev. Lett. 67, 1486 (1991). [42] K. S. Thorne, Phys. Rev. D 45, 520 (1992). [43] R. J. Gleiser and A. E. Domı́nguez, Phys. Rev. D 68, 104018 (2003), arXiv:gr-qc/0306085. [44] W. B. Bonnor, Communications in Mathematical Physics 13, 163 (1969). [45] P. C. Aichelburg and R. U. Sexl, Gen. Rel. Grav. 2, 303 (1971). [46] K. Hayashi and T. Samura, Phys. Rev. D 50, 3666 (1994), arXiv:gr-qc/9404027. [47] J. T. Mendonça and V. Cardoso, Phys. Rev. D 66, 104009 (2002), arXiv:gr-qc/0209076. [48] M. Servin, M. Marklund, G. Brodin, J. T. Mendonça, and V. Cardoso, Phys. Rev. D 67, 087501 (2003), arXiv:astro- ph/0303412. [49] J. T. Mendonça, V. Cardoso, M. Marklund, M. Servin, and G. Brodin, Phys. Rev. D 68, 084025 (2003), arXiv:gr-qc/0307031. [50] E. Nakar, A. Gal-Yam, and D. B. Fox, ApJ 650, 281 (2006), arXiv:astro-ph/0511254. [51] R. M. O’Leary, F. A. Rasio, J. M. Fregeau, N. Ivanova, and R. O’Shaughnessy, ApJ 637, 937 (2006), arXiv:astro-ph/0508224. [52] S. F. Portegies Zwart and S. L. W. McMillan, ApJ 528, L17 (2000), arXiv:astro-ph/9910061. [53] J. S. B. Wyithe and A. Loeb, ApJ 610, 117 (2004), arXiv:astro-ph/0401554. [54] S. F. Portegies Zwart, H. Baumgardt, S. L. W. McMil- lan, J. Makino, P. Hut, and T. Ebisuzaki, ApJ 641, 319 (2006), arXiv:astro-ph/0511397. [55] M. Micic, K. Holley-Bockelmann, S. Sigurdsson, and T. Abel, ArXiv Astrophysics e-prints (2007), astro- ph/0703540. [56] J. S. B. Wyithe and A. Loeb, ApJ 590, 691 (2003), arXiv:astro-ph/0211556. [57] A. Sesana, F. Haardt, P. Madau, and M. Volonteri, Clas- sical and Quantum Gravity 22, 363 (2005), arXiv:astro- ph/0502462. [58] B. Kocsis, M. E. Gáspár, and S. Márka, ApJ 648, 411 (2006), arXiv:astro-ph/0603441. [59] J. G. Baker, M. Campanelli, F. Pretorius, and Y. Zlo- chower, Classical and Quantum Gravity 24, 25 (2007), arXiv:gr-qc/0701016. [60] F. Löffler, L. Rezzolla, and M. Ansorg, Phys. Rev. D 74, 104018 (2006), arXiv:gr-qc/0606104. [61] M. Campanelli, C. O. Lousto, and Y. Zlochower, Phys. Rev. D 74, 084023 (2006), arXiv:astro- ph/0608275. [62] W. H. Lee and W. L. Kluźniak, MNRAS 308, 780 (1999), arXiv:astro-ph/9904328. [63] W. H. Lee and W. L. Kluźniak, ApJ 526, 178 (1999), arXiv:astro-ph/9808185. [64] U. Sperhake, B. Kelly, P. Laguna, K. L. Smith, and E. Schnetter, Phys. Rev. D 71, 124042 (2005), arXiv:gr- qc/0503071. [65] M. Alcubierre, W. Benger, B. Brügmann, G. Lan- fermann, L. Nerger, E. Seidel, and R. Takahashi, Phys. Rev. Lett. 87, 271103 (2001), arXiv:gr-qc/0012079. [66] P. D. D’Eath and P. N. Payne, Phys. Rev. D 46, 694 (1992). [67] V. Cardoso and J. P. S. Lemos, Phys. Rev. D 67, 084005 (2003), arXiv:gr-qc/0211094. [68] R. M. Wald, General relativity (Chicago, University of Chicago Press, 1984). [69] V. A. Fock, Theory of space time and gravitation (New York, Pergamon Press, 1959). [70] R. A. Isaacson and J. Winicour, Physical Review 168, 1451 (1968). [71] J. Madore, Annales de l’institut Henri Poincaré (A) Physique théorique 12, 285 (1970). [72] L. Lehner and O. M. Moreschi, ArXiv e-prints 706 (2007), 0706.1319. [73] R. N. Lang and S. A. Hughes, Phys. Rev. D 74, 122001 (2006), arXiv:gr-qc/0608062. [74] R. N. Lang and S. A. Hughes, Phys. Rev. D 75, 089902(E) (2007). [75] D. E. Holz and S. A. Hughes, ApJ 629, 15 (2005), arXiv:astro-ph/0504616. [76] http://www.ligo.caltech.edu/ [77] http://www.virgo.infn.it/ [78] http://tamago.mtk.nao.ac.jp/ [79] http://geo600.aei.mpg.de/ [80] http://www.ligo.caltech.edu/advLIGO/ [81] http://wwwcascina.virgo.infn.it/advirgo/ [82] http://www.icrr.u-tokyo.ac.jp/gr/LCGT.html [83] http://www.lisa-science.org/ [84] Throughout this paper we assume that the metric at r → ∞ is the Minkowski metric, neglecting cosmologi- cal effects. This is justified by the orders of magnitudes or radii where the radiation shell distortion is relevant r < 103M0 (see Fig. 2) is much less than cosmological scales r ∼ > 1018M0.

0704.1150

Fermionic approach to the evaluation of integrals of rational symmetric functions

CRM-3xxx(2006) Fermionic approach to the evaluation of integrals of rational symmetric functions 1 J. Harnad†‡2 and A. Yu. Orlov⋆3 † Centre de recherches mathématiques, Université de Montréal C. P. 6128, succ. centre ville, Montréal, Québec, Canada H3C 3J7 ‡ Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montréal, Québec, Canada H4B 1R6 ⋆ Nonlinear Wave Processes Laboratory, Oceanology Institute, 36 Nakhimovskii Prospect Moscow 117851, Russia Abstract We use the fermionic construction of two-matrix model partition functions to evaluate inte- grals over rational symmetric functions. This approach is complementary to the one used in the paper “Integrals of Rational Symmetric Functions, Two-Matrix Models and Biorthogonal Poly- nomials” [1], where these integrals were evaluated by a direct method. Using Wick’s theorem, we obtain the same determinantal expressions in terms of biorthogonal polynomials as in [1]. 1 Introduction In this work, we consider the following integral N (ξ, ζ, η, µ) := dµ(x1, y1) . . . dµ(xN , yN)∆N(x)∆N (y) 1Work of (J.H.) supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds FCAR du Québec; that of (A.O.) by the Russian Academy of Science program “Mathematical Methods in Nonlinear Dynamics” and RFBR grant No 05-01-00498. [email protected] [email protected] http://arxiv.org/abs/0704.1150v1 α=1(ξα − xa) β=1(ζα − ya) j=1(ηj − xa) k=1(µk − ya) (1.1) where ∆N(x) := (xi − xj), ∆N (y) := (yi − yj) (1.2) are Vandermonde determinants, and dµ(x1, y1) . . . dµ(xN , yN)∆N (x)∆N(y), (1.3) is a normalization constant, which in the context of two-matrix models is interpreted as the partition function. Here dµ(x, y) is a measure (in general, complex), supported on a finite set of products of curves in the complex x and y planes. It is known (see for instance [5]) that this integral can be presented in the form of a determinant of an N ×N matrix; namely, N (ξ, ζ, η, µ) = α=1(ξα − x) β=1(ζβ − y) j=1(ηj − x) k=1(µk − y) dµ(x, y)(x, y)xjyk 0≤j,k≤N−1 (1.4) However for some purposes, it is more useful to express it as the determinant of a matrix whose size does not depend on N . The problem of evaluation of integrals of such symmetric functions of N variables is of importance in the context of matrix models [4], where N is the number of eigenvalues. Expressing integrals that determine correlation functions and expectation values as de- terminants of matrices whose size is independent of N is of importance e.g., in the study of N → ∞ limits. For one-matrix models such integrals were studied in [3], [6], [7], [8] and [9]. In the case of complex, normal or two-matrix models [10] the problem was considered in [8], [2], [1]. Remark 1.1. In order to compare with the results in [8] and [2] on complex integrals the variables {xa, ya}a=1,...N must be replaced by complex conjugate pairs {za, z̄a}a=1,...N , and the integration domains taken as N copies of the complex plane. The present paper is complementary to ref. [1], where such integrals were evaluated by a “direct” method based on partial fraction expansions and the Cauchy-Binet identity. Ref. [1] and the present work are intended to provide concise presentations and new derivations of previously known as well as new results, using two distinct methods. The approach used here is based on constructing a fermionic representation of integrals of such rational symmetric functions, introduced in [19], using two–component fermions. This representation is different from the ones used previously in the context of matrix models in [16], [17] and [18], which were based on one-component fermions. Remark 1.2. Although there is a close connection between matrix integrals, orthogonal poly- nomials and the spectral transform approach to the theory of integrable systems, in the present work we do not develop this latter aspect, which concerns the deformation theory of the mea- sures involved. Our starting point however is similar to the previous paper [19], where the deformation theory is addressed, and the fermionic approach to integrable systems developed in [14] is utilized. The relation of these results to the objects appearing in the spectral transform approach is, however, briefly explained in the Appendices, where the biorthogonal polynomials and their Hilbert transforms are interpreted as Baker functions and adjoint Baker functions. The language of free two-component fermions used here is borrowed from [14]. The integrals that we are interested in are expressed as vacuum state expectation values of operator products formed from free fermion generators. Besides successive application of Wick’s theorem, the main computational device used consists of applying canonical (“dressing”) transformations to pass from free fermions fields, whose Laurent coefficients are the free Fermi creation and annihilation operators, to “dressed’ ones, where the mono- mials are replaced by biorthogonal polynomials. Wick’s theorem then serves to express the same operator product vacuum state expectation value (VEV) both as a multiple ratio- nal integral and as the determinant of a matrix formed from elementary factors involving orthogonal polynomials and their Hilbert transforms. Let dµ(x, y) be a measure (in general, complex), supported on a finite set of products of curves in the complex x and y planes, for which the semi-infinite matrix of bimoments is finite: Bjk := dµ(x, y)xjyk <∞, 0, ∀j, k ∈ N. (1.5) The integrals are understood to be evaluated on a specified linear combination of products of the support curves. Assuming that, for all N ≥ 1, the N×N submatrix (Bjk)0≤j,k,≤N−1 is nonsingular, the Gram-Schmidt process may be used to construct an infinite sequence of pairs of biorthogonal polynomials {Pj(x), Sj(y)}j=0...∞, unique up to signs, satisfying dµ(x, y)Pj(x)Sk(y) = δjk, (1.6) and normalized to have leading coefficients that are equal: Pj(x) = +O(xj−1), Sj(x) = +O(yj−1). (1.7) We will also assume that the Hilbert transforms of these biorthogonal polynomials, P̃n(µ) := dµ(x, y) Pm(x) , S̃n(η) := dµ(x, y) Sm(x) η − x , (1.8) exist for all n ∈ N. The following expression for Z N in terms of the leading term normalization factors hn is then easily shown to hold (see, e.g., [4]). N = N !det (Bjk) |j,k=1,...,N = N ! hn (1.9) Defining N1 := N + L1 −M1, N2 := N + L2 −M2, (1.10) we consider different cases. In each case the answer is written in form of the determinant of a matrix G which is different for different cases. Results are given (1) N2 ≥ N1 ≥ 0: by formulae (5.9)-(5.15). G is a (L2 +M1)× (L2 +M1) matrix (2) N1 ≤ 0 ≤ N2: by formulae (5.34)-(5.35). G is a (L2 +M1)× (L2 +M1) matrix (3) N1 ≤ N2 ≤ 0: by formulae (5.44)-(5.45). G is a (M1 +M2 −N)× (M1 +M2 −N) matrix The cases N1 ≥ N2 ≥ 0, N2 ≤ 0 ≤ N1 and N2 ≤ N1 ≤ 0 are related to previous ones by interchanging quantities L1 ↔ L2, M1 ↔ M2, N1 ↔ N2, ξ ↔ ζ , µ ↔ η, Pn ↔ Sn and P̃n ↔ S̃n in the final formulae. (See respectively (5.9)-(5.15) vs. (5.22)-(5.26), (5.34)- (5.35) vs. (5.36)-(5.37) and (5.44)-(5.45) vs. (5.46)-(5.47).) For instance, for the case N1 ≥ N2 ≥ 0 the answer is N (ξ, ζ, η, µ) = (−1) (M1+M2)(M1+M2−1)(−1)L1M2 N+L2−M2−1 N+L1−M1−1 j=1(ξα − ηj) k=1(ζβ − µk) ∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ) detG, (1.11) where G is a (L1+M2)×(L1+M2) matrix whose entries are expressed in terms of the values of Pn, Sn, P̃n and S̃n for n = 0, . . . ,max(N1, N2) evaluated at the points {ξα, ζβ, ηj , µk}. Remark 1.3. It is assumed that the support curves for the integrals involved do not pass through the points {ηj , µk}j=1,...M1,k=1,...M2 . The definition of the integrals may be extended throughout the complex plane of these parameters by analytic continuation, but might then take on multiple values. However, if the measures dµ(x, y) vanish with sufficient rapidity as these points approach the support curves, the integrals considered in the present paper become single valued. If the replacements {xa, ya}a=1,...N→{za, z̄}a=1,...N , are made, in order to compare with the results of [8], [2], it must be assumed that the corresponding measures dµ(z, z̄ in the complex plane vanish sufficiently rapidly at the points {z = ηj , z̄ = µk} for the integrals to converge. In the following sections, we present the integral (1.1) as a vacuum state expectation value of a certain fermionic expression, using two-component fermions (see (3.9) below). To evaluate this expectation value we use Wick’s theorem, which expresses the vacuum state expectation value of products of linear combinations of fermions as a determinant of the matrix formed by evaluating the pair-wise VEV’s. Since the expression (3.9) is not as yet of the form of a vacuum expectation value of a product of fermions, we use a set of tricks (namely, canonical transformations of fermions and re-writing of charged vacuum vectors in a suitable form) which serve also to reduce the final number of fermions inside the vacuum expectation value, and therefore to reduce as much as possible the size of the matrix G. The structure of the paper is the following. First we recall some facts about free fermions, including Wick’s theorem. Then we equal I N (ξ, ζ, η, µ) to a certain vacuum expectation value, see (3.12). After certain preliminaries, via (4.25), we introduce dressed fermions bα, b̄α to adapt formula (3.12) for a usage of Wick’s theorem in its determinantal form. Then we consider each of the numbered cases separately. For the case (1) (which is the most involved case) we introduce fermionic operators aα, āα, obtained from bα, b̄α via a sort of conjugation (see (5.4)-(5.7)), and whose vacuum expectation evaluated via Wick’s theorem yields the final result given by formulae (5.9)-(5.15) and are illustrated by six examples. For the case (2), instead of aα, āα, we similarly use operators bα, b̄α. At last, for the case (3), instead of aα, āα, we exploit operators cα, c̄α introduced in (5.39)-(5.42). Remark 1.4. One can obtain the corresponding results for N -fold integrals arising in one- matrix models by specifying that the measure be proportional to Dirac delta function, say δ(x − y). As this specialization will not be consider detailed here we shall write IN instead of 2 Summary of free fermions The following is a summary regarding the one and two-component free fermion algebra based on the introductory section of [19]. The reader may refer to [19], or [15], [14] for further details. 2.1 One component fermions In the following, A denotes the complex Clifford algebra over C generated by charged free fermions {fi, f̄i}i∈Z, satisfying the anticommutation relations [fi, fj]+ = [f̄i, f̄j]+ = 0, [fi, f̄j ]+ = δij . (2.1) where [, ]+ denotes the anticommutator. Elements of the linear part W := (⊕m∈ZCfm)⊕ ⊕m∈ZCf̄m (2.2) will be referred to as a free fermions. The fermionic free fields f(x) := k, f̄(y) := −k−1, (2.3) may be viewed as generating functions for the fj , f̄j’s. This Clifford algebra has a standard Fock space representation F and dual space F̄ (see [19]) which contain unique vacuum states |0〉 and 〈0| respectively satisfying the properties fm|0〉 = 0 (m < 0), f̄m|0〉 = 0 (m ≥ 0), 〈0|fm = 0 (m ≥ 0), 〈0|f̄m = 0 (m < 0). (2.4) The Fock spaces F and F̄ are mutually dual, with the hermitian pairing defined via the linear form 〈0||0〉 on A called the vacuum expectation value. This satisfies 〈0|1|0〉 = 1; 〈0|fmf̄m|0〉 = 1, m < 0; 〈0|f̄mfm|0〉 = 1, m ≥ 0, (2.5) 〈0|fn|0〉 = 〈0|f̄n|0〉 = 〈0|fmfn|0〉 = 〈0|f̄mf̄n|0〉 = 0; 〈0|fmf̄n|0〉 = 0, m 6= n, . (2.6) Wick’s theorem implies that for any finite set of elements {wk ∈ W}, we have 〈0|w1 · · ·w2n+1|0〉 = 0, 〈0|w1 · · ·w2n|0〉 = σ∈S2n sgnσ〈0|wσ(1)wσ(2)|0〉 · · · 〈0|wσ(2n−1)wσ(2n)|0〉. (2.7) Here σ runs over permutations for which σ(1) < σ(2), . . . , σ(2n− 1) < σ(2n) and σ(1) < σ(3) < · · · < σ(2n− 1). If {wi}i=1,...,N , are linear combinations of the fj ’s only, j ∈ Z, and {w̄i}i=1,...,N linear combinations of the f̄j ’s, j ∈ Z, then(2.7) implies 〈0|w1 · · ·wN w̄N · · · w̄1|0〉 = det (〈0|wiw̄j|0〉) |i,j=1,...,N (2.8) Following [14], [15], for all N ∈ Z, we also introduce the states 〈N | := 〈0|CN (2.9) where CN := f̄0 · · · f̄N−1 if N > 0 (2.10) CN := f−1 · · · fN if N < 0 (2.11) CN := 1 if N = 0 (2.12) |N〉 := C̄N |0〉 (2.13) where C̄N := fN−1 · · · f0 if N > 0 (2.14) C̄N := f̄N · · · f̄−1 if N < 0 (2.15) C̄N := 1 if N = 0 (2.16) The states (2.9) and (2.13) are referred to as the left and right charged vacuum vectors, respectively, with charge N . From the relations 〈0|f̄N−kf(xi)|0〉 = xN−ki , 〈0|f−N+k−1f̄(yi)|0〉 = yN−ki , k = 1, 2, . . .N, (2.17) and (2.8), it follows that 〈N |f(xn) · · ·f(x1)|0〉 = δn,N∆N(x), N ∈ Z, (2.18) 〈−N |f̄(yn) · · · f̄(y1)|0〉 = δn,N∆N(y), N ∈ Z. (2.19) For free fermion generators with |x| 6= |y|, 〈0|f(x)f̄(y)|0〉 = 1 (2.20) Note that the expression on the right hand side is actually defined, by (2.5), as the infinite series n=0 y nx−n−1 which converges only inside |x| < |y|. However one can consider expression (2.20) for the whole region of x and y (when |x| 6= |y|) in the sense of analytical continuation. From Wick’s theorem it follows that 〈n−m|f(xn) · · ·f(x1)f̄(ym) · · · f̄(y1)|0〉 = ∆n(x)∆m(y) i=1,...,n j=1,...,m (xi − yj) (2.21) 2.2 Two-component fermions The two-component fermion formalism is obtained by relabelling the above as follows. f (α)n := f2n+α−1 , f̄ n := f̄2n+α−1 , (2.22) (α)(z) := k , f̄ (α)(z) := k , (2.23) where α = 1, 2. Then (2.1) is equivalent to [f (α)n , f m ]+ = [f̄ n , f̄ m ]+ = 0, [f n , f̄ m ]+ = δα,βδnm. (2.24) We denote the right and left vacuum vectors respectively as |0, 0〉 := |0〉, 〈0, 0| := 〈0|. (2.25) Relations (2.4) then become, for α = 1, 2, f (α)m |0, 0〉 = 0 (m < 0), f̄ (α)m |0, 0〉 = 0 (m ≥ 0), (2.26) 〈0, 0|f (α)m = 0 (m ≥ 0), 〈0, 0|f̄ (α)m = 0 (m < 0). (2.27) For n,m ∈ Z, i, j = 1, 2 it follows from (2.5)-(2.6) that 〈0, 0|f (i)n f (j)m |0, 0〉 = 〈0, 0|f̄ (i)n f̄ (j)m |0, 0〉 = 0, (2.28) 〈0, 0|f (i)n f̄ (j)m |0, 0〉 = δijδnm, n < 0 (2.29) Following [14], [15], we also introduce the states 〈n(1), n(2)| := 〈0, 0|Cn(1)Cn(2), (2.30) where Cn(α) := f̄ 0 · · · f̄ n(α)−1 if n(α) > 0 (2.31) Cn(α) := f −1 · · · f if n(α) < 0 (2.32) Cn(α) := 1 if n (α) = 0 (2.33) |n(1), n(2)〉 := C̄n(2)C̄n(1)|0, 0〉 (2.34) where C̄n(α) := f n(α)−1 · · ·f (α)0 if n(α) > 0 (2.35) C̄n(α) := f̄ · · · f̄ (α)−1 if n(α) < 0 (2.36) C̄n(α) := 1 if n (α) = 0 (2.37) The states (2.30) and (2.34) will be referred to, respectively, as left and right charged vacuum vectors with charges (n(1), n(2)). It follows that f (1)m |n, ∗〉 = 0 (m < n), f̄ (1)m |n, ∗〉 = 0 (m ≥ n), (2.38) 〈n, ∗|f (1)m = 0 (m ≥ n), 〈n, ∗|f̄ (1)m = 0 (m < n), (2.39) f (2)m |∗, n〉 = 0 (m < n), f̄ (2)m |∗, n〉 = 0 (m ≥ n), (2.40) 〈∗, n|f (2)m = 0 (m ≥ n), 〈∗, n|f̄ (2)m = 0 (m < n). (2.41) Remark 2.1. Note that if we shift the charges of the vacuum vectors 〈∗, ∗| → 〈∗, ∗+ n|, |∗, ∗〉 → |∗, ∗ + n〉 (2.42) and, at the same time, re-label i → f i+n, f̄ i → f̄ i+n, i, n ∈ Z, (2.43) then the vacuum expectation values remain invariant. Remark 2.2. Wick’s theorem will be used in the form (2.8) in the following. In the two component notation, there are two possible ways to do this; either (1) Use formula (2.8), remembering that 2-component fermions consist just of the usual even and odd ones (see (2.22)), or (2) Use formula (2.8) separately for each component. To calculate the vacuum state expec- tation value of an operator O, we first present it in the form i . (2.44) 〈0, 0|O|0, 0〉 = 〈0, 0|O(1)i O i |0, 0〉 = 〈0, 0|O(1)i |0, 0〉〈0, 0|O i |0, 0〉 (2.45) where Wick’s theorem in the form (2.8) may be applied to each factor 〈0, 0|O(α)i |0, 0〉 The version (2) is used in the Section 3, and the version (1) is used in the Section 5. Now define F (j)(x(j)) := f (j)(x(j)nj ) · · ·f (j)(x F̄ (j)(y(j)) := f̄ (j)(y(j)mj ) · · · f̄ (j)(y 1 ), j = 1, 2 (2.46) Combining (2.21) and (2.45) we obtain 〈n1 −m1, n2 −m2|F (2)(x(2))F (1)(x(1))F̄ (1)(y(1))F̄ (2)(y(2))|0, 0〉 = (−1)m2(m1+n1) ∆n1(x (1))∆m1(y (1))∆n2(x (2))∆m2(y i=1,...,n1 j=1,...,m1 i − y i=1,...,n2 j=1,...,m2 i − y . (2.47) expectation value 3 The integral of rational symmetric functions as a certain expectation value We shall use the following notations η = (η1, . . . , ηM1), ξ = (ξ1, . . . , ξL1), µ = (µ1, . . . , µM2), ζ = (ζ1, . . . , ζL2) (3.1) Nj = N + Lj −Mj, j = 1, 2, (3.2) dµ(x,y)(·) = dµ(x1, y1) . . . dµ(xN , yN)(·) (3.3) F (1)(ξ) := f (1)(ξL1) · · ·f (1)(ξ1), (3.4) F (2)(µ) := f (2)(µM2) · · ·f (2)(µ1), (3.5) F̄ (1)(η) := f̄ (1)(ηM1) · · · f̄ (1)(η1), (3.6) F̄ (2)(ζ) := f̄ (2)(ζL2) · · · f̄ (2)(ζ1), (3.7) g := eA, A := f (1)(x)f̄ (2)(y)dµ(x, y), (3.8) and consider the expression JN(ξ, ζ, η, µ) := RN 〈N1,−N2|F (2)(µ)F (1)(ξ) g F̄ (1)(η)F̄ (2)(ζ)|0, 0〉 (3.9) where RN = RN (ξ, ζ, η, µ) := s(L1, L2,M1,M2) n=0 hn j=1(ξα − ηj) k=1(ζβ − µk) ∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ) (3.10) and s(L1, L2,M1,M2) is the sign factor s(L1, L2,M1,M2)) = (−1) N(N+1)+L2(L1+M1)+N(L1+M1)+M2L2 (3.11) Remark 3.1. Because of the form (3.8), g commutes with both F (1)(ξ) and F̄ (2)(µ). Moreover he conditions given in Remark (1.3) also imply that g commutes with F̄ (1)(η) and F (2)(ζ). The main result of this subsection is the equality IN(ξ, ζ, η, µ) = JN(ξ, ζ, η, µ) (3.12) Proof. Inserting into (3.9), we note that only the N -th power contributes, giving JN(ξ, ζ, η, µ) = 〈N1,−N2|F (2)(µ)F (1)(ξ)AN F̄ (1)(η)F̄ (2)(ζ)|0, 0〉 (3.13) Collecting the fermion terms of the same types (which gives rise to a sign factor), the right hand side of (3.13) may be expressed as N(N−1)+NM1 dµ(x,y)〈N1,−N2|F (2)(µ)F (1)(ξ, x)F̄ (1)(η)F̄ (2)(y, ζ)|0, 0〉 where (ξ, x) = (ξ1, . . . , ξL1, x1, . . . , xN ) and (y, ζ) = (y1, . . . , yN , ζ1, . . . , ζL2). Using (2.47) to evaluate the expectation value in the integrand we get JN(ξ, ζ, η, µ) = dµ(x,y)∆N(x)∆N (y) α=1(ξα − xa) β=1(ζα − ya) j=1(ηj − xa) k=1(µk − ya) = IN(ξ, ζ, η, µ). Remark 3.2. From (3.9) and (3.12) in the absence of the F (1), F (2), F̄ (1), F̄ (2) terms, we obtain the representation which we used in [19] = (−1) N(N+1)N !〈N,−N |g|0, 0〉. Having the representation (3.12) we evaluate IN(ξ, ζ, η, µ) via Wick’s theorem. Before we need certain preliminary relations. 4 Biorthogonal polynomials and dressed fermions In this section we introduce transformations Ω and Q and dressed fermions d(α), d̄(α) and bα, b̄α. We re-write the v.e.v. (3.9) in a form suitable for further calculation by means of Wick’s theorem. (a) Biorthogonal polynomials. Consider the sequence of biorthogonal polynomials associated to the measure dµ(x, y): Pn(x)Sm(y)dµ(x, y) = δn,m, n,m ≥ 0 (4.1) It is convenient to write down the biorthogonal polynomials in the following form Pn(x) = m, Sn(y) = ym(K̄−1)mn, n ≥ 0 (4.2) where Knm and (K̄ −1)mn are respectively viewed as entries of semi-infinite matrices K and (K̄−1). K is a lower triangular matrix and K̄ is an upper triangular one, both have units on the diagonal: Knn = K̄nn = 1, n = 0, 1, 2, . . . . As is well-known, the orthogonality relation (4.1) implies HK̄ = KB, (4.3) where H = diag(hn) and B is the bi-moment matrix, Bnm = xnymdµ(x, y), m, n ≥ 0, (4.4) In time (see [11]), (4.3) was made good use to relate matrix models to integrable systems via the factorization method widely used in soliton theory starting from the basic paper [23]. Then K and K̄ may be identified with the Mikhailov-Ueno-Takasaki dressing matrices (see [20], [22]), whose rows give rise to a pair the so-called Baker functions, while columns of K−1and of K̄−1 give rise to a pair of adjoint Baker functions. In case the dressing matrices are semi-infinite, the first of the Baker functions (related to rows of K) and the second of the adjoint Baker function (related to columns K̄−1) take the form of quasi-polynomials. Here we shall use (4.3) differently. (b) Canonical transformation Ω. First, we introduce S̃n(x) := x−m−1(K−1)mn hn, P̃n(y) := K̄nmy hn, n ≥ 0 (4.5) Properties of S̃n(x) and P̃n(y) are described in the Appendix C. The main is that they coincide with Hilbert transforms of biorthogonal polynomials (1.8). Now, let us define matrices ω(1) and ω(2) via = K, eω = K̄ (4.6) (Respectively, strictly upper and strictly lower) triangular matrices ω(1),(2) may be defined in a unique way by a recursion procedure. We introduce Ω = exp n>m≥0 ω(1)n,mf m + ω −m−1f̄ (4.7) Note that, for any N1 and N2, 〈N1,−N2|Ω−1 = 〈N1,−N2| , Ω|0, 0〉 = |0, 0〉 (4.8) The first equality in (4.8) follows from (2.39),(2.41) and from the restriction n > m in the summation in (4.7). The second equality in (4.8) follows from the restriction m ≥ 0 and from (2.39),(2.41). Consider d(α)(x) = Ωf (α)(x)Ω−1, d̄(α)(x) = Ωf̄ (α)(x)Ω−1, α = 1, 2 (4.9) which we refer as dressed fermions. We have (see Appendix B) d(1)(x) = f (1)n Pn(x) hn (4.10) d̄(1)(x) = f̄ (1)n S̃n(x)√ (4.11) d(2)(y) = P̃n(y)√ (4.12) (2)(y) = −n−1Sn(y) hn (4.13) where for n < 0 we use the following notational convention Pn(x) = Sn(x) = x n, S̃n(x) = P̃n(x) = x −n−1, hn = 1, n < 0 (4.14) A kind of dressed fermionic operators (where powers of x were replaced by Baker functions similar to (4.10)-(4.13)), in a different context, were also introduced in [30] where they were called Krichever-Novikov fermions (see Appendix to [30]). If we write d(α)(x) = d(α)n x n, d̄(α)(x) = d̄(α)n x −n−1, α = 1, 2 where d(α)n = Ωf −1, d̄(α)n = Ωf̄ n≥i≥0 f (1)n Kni, i ≥ 0, d i = f i , i < 0 (4.15) i≥n≥0 (K−1)inf̄ n , i ≥ 0, d̄ i = f̄ i , i < 0 (4.16) −i−1 = i≥n≥0 −n−1K̄ni, i ≥ 0, d −i−1 = f −i−1, i < 0 (4.17) −i−1 = n≥i≥0 (K̄−1)inf̄ −n−1, i ≥ 0, d̄ −i−1 = f̄ −i−1, i < 0 (4.18) (c) Useful formulae for charged vacuum vectors. First, we introduce gn := e −n−1 = 1 + f (1)n f̄ −n−1, (4.19) en := e −n−1 = 1 + f̄ (1)n f −n−1 (4.20) We see that gn, en ∈ ĜL∞. We shall also consider powers of these operators p = epf −n−1 = 1 + pf (1)n f −n−1, (4.21) p = epf̄ −n−1 = 1 + pf̄ (1)n f −n−1 (4.22) where n ∈ Z, p ∈ C. Now, due to 〈0, 0|engn = 〈0, 0|(1+f̄ (1)n f −n−1)(1+f −n−1) = 〈0, 0|(1−1+f̄ (1)n f −n−1) = 〈0, 0|f̄ (1)n f −n−1, which is true for n > 0, and due to (A.1), (A.2), we conclude that for N > 0 〈N,−N | = (−1) N(N−1)〈0, 0| where 〈N,−N | was defined in (2.30)-(2.32). In the similar way we obtain a representation we shall need later N2 > N1 ≥ 0 : 〈N1,−N2| = (−1) N1(N1−1)〈0, 0|f (2)−N1−1 · · · f gn (4.23) (d) Evaluation of Ωg|0, 0〉. Taking into account that f −n−1|0, 0〉 = f̄ n |0, 0〉 = 0 for n > 0, one obtains Ωg|0, 0〉 = e i,j,n,m≥0 KinBnmK̄mjf −j−1 |0, 0〉 = Q−1|0, 0〉, Q−1 := hn (4.24) where we used (4.15),(4.18) and (4.3). (e) Fermionic operators bα, b̄α. We shall also need the following fermionic operators bα(x) := Qd (α)(x)Q−1 = QΩ f (α)(QΩ)−1, b̄α(x) := Qd̄ (α)(x)Q−1 = QΩ f̄ (α)(QΩ)−1 (4.25) Using (A.3)-(A.6) we write down b1(ξ) = f (1)n Pn(ξ) hn = d (1)(ξ) , (4.26) b̄1(η) = f̄ (1)n S̃n(η)√ −n−1S̃n(η) hn , (4.27) b2(µ) = P̃n(µ)√ f (1)n P̃n(µ) hn , (4.28) b̄2(ζ) = −n−1Sn(ζ) hn = d̄ (2)(ζ) (4.29) Notice that each of b̄1 and b2 contains both component fermions. The fermionic operators bα, b̄α may be also called dressed fermions. Similarly to (3.4)- (3.7), for their products, we shall use large letters, namely, we define B1(ξ) := QΩ F (1)(ξ) (QΩ)−1, B2(µ) := QΩ F (2)(µ) (QΩ)−1, B̄1(η) := QΩ F̄ (1)(η) (QΩ)−1, B̄2(ζ) := QΩ F̄ (2)(ζ) (QΩ)−1 (f) IN as a re-written vacuum expectation value Now we restate (3.12) as IN(ξ, ζ, η, µ)R N = 〈N1,−N2|Q −1 B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 (4.30) where RN was defined by (3.10). Now we have to consider the cases (1),(2),(3) separately. 5 Determinantal expression for integrals of rational symmetric functions Below we apply Wick’s theorem to evaluate v.e.v. in the right hand side of (4.30), getting answers for I N (ξ, ζ, η, µ) for all the cases listed in the Introduction. 5.1 Determinantal representation in the case (1): N2 ≥ N1 ≥ 0 In the case (1) we have the following formulae for 〈N1,−N2|Q−1: N2 ≥ N1 ≥ 0 : 〈N1,−N2|Q−1 = (−1) N1(N1+1)〈0, 0|f (2)−N1−1 · · · f hn , (5.1) where n (5.2) Formula (5.1) follows from relations 〈0, 0|eng1+hnn = 〈0, 0|(1 + f̄ (1)n f −n−1)(1 + (1 + hn)f −n−1) = 〈0, 0|(−hn + f̄ (1)n f −n−1) = −hn〈0, 0|e n , n ≥ 0 and from (4.23). It is important that E|0, 0〉 = 0 , (5.3) which is true as each epn|0, 0〉 = 0, n ≥ 0. Using (A.3)-(A.6), we may evaluate a1(ξ) := Eb1(ξ)E f (1)n Pn(ξ) Pn(ξ)√ , (5.4) ā1(η) := Eb̄1(η)E f̄ (1)n S̃n(η)√ f̄ (1)n S̃n(η)√ −n−1S̃n(η) hn , (5.5) a2(µ) := Eb (2)(µ)E−1 = P̃n(µ)√ P̃n(µ)√ f (1)n P̃n(µ) hn , (5.6) ā2(ζ) := Eb̄ (2)(ζ)E−1 = −n−1Sn(ζ) f̄ (1)n Sn(ζ)√ (5.7) Using notations A1(ξ) := EQΩ F (1)(ξ) (EQΩ)−1, A2(µ) := EQΩ F (2)(µ) (EQΩ)−1, Ā1(η) := EQΩ F̄ (1)(η) (EQΩ)−1, Ā2(ζ) := EQΩ F̄ (2)(ζ) (EQΩ)−1 for the products, by analogy with (3.4)-(3.7), by (5.1) and (5.3), we arrive at IN(ξ, ζ, η, µ)R N = (−1) N1(N1+1) hn〈0, 0|f (2)−N1−1 · · · f A2(µ)A1(ξ)Ā1(η)Ā2(ζ)|0, 0〉 (5.8) Finely, this form is applicable to apply the Wick’s formula (2.8). Indeed, each ai is a linear combination of fermions f (1) and f (2), while each āi is a linear combination of fermions f̄ (1) and f̄ (2). Then, by (2.8), the vacuum expectation value in formula (5.8) is equal to the deter- minant of a L1 +M2 by L1 +M2 matrix, which consists of six blocks formed by pair-wise v.e.v.: 〈a1(ξα)ā1(ηj)〉 〈a2(µk)ā1(ηj)〉 〈f (2)i−1−N−L2+M2 ā1(ηj)〉 〈a1(ξα)ā2(ζβ)〉 〈a2(µk)ā2(ζβ)〉 〈f (2)i−1−N−L2+M2ā2(ζβ)〉 where α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2 i = 1, . . . ,M1 + L2 − L1 −M2 Now, taking into account (2.29)-(2.29), from the explicit formulae (5.4)-(5.7), and also from (C.3) and (C.4), we compute all relevant pair-wise vacuum expectation values 〈a1(ξi)ā1(ηj)〉 = ξi − ηj Pn(ξi)S̃n(ηj), 〈a2(µi)ā2(ζj)〉 = Sn(ζj)P̃n(µi) = − ζj − µi Sn(ζj)P̃n(µi), 〈a1(ξi)ā2(ζj)〉 = Pn(ξi)Sn(ζj), 〈a2(µi)ā1(ηj)〉 = S̃n(ηj)P̃n(µi) = dµ(x, y) (ηj − x)(µi − y) S̃n(ηj)P̃n(µi), 〈f (2)i−1−N−L2+M2 ā1(ηj)〉 = hN+L2−M2−iS̃N+L2−M2−i(ηj), 〈f (2)i−1−N−L2+M2 ā2(ζj)〉 = hN+L2−M2−iSN+L2−M2−i(ζj) After trivial manipulations with rows and columns of the matrix of pair-wise v.e.v. we obtain the answer: IN(ξ, ζ, η, µ) = (−1) (M1+M2)(M1+M2−1)(−1)L2M1 N+L1−M1−1 N+L2−M2−1 j=1(ξα − ηj) k=1(ζβ − µk) ∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ) detG, (5.9) where the matrix G is (L2 +M1)× (L2 +M1) matrix which consists of six blocks: K11(µk, ζβ) K21(ξα, ζβ) SN+L1−M1(ζβ) . . . SN+L2−M2−1(ζβ) K12(µk, ηj) K22(ξα, ηj) S̃N+L1−M1(ηj) . . . S̃N+L2−M2−1(ηj) where α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2 and where N1 = N + L1 −M1 and K11(µ, ζ) = P̃n(µ)Sn(ζ) + ζ − µ (5.10) K22(ξ, η) = Pn(ξ)S̃n(η) + ξ − η (5.11) K21(ξ, ζ) = Pn(ξ)Sn(ζ) (5.12) K12(µ, η) = P̃n(µ)S̃n(η)− dµ(x, y) (η − x)(µ− y) (5.13) SN+L1−M1+i−1(ζj), i = 1, . . . ,M1 − L1 −M2 + L2, j = 1, . . . , L2 (5.14) S̃N+L1−M1+i−1(ηj), i = 1, . . . ,M1 − L1 −M2 + L2, j = 1, . . . ,M1, (5.15) Now we shall consider six examples, related to the six-block structure, where, in each case, only one entry contributes. Example 1. M1 = 1 and L1 = L2 = M2 = 0, thus N2 > N1 ≥ 0. We put η1 = η. In this case the matrix has only one non-vanishing element, giving the well-known formula IN(η) = S̃N−1(η) (5.16) Example 2. L2 = 1 and L1 = M1 = M2 = 0, thus N2 > N1 ≥ 0. We put ζ1 = ζ . In this case the matrix has only one non-vanishing element, giving the well-known formula IN(ζ) = hNSN(ζ) (5.17) Examples below are related to the equality N2 = N1 ≥ 0. Example 3. M1 = L1 = 1 and L2 = M2 = 0. We put ξ1 = ξ and η1 = η. In this case the matrix has only one non-vanishing element and we obtain IN(ξ, η) = 1 + (ξ − η) Pn(ξ)S̃n(η) (5.18) Similarly, we have Example 4. L2 = M2 = 1 and L1 = M1 = 0. In this case the matrix has only one non-vanishing element, and we obtain IN(ζ, µ) = −(ζ − µ) Sn(ζ)P̃n(µ) = 1 + (ζ − µ) Sn(ζ)P̃n(µ) (5.19) Example 5. M1 = M2 = 1 and L1 = L2 = 0. In this case the matrix has only one non-vanishing element. We obtain IN(η, µ) = n=N−1 S̃n(η)P̃n(µ) = dµ(x, y) (η − x)(µ − y) S̃n(η)P̃n(µ) (5.20) Example 6. L1 = L2 = 1 and M1 = M2 = 0. In this case the matrix has only one non-vanishing element and we obtain IN(ξ, ζ) = hN Pn(ξ)Sn(ζ) (5.21) Evaluation for the case N1 ≥ N2 ≥ 0 This case may be obtained from the previous one, by interchanging subscripts L1 ↔ L2, M1 ↔ M2, N1 ↔ N2, and ξ ↔ ζ , µ↔ η and also Pn ↔ Sn, P̃n ↔ S̃n. We obtain the answer which coincides with the answer given by formulae (1.8)-(1.16) of [1]: IN(ξ, ζ, η, µ) = (−1) (M1+M2)(M1+M2−1)(−1)L1M2 N+L2−M2−1 N+L1−M1−1 j=1(ξα − ηj) k=1(ζβ − µk) ∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ) detG, (5.22) where (L1 +M2)× (L1 +M2) matrix G consists of six blocks: K11(ξα, ηj) K12(ξα, ζβ) PN+L2−M2(ξα) . . . PN+L1−M1−1(ξα) K21(µk, ηj) K22(µk, ζβ) P̃N+L2−M2(µk) . . . P̃N+L1−M1−1(µk) where α = 1, . . . , L1; β = 1, . . . , L2; ; j = 1, . . . ,M1; k = 1, . . . ,M2 and where N2 = N + L2 −M2 and K11(ξ, η) = Pn(ξ)S̃n(η) + ξ − η (5.23) K22(µ, ζ) = P̃n(µ)Sn(ζ) + ζ − µ (5.24) K12(ξ, ζ) = Pn(ξ)Sn(ζ) (5.25) K21(µ, η) = P̃n(µ)S̃n(η)− dµ(x, y) (η − x)(µ− y) (5.26) Example 7. M2 = 1 and L2 = L1 =M1 = 0, thus N1 > N2 ≥ 0. We put µ1 = µ. Then IN(µ) = P̃N−1(µ) (5.27) Example 8. L1 = 1 and L2 = M2 = M1 = 0, thus N1 > N2 ≥ 0. We put ξ1 = ξ. We obtain the well-known formula IN(ξ) = hNPN(ξ) (5.28) 5.2 When N1 < 0 In cases listed in the Introduction as (2) and (3) we have N1 < 0. Let us explicitly write N1 ≤ 0 ≤ N2 : 〈N1,−N2| = 〈0, 0|f (1)−1 · · · f −1 · · · f , (5.29) N1 ≤ N2 ≤ 0 : 〈N1,−N2| = 〈0, 0|f (1)−1 · · · f 0 · · · f̄ −N2−1 (5.30) Then, as it follows from (A.3)-(A.4), in either case 〈N1,−N2|Q−1 = 〈N1,−N2| (5.31) Thus, in the both cases, we re-write (4.30) as IN(ξ, ζ, η, µ)R N = 〈N1,−N2|B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 (5.32) 5.3 Evaluation for the case (2): N1 ≤ 0 ≤ N2. Thus, by (5.32) and (5.29) we have IN(ξ, ζ, η, µ)R N = 〈0, 0|f −1 · · · f −1 · · · f B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 (5.33) By Wick’s formula (2.8) the right hand side is the determinant of a L2+M1 by L2+M1 matrix which consists of eight blocks: 〈b1(ξα)b̄1(ηj)〉 〈b2(µk)b̄1(ηj)〉 〈f (2)b−1−N−L2+M2 b̄1(ηj)〉 〈f m−1+N+L1−M1 b̄1(ηj)〉 〈b1(ξα)b̄2(ζβ)〉 〈b2(µk)b̄2(ζβ)〉 〈f (2)b−1−N−L2+M2 b̄2(ζβ)〉 〈f m−1+N+L1−M1 b̄2(ζβ)〉 where α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2; b = 1, . . . , N + L2 −M2; m = 1, . . . ,−N − L1 +M1 Then using (4.26)-(4.29), we evaluate all pair-wise vacuum expectation values: 〈b1(ξα)b̄1(ηj)〉 = 1− ηj ξα − ηj 〈b2(µk)b̄2(ζβ)〉 = P̃n(µk)Sn(ζβ) = µk − ζβ 〈b2(µk)b̄1(ηj)〉 = P̃n(µk)S̃n(ηj) = H(µk, ηj) 〈b1(ξα)b̄2(ζβ)〉 = 0, 〈f (2) b−1−N−L2+M2 b̄1(ηj)〉 = hN+L2−M2−bS̃N+L2−M2−b(ηj), b = 1, . . . , N + L2 −M2, 〈f (2) b−1−N−L2+M2 b̄2(ζβ)〉 = hN+L2−M2−bSN+L2−M2−b(ζβ), b = 1, . . . , N + L2 −M2, 〈f (1)m−1+N+L1−M1 b̄1(ηj)〉 = η −N−L1+M1−m j , m = 1, . . . ,−N − L1 +M1, 〈f (1)m−1+N+L1−M1 b̄2(ζβ)〉 = 0, m = 1, . . . ,−N − L1 +M1 After trivial manipulations with rows and columns of the matrix of pair-wise v.e.v. we obtain the answer IN(ξ, ζ, η, µ) = N+L1−M1−1 N+L2−M2−1 j=1(ξα − ηj) k=1(ζβ − µk) ∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ) detG, (5.34) ǫ = (−1) N(N−1)+N(L1−M1)+ M1(M1−1)+ L2(L2−1)+L1(L2−M2)+L2M2−M2 where the (L2 +M1)× (L2 +M1) matrix G consists of eight blocks µk−ζβ 0 Sb(ζβ) 0 H(µk, ηj) ξα−ηj S̃b(ηj) Sm(ηj) (5.35) where α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2; b = 1, . . . , N + L2 −M2; m = 1, . . . ,−N − L1 +M1 Evaluation for the case N2 ≤ 0 ≤ N1. The answer for this case may be obtained by the answer for the previous case, if we inter-change subscripts L1 ↔ L2, M1 ↔ M2, N1 ↔ N2, and the variables ξ ↔ ζ , µ ↔ η and also Pn ↔ Sn, P̃n ↔ S̃n, and (due to the definition (C.4)) H(µ, η) → H(µ, η) . The answer coincides with the answer of [1] (see (3.21)-(3.22) there): IN(ξ, ζ, η, µ) = N+L2−M2−1 N+L1−M1−1 (ξα − ηj) (ζβ − µk) ∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ) detG, (5.36) ǫ = (−1) N(N−1)+N(L2−M2)+ M2(M2−1)+ L1(L1−1)+L2(L1−M1)+L1M1−M1 where the (L1 +M2)× (L1 +M2) matrix G consists of eight blocks ξα−ηi 0 Pb(ξα) 0 H(µk, ηi) µk−ζβ P̃b(µk) Sm(µk) (5.37) where α = 1, . . . , L1; β = 1, . . . , L2; i = 1, . . . ,M1; k = 1, . . . ,M2; b = 1, . . . , N + L1 −M1; m = 1, . . . ,M2 − L2 −N 5.4 Evaluation for the case (3): N1 ≤ N2 ≤ 0 In this case we have (5.32). Using transformation (2.42)-(2.43) which leaves vacuum expectation values invariant, in form: 〈∗, ∗| → 〈∗, ∗+N2|, |∗, ∗〉 → |∗, ∗+N2〉, i → f i → f̄ , i ∈ Z, we write IN(ξ, ζ, η, µ)R N = 〈N1,−N2|B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 = 〈N1, 0|C2(µ)C1(ξ)C̄1(η)C̄2(ζ)|0, N2〉 (5.38) where we shift the charges of the second component of the vacuum vector by N2 and replaced fermionic operators b1(ξ), b̄1(η), b2(µ) and b̄2(ζ) given by (4.26)-(4.29), by c1(ξ) = n Pn(ξ) hn , (5.39) c1(η) = S̃n(η)√ −n−1+N2 S̃n(η) hn , (5.40) c2(µ) = −n−1+N2 P̃n(µ)√ f (1)n P̃n(µ) hn , (5.41) c̄2(ζ) = −n−1+N2 Sn(ζ) hn (5.42) We produced this shift of the vacuum charge and the fermion numbering in order to come to the form of v.e.v., where all fermions with bar are situated to the right: 〈0, 0|f (1)−1 · · · f c2(µn) c1(ξn) c̄1(ηn) c̄2(ζn)f̄ · · · f̄ (2)−1 |0, 0〉, (5.43) where formula (2.8) is applicable. We have nine-block matrix: 〈c1(ξα)c̄1(ηj)〉 〈c1(ξα)c̄2(ζβ)〉 〈c1(ξα)f̄ (2)m−1+N+L2−M2〉 〈c2(µk)c̄1(ηj)〉 〈c2(µk)c̄2(ζβ)〉 〈c2(µk)f̄ (2)m−1+N+L2−M2〉 〈f (1) ℓ−1+N+L1−M1 c̄1(ηj)〉 〈f (1)ℓ−1+N+L1−M1 c̄2(ζβ)〉 0 where α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2; ℓ = 1, . . . ,−N − L1 +M1; m = 1, . . . , N + L2 −M2 Pair-wise expectation values are 〈c1(ξ)c̄1(η)〉 = ξ − η 〈c2(µ)c̄2(ζ)〉 = P̃n(µ)Sn(ζ) = 〈c2(µ)c̄1(η)〉 = P̃n(µ)S̃n(η) = H(µ, η), 〈c1(ξ)c̄2(ζ)〉 = 0, 〈f (1) ℓ−1+N+L1−M1 c̄1(η)〉 = S̃ℓ−1+N+L1−M1(η) hℓ−1+N+L1−M1 = η −N−L1+M1−ℓ, ℓ = 1, . . . ,−N−L1+M1, 〈f (1) ℓ−1+N+L1−M1 c̄2(ζ)〉 = 0, ℓ = 1, . . . ,−N − L1 +M1, 〈c1(ξ)f̄ (2)m−1+N+L2−M2〉 = 0, m = 1, . . . , N + L2 −M2, 〈c2(µ)f̄ (2)m−1+N+L2−M2〉 = P̃−m(µ) h−m = µ ℓ−1, m = 1, . . . , N + L2 −M2 After trivial manipulations with rows and columns of the matrix of pair-wise v.e.v. we obtain the answer which coincides with the answer of [1] (given by (3.35)-(3.36) there): IN(ξ, ζ, η, µ) = N+L2−M2−1 N+L1−M1−1 j=1(ξα − ηj) k=1(ζβ − µk) ∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ) detG, (5.44) ǫ = (−1) L2(L2−1)+ M1(M1−1)+ N(N+1)+L2(L1+M1)+N(L1+M1)+M2L2 where G is (M1 +M2 −N)× (M1 +M2 −N) matrix which consists of nine blocks: ξα−ηj H(µk, ηj) µk−ζβ Sm(µk) Pℓ(ηj) 0 0 (5.45) where α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2; ℓ = 1, . . . ,M1 − L1 −N ; m = 1, . . . , N + L2 −M2 Evaluation for the case N2 ≤ N1 ≤ 0 After replacements we obtain IN(ξ, ζ, η, µ) = N+L2−M2−1 N+L1−M1−1 j=1(ξα − ηj) k=1(ζβ − µk) ∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ) detG, (5.46) ǫ = (−1) L1(L1−1)+ M2(M2−1)+ N(N+1)+L1(L2+M2)+N(L2+M2)+M1L1 where G is (M1 +M2 −N)× (M1 +M2 −N) matrix which consists of nine blocks ζβ−µk H(µk, ηj) ηj−ξα Pm(ηj) Sℓ(µk) 0 0 (5.47) where α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2; ℓ = 1, . . . ,M2 − L2 −N ; m = 1, . . . ,M1 − L1 −N Acknowledgements The authors would like to thank T. Shiota and J. van de Leur for helpful discussions, and (A.O.) thanks A. Odzijevicz for kind hospitality during his stay in Bialystok in June 2005. A Appendix A By (2.24) we have for p, q ∈ C: [(gn) , (gm) q] = 0, [(en) , (em) q] = 0, n,m ∈ Z (A.1) [(gn) p, (em) q] = 0, n 6= m (A.2) where [, ] denotes the commutator. Also: [(gn) p, f (1)m ] = [(gn) p, f̄ (2)m ] = [(en) p, f̄ (1)m ] = [(en) p, f (2)m ] = 0, n,m ∈ Z, (A.3) [(gn) p, f̄ (1)m ] = [(gn) −m−1] = [(en) p, f (1)m ] = [(en) p, f̄ −m−1] = 0, n 6= m, (A.4) −pf̄ (1)n (gn) p = f̄ (1)n + pf̄ −n−1, (gn) −n−1(gn) p = f −n−1 − pf (1)n , n ∈ Z, (A.5) pf (1)n (en) −p = f (1)n − pf −n−1, (en) −n−1(en) −p = f̄ −n−1 + pf̄ n n ∈ Z (A.6) B Appendix B For ω̂ := ωn,mfnf̄m we have adω̂ fi = fnωn,i , adω̂ f̄i = − ωi,nf̄n Then it follows Adω̂ fi = fn (e , Adω̂ f̄i = This yields (4.15) and (4.16) which are equivalent respectively to (4.10) and (4.11). In the same way we prove (4.18) and (4.17) which are equivalent respectively to (4.13) and (4.12). C Appendix C Now, let us show that S̃n(η) = Sn(y) η − x dµ(x, y), P̃n(µ) = Pn(x) dµ(x, y), n ≥ 0 (C.1) Proof I. −n−1]+ = Pn(x)S̃n(η) (C.2) where the first equality follows from the definitions (4.15) and (4.16), while the equality of the anti-commutator to the last member follows from (4.10)-(4.11). Multiplying both sides of the second equality by Sn(y)dµ(x, y) and integrating we come to the first equality (C.1). We obtain the second equality (C.1) by the similar integration of −n−1µ −n−1y n]+ = Sn(y)P̃n(µ) (C.3) (These equalities results from definitions (4.17) and (4.18), and from (4.12),(4.13)). We shall also use (in (5.20) below) the corollary of these equalities: dµ(x, y) (η − x)(µ− y) S̃n(η)P̃n(µ) =: H(µ, η) (C.4) which is obtained by multiplying of the right hand sides of the relations (C.2) and (C.3), integrating and using the orthogonality of Pn and Sn. Proof II. Considering the sum of entries weighted with powers of x−m−1 (where m is the row-number of the entry) of the n-th column of relation K−1H = BK̄−1 (which follows from the known factorization relation HK̄ = KB), we obtain the first equality of (C.1) from the first of (4.5) and from the definition of bi-moments. (The second relation in (C.1) is proved similarly). D Appendix: Links with soliton theory In this appendix, we discuss the links between the evaluation of such symmetric rational integrals and soliton theory. There are two problems: (A) To find which measure dµ is related to a given matrix integral? We do not know the general answer to this problem. A partial answer can be found in the papers [24], [25] and the Appendices to [26] and to [19]. (B) To find which measure dµ is related to soliton theory. This problem is addressed in [19] and [28]. We restrict ourselves to the relation to the usual (one-component) TL hierarchy. Indeed, one can easily show that the expression (1), t(2)) = 〈N,−N |g(t(1), t(2))|0, 0〉, (D-1) where g(t(1), t(2)) = eA(t (1),t(2)), and A is of form A(t(1), t(2)) = i,j=1,2 eV (x,t (1))−V (y,t(2))f (1)(x)f̄ (2)(y)dµ0(x, y) (D-2) where V (x, t(1)) = t(1)n x n, V (y, t(2)) = t(2)n y n (D-3) and where dµ0(x, y) is a rather arbitrary measure, fits into a general expression for tau functions of the two-component KP hierarchy and also the TL hierarchy developed in [14] (see also [22], [21]). Remark: the notations t m and t m are related respectively to the notations −umm and of [19]. Our V (x, t(j)) here is Vj(x) of [19]. In [19] it is shown that the expression (D-1) on the one hand gives rise to multiple integrals, which, in turn, for certain choices of the measure dµ0(x, y) (see Problem (A) above) can be identified with a partition function of some model of random matrices. On the other hand it is also an example of TL tau function. This means that if we choose dµ(x, y) = dµ(x, y, t(1), t(2)) = eV (x,t (1))−V (y,t(2))dµ0(x, y) (D-4) then the expectation value (D-1) is a TL (and also two-component KP) tau function. Remark: if dµ0 solves this or that differential equation, than, one can write a corre- spondent ”string equation”. Notice that the first non-trivial member of the TL hierarchy was introduced and in- tegrated in [20] and called the relativistic two-dimensional Toda lattice. In this paper a factorization problem similar to (4.3) (but with matrices that are infinite in both direc- tions) was considered (see also [22]). Notational Remark. In formulas below we shall denote the pair of sets of times (t(1), t(2)) by a single letter t. The only quantity which depends on a single such set (either t(1) or t(2)) is V , where we shall point out which set it depends on. Then from general consideration in [14] (see also [22], [21]) one finds that the inte- grals of rational functions IN (η, ξ, µ, ζ), see (3.9), may be obtained via the bosonization procedure as follows IN(ξ, ζ, η, µ; t) = τN(t) ξNi D1(ξi) η−Ni D1(ηi) µ−Ni D2(µi) ζNi D2(ζi) −1τN (t), (D-5) where Dj(z) = exp (D-6) is a vertex operator. Notational Remark. In [1] we use the notations Pn(x) = pn(x), (D-7) S̃n(x) = n(x) (D-8) P̃ (y) = n(y, t), (D-9) Sn(y) = sn(y) (D-10) The functions n (x, t) := e V (x,t(1)) Pn(x, t) hn = e V (x,t(1)) pn(x, t), (D-11) ψ∗(1)n (x, t) := e −V (x,t(1)) S̃n−1(x, t) = e−V (x,t (1))p∗n−1(x, t) (D-12) are to be interpreted as first component Baker functions (respectively, adjoint first com- ponent Baker functions) which depend on a spectral parameter x, while ψ(2)n (y, t) := e V (y,t(2)) P̃n−1(y, t) = eV (y,t (2))s∗n−1(y, t), (D-13) ψ∗(2)n (y, t) := e −V (y,t(2))Sn(y, t) hn = e −V (y,t(2))sn(y, t) (D-14) are to be interpreted as second component Baker functions (respectively, adjoint second component Baker functions) which depend on a spectral parameter y. This fact is due to the formulae (notice the factor (−1)N which result from re-ordering the fermions of two different types to achieve true sign according to (2.30)) ψ(1)n (x, t) = (−1)NeV (x,t (1)) 〈N + 1,−N |f (1)(x)g(t)|0, 0〉 〈N,−N |g(t)|0, 0〉 = xNeV (x,t (1))D1(x)τN (t) τN(t) (D-15) ψ∗(1)n (x, t) = (−1)Ne−V (x,t (1)) 〈N − 1,−N |f̄ (1)(x)g(t)|0, 0〉 〈N,−N |g(t)|0, 0〉 = x−Ne−V (x,t (1))D1(x) −1τN (t) τN (t) (D-16) ψ(2)n (y, t) = e V (y,t(2)) 〈N, 1−N |f (2)(y)g(t)|0, 0〉 〈N,−N |g(t)|0, 0〉 = y−NeV (y,t (2))D2(y)τN(t) τN(t) (D-17) ψ∗(2)n (y, t) = e −V (y,t(2)) 〈N,−1−N |f̄ (2)(y)g(t)|0, 0〉 〈N,−N |g(t)|0, 0〉 = yNe−V (y,t (2))D2(y) −1τN (t) τN (t) (D-18) The first equality in each of the relations (D-15),(D-16),(D-17) and (D-18) is an example of the evaluation of IN . Formulae (D-16) and (D-18) fit into the case N2 ≥ N1 ≥ 0 and are just particular cases of formula (5.9, see the Examples following (5.9)). The second equalities in each of the relations (D-15)-(D-18) are examples of bosonization formula (D-5). Notice that different examples of IN , K11(ξ, η, t) = 〈N,−N |f (1)(ξ)f̄ (1)(η)g(t)|0, 0〉 〈N,−N |g(t)|0, 0〉 ξNη−ND1(ξ)D 1 (η)τN(t) τN (t) , (D-19) K22(µ, ζ, t) = 〈N,−N |f (2)(µ)f̄ (2)(ζ)g(t)|0, 0〉 〈N,−N |g(t)|0, 0〉 µ−NζND2(µ)D 2 (ζ)τN(t) τN (t) , (D-20) K12(ξ, ζ, t) = 〈N + 1,−N − 1|f (1)(ξ)f̄ (2)(ζ)g(t)|0, 0〉 〈N,−N |g(t)|0, 0〉 ξNζND1(ξ)D 2 (ζ)τN(t) τN (t) (D-21) K21(µ, η, t) = 〈N − 1,−N + 1|f (2)(µ)f̄ (1)(η)g(t)|0, 0〉 〈N,−N |g(t)|0, 0〉 µ−Nη−ND2(µ)D 1 (η)τN(t) τN (t) (D-22) (where for the sake of brevity we omit t(1), t(2)-dependence in the l.h. sides), may be con- sidered as 2-component analogue of a modified Cauchy-Baker-Akhiezer kernel, introduced in [30], [31] to present an explicit version of Segal-Wilson construction to study Virasoro deformations of tau functions for the quasi-periodical solutions of the KP (and actually for the TL) hierarchies. As we have obtained (see Examples following (5.9)): K11(µ, ζ) = P̃n(µ)Sn(ζ) + ζ − µ (D-23) K22(ξ, η) = Pn(ξ)S̃n(η) + ξ − η (D-24) K21(ξ, ζ) = Pn(ξ)Sn(ζ) (D-25) K12(µ, η) = P̃n(µ)S̃n(η)− dµ(x, y) (η − x)(µ− y) (D-26) References [1] J. Harnad and A.Yu. Orlov, “Integrals of Rational Symmetric Functions, Two–Matrix Models and Biorthogonal Polynomials” preprint CRM-.... (2005), math-phys/0512... [2] M. Bergère, “Biorthogonal polynomials for potentials of two variables and external sources at the denominator”, hep-th/0404126 [3] V.B. Uvarov, “The connection between systems of polynomials orthogonal with re- spect to different distribution functions”, U.S.S.R. Comput. Math. and Math. Phys. 9, 25–36 (1969). [4] Mehta, M. L., Random Matrices, 2nd edition (Academic, San Diego, 1991). [5] L.-L. Chau and O. Zaboronsky, “On the Structure of Correlation Functions in the normal Matrix Models”, Commun. Math. Phys. 196 (1998) 203–247; hep-th/9711091 [6] E. Brezin and S. Hikami, “Characteristics polynomials of random matrices”, Com- mun. Math. Phys. 214, 111–135 (2000). [7] Y.V. Fyodorov and E. Strahov, “An exact formula for spectral correlation functions of random matrices”, J. Phys. A 36, 3203–3213 (2003). http://arxiv.org/abs/hep-th/0404126 http://arxiv.org/abs/hep-th/9711091 [8] G. Akemann and G. Vernizzi, “Characteristic polynomials of complex random matrix models”, Nucl. Phys. B 660, 532–556 (2003). [9] J. Baik, P. Deift and E. Strahov, “Products and ratios of characteristic polynomials of random Hermitian matrices”, math-ph/0304016. [10] C. Itzykson and J.-B. Zuber, “The planar approximation. II”, J. Math. Phys. 21, 411–421 (1980). [11] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A.Yu.Orlov, “Matrix Models of 2D Gravity and Toda Theory”, Nucl. Phys. B 357, 565–618 (1991). [12] M. Bertola, B. Eynard and J. Harnad, “Duality, Biorthogonal Polynomials and Multi–Matrix Models”, Commun. Math. Phys. 229, 73–120 (2002). [13] M. Bertola, B. Eynard and J. Harnad, “Differential systems for biorthogonal polyno- mials appearing in 2-matrix models and the associated Riemann-Hilbert problem”, Commun. Math. Phys. 243, 193–240 (2003). [14] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, “Transformation groups for soliton equations”, in Nonlinear integrable systems-classical theory and quantum theory, eds. M. Jimbo, and T. Miwa, World Scientific, pp. 39–120, (1983). [15] Jimbo,M and Miwa,T,: Solitons and Infinite Dimensional Lie Algebras, Publ. RIMS Kyoto Univ. 19, 943–1001 (1983) [16] Zabrodin, A., Kharchev, S., Mironov, A., Marshakov, A. and Orlov, A.: Matrix Models among Integrable Theories: Forced Hierarchies and Operator Formalism, Nuclear Physics B 366 (1991) 569–601 [17] J. Harnad and A.Yu. Orlov, “Scalar products of symmetric functions and matrix integrals”, Theoretical and Mathematical Physics 137, 1676–1690, (2003). [18] J. Harnad and A. Yu. Orlov, “Matrix integrals as Borel sums of Schur function expansions”, In: Symmetries and Perturbation theory SPT2002, eds. S. Abenda and G. Gaeta, World Scientific, Singapore, (2003). [19] J. Harnad and A.Yu. Orlov, Fermionic construction of partition functions for two- matrix models and perturbative Schur function expansions, preprint CRM-3195 (2005), math-phys/0512056 http://arxiv.org/abs/math-ph/0304016 http://arxiv.org/abs/math-phys/0512056 [20] Mikhailov, A.V.: On the Integrability of two-dimensional Generalization of the Toda Lattice, Letters in Journal of Experimental and Theoretical Physics 30 (1979) 443– [21] V.G. Kac and J.W. van de Leur. The n-component KP hierarchy and representa- tion theory, in Important developments in soliton theory, eds. A.S. Fokas and V. E. Zakharov, Springer Series in Nonlinear Dynamics (1993), 302 – 343 [22] K. Ueno and K. Takasaki, Adv. Stud. Pure Math. 4, 1-95, (1984). [23] V.E. Zakharov and A.B. Shabat, J. Funct. Anal. Appl. 8 226, (1974). [24] P. Zinn-Justin and J.-B. Zuber, “On some integrals over the U(N) unitary group and their large N limit”, J. Phys. A 36, 3173-3193 (2003), math-ph/0209019. [25] A. Yu. Orlov, ”New Solvable Matrix Integrals”, nlin. SI/0209063, Intern. Journ. of Modern Physycs A, Vol. 19, Supplement (2004) 276-293 [26] A. Yu. Orlov and T. Shiota, “Schur function expansion for normal matrix model and associated discrete matrix models”, Phys. Lett. A 343, 384-396 (2005). [27] J. Harnad and A.Yu. Orlov, in progress [28] J. Harnad and A.Yu. Orlov, “ Fermionic construction of partition function for multi- matrix models and multi-component TL hierarchy”, preprint [29] J. Harnad and A.Yu. Orlov, “ Fermionic construction of grand partition function for supersymmetric matrix models and BKP hierarchies”, preprint [30] P.G.Grinevich and A.Yu.Orlov, ”Flag space in Toda lattice theory and Virasoro action on det∂̄j” 1989, Cornell university preprint ...; Arxiv ... based on the ear- lier paper: P.G.Grinevich and A.Yu.Orlov, “Virasoro Action on Riemann Surfaces, Grassmannians, det∂̄j and Segal-Wilson tau-function”, in Problems of Modern Quan- tum Field Theory pp. 86-106, Springer, 1989 (editors: A.A.Belavin, A.U.Klimyk, A.B.Zamolodchikov) [31] P.G.Grinevich and A.Yu.Orlov, ”Variations of the complex structure of Riemann surfaces by vector fields on the circle and objects in KP theory. The Krichever- Novikov problem of the action by the Baker-Akhiezer function”, Funct. Anal. Appl. 24 (1990), no.1, 61–63 http://arxiv.org/abs/math-ph/0209019 Introduction Summary of free fermions One component fermions Two-component fermions The integral of rational symmetric functions as a certain expectation value Biorthogonal polynomials and dressed fermions Determinantal expression for integrals of rational symmetric functions Determinantal representation in the case (1): N2N10 When N1<0 Evaluation for the case (2): N10N2. Evaluation for the case (3): N1N20 Appendix A Appendix B Appendix C Appendix: Links with soliton theory

0704.1151

Old Main-Sequence Turnoff Photometry in the Small Magellanic Cloud. I. Constraints on the Star Formation History in Different Fields

OLD MAIN-SEQUENCE TURNOFF PHOTOMETRY IN THE SMALL MAGELLANIC CLOUD. I. CONSTRAINTS ON THE STAR FORMATION HISTORY IN DIFFERENT FIELDS Noelia E. D. Noël Instituto de Astrof́ısica de Canarias. 38200 La Laguna. Tenerife, Canary Islands. Spain. [email protected] Carme Gallart Instituto de Astrof́ısica de Canarias. 38200 La Laguna. Tenerife, Canary Islands. Spain. [email protected] Edgardo Costa Departamento de Astronomı́a, Universidad de Chile, Casilla 36-D, Santiago, Chile. [email protected] René A. Méndez Departamento de Astronomı́a, Universidad de Chile, Casilla 36-D, Santiago, Chile. [email protected] ABSTRACT We present ground-based B and R-band color-magnitude diagrams (CMDs), reaching the oldest main-sequence turnoffs with good photometric accuracy for twelve fields in the Small Magellanic Cloud (SMC). Our fields, located between ∼1◦ and ∼4◦ from the center of the galaxy, are situated in different parts of the SMC such as the “Wing” area, and towards the West and South. In this paper we perform a first analysis of the stellar content in our SMC fields through com- parison with theoretical isochrones and color functions (CFs). We find that the underlying spheroidally distributed population is composed of both intermediate- age and old stars and that its age composition does not show strong galacto- centric gradients. The three fields situated toward the east, in the Wing region, http://arxiv.org/abs/0704.1151v1 – 2 – show very active current star formation. However, only in the eastern field closest to the center do we find an enhancement of recent star formation with respect to a constant SFR(t). The fields corresponding to the western side of the SMC present a much less populated young MS, and the CF analysis indicates that the SFR(t) greatly diminished around 2 Gyr ago in these parts. Field smc0057, the closest to the center of the galaxy and located in the southern part, shows recent star formation, while the rest of the southern fields present few bright MS stars. The structure of the red clump in all the CMDs is consistent with the large amount of intermediate-age stars inferred from the CMDs and color functions. None of the SMC fields presented here are dominated by old stellar populations, a fact that is in agreement with the lack of a conspicuous horizontal branch in all these SMC CMDs. This could indicate that a disk population is ruling over a possible old halo in all the observed fields. Subject headings: local group galaxies: evolution — galaxies: individual (Small Magellanic Cloud) — galaxies: photometry — galaxies: stellar content — Local Group 1. INTRODUCTION Dwarf galaxies are believed to represent the dominant population, by number, of the present day universe, and a major constituent of groups (Côté et al. 1997) and clusters (e.g. Ferguson & Sandage 1991). Studying their star formation and chemical enrichment histories is key to understanding the evolution of galaxies on cosmological timescales (e.g. Madau et al. 1998). Local Group galaxies are ideal laboratories for detailed studies of dwarf galaxy properties: we can resolve their individual stars and thus learn about their star formation histories (SFHs) by exploring ages, metallicities, and the spatial distribution of the stellar populations they contain. The color-magnitude diagram (CMD) is the best tool to retrieve the SFH of a stellar system. The ideal situation is when it reaches the oldest main-sequence (MS) turnoffs with good photometric accuracy. In this case, the information on the SFH can be obtained directly and with little ambiguity from the distribution of stars on the MS, and its comparison with that predicted by stellar evolution models in this relatively well known phase of stellar evolution. In particular, the range of ages and metallicities present can be determined through comparison with theoretical isochrones. To quantitatively determine the SFH, it is necessary to compare the observed density distribution of stars with that predicted by stellar evolution models (see Gallart, Zoccali & Aparicio 2005). Shallower CMDs (e.g. those – 3 – reaching just below the horizontal-branch (HB) or a couple of magnitudes below the tip of the red giant branch (RGB)) still contain stars born through the whole galaxy’s lifetime, but the interpretation of the distribution of stars in terms of the SFH is progressively less detailed and uncertain as the CMD is shallower. The reason for this is twofold: on one hand, the stellar evolution models are less accurate for more advanced stellar evolution phases such as the RGB or the HB, because the corresponding physics is more complicated or uncertain. On the other hand, in these stellar evolution phases, stars of very different ages are packed together in the CMD in a small interval of color and/or magnitude, and suffer from important age-metallicity degeneracies (see Gallart 2000 and Gallart et al. 2005 for detailed discussions of all these issues). For all the Milky Way satellites, it is possible to obtain CMDs reaching the oldest MS turnoffs using ground based telescopes, while for the rest of the Local Group, this is possible using the ACS on board the HST. The Magellanic Clouds (MCs) are particularly interesting among the Milky Way satellites since they are actively star forming galaxies which SFH can shed light on the role played by interactions in galaxy evolution. But in spite of their proximity and intrinsic interest, it is remarkable that there are still important gaps in the knowledge of the SFH of the MCs. This may be explained by their huge projected size and the big number of stars to analyze. In this paper we will focus on the Small Magellanic Cloud (SMC). The SMC is a dwarf irregular satellite of our Galaxy, with low mean metallicity and a high mass fraction remain- ing in gaseous form (van den Bergh 1999). These characteristics suggest that the SMC is in a more primitive evolutionary stage than its larger neighbor, the Large Magellanic Cloud (LMC). Historically, the study of this galaxy has been in general neglected in favor of the LMC. The uncertainty about its line of sight depth, its more complicated shape (elongated towards the Northeast), and its larger distance from us are factors at play. To our knowledge, only two papers have presented CMDs reaching the oldest MS turnoffs for a small field of view each: Dolphin et al. (2001) and McCumber et al. (2005). Dolphin et al. presented a combination of HST and ground-based V and I images of a SMC field situated 2◦ Northeast of NGC 121 (avoiding the contamination from 47 Tucanae) in order to derive the SFH of this SMC field. From their ground-based images, they inferred a peak of star formation between 5 and 8 Gyr ago with a medium age of 7.5 Gyr at the 1 σ level, and that 14±5% of the stars were formed more than 11 Gyr ago. The SFH they derived is also consistent at the 2 σ level with a continuous star formation from ancient times until ∼2 Gyr ago, when the star formation rate dropped to finally stop entirely in the past 0.5 Gyr. More recently, McCumber et al. (2005) analyzed the stellar populations of a SMC field located in – 4 – the “Wing”1 area with observations from the HST WFPC2. They compared the luminosity function from their observed CMD with those obtained from two different model CMDs, one with constant SFR(t) and another with bursts of star formation at ∼2 and at ∼8 Gyr. They found that the population appears to have formed largely in a quasi-continuous mode, with a main period of star formation between 4-12 Gyr ago and very recent star formation with bright stars as young as 100±10 Myr. Two main wide field studies of the SMC exist: the pioneer work by Gardiner & Hatzidim- itriou (1992) and a recent study from Harris & Zaritsky (2004; hereafter HZ04). Gardiner & Hatzidimitriou (1992) presented the first large area study of the SMC, and mainly con- centrated their analysis in the outer regions, beyond 2◦ from the SMC center. They took six photographic plates with the UK Schmidt Telescope covering a total area of 130◦×130◦. Since their photometry is rather shallow (reaching the HB level at R∼20 mag), they mainly gave information about the young populations (age ≤2 Gyr). From their CMDs and con- tour plots of the surface distribution of MS stars with B-R<0.1 and R<20, they noticed the almost complete absence of bright MS stars in the Northwestern part, while a considerable bright MS population was present in the Eastern and Southern area. With the aid of lu- minosity functions they found that young populations (<0.6 Gyr in age) are concentrated towards the center of the SMC and in the “Wing” region. Using an index defined as the difference between the median color (in (B-R)) of the RC and the color of the RGB at the level of the HB, the authors inferred that the bulk of the field population has a median age around 10-12 Gyr. They were the first to notice the different distribution of young and old populations in the SMC. Zaritsky et al. (2000) and Cioni et al. (2000) confirmed that the asymmetric appearance of the SMC, similar to that of the HI (Stanimirović et al. 1999), is primarily caused by the distribution of young stars (upper MS stars, younger than ∼0.2 Gyr), and that the older stars have a spheroidal distribution. HZ04 presented a study of the SMC based on the Magellanic Clouds Photometric Survey (MCPS) UBVI catalog (Zaritsky et al. 1997), which covers a 4◦×4.5◦ area of the SMC to a depth of V≤21. This work represents the most complete analysis of the spatially resolved SFH of the central part of the SMC. They divided the SMC survey in a grid of 351 cells and obtained the SFH, through a χ2 minimization between the star counts in the observed CMD and those in the model CMDs based on isochrones by Girardi et al. (2002). They inferred that a significant fraction (∼50%) of all stars in the SMC were formed more than 8.4 Gyr ago. Between 3 and 8.4 Gyr ago they found a quiescent period during which the 1We call “Wing” area to the Eastern part of the SMC. Some authors (in particular, Gardiner & Hatzidim- itriou (1992) and papers in the same series) also distinguish between the inner and outer “Wing” and the “Arm”. – 5 – SMC formed few stars. They also found a recent active time from 3 Gyr ago until now, with bursts at 2.5 Gyr, 400 Myr, and 60 Myr. Although deeper than the one from Gardiner & Hatzidimitriou (1992), this study is still limited by its shallowness. In fact, the oldest stars that they can observe on the MS are ∼2 Gyr old, and this implies a necessarily poor temporal resolution and larger uncertainties in their SFH at ages older than that. In spite of the increasing interest in studying the SMC, reflected by the recent works mentioned before, there are still many questions unanswered which require the information provided by CMDs reaching the oldest MS turnoffs. What is the age distribution of the old and intermediate-age population?, are there gradients in the composition of this underlying population?. Shallower studies inform about the existence of a large amount of young popu- lation in certain areas, but does this young population reflect an exceptional increase of the star formation at the present time with respect to the average SFR?. To shed light on these aspects, in this article we present twelve unprecedented deep BR-based SMC CMDs with positions ranging from ∼1◦ to ∼4◦ from the SMC center. Our fields are distributed in differ- ent parts of the SMC, avoiding the central area, where crowding may not allow us to reach the oldest MS turnoffs from the ground. Three of our fields are located in the “Wing” area, near the field studied by McCumber et al. (2005); two fields are in the Western part and one in the Northwestern area, near the field from Dolphin et al. (2001) located 2◦ northeast of NGC121; finally, six fields are placed towards the South at different galactocentric distances. This strategic selection of fields, in spite of their small size, permit us to sample several regions of the SMC with high temporal resolution in order to address the above issues. The depth of our fields, reaching the oldest MS turnoffs with very good photometric accuracy, will allow us to obtain detailed SFHs from all of them, and to investigate, therefore, the variation of the SFH across the SMC field. This will be the subject of a forthcoming paper. Here we will focus the analysis in the inspection of the CMDs with the aid of theoretical isochrones and color functions. This paper is organized as follows. In § 2 we present the observations and data reduction. In § 3 we discuss the photometry of the SMC fields, obtained using DAOPHOT II/ALLSTAR and ALLFRAME programs. In § 4 the CMDs are discussed with the help of isochrones and color functions. In § 5 and § 6 our results are summarized and discussed. In a future paper (Noël et al., in preparation) we will derive the SFH of the SMC through comparison of the observed CMDs with synthetic CMDs. – 6 – 2. OBSERVATIONS AND DATA REDUCTION The present paper is a result of a more comprehensive program to study the SMC, which also includes the determination of its Absolute Proper Motion by observing SMC fields in which background quasars have been identified. These quasars provide a quasi-inertial refer- ence frame with respect to which (in conjunction with existing radial velocity measurements) the absolute total space velocity vector of the SMC can be precisely determined (see e.g. Pedreros, Costa & Méndez 2006). The imaging strategy was therefore designed in a way that would satisfy the needs of the stellar populations program as well as the astrometry program. Given that for the latter it was mandatory to obtain R-band images, this bandpass together with B imaging was chosen to construct the CMDs. The observations were made with a 24 µm/pixel Tektronic 2048x2048 CCD detector attached to the Cassegrain focus of the 100-inch Irénée du Pont telescope (C100) at Las Campanas Observatory, in Chile. This combination gives a field size of 8.85′×8.85′ and a scale of 0.259′′/pixel. Inverse gain was 3e−/ADU with a read noise of 7e−. Basic reduction of the CCD frames was done using standard IRAF2 tasks. For this purpose, bias exposures, sky flats, and dome flats were taken every night. Throughout our four year campaign (2001-2004), B and R band images of thirteen fields in the SMC were obtained. The coordinates of these fields and the data obtained for each of them are detailed in Table 1, where the first column denotes the field, the second and third columns the right ascension and the declination respectively, fourth and fifth columns the galactocentric distance (r) and the position angle (p), and the sixth and seventh columns the integration times in B and R. Seeing was typically between 0.′′7 and 1.′′2 in all epochs. Figure 1a shows a HI column-density image of the SMC taken from Stanimirović et al. (1999) with our SMC fields (black squares) and the fields from Dolphin et al. (2001) and McCumber et al. (2005) (black circles) overlapped. The black rectangle denotes the area covered by the MCPS (Zaristky et al. 1997). The actual figure from Stanimirović et al. (1999) includes the range in declination from ∼-75◦ to ∼-70◦. Those squares denoted with “qj” represent the fields centered in quasars observed for astrometric purposes. We followed the naming convention by Tinney et al. (1997), in which the numbers following “qj” refer to the right ascension of the quasar (the declination, includded in the full Tinney naming convention, was ommitted). The fields denoted with “smc” (and followed by the right ascension, for consistency with the naming convention) were specifically selected for this project and were chosen to span a wide range of galactocentric distances, from ∼1◦ to ∼4◦. The combination of “qj” and “smc” fields results in a sampling of the SMC that covers 2version 2.11.3, NOAO, University of Arizona – 7 – Fig. 1.— Panel (a) shows the spatial distribution of our SMC fields (black squares) over- lapped on an HI column-density image of the SMC from Stanimirović et al. (1999). The big rectangle denotes the area covered by the MCPS (Zaritsky et al. 1997). The position of the fields studied by Dolphin et al. (2001) and McCumber et al. (2005) is also superimposed. The actual figure from Stanimirović et al. (1999) is delimitated by the labelled box and includes the range in declination from ∼-75◦ to ∼-70◦. The grey-scale intensity range is 0 to 1.03×1022 atom/cm2 with a linear transfer function. The maximum HI column-density is 1.43×1022 atom/cm2. Panels (b) and (c), are taken from Cioni et al. (2000), and show the distribution of young and RGB stars in the SMC respectively. The position of our fields has also been overlapped. The fields superimposed are the same in the three figures but the name of each field in panels (b) and (c) have been omitted for clarity. – 8 – both a good range of galactocentric distances and position angles. It is important to note that given the elongation of the SMC, with a position angle of ∼45◦ (Cioni et al. 2000), a certain galactocentric distance in the Northwest or Southeast directions is different than in the Southwest or Northeast directions in which the stellar densities are larger. Figure 1b show the distribution of young MS stars and Figure 1c the layout of the RGB (intermediate- age and old population) stars in the SMC from Cioni et al. (2000) with our SMC fields superimposed. The name of each field was omitted for clarity. 3. THE PHOTOMETRY 3.1. Photometry of SMC fields We performed our photometry using the DAOPHOT II/ALLSTAR software package (Stetson 1987, and updates: Stetson 1990, 1992) to find stars and to determine an empirical PSF for each frame and then simultaneously fit it to all detected stars. After testing various PSF models, we adopted a Moffat function with β=2.5 (Stetson et al. 2003) as the best analytical model. The Moffat function is usually a much better representation of a star image than a Gaussian function, because it has more prominent and extended wings, like real star images. Also, unlike other functions (e.g. the Lorentz function), the Moffat function offers the option of tuning the radial falloff of the wings to match observations. Between 50 and 100 stars were used to construct the PSF of each image. Quadratic spatial variation was included because it reduced the fitting errors. After the DAOPHOT/ALLSTAR analysis of all the original frames was performed, an optimum, complete star list was achieved by cross matching the ALLSTAR results of the individual frames with DAOMASTER (Stetson 1993). ALLFRAME (Stetson 1994) was then used to obtain simultaneous photometry of stars in all CCD images of each SMC field. After ALLFRAME, we ran MONTAGE2 for each field. MONTAGE2 takes the images from which all the stars have been subtracted according to their final positions and bright- nesses, and applies the known geometric transformations that relate their coordinate systems and the known differences in their photometric zero-points, to produce a median image of each field. We tested in each of the 13 SMC fields that the number of stars detected on the median of the subtracted images, and considered real stars after running DAOPHOT, ALLSTAR, and ALLFRAME a second time, was less than 1%, i.e. negligible for our goal. ALLFRAME produces an image-quality index, CHI, which is a dimensionless measure of the agreement between the brightness profile of any given object and that of the model PSF for the frame in which it is measured. Similarly, ALLFRAME gives the SHARP index, – 9 – which is a first-order estimate of the intrinsic angular radius of a source. Another index provided by ALLFRAME is σ, which represent the standard error of the star’s magnitude and is representative of the internal errors of the photometry. Stars with at least one valid measurement in each band (B and R) were selected, and their final magnitudes were obtained using DAOMASTER, which combines the magnitudes measured for each star in each image to provide the “mean weighted” magnitude. DAOMASTER also provides σ, CHI, and SHARP parameters for each stars, which are a combination of the corresponding parameters for each image. We used the following limits for the error and shape parameters given by DAOMASTER: σ(B−R) 260.15, -0.66|SHARP|60.6, and |CHI|62.5. The final number of stars we kept, measured in (B-R) of each field (and consequently, in each CMD) is given in Table 2. A total of 215,121 stars down to R≈24 were measured. The so-called aperture corrections, i.e., the corrections that place the relative profile- fitting magnitudes on the system of the “total” instrumental magnitudes for a particular frame, were obtained from synthetic aperture photometry by measuring several isolated, bright stars through a series of increasing apertures and the construction of growth curves (Stetson 1990). We used a growth sequence which consists in twelve apertures from r1 to r12 in such a way that we considered the first radius r1 about half of the one we used for the aperture photometry of our objects and r12=30 ′′. The sequence is rk= r12/r1× rk−1 , k = 2, ..., 12. The aperture corrections were derived using the program DAOGROW (Stetson 1990). In each image we selected the brightest and most isolated 30 to 80 stars among those used in the derivation of the PSF for that frame. The aperture photometry results were then provided to DAOGROW which returns the “total” instrumental magnitude and its standard error for each of the selected stars. The aperture correction for a particular frame was obtained as the median of the differences between the “total” magnitude and the profile- fitting ALLFRAME magnitude for all selected stars on that frame. Errors in the aperture corrections were calculated as the standard error of the mean of these differences, and were typically between ±0.001 and ±0.003. 3.2. Standard Stars Photometry Our instrumental photometric system was defined by the use of the Harris BVRI filter set, which constitutes the default option on the C100 for broad-band photometry on the standard Johnson-Kron-Cousins system. In photometric nights, typically six BVRI standard star areas from the catalog of Landolt (1992) were observed several times to determine the transformation of our instrumental magnitudes to the standard BR system. Thirty-one different standard stars with colors: -0.49<(B-R)<5.00 were observed, and a total of 272 – 10 – Table 1. Data obtained in the SMC Field α2000 δ2000 r( ′)a p(◦) B-band exposures (sec) R-band exposures (sec) smc0057 00:57 -73:53 65.7 164.4 1×60+1×600+2×800+2×1000+1×1200 2×60+2×600+3×800+1×1200 qj0037 00:37 -72:18 78.5 294 1×60+5×800 7×60+16×600 qj0036 00:36 -72:25 79.8 288 2×60+10×600+12×800 15×60+26×500+22×600 qj0111 01:11 -72:49 80.9 89.5 1×60+3×800 8×60+4×500+17×600+1×700+1×800 qj0112 01:12 -72:36 87.4 81 1×60+1×600+6×800 9×60+7×500+15×600 qj0035 00:35 -72:01 95.5 300.6 4×600+1×700 4×60+1×100+3×500+15×600+3×800 qj0116 01:16 -72:59 102.5 95.2 1×60+13×600+2×800 7×60+14×500+20×600 smc0100 01:00 -74:57 130.4 167.5 1×60+7×800+3×900 1×60+7×600+3×700 qj0047 00:47 -75:30 161.7 187.7 1×800+3×1000 6×60+16×600+2×800 qj0033 00:33 -70:28 172.9 325 1×60+5×600+1×700 6×60+3×500+20×600 smc0049 00:49 -75:44 174.8 184.6 1×60+1×600+4×800 1×60+5×600 qj0102 01:02 -75:46 179.5 169.4 3×60+6×800 8×60+3×500+15×600+1×700+2×800 smc0053 00:53 -76:46 236.3 179.4 1×60+8×800+2×900 1×60+8×600+2×700 aDistance from the SMC center, α2000 = 00:52.7, δ2000 = -72:49 Table 2. Final number of stars measured in (B-R) Field N† smc0057 21246 qj0037 21579 qj0036 23756 qj0111 25845 qj0112 47548 qj0035 24114 qj0116 19875 smc0100 7801 qj0047 9762 qj0033 3797 smc0049 3580 qj0102 4030 smc0053 2188 †Selected stars with σ60.15, CHI62.5 and -0.66SHARP60.6. – 11 – observations were made; 178 in 2001 and 94 in 2002. Most of these areas include stars of a wide variety of colors. A few of them were followed up to about 2.0 airmasses to determine atmospheric extinction optimally. During these nights we also obtained short exposure BR frames of all our fields of interest, which served the purpose of calibrating all frames taken with non-photometric sky. Field qj0035 is not considered in this paper because we do not have yet a secure calibration for it. The Landolt standard images are uncrowded fields, thus no profile-fitting photometry was necessary for them. Instead, DAOGROWwas used in an identical way as for the program stars to directly derive the total instrumental magnitudes for the standard stars from their aperture photometry. To put our observations into the standard system, we used the following transformation equations: B = b+ αb + βb(B −R) + γbXb R = r + αr + βr(B −R) + γrXr where (b, r) and (B,R) are the instrumental and standard magnitudes respectively, and (Xb, Xr) are the airmasses. No time dependence terms were added since a preliminary fit showed no trends in the residuals of both B and R with time. The color terms (βb, βr) and extinction coefficients (γb, γr), as well as the zero-points (αb, αr) are unknown, presumably constant, transformation coefficients and are to be calculated. Both the color-dependent term and the zero-point in the transformation, are expected to be reasonably constant properties of the telescope/filter/detector combination and, in fact, the night-to-night differences for a given year in their computed values were found to be consistent with the uncertainty of the individual determinations. The above equations were applied to the Landolt standard star magnitudes, and solved for αb, αr, βb, βr, γb and γr, for each night, using a custom program. Then, we followed an iterative procedure to refine our photometric transformation. First, a new set of unique (αb, αr) and (βb, βr) values for each campaign was obtained by imposing the extinction coefficients (γb, γr) corresponding to each night. Then, we applied the resultant zero-points and color terms to each night of each year and new extinction coefficients were derived for each night. In this way, we have a set of (αb, αr) and (βb, βr) values for each year and (γb, γr) values for each night of each year. In Table 3 we present the zero-points and the color coefficients. The first column denotes the year, the second – 12 – and third columns indicate the (αb, αr) coefficients, and fourth and fifth columns the (βb, βr) coefficients. The final values for the extinction coefficients are shown in the second and third columns of Table 4, where the first column shows the night to which the coefficients correspond. Figure 2 shows the standard distributions for each filter and each year; the slope represents the extinction coefficient. Fitting errors in the zero-points correspond to (N − 2), where N is the number of measurements we have. They turned out to be between ±0.001 and ±0.002. The total zero-point errors of the photometry, including the error in the extinction, in the aperture corrections, and the uncertainties in the calibrations, are ∼0.02 mag in both B and R. These values are consistent with the systematic errors derived comparing photometry of different epochs. Using those SMC fields for which we have standard star observations from more than one night, we tested the photometric differences resulting from the use of different standard sets corresponding to different nights. We found that these differences are between ±0.001 and ±0.03 in B and between ±0.001 and ±0.04 in R (standard errors of the mean). 4. THE SMC STELLAR CONTENT We present unprecedented deep SMC CMDs obtained using a medium aperture telescope (100-inch). The quality of the CMDs is comparable to that of the CMDs obtained for the MCs using the HST (e.g. McCumber et al. 2005). Given the depth of our diagrams, which reach the oldest turnoffs, accurate information about age can be derived from the MS turnoff luminosities. As discussed by Gallart, Zoccali, & Aparicio (2005), reaching the oldest turnoffs will allow us to break the age-metallicity degeneracy affecting most methods to obtain the Figures 3 to 5 show the [(B-R),R] CMDs for the twelve SMC fields in order of increasing galactocentric distance. The fields range from galactocentric radius ∼1◦ to ∼4◦, and sample different regions of the SMC, as shown in Figure 1, where the fields are depicted in the context Table 3. Transformation equation coefficients Campaign αb αr βb βr 2001 -0.709±0.002 -0.423±0.001 +0.040±0.002 +0.001±0.001 2002 -0.892±0.002 -0.549±0.002 +0.045±0.002 +0.002±0.002 – 13 – 1 1.5 2 2.5 std2002B 1 1.5 2 2.5 std2001B 1 1.5 2 2.5 std2001R 1 1.5 2 2.5 std2002R Fig. 2.— The upper panels show the [B-b-βb(B-R)] vs. Xb (for standards in B-band) and the lower panels show the [R-r-βr(B-R)] vs. Xr (standards in R-band) for 2001 and 2002. The slope correspond to the extinction coefficient value. Open triangles correspond to 10/16/2001, open squares to 10/17/2001, diagonal crosses to 10/18/2001 and open circles to 10/19/2001. For 2002, open squares correspond to 10/10 and filled triangles to 10/13. – 14 – Fig. 3.— CMDs corresponding to the fields on the Eastern side of the SMC, in order of increasing distance from the SMC center (r) and with different position angles (p). – 15 – Fig. 4.— CMDs corresponding to the Western side of the SMC, in order of increasing distance from the SMC center (r) and for different position angles (p). – 16 – Fig. 5.— CMDs corresponding to the Southern side of the SMC, in order of increasing distance from the SMC center (r) with different position angles (p). – 17 – of the HI distribution (Figure 1a), and of the young (Figure 1b) and intermediate-age and old (RGB stars) populations (Figure 1c). All the CMDs reach a couple of magnitudes below the old MS turnoff (even the most populated), with great photometric accuracy. Figure 3 shows the CMD of fields qj0111, qj0112, and qj0116, which have a galactocentric radius ranging from 1.4◦ to 1.7◦. These fields correspond to the Eastern side of the galaxy (facing the LMC) as seen in Figure 1a, from which we can notice that they are located in an area of high HI concentration, inside the Supergiant Shell 304A with HI mass of 5.7×107 M⊙ (the total HI mass in the area observed by Stanimirović et al. is ∼4×10 8 M⊙). These fields are also in the region of the bridge which connects the so-called “stellar bar”3 with the “Wing”. From Figure 1b and Figure 1c, it can be noticed that these fields are located in very densely populated isopleths of young, intermediate-age and old population. Indeed, the CMDs of these fields show a conspicuous MS, well populated from the oldest turnoff at R∼22 to R∼16, which implies the presence of a large number of very young stars in all the CMDs. The three Western fields are also shown in Figure 1a and the corresponding CMDs are presented in order of increasing galactocentric distance from 1.3◦ to 2.9◦ in Figure 4. These fields are situated in the opposite side of the LMC, in a region with low concentration of HI. From Figure 1b it is evident that they are located in an area of low young stellar density, but still high density of intermediate and old population similar to that of the Eastern fields. The CMDs of these Western fields show a less significant MS, even at similar distances from the center as the Eastern fields. This fact is in agreement with the low density of the HI column in this part of the galaxy (see Figure 1a). We would like to address if the differences seen in the young population can be extended to the intermediate and old population. Figure 5 shows the CMDs of the six Southern fields, whose galactocentric distances 3The SMC is not a type of barred Irr galaxy, for which the LMC is the prototype, but sometimes it is useful to call bar to the brightest portion of its chaotic major axis. Table 4. Atmospheric extinction coefficients Night γb γr 16/10/2001 -0.218±0.007 -0.078±0.003 17/10/2001 -0.219±0.007 -0.077±0.007 18/10/2001 -0.219±0.006 -0.070±0.009 19/10/2001 -0.211±0.005 -0.065±0.005 10/10/2002 -0.207±0.008 -0.066±0.009 13/10/2002 -0.215±0.008 -0.076±0.008 – 18 – range from 1.1◦ to 4◦. As seen in Figure 1a these fields are located in regions in which the HI column-density is very low. Field smc0057, the closest to the center, presents a conspicuous young MS. In Figure 1b is shown that the stellar density of young stars in this field is higher than in the rest of the Southern fields and similar to the Eastern field qj0111. Field smc0100 still shows many bright young MS stars but further on 2.7◦ the CMDs have less populated young MS, and present a similar distribution of stars (with less statistics while going further South). The four furthest fields are located in an area in which the stellar density is similar, as shown in Figures 1b and 1c. Depending on their mass and metallicity, the core He-burning stars produce the horizon- tal branch (HB) and the RC. The presence of a prominent blue HB points out the existence of an old, metal poor population; the lack of such HB and the presence of a RC may indicate that core He burners are not so old, or more metal rich, or both (see Gallart et al. 2005 for details). A shared feature in all our SMC CMDs is the absence of a well populated, blue extended HB, pointing out that the amount of field stellar population as old and metal poor as that of the Milky Way halo globular clusters and dwarf spheroidal galaxies is small in the SMC. The RC is well populated and extended in luminosity, with a width of up to △R∼1 mag, denoting the presence of a large intermediate-age population. The fields with a prominent bright MS, as the Eastern fields and smc0057, for example, show the vertical extension of the RC feature corresponding to non-degenerate core He-burners (Gallart 1998). The older stars in the core He-burning phase lie in the lower part of the observed RC, while younger stars are brighter. In the following sections, we discuss the main features of the SMC CMDs using isochrones (sec § 4.1) and Color Functions (sec § 4.2). This will allow us to describe the SMC stellar populations, as a starting point to a study (Noël et al., in preparation) in which we will ad- dress the SMC SFH by comparing in detail the distribution of stars in the observed CMDs with a set of model CMDs. 4.1. Theoretical isochrones The interpretation of CMDs of composite stellar population strongly relies on the stellar evolution models adopted (see Gallart et al. 2005). For the present work, we used the Teramo stellar evolution models (Pietrinferni et al. 2004) as the reference. In Figures 6, 7, and 8 we have superimposed Teramo isochrones to our CMDs for three different metallicities suitable for the SMC stellar populations: Z=0.001 ([Fe/H]=-1.27), Z=0.002 ([Fe/H]=-0.96), and Z=0.004 ([Fe/H]=-0.66). The final selection of the metallicities was done taking into account the RGB and MS star colors and its best match by Teramo isochrones of given ages – 19 – and metallicities. The observed CMDs have been transformed to absolute magnitudes using a distance modulus of (m − M)0 = 18.9 (see van den Bergh 1999) and de-reddened as follows. For the two outermost fields, the reddening values given by the IRAS/COBE extinction maps (Schlegel, Finkbeiner & Davis 1998) were used. For the innermost regions, Schlegel et al. (1998) estimate a typical reddening of E(B-V)=0.037, from the median dust emission in surrounding annuli, since reddenings through the SMC cannot be calculated because its tem- perature structure is not sufficiently well resolved by DIRBE. Their quoted value, therefore, is not accurate enough and we have estimated a mean value of the reddening for each of the fields smc0057, qj0037, qj0036, and qj0111 by requiring a good fit of the CMD by the same isochrones that were overlapped to the other fields, for which the IRAS/COBE reddening values are reliable. This assumes the same age-metallicity relation for all fields, which is in agreement with the results found by Carrera (2006) and Carrera et al. (2006, in preparation) using CaII triplet spectroscopy. In all cases, the relations AB = 1.316E(B-V), and AR = 0.758 E(B-V) (Schlegel et al. 1998) have been assumed to calculate the extinction in the B and R bands. Canonical isochrones of 6, 9, and 13 Gyr with Z=0.001, isochrones with overshooting of 1, 3, and 4 Gyr with Z=0.002, and of 0.1 Gyr with Z=0.004 were overlapped. In the Eastern fields and in field smc0057 a 0.03 Gyr isochrone with overshooting and Z=0.004 was also superimposed. Figure 6 shows the CMDs of the Eastern fields with the superposition of Teramo isochrones. In these CMDs the conspicuous MS is well populated from the oldest turnoff at MR=3.5 (13 Gyr isochrone) up to the 0.03 Gyr isochrone. All the CMDs show a large number of young stars well represented by the 0.03 Gyr and 0.1 Gyr isochrones. The areas around the 1 to 6 Gyr isochrones are densely populated, indicating a strong presence of intermediate-age stars. No obvious differences between the MS of these fields can be inferred from the comparison with isochrones alone. Figure 7 shows the CMDs of the Western fields with the isochrones overlapped. In these fields, the intermediate-age population (∼3-4 to 9 Gyr) seems dominant. Fields qj0036 and qj0037 have a small fraction of stars born up to 0.1 Gyr ago, while qj0033, the most remote of the three, seems to have no stars younger than 1 Gyr (though this could be an effect of small number statistics). Figure 8 shows the CMDs of the six Southern fields with the superposition of isochrones. Field smc0057, the closest to the center, presents an important MS where the 0.1 Gyr isochrone is still quite populated. In field smc0100 there is still a substantial population – 20 – Fig. 6.— Fields located to the East side of the SMC, with Teramo isochrones (Pietrinferni et al. 2004) superimposed. The CMDs were transformed to absolute magnitudes using (m- M)0=18.9, and de-reddened with the reddening labelled, obtained using the criteria explained in the text. – 21 – Fig. 7.— As in Figure 6, but showing the CMDs of fields located to the West side of the – 22 – Fig. 8.— As in Figure 6, but showing the CMDs of fields located to the South of the SMC. – 23 – between 1 and 3 Gyr old, but fields qj0047 to smc0053 show few stars younger than 3-4 Gyr. The population of these relatively far-off fields is not purely old but contains an important amount of intermediate-age stars that inhabit the zones of the MS and subgiant branch around the 3, 4, 6, and 9 Gyr isochrones, in addition to the several old stars around the 13 Gyr isochrones. 4.2. Color Functions Gallart et al. (2005) have discussed the potential of MS Color Functions (CFs) to provide information on the stellar populations present in a galaxy. They concluded that, in fact, the CF may be a better tracer of the SFH than the LF. The reason is that, in CMDs like [(B-R), R] or [(V-I), I], the MS of isochrones of different age follow roughly vertical paths up to the MS turnoff where they turn to the red, almost perpendicularly to the MS, to form the subgiant branch or, in the case of more massive stars, the blue loops. This effect is clear from the CMD in panel (a) of Figure 9 (we will return to this figure later). The fact that the turnoff of a stellar population gets redder as the populations gets older implies a general dependency on age of the CF of the MS and subgiant branch stars brighter than the oldest MS turnoff. Here we will discuss CFs that will include the whole CMD within a region comprised between: -16MR63.5 and -0.56(B-R)61.5. The star counts have been normalized to the total stars present in the integrated range. Most of the information belongs in any case to the MS and subgiant branch. The fraction of CF that belongs to the RGB and RC serves as a check of the metallicity assumed for the synthetic CMD, and contributes to the overall normalization of the CF. Comparison of the stellar content of a galaxy with isochrones and CFs is complementary in terms of the information on the stellar population they provide. The isochrones give us the range of ages present in a given CMD; the comparison between theoretical and observed CFs is a good tool to assess the relative amounts of stars of different ages present in a stellar population. This is a particular case of a comparison of the number of stars in a set of boxes in the observed and synthetic CMDs respectively (e.g. Gallart et al. 1999). However, the CF is particularly useful for a first qualitative assessment of the SFH because it is a one- dimensional representation of the content of the CMD in which there is a relatively direct correlation between range of ages and color. All the models used here have been computed using the synthetic CMD algorithm iac-STAR (Aparicio & Gallart 2004). Panel (a) in Figure 9 shows a synthetic CMD calculated assuming a constant SFR(t) from 13 Gyr ago to date and metallicities: Z=0.004 in the range 0 Gyr6t61 Gyr, Z=0.002 in the range 1 Gyr6t65 Gyr, and Z=0.001 in the range 5 Gyr6t613 Gyr (see inset). Stars – 24 – Fig. 9.— The panel (a) shows a synthetic CMD showing the position of stars in different ages intervals. The Teramo stellar evolution models have been used in the computation using the tool iac- STAR (http://iac-star.iac.es/iac-star/). Constant SFR(t) from 13 Gyr ago to the present time has been assumed. In the inset on top of panel (a) the metallicity law used is shown: Z=0.004 ([Fe/H]=- 0.66) between 0 and 1 Gyr ago, Z=0.002([Fe/H]=-0.96) between 1 and 5 Gyr ago, and Z=0.001 ([Fe/H]=-1.27) between 5 and 13 Gyr ago. In panel (b) different CF distributions, corresponding to synthetic populations with different SFR(t) and integrated in the range -1<MR<3.5 are depicted (see inset). Black CF correspond to the CMD in panel (a). The CF for a population with an enhancement in the SFR(t) 3 Gyr ago is depicted in blue. The green CF represents a population in which the SFR(t) decreased by 50% between 8 and 3 Gyr ago. The CF in red represents a population with a gap in the SFR(t) between 8 and 3 Gyr ago. Panel (c) shows the CF corresponding to the CMD in panel (a) with limited age ranges, as described in the labels. http://iac-star.iac.es/iac-star/ – 25 – corresponding to each age interval are depicted in different colors. A 30% of binary stars have been assumed; the parallel feature to the red of the MS between 3.56MR65 corresponds to binary stars. Some of these binaries between 8-13 Gyr old are noticeably brighter than single stars at the same age and mass (red points in the synthetic CMD between 26MR63). Panel (b) in Figure 9 shows the composite CFs of populations with different SFR(t) (shown in the inset). Three main features characterize the CF: a local elevation in the blue part between -0.36(B-R)60.2, a central peak between 0.26(B-R)60.9, and a second local maximum in the red part. Panel (c) of Figure 9 shows which populations contribute to each feature in the CF for the synthetic CMD in panel (a). By comparing these, we find the following: the elevation in -0.3.(B-R).0.2 in the composite CF mostly corresponds to stars in the age range 06t.1.1 Gyr; in the central peak (0.3.(B-R).0.9) there are populations of all ages, predominating the intermediate-age and old stars (∼2-13 Gyr). The second local maximum corresponds to the RC and the red part of the subgiant branch and is composed by stars with age &1 Gyr as seen in panel (c) in Figure 9. Going back to panel (b), the CF in black corresponds to a galaxy with a constant SFR(t) (i.e., the CF of the CMD in the panel (a)). The blue CF is from a synthetic CMD computed assuming a SFR(t) that increased from 3 Gyr ago to the present. This CF presents a higher blue elevation due to the bigger number of young blue stars; also the central peak is somewhat blueshifted. The green CF represents a stellar population in which the SFR(t) dropped by 50% between 3 and 8 Gyr ago. This is reflected in the shape of the absolute maximum, where a depression appears as a result of the assumed SFH. This kind of feature would be difficult to detect in a real population. Finally, the red CF corresponds to a SFR(t) with a totally quiescent period between 3 and 8 Gyr ago. The deep hollow shown in the absolute maximum of this CF is due to the gap at intermediate ages in the SFH. The last two CFs show a higher blue elevation as compared to a CF of a CMD with constant SFR(t) due to the higher fraction of young stars over the whole population. Prior to present a discussion based on CFs, we will briefly talk about the crowding effects and, in general, the observational errors that are present in our photometry. A careful study of the extent and nature of these crowding effects is key to study the stellar populations present in the SMC. We will understand and quantify the crowding effects using the artificial- star tests procedure. These artificial-star tests have been performed in a way similar to that described in Gallart et al. (1999) and will be presented in detail in a forthcoming paper (Noël et al. in preparation). In short, a number of artificial stars of known magnitudes and colors are injected in the original frames, using the ADDSTAR algorithm of DAOPHOT II (Stetson 1993). The photometry is re-done following the same procedure used to obtain the photometry for the original frames. This process is repeated several times in order to test a large number of artificial stars (60,000 in the case of each of our SMC fields). The injected – 26 – and recovered magnitudes of the artificial stars, together with the information from those that were lost, provide the necessary information about all the observational effects present in the photometry. We will discuss here how the observational errors affect the CF of a few representative fields. Those located closer to the SMC center (e.g. smc0057 and qj0111) are more affected by crowding than the rest of the fields. To quantify these observational errors we have simulated them for different SMC fields in the synthetic CMD of Figure 9. Following Gallart et al. (1999), we will call the latter “model CMDs”. The model CFs (corresponding to the model CMDs) corresponding to fields smc0057, qj0111, and smc0049 are shown in Figure 10, together with the original CF (the CF obtained from the synthetic CMD). The observational errors may produce significant variations in the appearance of the CF of our more central, crowded field smc0057. Simulating the observational errors of this field in the synthetic CMD, we obtain a model CF (dotted line) which differs substantially from the CF of the original synthetic CMD: the absolute maximum in its CF is smaller than in the CF of the synthetic CMD, and the blue elevation is bigger because faint stars on the low MS are lost much more frequently than the brighter blue stars that constitute the young MS. The dispersion introduced in the CMD when the observational errors are simulated produces an absolute maximum that is slightly wider as shown in the dotted line in Figure 10. This shows how crowding could simulate a non-constant SFH, mimicking a recent enhancement of the star formation, similar to the one shown in blue in panel (b) of Figure 9. Although still present, the effects of crowding are less severe in the case of field qj0111 since the CF of the corresponding model CMD (dashed line) resembles quite closely the CF for the synthetic CMD. In this field, confusion is mainly affecting the yellow stars so the main maximum is shorter than in the synthetic CF. In the case of smc0049, it can be seen that the CF of the model CMD (dot-long dashed line) is similar to the one of the synthetic CMD (solid line), indicating that crowding is affecting very little the CMDs of this part of the galaxy (at ∼2.9◦). In conclusion, only the CF of field smc0057 is substantially affected by observational errors. The effects of crowding on field qj0111 can be considered as an upper limit for fields qj0112 and qj0116. For fields smc0100, qj0047, qj0102, smc0053, qj0037, qj0036, and qj0033 the crowding level is similar to that of smc0049. Therefore, we can safely discuss the stellar content of all of our fields (except smc0057) by comparing their CFs straight with that of synthetic CMDs. Let’s concentrate now on the observed CFs of each of our fields. In Figures 11 to 13 the CF corresponding to the synthetic CMD (dashed lines) from panel (a) in Figure 9 and the CFs (solid lines) from the observed CMDs were overlapped. Note that the shape of the synthetic CF depends on the selection of metallicities and stellar evolution libraries. However, small changes in the metallicity law, which result in synthetic CMDs with a color – 27 – 0 1 2 (B-R) Fig. 10.— Color functions of a synthetic CMD integrated over the magnitude range - 16MR63.5 assuming different levels of crowding in the data. The solid line represents the CF of the original synthetic CMD, the dotted line corresponds to the CF of the synthetic CMD, in which the observational errors of field smc0057 have been simulated, the dashed line is the CF of the synthetic CMD with the simulation of errors from field qj0111, and dot-long dashed line is the CF of the synthetic CMD with errors in field smc0049 simulated. – 28 – Fig. 11.— CF for the observed CMDs corresponding to the Eastern SMC fields (solid lines), overlapped with the synthetic CF (dashed lines) from panel (a) in Figure 9, normalized to the number of stars in the region between -0.5<(B-R)<1.5 and -1<MR<3.5. In the first panel, corresponding to field qj0111, the CFs of synthetic populations with burst of star formation of 30% and 50% from 1 Gyr ago to now (dotted and long-dashed line respectively) and of 50% from 2 Gyr ago to now (dot-long dashed line) were also overplotted, showing that there was a recent enhancement of the star formation around 1 Gyr ago in this field. – 29 – qj0037 (B-R ) qj0036 (B-R ) (B-R ) qj0033 Fig. 12.— CF for the observed CMDs corresponding to the Western SMC fields (solid lines), overlapped with the synthetic CF (dashed lines) from panel (a) in Figure 9, normalized to the number of stars in this region between -0.5<(B-R)<1.5 and -1<MR<3.5. – 30 – smc0057 smc0100 qj0047 (B-R ) smc0049 (B-R ) qj0102 (B-R ) smc0053 Fig. 13.— CF for the observed CMDs corresponding to the Southern SMC fields (solid lines), overlapped with the synthetic CF (dashed lines) from panel (a) in Figure 9, normalized to the number of stars in this region between -0.5<(B-R)<1.5 and -1<MR<3.5. In field smc0057, was also superposed the CF of a model CMD (dotted lines) in which observational errors from this field were simulated (same as in Figure 10). – 31 – distribution compatible with the data have also very similar CFs. Comparison between the observed and synthetic CFs computed assuming constant SFR(t) lets us to qualitatively analyze the SFH of each field. Figure 11 shows the CFs of the Eastern SMC fields. In the first panel, corresponding to field qj0111, synthetic CFs of synthetic populations with bursts of 50% 1 Gyr ago (long dashed line), of 30% 1 Gyr ago (dotted line), and of 50% 2 Gyr ago (dot-long dashed line) were also overplotted. The CF of field qj0111 shows some differences with respect to the CF of a population with constant SFR(t). In particular, the observed CF has higher values than the synthetic one in the blue elevation (-0.3.(B-R).0.2; age.1.1 Gyr) which indicates a SFH with enhanced star formation at young ages. This fact is also shown by comparing with the CF of synthetic populations with bursts. In fact, the blue maximum and the main peak of the CF with a burst of 30% 1 Gyr ago and the ones of the observed CF match very well, pointing out that there was an episode around 1 Gyr ago of enhanced star formation in this field which may have continued to the present time. Though a comprehensive analysis of the CMDs will be presented in a forthcoming paper, this superposition allows us to say that the observed CF is compatible with a SFH with an increase in the star formation of ∼30% in the last 1 Gyr. Also note that crowding effects are not very significative in this field, as seen from the artificial star tests. The CF of field qj0112 is a remarkable case where the CF corresponds almost exactly with that of a population with constant SFR(t) (at least in average). The CF of qj0116 presents a somewhat greater values of the CF in 0.(B-R).0.3 as compared with the CF of a CMD with constant SFR(t), which may correspond to stars formed between ∼1-2 Gyr ago, possibly indicating some enhancement in the SFR(t) at those ages in comparison with more recent ages. Note that these three fields have a very similar CMD, and that simple comparison with isochrones gives the same information for all three: presence of stars of all ages. The differences in their CFs are a good illustration of the complementary information that the CF can provide. The presence in the observed CFs of all the features that characterize the synthetic CF is the counterpart, in this representation of the data, of the fact that the CMDs are well populated over all age ranges. The additional information brought forward by the CFs is the relative weight of the different age ranges in the population. In Figure 12, the CFs of the Western SMC fields are shown. The almost complete absence of the blue elevation in the color interval -0.3.(B-R).0.2, reflects the fact that there are few stars younger than ∼1 Gyr. This is another example of the usefulness of the CFs: while in fields qj0036 and qj0037 there are stars until around the 0.1 Gyr isochrone, the CFs indicate that the relative amount of stars between 0 Gyr and 2 Gyr is substantially less than in the case of a field with constant SFR(t). The central maximum is slightly shifted to the red, possibly indicating that the SFR(t) started to decline around 3 Gyr ago. – 32 – Note that this shift is unlikely due to an overall shift of the observed CMD with respect to the synthetic one, since the shift does not affect the red relative maximum of the CF. In fact, this maximum is slightly shifted to the blue and higher in the CF of the observed CMD as compared with that of the synthetic one. This is possibly due to field stars, which contribute more in relative numbers to the poorly populated CMDs. Actually, the observed red maximum is higher with respect to the theoretical one in the less populated qj0033. The composition of the population in these three Western CMDs seems to be very similar, even in the outermost field, qj0033. Finally, Figure 13 shows the CFs of the Southern SMC fields. In the case of field smc0057, the relative amount of young stars might be overestimated due to crowding effects if these were not taken into account. The overplot of the observed CF (solid line) with the synthetic (dashed line) and the model (i.e. after taking into account crowding effects, dotted line) ones shown in Figure 13 indicate that part of the enhancement of the blue elevation in field smc0057 is produced by observational errors but that there is still a true recent enhancement (06t63 Gyr) in the star formation. The CF of field smc0100 is still almost compatible with a constant SFR(t) on average, with possibly a truncation of the SFR(t) at recent ages, as also hinted by the comparison of its CMD with isochrones in Figure 8. Moving further away from the SMC center, the elevations corresponding to the first structure in the CFs are less conspicuous, indicating that young stars are less common and intermediate- age stars dominate the fields. The fact that the main peak is redder from field qj0047 on (except, maybe, in the case of field qj0102 which is situated more towards the East than the other Southern fields), points out that there is a larger amount of old stars and, like in the Western fields, that possibly the SFR(t) started to decline around 3 Gyr ago. As in the case of the Western field qj0033, the second, red relative maximum in the CFs gets bigger with increasing galactocentric distance. The reasons for this are twofold. First, the amount of stars in the RC and in the RGB, relative to the total stellar content is larger at larger distances from the center (even though not large differences among fields seem to be present beyong r≈2.7◦), where the age of the stellar population is older on average. Second, the contamination by foreground stars is larger relative to the total number of stars from the galaxy. This last factor, may be the main player in the case of field smc0053. 5. SUMMARY We obtained B and R-band photometry of stars from 12 SMC fields observed during 4 different campaigns with the C100 telescope at Las Campanas Observatory. The spatial distribution of the fields samples different parts of the galaxy, both in areas with large – 33 – amounts of recent star formation such as the “Wing” area and “undisturbed” regions at the West and towards the South part of the galaxy, in which the stellar populations may be representative of the underlying population of the SMC formed prior to the events of star formation that shaped its current irregular morphology. The stellar content present in these SMC fields has been analyzed by means of a set of [B-R, R] CMDs that reach the oldest MS turnoffs (MR∼3.5) with an excellent photometric accuracy. In this first aproach to study the SFH of our SMC fields, we found the following. The fields in the “Wing” area contain very young blue stars and show a broad MS turnoff/subgiant region and a wide range in luminosity of the RC, pointing out that star formation in these parts has extended from at least ∼13 Gyr ago to the present with no substantial gaps, as the areas around all the overlapped isochrones are well populated. Field qj0111 (the closest to the center of the SMC in this “Wing” area) seems to have experienced an enhancement in its recent star formation, compatible with an enhancement of 30% in the SFR(t) over the last 1 Gyr. This is indicated by the height of the blue elevation its CF (color interval in the range -0.3.(B-R).0.2) as compared with that of CFs corresponding to synthetic populations with different SFR(t). The other two fields analyzed in this direction seem to have had a SFR(t) close to constant on average. Very little star formation has been going on from ∼1 Gyr ago until now in the Western fields, as indicated by the much lower amount of young blue stars and the lack of the blue elevation in the CFs. Considering the RC (formed by stars older than ∼1 Gyr) of fields qj0036 and qj0111, both located at the same galactocentric distance, we found that the number of stars is comparable, indicating that the integrated SFR(t) from old ages and until ∼1 Gyr ago in both fields is similar, and that there was a mechanism in the “Wing” area that triggered the star formation around 1 Gyr ago, in particular, in the location of field qj0111. Field qj0033, though located further away from the center in the Northwest, presents a distribution of stellar ages similar to that of the two other Western fields, as seen from the resemblance of the three CFs and from its CMD, in which the area below the turnoff of the 4 Gyr isochrone is still quite populated. In fact, the apparent differences between the CMD of field qj0033 and the other two are due to the much larger number of stars populating those. If only a subset of stars of those CMDs are represented, to equal number of stars in the CMD of qj0033, the distribution of stars around the different isochrones is very similar. In these three fields, the bulk of stars seems to be older than 3-4 Gyr. A similar conclusion was reached by Dolphin et al. (2001) for a field located in the Northwestern part and relatively close to field qj0033 (see Figure 1), where they found a greatly decreased star formation in the past 2 Gyr. Regarding the fields observed toward the South, the young population is less important – 34 – from r=2.7◦ on. The blue elevation in the CFs of the Southern fields is only found in the two fields closer to the center. In the case of smc0057, its shape is consistent with some enhancement of the recent SFH as compared to a constant SFR(t). From the CFs it is evident that field smc0100 has many more stars younger than 2 Gyr (relative to the total population of the field) than the Western fields qj0036 and qj0037, even when it is almost 1◦ further from the SMC center than the latters. Note that this field is located at the border of a more densely populated isopleth than qj0036 and qj0037 and in the part in which the trumpet-like distribution of the young population fans out. Beyond ∼2.7◦, there are only few stars younger than 3 Gyr old and the differences between the fields are small (except for the fact that the density of stars in field smc0049 is larger than in the other two). Finally, except for Eastermost field qj0102, the further South from the SMC center we go, the more is the main peak of the CF shifted to the red. This, together with the above facts, may indicate that the main epoch of star formation beyond ∼2.2◦ in this direction ended around 3-4 Gyr ago. 6. CONCLUSIONS The SMC shows a clear morphologic dichotomy between its young and old population. The youngest component has an asymmetric distribution which fans out towards the “Wing” area in the Northeast, similarly to the HI distribution (Stanimirović et al. 1999), while the RGB and AGB stars have a more symmetric, spheroidal distribution. Using deep (down to MR = 5.5) CMDs in 12 small fields that strategically sample different SMC regions, we are trying to shed some new light into our knowledge of the SMC evolution, and answer some of the questions posed in the Introduction. We found that the underlying spheroidally distributed population is mainly composed by intermediate-age and old stars, and its distribution does not show strong galactocentric gradients. Our fields situated toward the Northwest of the bar, at galactocentric distances from 1.3◦ to 2.9◦ contain very few stars younger than ≃ 3 Gyr. The comparison of their CFs with that of a synthetic population with constant SFR(t) indicates there has been an actual drop of the SFR(t) at around this age until the present. Even though, some residual star formation seems to have continued up to now. No strong differences can be noticed between the CMDs and CFs of the fields at 1.3◦ and 1.4◦ and the one at 2.9◦. The apparent differences between these CMDs are mainly due to the different number of stars in them, rather to a fundamentally different composition of the stellar populations. Something similar occurs in the case of the Southern fields beyond ≃ 2.2◦. In addition, we estimated that the SFR(t) at intermediate to old ages in fields located at the same distance from the SMC center and – 35 – at either side (e.g. qj0111 and qj0036) is similar. With the current analysis of the data we cannot provide a detailed SFH for these fields, and therefore we cannot confirm nor rule out periods of quiescence as found in other studies that used shallower data. This will be the subject of a forthcoming paper. The three fields situated towards the East, in the “Wing” region, show very active current star formation. In the case of the field closer to the center, qj0111, we estimated that the current star formation rate (from around 1 Gyr ago to the present) in this field is ∼30% larger than the SFR(t) averaged over the galaxy’s life. However, the other two northeastern fields, situated just slightly further away from the center, have a CF which is compatible with that of a constant SFR(t) over the whole galaxy’s history. Therefore, only the parts of the Wing closer to the center, seem to be undergoing an exceptionally intense episode of star formation at the present time. At the radius of the innermost field (∼1 kpc), we derived a crossing time of ∼108 yr, obtained adopting a measured velocity dispersion of 36 km/s (Carrera 2006, using the stars below the RGB tip corresponding to our SMC fields). In the case of our outermost field (∼4 kpc) the crossing time is ∼2×108 yr. Large scale asymmetries should disappear within a few crossing times, and the population older than ∼1 Gyr is then expected to be well mixed at a radius of ∼1 kpc, while the population older than ∼2 Gyr is well mixed at ∼4 kpc. Hence, although our fields only cover little isolated parts of the SMC (as compared with the huge area embraced by HZ04), they are representative of the SMC populations older than ∼2 Gyr. The larger amount of young stars present in the Eastern fields may be related to a recent interaction between the SMC, the LMC, and the Milky Way as is suggested by numerical models (e.g. Yoshizawa & Noguchi 2003). Note that the presence of a considerable young population in the eastern fields and lack thereof in the western ones is in good correspon- dence with the existence or absence of large amounts of HI at the corresponding locations (Stanimirović et al. 1999). It is interesting to notice that this trend is not maintained in the closest Southern field, smc0057, in which there is a considerable amount of young population but the amount of HI is small. The presence of a substantial amount of intermediate-age population in our outermost SMC fields, could be a sign that a few gigayears ago HI gas extended as far as ∼4◦ and has since gradually receded to the more central regions. Another possible explanation for this is that tidal interactions happened periodically, and then these stars dynamically mixed, resulting in old stars (those from previous passages) being symmetrically distributed and the younger ones only in the current tidal feature. None of the studied fields is dominated by an old stellar component, as one would expect for an old stellar halo similar to the one – 36 – of the Milky Way. This may mean that a disk population is still dominating over a possible old halo, if it does exist in the SMC. We want to thank Antonio Aparicio and Santi Cassisi for many very useful opinions and discussions and a careful reading of the manuscript. We are very grateful to the anony- mous referee for a critical and useful report that helped to improve the paper. The authors acknowledge support by the Plan Nacional de Investigación Cient́ıfica, Desarrollo, e Inves- tigación Tecnológica, (AYA2004-06343). E. C. and R. A. M. acknowledge support by the Fondo Nacional de Investigación Cient́ıfica y Tecnológica (No. 1050718, Fondecyt) and by the Chilean Centro de Astrof́ısica FONDAP (No. 15010003). This project has made generous use of the 10% Chilean time, and continuous support from the CNTAC and Las Campanas staff is greatly appreciated. REFERENCES Aparicio, A., & Gallart, C. 2004, AJ, 128, 1465 Carrera, R.J. 2006, PhD thesis Cioni, M.-R. L., Habing, H. J., Israel, F. P. 2000, A&A, 358, 9L Côté, S., Freeman, K. C., Carignan, C. & Quinn, P. J. 1997, AJ, 114, 1313 Dolphin, A. E., Walker, A. R., Hodge, P. W., Mateo, M., Olszewski, E. W., Schommer, R. A., & Suntzeff, N. B. 2001, ApJ, 562, 303 Ferguson, H. C., & Sandage, A. 1991, AJ, 101, 765 Gallart, C. 1998, ApJ, 495, 43L Gallart, C., Freedman, W. L., Aparicio, A., Bertelli, G. 1999, AJ, 118, 2245 Gallart, C. 2000, in ASP Conf. Proc. 221, Stars, Gas and Dust in Galaxies: Exploring the Links, ed. D. Alloin, K. Olsen, & G. Galaz (San Francisco, ASP), 133 Gallart, C., Zoccali, M., Aparicio, A. 2005, ARA&A, 43, 10 Gardiner, L. T., & Hatzidimitriou, D. 1992, MNRAS, 257, 195 Girardi, L., Bertelli, G., Bressan, A., Chiosi, C., Groenewegen, M. A. T., Marigo, P., Salas- nich, B., & Weiss, A. 2002, A&A, 391, 195 – 37 – Harris, J., & Zaritsky, D. 2004, AJ, 127, 1531 (HZ04) Holtzman, J. A., Gallagher, J. S. III, Cole, A. A., et al. 1999, AJ, 118, 2262 Landolt, A. U. 1992, AJ, 104, 340 Madau, P., Pozzetti, L., Dickinson, M. 1998, ApJ, 498, 106 McCumber, M. P., Garnett, D. R., & Dufour, R. J. 2005, AJ, 130, 1083 Pedreros, M. H., Costa, E., & Méndez, R. A. 2006, AJ, 131, 1461 Pietrinferni, A., Cassisi, S., Salaris, M., & Castelli, F. 2004, ApJ, 612, 168 Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525 Stanimirović, S., Staveley-Smith, L., Dickey, J. M., Sault, R. J. & Snowden, S. L. 1999, MNRAS, 302, 417 Stetson, P. B. 1987, PASP, 99, 191 Stetson, P. B. 1990, PASP, 102, 932 Stetson, P. B. 1992, in ASP Conf. Ser. 25, Astronomical Data Analysis Software and Systems I, ed. D. M. Worrall, C. Biemesderfer, & J. Barnes (San Francisco, ASP), 297 Stetson, P. B. 1993, User’s Manual for DAOPHOT II Stetson, P. B. 1994, PASP, 106, 250 Stetson, P. B., Bruntt, H., & Grundahl, F. 2003, PASP, 115, 413 Tinney, C. G., Da Costa, G. S., & Zinnecker, H. 1997, MNRAS, 285, 111 van den Bergh, S. 1999, ARA&A, 9, 273 Yoshizawa, A. M., Noguchi, M. 2003, MNRAS, 339, 1135 Zaritsky, D., Harris, J., Grebel, E. K., & Thompson, I. B. 2000, ApJ, 534, 53L Zaritsky, D., Harris, J., & Thompson, I. B. 1997, AJ, 114, 1002 This preprint was prepared with the AAS LATEX macros v5.0. INTRODUCTION OBSERVATIONS AND DATA REDUCTION THE PHOTOMETRY Photometry of SMC fields Standard Stars Photometry THE SMC STELLAR CONTENT Theoretical isochrones Color Functions SUMMARY CONCLUSIONS

0704.1152

Curvature perturbations from ekpyrotic collapse with multiple fields

Curvature perturbations from ekpyrotic collapse with multiple fields Kazuya Koyama† 1 Shuntaro Mizuno‡ 2 and David Wands† 3 † Institute of Cosmology and Gravitation, Mercantile House, University of Portsmouth, Portsmouth PO1 2EG, United Kingdom ‡ Research Center for the Early Universe (RESCEU), School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan Abstract A scale-invariant spectrum of isocurvature perturbations is generated during col- lapse in the ekpyrotic scaling solution in models where multiple fields have steep neg- ative exponential potentials. The scale invariance of the spectrum is realized by a tachyonic instability in the isocurvature field. This instability drives the scaling solu- tion to the late time attractor that is the old ekpyrotic collapse dominated by a single field. We show that the transition from the scaling solution to the single field domi- nated ekpyrotic collapse automatically converts the initial isocurvature perturbations about the scaling solution to comoving curvature perturbations about the late-time attractor. The final amplitude of the comoving curvature perturbation is determined by the Hubble scale at the transition. 1 Introduction The existence of an almost scale-invariant spectrum of primordial curvature perturbations on large scales is one of the most important observations that any model for the early universe should explain. An inflationary expansion in the very early universe is most commonly assumed to achieve this, but it is important to consider whether there is any alternative model. In this paper we focus on the ekpyrotic scenario as an alternative [1] (see also [2, 3]). In the old ekpyrotic scenario, the large scale perturbations are supposed to be generated during a collapse driven by a single scalar field with a steep negative exponential potential. It was shown that the Newtonian potential acquires a scale-invariant spectrum, but the comoving curvature perturbation has a steep blue spectrum [4]. In this scenario we need a mechanism to convert contraction to expansion, and for a regular four-dimensional bounce, the scale-invariant Newtonian potential is matched to the decaying mode in an expanding universe, and the growing mode of curvature perturbations acquires a steep blue spectrum [5, 6]. It has been suggested that this conclusion might be altered by allowing a singular matching between collapse and expanion [7], but the general rule that the comoving curvature perturbation remains constant still holds for adiabatic perturbations on large scales [8]. In 1 E-mail: [email protected] 2 E-mail: [email protected] 3E-mail: [email protected] http://arxiv.org/abs/0704.1152v3 a braneworld context, the conversion from contraction to expansion might be accomplished by a collision of two branes where one of extra-dimensions disappears [9]. It was argued that the scale-invariant Newtonian potential can be transferred to the comoving curvature perturbations by this singular bounce [10]. However, without having a concrete theory to describe the singularity, it is difficult to have a definite conclusion on how perturbations pass through the singularity. Recently, there has been some progress in generating a scale invariant spectrum for curvature perturbations in the ekpyrotic scenario [11, 12, 13], by considering non-adiabatic perturbations which has been suggested previously by Ref. [15]. In this case we require two or more fields. If these have steep exponential potentials then there exists a scaling solution where the energy density of the fields grow at the same rate during collapse [16, 17]. The isocurvature perturbations then have a scale-invariant spectrum [16]. These isocurvature perturbations can be converted to curvature perturbations if there is a sharp turn in the trajectory in field space [11, 12, 13]. For example Ref. [11] considered a situation where one of the fields changes its direction in field space, which corresponds to a time when a negative tension brane is reflected by a curvature singularity in the bulk, in the context of the heterotic M-theory. Refs. [12, 13] considered a regular bounce realized by a ghost condensate. One of the fields exits the ekpyrotic phase and hits the transition to the ghost condensate phase that creates a sharp turn in the trajectory in field space and curvature perturbations can be generated [12]. It is still necessary to match curvature perturbations in a contracting phase to those in an expanding universe, but it is shown that the comoving curvature perturbation is conserved on large scales resulting in an almost scale-invariant spectrum observed today for a regular bounce like a ghost condensate model [12]. The isocurvature perturbations behave like δs ∝ H on large scales. As the Hubble parameter is rapidly increasing in a collapsing universe, this signals an instability. In fact it is easy to see that we always require an instability of this form in order to generate a scale- invariant spectrum with a canonical scalar field during collapse4. As the amplitude of field perturbations at Hubble exit are of order H , we require the super-Hubble perturbations to grow at the same rate to maintain a scale-invariant spectrum. Ref. [19] studied this instability in a phase space analysis. The multi-field scaling solution was shown to be a saddle point in field space and the late-time attractor is the old ekpyrotic collapse dominated by a single- field. A tachyonic instability drives the scaling solution towards the late-time attractor (see also [11, 12]). In Ref. [19], we pointed out that the natural turning point in the field space trajectory due to the instability of the scaling solution might itself offer the possibility of converting the scale invariant spectrum of isocurvature field perturbations into a scale invariant spectrum of curvature perturbations. In this paper, we confirm this expectation by explicitly solving the evolution equations for perturbations in a two field model. We find that the ratio between curvature perturbations and isocurvature perturbations at the final old ekpyrotic phase is solely determined by the ratio of exponents of the two exponential potentials, and the amplitude is set by the Hubble rate at the transition time. 4The only way to produce a scale-invariant spectrum without the presence of an instability seems to be due to non-canonical kinetic terms, as in the case of axion-type fields which can acquire scale-invariant perturbation spectra while remaining massless [18]. 2 Homogeneous field dynamics We first review the background dynamics of the fields. During the ekpyrotic collapse the contraction of the universe is assumed to be described by a 4D Friedmann equation in the Einstein frame with scalar fields with negative exponential potentials 3H2 = V + φ̇2i , (1) where V = − −ciφi , (2) and we take Vi > 0 and set 8πG equal to unity. The authors of [11] found a scaling solution (previously studied in [16, 17]) in which both fields roll down their potential as the universe approaches a big crunch singularity. In this ekpyrotic scaling collapse we find a power-law solution for the scale factor a ∝ (−t)p , where p = , (3) where −Vie−ciφi −Vje−cjφj . (4) As we will see in the next section, it is possible to generate scale-invariant isocurvature per- turbations around this background. However, the ekpyrotic scaling solution (4) is unstable. In addition to the scaling solution we have fixed points corresponding to any one of the original fields φi dominating the energy density where the other fields have negligible energy density. These correspond to the original ekpyrotic power-law solutions where a ∝ (−t)pi , where pi = , (5) for c2i > 6. We find that any of these single field dominated solutions is a stable local attractor at late times during collapse. In Ref. [19], the ekpyrotic scaling solution (4) was shown to be a saddle point in the phase space. We briefly review the phase space analysis. Introducing phase space variables [20, 21, 17] , (6) Vie−ciφi√ . (7) the first order evolution equations for the phase space variables are given by = −3xi(1− x2j)− ci y2i , (8) x2j − ci  , (9) where N = log a. The Friedmann equation gives a constraint x2j − y2j = 1. (10) There are (n + 2) fixed points of the system where dxi/dN = dyi/dN = 0. x2j = 1, yj = 0. (11) Bi : xi = , yi = − − 1, xj = yj = 0, (for j 6= i), (12) B : xj = , yj = − . (13) In this paper, we focus on the fixed points B and Bi assuming c i > 6 and c−2i < 1/6. The linearized analysis shows that the multi-field scaling solution, B, always has one unstable mode. On the other hand, the single field dominated fixed points, Bi, are always stable. From now on we concentrate our attention on two fields case. Then we have three fixed points B,B1 and B2. It is interesting to note that in the (x1, x2) plane, the fixed points B, B1 and B2 are connected by a straight line, which is given by c2x1 + c1x2 = c1c2√ . (14) The eigenvector associated with the unstable mode around the scaling solution B lies in the same direction as the line (14). Thus this is an attractor trajectory, which all solutions near B approach. Fig. 1 shows numerical solutions for the evolution of x1. Initial positions in the phase space are perturbed away from B along the line (14). The solutions go to B1 or B2, depending on the initial position in the phase space. An important observation is that, as we follow phase space trajectories during the tran- sition from the scaling solution B to the attractor solutions B1 or B2, the solutions obey the relation Eq. (14), even far away from the saddle point, B. Using the Friedmann equation and the field equations, we can show that H − φ̇1 − φ̇2 H − φ̇1 − φ̇2 = 0 , (15) and hence H − φ̇1 − φ̇2 , (16) where C is an integration constant. In terms of φ1 and φ2, Eq. (14) can be rewritten as = H. (17) and hence we see that for trajectories starting from point B we have C = 0, which is the late time attractor. Thus we see that Eq. (17) holds even during the transition caused by the tachyonic instability from point B to B1 or B2 and in the final single-field dominated phase. This fact will be important when we study perturbations. We will study the behaviour of perturbations during the transition in the next section. 2.5 5 7.5 10 12.5 15 0.01 0.02 0.03 0.04 0.05 Figure 1: Left: Numerical solutions for x1(N). The horizontal axis is N = log a and we take c1 = 40 and c2 = 30. The initial time is N = 0.05. Note that N decreases towards the future in a collapsing universe. Right: The corresponding phase space trajectories in (x1, x2) plane. 3 Generation of quantum fluctuations In this section, we consider inhomogeneous linear perturbations around the background solution. We consider the scalar field perturbations on spatially flat hypersurfaces. Then the scalar field perturbations are given by [22, 23, 24, 25] δ̈φi + 3H ˙δφi + δφi − c2iVi exp(−ciφi)δφi − φ̇iφ̇j δφj = 0. (18) We can decompose the perturbations into the instantaneous adiabatic and entropy field perturbations as follows [24]: φ̇1δφ1 + φ̇2δφ2 φ̇21 + φ̇ , δs = φ̇2δφ1 − φ̇1δφ2 φ̇21 + φ̇ . (19) The adiabatic field perturbation δr is the component of the two-field perturbation along the direction of the background fields’ evolution while the entropy perturbation δs represents fluctuations orthogonal to the background classical trajectory. The adiabatic field perturba- tion leads to a perturbation in the comoving curavture perturbation: , (20) whereas the entropy field perturbations correspond to isocurvature perturbations. Their evolution equations are given by [24] δ̈r + 3Hδ̇r + V,rr − θ̇2 − a3ṙ2 δr = 2θ̇δ̇s+ 2 δs, (21) δ̈s+ 3Hδ̇s+ δs+ (V,ss − θ̇2)δs = −2 ṙδ̇r − , (22) where the angle θ is defined as cos θ = φ̇21 + φ̇ , sin θ = φ̇21 + φ̇ , (23) such that ṙ = (cos θ)φ̇2 + (sin θ)φ̇1, (24) θ̇ = −V,s , (25) V,r = (sin θ)c1V1 exp(−c1φ1) + (cos θ)c2V2 exp(−c2φ2), (26) V,s = (cos θ)c1V1 exp(−c1φ1)− (sin θ)c2V2 exp(−c2φ2), (27) V,rr = −(sin θ)2c21V1 exp(−c1φ1)− (cos θ)2c22V2 exp(−c2φ2), (28) V,ss = −(sin θ)2c22V2 exp(−c2φ2)− (cos θ)2c21V1 exp(−c1φ1). (29) For the multi-field scaling solution, B, we have θ = arctan , (30) and for the single field scaling solutions we have (B1) , θ = 0 (B2) . (31) Thus we have θ =constant for the fixed points and the adiabatic and the entropy fields are decoupled. This allows us to quantise the independent fluctuations in the two fields. For the multi-field scaling solutionB, the spectrum of quantum fluctuations of the entropy field is given on large scales (k ≪ aH) by Pδs ≡ |δs2| = C2ν (−kτ)1−2ν , (32) where τ < 0 is conformal time, and (ǫ− 1)2 , ǫ ≡ −Ḣ/H2 = 1/p, (33) and Cν = 2 ν−3/2Γ(ν)/π3/2 [16]. The spectral tilt is given by ∆nδχ ≃ , (34) to leading order in a fast-roll expansion (ǫ ≫ 1) [11, 12, 13]. In this limit, the spectrum (32) can be written as P1/2δs = ǫ . (35) Note that |H| is rapidly increasing and thus δs is also growing on super-Hubble scales due to the tachyonic instability. This instability is essential in order to realize the scale invariance of the spectrum (35). The amplitude of field perturbations at Hubble exit is of order H and thus we require the super-Hubble perturbations to grow at the same rate in order to maintain a scale-invariant spectrum. Note that in the simplest model, the spectrum is slightly blue [11, 12, 13]. However, any deviations from an exponential potential for adiabatic field can introduce the corrections to the spectral tilt and thus it becomes model dependent [11, 12]. The spectrum of quantum fluctuations in the adiabatic field about the scaling solution has the same power-law form on large scales Pδr = C2µ (−kτ)1−2µ , (36) where to µ ≃ 1/2 to leading order in 1/ǫ. Thus the adiabatic field perturbations become constant in the large scale limit and the spectral tilt is given by ∆nδr ≃ 2 . (37) Thus we have ∝ (−kτ)2 , (38) and hence in what follows we can neglect the adiabatic field fluctuations in the large-scale limit. By contrast, for the single field dominated scaling solutions, the adiabatic and entropy field perturbations are both frozen on super-Hubble scales: δs , δr = const. (39) These perturbations both have a steep blue spectrum if they cross the horizon when the background solutions are described by the single field dominated solution [19]. 4 Generation of curvature perturbations Now let us consider the evolution of perturbations in a situation where the classical solution starts from near the saddle point, B. As emphasized in Ref. [19] this requires an additional preceding mechanism that drives the classical background solution to the unstable saddle point throughout our observable part of the universe. In this paper, we will not discuss the mechanism required to bring the classical solution to the saddle point and we just assume that the classical solution stays near the saddle point for long enough to ensure that a scale- invariant spectrum of isocurvature perturbations is generated over the relevant scales for the observed large scale structure of our Universe. Then the initial conditions for the adiabatic and the entropy field perturbations can be set from the amplitude of quantum fluctuation as described in the previous section: δr = 0, δs = ǫ , (40) on sufficiently large scales and for 1/ǫ ≪ 1. Unless the spatially homogeneous background solution is located exactly at the fixed point, the tachyonic instability drives the background solution away from the multi-field scaling solution, B, to one of the single field dominated solutions, B1 or B2, depending on the initial conditions. During the transition θ is not constant and the adiabatic and entropy field perturbations mix, so it is possible to generate perturbations in the adiabatic field, and hence comoving curvature perturbations (20), from initial fluctuations in the entropy field. We can solve the evolution equations (21) and (22) numerically for any given classical background. Figure 2 shows the behaviour of δr and δs. Due to the coupling between δr and δs during the transition, curvature perturbations are generated during the transition. On the other hand, δs shows a tachyonic instability according to Eq. (40) close to the scaling solution, but when the background solution goes to Bi, the entropy field perturbation becomes constant. The final amplitude of δr depends on when the transition from B to Bi occurs, but, interestingly, the final ratio between δr and δs does not depend on the details of the transition. We find that the ratio is determined solely by the parameters c1 and c2 as , at B1, (41) = −c2 , at B2, (42) as is shown in Fig. 2. We will explain later why such a simple result is found. 0.01 0.02 0.03 0.04 0.05 -0.002 0.002 0.004 0.006 0.008 0.01 0.02 0.03 0.04 0.05 Figure 2: Left: Solutions for δr(N), using the same parameters as in Figure 1. The corre- sponding background solutions are shown in Figure 1. Right: The ratio between δr and δs. The ratio approaches a constant given by Eqs. (41) and (42). In this case 1.3333 for B1 and −0.75 for B2. The resulting curvature perturbation on a comoving hypersurface in the final single-field dominated phase is thus given by δr, at B1, (43) δr, at B2. (44) The equation for the entropy field perturbation (22) can be rewritten as [24] δ̈s+ 3Hδ̇s+ + Vss + 3θ̇ δs = 4 Ψ , (45) where Ψ is the curvature perturbations in Newtonian gauge. The change of the curvature perturbations is determined by [24] Ṙc = θ̇δs. (46) On large scales, we can neglect (k2/a2)Ψ and it is possible to reproduce the previous results from Eq. (46). Although the physical meaning of the instantaneous adiabatic and entropy field perturba- tions is clear, the dynamics of the perturbations during the transition are rather complicated in this basis. We find it is much easier to work in terms of new variables [19] c2φ1 + c1φ2 c21 + c , χ = c1φ1 − c2φ2 c21 + c , (47) corresponding to a fixed rotation in field space. The potential Eq. (2) can then be simply re-written as [29, 16, 19] V = −U(χ) e−cϕ , (48) where , (49) and the potential for the orthogonal field is given by − U(χ) = −V1 e−(c1/c2)cχ − V2 e(c2/c1)cχ , (50) which has a maximum at χ = χ0 = c21 + c c21V1 c22V2 . (51) The multi-field scaling solution corresponds to χ = χ0, while ϕ is rolling down the exponential potential. The potential for χ has a negative mass-squared around χ = χ0, and thus χ represents the instability direction. If the initial condition for χ is slightly different from χ0 or χ̇ is not zero, then χ starts rolling down the potential and the solution approaches a single-field dominated solution. Note that perturbations δϕ and δχ coincide with the instantaeous adiabatic and entropy field perturbations respectively, defined in Eq. (19), at the scaling solution, B. Thus we can use the initial perturbations (40) due to vacuum fluctuations about the scaling solution previously calculated. However as we follow the evolution away from this saddle point we can no longer identify ϕ and χ with the adiabatic and entropy perturbations. Nevertheless, the dynamics of perturbations turns out to be much simpler using these fields. In terms of ϕ and χ the equations for perturbations are given by δ̈ϕ+ 3H ˙δϕ+ δϕ+Mϕϕδϕ+Mϕχδχ = 0, (52) δ̈χ+ 3H ˙δχ+ δχ+Mχχδχ+Mϕχδϕ = 0, (53) where Mϕϕ = V,ϕϕ − , (54) Mϕχ = V,ϕχ − , (55) Mχχ = V,χχ − . (56) A key observation is that the phase space trajectory of the background fields during the transition, Eq. (17), can be re-written as = c. (57) Thus even away from the multi-field scaling solution, ϕ obeys a scaling relation. We can then show that two of the effective mass terms become Mϕϕ = V,ϕϕ + cV,ϕ = 0 , (58) Mϕχ = V,ϕχ + cV,χ = 0 , (59) independently of the form of U(χ) since V ∝ exp(−cϕ). Thus on large scales δϕ is constant. As we take δϕ = 0 as our initial condition, this remains so even during the transition and in the final single field dominated phase. Thus from Eq. (47) we have a relation between δφ1 and δφ2, c2δφ1 + c1δφ2 = 0, (60) and δφ1 and δφ2 are given by δφ1 = c21 + c δχ, (61) δφ2 = − c21 + c δχ. (62) In the single field dominated solutions, ϕ is no longer the adiabatic field. The adiabatic field is simply φ1 (or φ2) in B1 (or B2) and the entropy field is −φ2 (or φ1) in B1 (or B2). Thus at the late-time attractor we have δr = δφ1, δs = −δφ2, at B1, (63) δr = δφ2, δs = δφ1, at B2. (64) Thus the ratio between δr and δs is determined by the ratio between δφ1 and δφ2 and we can easily find the ratio δr/δs as given in Eqs.(41) and (42). The amplitude of the curvature perturbations can be estimated from δχ. In the initial stage, δχ coincides with the entropy field perturbations δs, and thus its initial amplitude on super-Hubble scales is given by . (65) where we used ǫ = c2/2 in Eq. (40) and c2 is given by Eq. (49). After the transition, δχ becomes massless and the amplitude becomes frozen on large scales. In the single field dominated solutions, the comoving curvature perturbation Rc is given by |Rc| = c21 + c δχ. (66) Assuming the transition occurs suddenly, the final amplitude of the comoving curvature perturbation is given by |Rc| = c21 + c , (67) where the subscript T means that the quantity is evaluated at the transition time. On the other hand the amplitude of the entropy field perturbation is given by c21 + c , at B1, (68) c21 + c , at B2. (69) From the numerical solutions for δr and δs, we can reconstruct HT . We confirm that HT constructed in this way agrees with the Hubble scale at the transition as is shown in Figure 3. 5 Conclusion In this paper we have studied the generation of curvature perturbations during an ekpyrotic collapse with multiple fields. We must assume that the classical background solution starts from a state very close to a saddle point in the phase space that corresponds to the multi- field ekpyrotic scaling solution. If the solution stays at this fixed point for long enough, a scale-invariant spectrum of isocurvature perturbations is generated over the range of scales that is relevant for large scale structure in the Universe. Even if the background solution deviates only slightly from the multi-field scaling solution initially, the tachyonic instability eventually drives the solution to the old ekpyrotic collapse dominated by a single field. During this transition, the initial isocurvature field perturbations generate a scale-invariant spectrum of comoving curvature perturbations. First we studied the perturbations by decomposing them into the instantaneous adiabatic and the entropy field perturbations. These fields are decoupled at the fixed points. The adia- batic field perturbations are effectively massless around both the multi-field scaling solutions 0.01 0.02 0.03 0.04 0.05 0.00025 0.0005 0.00075 0.001 0.00125 0.0015 0.00175 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Log|H| Log|H |T Figure 3: Left: Solutions for δχ(N). We used the same parameters as Figures 1 and 2. Right: Solutions for log |H| with three different background solutions. We also show log |HT | that is determined from the numerical solutions for δχ. and the single field dominated solution, so become constant on large scales. On the other hand, the entropy field perturbations have a tachyonic mass around the multi-field scaling solution, where they grow like δs ∝ H on large scales, but they are effectively massless around the single field dominated solution. We set initial conditions for these perturbations from quantum fluctuations about the multi-field scaling solution. As the adiabatic field perturbations have a blue spectrum they can be neglected compared with the entropy field perturbations on large scales. However, during the transition to the single-field attractor, adiabatic field perturbations are generated from the entropy field perturbations. It turns out to be more convenient to use new fields ϕ and χ defined in Eqs. (47) to follow the perturbations through the transition. These fields coincide with the adiabatic and entropy fields around the multi-field scaling solution but not during the transition or at the final single field dominated phase. In terms of ϕ and χ, the potential is given by a product of the potential for χ, U(χ), and an exponential potential for ϕ. U(χ) has an extremum which corresponds to the multi-field scaling solution. A crucial observation is that even during the transition and in the final phase, ϕ satisfies the scaling relation Eq. (57). We can then show that the field perturbations for ϕ and χ are always decoupled. Since δϕ = 0 on large scales during the ekpyrotic scaling solution, this remains so. This determines the ratio between the adiabatic and entropy field perturbations at the final phase. On the other hand, δχ grows during the scaling solution and then becomes constant during the single field dominated solution. Our final results are Eqs. (67), (68) and (69) for the amplitude of comoving curvature and isocurvature field perturbations during the single field ekpyrotic collapse phase. The amplitude of the comoving curvature perturbation is determined by the Hubble scale at the transition. We still need to convert the ekpyrotic collapse to expansion (see, for instance, [12, 13]) and see how this curvature perturbation is matched to that in an expanding universe. For a regular bounce the comoving curvature perturbation is conserved for adiabatic pertur- bations and thus Eq. (67) is directly related to the amplitude of the observed primordial density perturbation. If the radiation and matter content in today’s universe comes solely from the single field that dominates the final ekpyrotic phase, we will have no isocurvaure perturbations in an expanding universe. It is interesting to compare this multi-field model with a single field model that gives a scale-invariant spectrum of comoving curvature perturbations during collapse [14, 26, 27]. The single-field model requires the correct exponent for a relatively flat and positive expo- nential potential in order to obtain a ∝ |t|2/3, whereas the ekyprotic model only requires sufficiently steep, negative exponential potentials to obtain a ∝ |t|1/ǫ with ǫ ≫ 1. On the other hand both models require fine-tuned initial conditions as it is the existence of an insta- bility that gives rise to the scale-invariant perturbation spectrum during collapse. In both models the amplitude of tensor perturbations is determined by the Hubble scale when the perturbations leave the horizon as tensor perturbations are then frozen on super-horizon scales. In the single field model the tensor metric perturbations thus acquire the same al- most scale-invariant spectrum [28] as the comoving curvature perturbation, with a similar amplitude, giving rise to a dangerously large tensor-scalar ratio which severely constrains the model [27]. But in the ekpyrotic scaling solution the tensor perturbations have a steep blue spectrum and are completely negligible at scales relevant for the cosmic microwave anisotropies. In summary, a simple ekpyrotic model with two steep, negative exponential potentials is capable of generating a scale-invariant spectrum of comoving curvature perturbations. A key ingredient is the instability of the multi-field scaling solution [16, 11, 12, 19]. This insta- bility generates a scale-invariant spectrum of isocurvature field perturbations from vacuum fluctuations about the scaling solution, and converts them to a scale-invariant spectrum of comoving curvature perturbations. We should emphasize that this conversion occurs auto- matically due to the dynamics of the fields in our simple model and does not require any change in the shape of the potential or any additional dynamics [11, 12, 13]. Thus ekpyrotic collapse with multiple fields can generate a scale-invariant spectrum for curvature perturbations in several different ways (see also [30] for a different idea). In all these models, we need some preceding phase that initially drives the classical background solution to the unstable multi-field scaling solution throughout our observable region of space. This problem cannot be solved within the simplest model with multiple exponential potentials that we considered in this paper and we would need to appeal to a more ambitious framework for the model such as the cyclic scenario [31, 32] to address this problem. Acknowledgments We would like to thank J-L. Lehners and P.J. Steinhardt for discussions. KK and DW are supported by STFC. SM is grateful to the ICG, Portsmouth, for their hospitality when this work was initiated. SM is supported in part by the Japan Society for Promotion of Science (JSPS) Research Fellowship. References [1] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, Phys. Rev. D 64, 123522 (2001) [arXiv:hep-th/0103239]. [2] R. Kallosh, L. Kofman and A. D. Linde, Phys. Rev. D 64, 123523 (2001) [arXiv:hep-th/0104073]. [3] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, arXiv:hep-th/0105212. [4] D. H. Lyth, Phys. Lett. B 524, 1 (2002) [arXiv:hep-ph/0106153]. [5] R. Brandenberger and F. Finelli, JHEP 0111, 056 (2001) [arXiv:hep-th/0109004]. [6] S. Tsujikawa, R. Brandenberger and F. Finelli, Phys. Rev. D 66, 083513 (2002) [arXiv:hep-th/0207228]. [7] C. Cartier, R. Durrer and E. J. Copeland, Phys. Rev. D 67, 103517 (2003) [arXiv:hep-th/0301198]. [8] E. J. Copeland and D. Wands, arXiv:hep-th/0609183. [9] J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Steinhardt and N. Turok, Phys. Rev. D 65, 086007 (2002) [arXiv:hep-th/0108187]. [10] A. J. Tolley, N. Turok and P. J. Steinhardt, Phys. Rev. D 69, 106005 (2004) [arXiv:hep-th/0306109]. [11] J. L. Lehners, P. McFadden, N. Turok and P. J. Steinhardt, arXiv:hep-th/0702153. [12] E. I. Buchbinder, J. Khoury and B. A. Ovrut, arXiv:hep-th/0702154. [13] P. Creminelli and L. Senatore, arXiv:hep-th/0702165. [14] D. Wands, Phys. Rev. D 60, 023507 (1999) [arXiv:gr-qc/9809062]. [15] A. Notari and A. Riotto, Nucl. Phys. B 644, 371 (2002) [arXiv:hep-th/0205019]. [16] F. Finelli, Phys. Lett. B 545, 1 (2002) [arXiv:hep-th/0206112]. [17] Z. K. Guo, Y. S. Piao and Y. Z. Zhang, Phys. Lett. B 568, 1 (2003) [arXiv:hep-th/0304048]. [18] E. J. Copeland, R. Easther and D. Wands, Phys. Rev. D 56, 874 (1997) [arXiv:hep-th/9701082]. [19] K. Koyama and D. Wands, JCAP 0704, 008 (2007) [arXiv:hep-th/0703040]. [20] E. J. Copeland, A. R. Liddle and D. Wands, Phys. Rev. D 57, 4686 (1998) [arXiv:gr-qc/9711068]. http://arxiv.org/abs/hep-th/0103239 http://arxiv.org/abs/hep-th/0104073 http://arxiv.org/abs/hep-th/0105212 http://arxiv.org/abs/hep-ph/0106153 http://arxiv.org/abs/hep-th/0109004 http://arxiv.org/abs/hep-th/0207228 http://arxiv.org/abs/hep-th/0301198 http://arxiv.org/abs/hep-th/0609183 http://arxiv.org/abs/hep-th/0108187 http://arxiv.org/abs/hep-th/0306109 http://arxiv.org/abs/hep-th/0702153 http://arxiv.org/abs/hep-th/0702154 http://arxiv.org/abs/hep-th/0702165 http://arxiv.org/abs/gr-qc/9809062 http://arxiv.org/abs/hep-th/0205019 http://arxiv.org/abs/hep-th/0206112 http://arxiv.org/abs/hep-th/0304048 http://arxiv.org/abs/hep-th/9701082 http://arxiv.org/abs/hep-th/0703040 http://arxiv.org/abs/gr-qc/9711068 [21] I. P. C. Heard and D. Wands, Class. Quant. Grav. 19, 5435 (2002) [arXiv:gr-qc/0206085]. [22] M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 95, 71 (1996) [arXiv:astro-ph/9507001]. [23] A. Taruya and Y. Nambu, Phys. Lett. B 428, 37 (1998) [arXiv:gr-qc/9709035]. [24] C. Gordon, D. Wands, B. A. Bassett and R. Maartens, Phys. Rev. D 63, 023506 (2001) [arXiv:astro-ph/0009131]. [25] J. c. Hwang and H. Noh, Phys. Lett. B 495, 277 (2000) [arXiv:astro-ph/0009268]. [26] F. Finelli and R. Brandenberger, Phys. Rev. D 65, 103522 (2002) [arXiv:hep-th/0112249]. [27] L. E. Allen and D. Wands, Phys. Rev. D 70, 063515 (2004) [arXiv:astro-ph/0404441]. [28] A. A. Starobinsky, JETP Lett. 30, 682 (1979) [Pisma Zh. Eksp. Teor. Fiz. 30, 719 (1979)]. [29] K. A. Malik and D. Wands, Phys. Rev. D 59, 123501 (1999) [arXiv:astro-ph/9812204]. [30] A. J. Tolley and D. H. Wesley, arXiv:hep-th/0703101. [31] P. J. Steinhardt and N. Turok, Phys. Rev. D 65, 126003 (2002) [arXiv:hep-th/0111098]. [32] P. J. Steinhardt and N. Turok, Science 296 (2002) 1436. http://arxiv.org/abs/gr-qc/0206085 http://arxiv.org/abs/astro-ph/9507001 http://arxiv.org/abs/gr-qc/9709035 http://arxiv.org/abs/astro-ph/0009131 http://arxiv.org/abs/astro-ph/0009268 http://arxiv.org/abs/hep-th/0112249 http://arxiv.org/abs/astro-ph/0404441 http://arxiv.org/abs/astro-ph/9812204 http://arxiv.org/abs/hep-th/0703101 http://arxiv.org/abs/hep-th/0111098 Introduction Homogeneous field dynamics Generation of quantum fluctuations Generation of curvature perturbations Conclusion

0704.1153

Asymmetric Di-jet Production in Polarized Hadronic Collisions

BNL-NT-07/15 RBRC-665 Asymmetric Di-jet Production in Polarized Hadronic Collisions Jian-Wei Qiu,1, 2 Werner Vogelsang,2 and Feng Yuan3 1Department of Physics and Astronomy, Iowa State University, Ames, IA 50011 2Physics Department, Brookhaven National Laboratory, Upton, NY 11973 3RIKEN BNL Research Center, Building 510A, Brookhaven National Laboratory, Upton, NY 11973 (Dated: October 25, 2018) Abstract Using the collinear QCD factorization approach, we study the single-transverse-spin dependent cross section ∆σ(S⊥) for the hadronic production of two jets of momenta P1 = P + q/2 and P2 = −P + q/2. We consider the kinematic region where the transverse components of the momentum vectors satisfy P⊥ ≫ q⊥ ≫ ΛQCD. For the case of initial-state gluon radiation, we show that at the leading power in q⊥/P⊥ and at the lowest non-trivial perturbative order, the dependence of ∆σ(S⊥) on q⊥ decouples from that on P⊥, so that the cross section can be factorized into a hard part that is a function only of the single scale P⊥, and into perturbatively generated transverse-momentum dependent (TMD) parton distributions with transverse momenta k⊥ = O(q⊥). PACS numbers: 12.38.Bx, 13.88.+e, 12.39.St Keywords: QCD factorization, single transverse-spin asymmetry, di-jet correlation http://arxiv.org/abs/0704.1153v1 1. Introduction. Single-transverse-spin asymmetries (SSAs) in high-energy hadronic reactions with one transversely polarized hadron were first observed more than three decades ago [1]. The SSA is defined as AN ≡ (σ(S⊥)− σ(−S⊥))/(σ(S⊥) + σ(−S⊥)), the ratio of the difference and the sum of (differential) cross sections when the hadron’s spin vector, S⊥, is flipped. Recent experimental measurements of SSAs both in polarized hadronic collisions [2, 3] and in semi-inclusive lepton-nucleon deep inelastic scattering (SIDIS) [4] have renewed the interest in investigating the origin of SSAs in Quantum Chromodynamics (QCD) [5]. It is believed that some SSAs are a consequence of the partons’ transverse motion inside the polarized hadron. The momentum scale of this transverse motion is a typical hadronic scale, 〈k⊥〉 ∼ 1/fm ∼ ΛQCD. For observables with only one hard scale Q≫ ΛQCD, the SSA should be proportional to 〈k⊥〉/Q [6, 7]. Such observables only probe an averaged effect of the partons’ transverse motion. However, for observables characterized by more than one physical scale, SSAs may directly probe the partons’ transverse motion. For example, in the case of Drell-Yan hadronic production of a lepton pair of large invariant mass Q and transverse momentum q⊥ ≪ Q, the pair probes the (anti-) quark’s transverse motion at the scale q⊥, while the invariant mass Q of the pair sets the hard scale of the collision [8]. In this letter, we study the SSA in hadronic production of two jets: A(PA, S⊥) + B(PB) → J1(P1) + J2(P2) +X , with the jet momenta P1 ≡ P + q/2 and P2 ≡ −P + q/2 [9, 10, 11]. Unlike in the Drell-Yan process or in SIDIS, the SSA in di-jet production can be generated by both initial- and final-state interactions. We emphasize that measurements of the SSA for di-jet production have begun at RHIC [12], complementing the measurements in SIDIS. We are interested here in deriving a QCD formalism for the single transverse-spin de- pendent cross section, ∆σ(S⊥) = (σ(S⊥)− σ(−S⊥))/2, that is valid in the kinematic region P⊥ ≫ q⊥ & ΛQCD, where P⊥ and q⊥ are the transverse components of the momenta P and q, respectively, so that the SSA provides direct information on the partons’ transverse motion. We first consider the region P⊥ ≫ q⊥ ≫ ΛQCD, where both observed momentum scales are much larger than the typical hadronic scale ΛQCD. We calculate ∆σ(S⊥) in terms of the collinear QCD factorization approach, which is expected to be valid in this region [13]. In this approach, the incoming partons are approximated to be collinear to the corresponding initial hadrons, and the leading-order partonic processes produce two back-to-back jets with zero momentum imbalance. The di-jet momentum imbalance, ~q⊥ = ~P1⊥ + ~P2⊥, has to be perturbatively generated by radiating an additional hard parton. PA, S⊥ PA, S⊥ PA, S⊥ kg kg kg (a) (b) (c) FIG. 1: Sample diagrams for quark-quark scattering contributing to ∆σ(S⊥) through an initial-state interaction (a), and through final-state interactions with jet P1 (b) and jet P2 (c). In this letter, we concentrate on the physics issues related to di-jet production, and re- strict ourselves to the case that the leading contribution in the expansion of the partonic scattering in q⊥/P⊥ involves a hard qq ′ → qq′ subprocess. We consider the jet imbalance q⊥ to be generated by gluon radiation off the initial quarks. We derive the corresponding leading-order contribution to ∆σ(S⊥), and demonstrate that the perturbatively calculated partonic parts can be further factorized into a single-scale (P⊥) hard part and perturbatively generated transverse-momentum dependent (TMD) parton distributions with transverse mo- menta k⊥ = O(q⊥). We find that the complete contributions from all other partonic sub- processes at the leading order have the same factorization property. These will be discussed in a forthcoming publication [14]. The factorization of the physics at scale P⊥ from that at scale q⊥ that we find when q⊥ ≪ P⊥, is consistent with a more general TMD factorization formula for the SSA in the di-jet momentum imbalance. 2. Single transverse-spin dependent cross section. When both P⊥ and q⊥ are much larger than ΛQCD, a nonvanishing single transverse-spin dependent cross section ∆σ(S⊥) is generated by the Efremov-Teryaev-Qiu-Sterman (ETQS) mechanism [6, 7] in the collinear factorization approach. The calculation of ∆σ(S⊥) then requires to evaluate partonic pro- cesses with 3-parton initial- and final-states [13]. In Fig. 1 we show generic diagrams for the quark-quark scattering channel that contribute to ∆σ(S⊥) through initial- and final- state interactions with the gluon of momentum kg, which is needed for generating the phase required for a nonvanishing SSA [6, 7]. Radiation of a hard gluon of momentum k′ into the final state generates the jet imbalance q⊥. The blob in the center represents tree-level Feynman diagrams with the given initial- and final-state partons. In the ETQS formalism, the contribution of the subprocess (g)qq′ → qq′g to ∆σ(S⊥), shown in Fig. 1, is generically given by d∆σ(S⊥)(qq′) dy1dy2dP dx1dx2 TF (x1, x2) q ′(x′) 16s(2π)4 δ((k′)2)H(g)qq′→qq′g , (1) where y1 and y2 are the rapidities of the two jets, s = (PA + PB) 2, and H represents a partonic hard part. q′(x′) is the usual quark distribution at momentum fraction x′ in the incoming hadron B. x1 and x2 are the momentum fractions of the quarks from the polarized hadron A on the two sides of the cut shown in Fig. 1, and TF (x1, x2) is the corresponding twist-three quark-gluon correlation function, extracted from the lower blob in the figure [7, 15]: TF (x1, x2) ≡ dζ−dη− ei(x1P η−+(x2−x1)P+A ζ × ǫβα⊥ S⊥β PA, S|ψ(0)L(0, ζ−)γ+ (2) × gFα+(ζ−)L(ζ−, η−)ψ(η−)|PA, S where L is the proper gauge link to make the matrix element gauge invariant, and where the sums over color and spin indices are implicit. In Eq. (1) and the rest of this paper, the dependence on factorization and renormalization scales is suppressed. Equation (1) applies when q⊥ ∼ P⊥. Our goal is now to investigate the leading structure that emerges from Eq. (1) when q⊥ ≪ P⊥. In this limit the gluon of momentum k′ is radiated either nearly collinearly from one of the external quark legs and/or is soft. In the present work, we only discuss collinear emission by one of the initial quarks. This radiation is the most interesting from the point of view of studying the factorization properties of the cross section at small q⊥, because TMD factorization can only hold if the initial-state collinear radiation leads to a certain specific structure, as we shall discuss below. Since we are considering the production of jets (as opposed to that of two specific hadrons), collinear radiation from final-state quarks becomes part of the jet and will not produce leading behavior in q⊥/P⊥. On the other hand, large-angle soft gluons produced by the interference of initial- and final state radiation may give leading contributions, through a so-called soft factor. We leave the detailed study of the soft factor to future work, but will briefly return to it later. As we mentioned above, the strong interaction phase necessary for a nonvanishing ∆σ(S⊥) arises from the interference between the imaginary part of the partonic scattering amplitude (4) (5) (6) (1) (2) (3) FIG. 2: Sample diagrams for the soft-pole (1-3) and hard-pole (4-6) contributions. We have indi- cated the momentum k′ of the radiated gluon that produces the jet imbalance q⊥, and the momentum kg of the additional gluon from the polarized proton. The pole of each diagram is taken from the propagator with a short bar. Diagram (1) is an example of an initial-state interaction, and (2) and (3) show final-state interactions. Diagrams (4-6) show the complete set of diagrams for one hard pole, indicated again by the short bar. with the extra polarized gluon of momentum kg = xgPA and the real scattering amplitude without the gluon in Fig. 1. The imaginary part comes from taking the pole of the parton propagator associated with the integration over the gluon momentum fraction xg = x2−x1. For a process with two physical scales, P⊥ and q⊥, tree scattering diagrams in Fig. 1 have two types of poles, corresponding to xg = 0 (“soft-pole”) [7] and xg 6= 0 (“hard-pole”) [8]. With the extra initial-state gluon attachment of momentum k′, there are many more diagrams that contribute to ∆σ(S⊥) in comparison to the spin-averaged cross section [14]. In Fig. 2, we show some sample diagrams that give the leading soft-pole (1-3) and hard pole (4-6) contributions to ∆σ(S⊥) at q⊥ ≪ P⊥. By using the power counting technique [16], we are able to classify all Feynman diagrams into different groups [14]. For example, diagrams (1-3) only make leading contributions when the momentum k′ of the radiated gluon is parallel to PA, whereas diagrams (4-6) can give a leading contribution when k ′ is either parallel to PA or to PB. Beyond that, the calculation of the soft-pole and hard-pole contributions to ∆σ(S⊥) follows the same procedure as introduced in Ref. [8] for the SSA in Drell-Yan and SIDIS, because the kinematic limit considered is similar. In order to extract the contributions to ∆σ(S⊥) in Eq. (1), we need to convert the extra gluon field operator in the hadronic matrix element of the polarized hadron in Fig. 1 to a field strength operator in the definition of TF (x1, x2) in Eq. (2). Working in Feynman gauge, we first give the initial-state collinear partons from the polarized hadron a small transverse momentum, ki = xiPA + ki⊥ with i = 1, 2, and then expand the calculated partonic scattering amplitudes around ki⊥ = 0, or equivalently, kg⊥ = k2⊥ − k1⊥ = 0. The contribution to ∆σ(S⊥) arises from terms linear in kg⊥. After summing up all contributions [14], we obtain the total leading-power contribution to ∆σ(S⊥) from the (g)qq ′ → qq′g partonic subprocess in the q⊥/P⊥ expansion: d∆σ(S⊥)(qq′) dy1dy2dP P⊥≫q⊥≫ΛQCD ǫαβSα⊥q (~q2⊥) HSiversqq′→qq′ q′(xb) A+ TF (xa, xa) , (3) where TF (x, x) 1 + ξ2 +TF (x, x) 2ξ3 − 3ξ2 − 1 +TF (x, ξx) 1 + ξ B = q′(x′)CF 1 + ξ′2 1− ξ′ , (5) where ξ = xa/x and ξ ′ = xb/x ′ with xa = (ey1 + ey2) and xb = (e−y1 + e−y2). The above relations are regularized at the integration limits by “plus”-distributions [8]. The single-scale partonic hard part is given by HSiversqq′→qq′(ŝ, t̂, û) = N2c − 5 2(ŝ2 + û2) , (6) where the use of the superscript “Sivers” will become clear in the next section, and where the partonic Mandelstam variables are given as ŝ = xaxbs, t̂ = −P 2⊥(ey2−y1 + 1), and û = −P 2⊥(ey1−y2 + 1). We note that to the leading power in q⊥/P⊥, we have P1 = P⊥ (e 2, e−y1/ 2, 1) and P2 = P⊥ (e 2, e−y2/ 2,−1) for the jet momenta in light-cone coordinates. The hard part for the partonic channel we have considered, HSiversqq′→qq′, is very similar to the spin-averaged partonic differential cross section dσ̂/dt̂ [14]. The only difference is the color factor in square brackets. In fact, the color factor for HSiversqq′→qq′ is equal to a sum of three color factors: CI + CF1 + CF2, corresponding to color factors of scattering amplitudes when the initial-state gluon of momentum kg is attached to the initial-state incoming quark of momentum kb, the final-state quark with P1, or the final-state quark with P2, respectively. For the (g)qq′ → qq′g subprocess, we have [14] CI = − 12N2 , CF1 = − 14N2 , and CF2 = N2c−2 As a result, the final-state interaction with the jet of momentum P2 dominates, and the overall color factor CI + CF1 + CF2 has a sign opposite to that of CI alone. 3. Factorization in terms of TMD distributions. When q⊥ ≪ P⊥, the di-jet production and Drell-Yan process at low transverse momentum (qT ≪ Q) considered in Ref. [8] share very similar kinematics. The difference is that the Drell-Yan process has only initial-state interactions while for di-jet production both initial- and final-state interactions are present. It is known [17, 18] that when q⊥ ≪ Q, the Drell-Yan cross section in leading order in q⊥/Q can be calculated from a generalized QCD factorization formula involving the TMD parton distributions. It is natural to ask if the generalized QCD factorization formula can be extended to the di-jet cross section for q⊥ ≪ P⊥. When q⊥ ≫ ΛQCD, TMD parton distributions can be calculated in perturbative QCD (pQCD) from hard radiation and parton splitting [8]. The unpolarized TMD quark distri- bution is well known and given by [8] q(xb, q⊥) = [B + · · · ] , (7) where B is given in Eq. (5), and where the ellipses denote terms proportional to δ(1−ξ) and terms generated by gluon splitting. The transverse-spin dependent TMD quark distribution, qT (xa, q⊥), known as the Sivers function [19], can also be calculated in pQCD and is given in terms of the twist-3 quark-gluon correlation function TF as [8]: qSIDIST (xa, q⊥) = − (~q2⊥) [A+ . . . ] , (8) where A is given in Eq. (4), MP is a hadron mass scale introduced to keep q(xb, q⊥) and qT (xa, q⊥) at the same dimension, and where the ellipses as above denote terms proportional to δ(1 − ξ) and terms not relevant to the following discussion. In Eq. (8), the superscript “SIDIS” indicates the Sivers function for the SIDIS process, which has an opposite sign from that for the Drell-Yan process given in Ref. [8], due to the difference in the directions of the gauge link that defines the TMD quark distributions [20, 21, 22, 23]. One of the important features of the di-jet cross section calculated above to leading order in q⊥/P⊥ is the separation of two observed physical scales, P⊥ and q⊥. We can use Eqs. (7) and (8) to rewrite the single transverse-spin dependent di-jet cross section in Eq. (3) in terms of a single-scale hard factor, HSivers, which is a function of P⊥, and of the q⊥-dependent perturbatively generated TMD parton distributions: d∆σ(S⊥)(qq′) dy1dy2dP P⊥≫q⊥≫ΛQCD = ǫαβSα⊥q Sivers qq′→qq′ ′(xb)) SIDIS T (xa, q⊥) (xb q ′(xb, q⊥)) (xa TF (xa, xa)) Using the leading order relation [23] d2k⊥ ~k SIDIS T (x, k⊥) = −TF (x, x) , (10) we find that our result in Eq. (9), when the contributions from all other partonic subprocesses at the same order [14] are added, is consistent with the leading-order term of a more general factorization formula in terms of TMD parton distributions: d∆σ(S⊥) dy1dy2dP ǫαβSα⊥q d2k1⊥d 2k2⊥d ~k1⊥ · ~q⊥ SIDIS Ta (xa, k1⊥) xb f SIDIS b (xb, k2⊥) (11) Sab→cd(λ⊥)H Sivers ab→cd(P δ(2)(~k1⊥ + ~k2⊥ + ~λ⊥ − ~q⊥) , where apart from the functions already given, fSIDISb denotes the unpolarized TMD quark distribution and Sab→cd is the soft factor mentioned above [8, 14]. Because of the color flow into the jets, the product of the soft and hard factors will involve a sum over separate color amplitudes in the full factorization formalism [24, 25], which has been represented by a trace [ ]c in color space in the above equation. Our calculation of initial-state collinear gluon radiation described above would not be sensitive to this complexity of the color flow, but we emphasize that the definition of the parton distributions cannot be affected by it [14, 25]. In Eq. (11), we have chosen TMD parton distributions defined in SIDIS because of the dominance of final-state interactions. Choosing TMD parton distributions defined according to the Drell-Yan process would change the sign of the partonic hard factors, but not affect the overall sign of the physical cross section. Based on our explicit calculation here and the generalized factorization property of the Drell-Yan process at low q⊥ [17, 18], and because of the similarity in kinematics between two processes, we expect the generalized factorization formula in Eq. (11) to be valid for describing the single-transverse-spin dependent cross section for the di-jet momentum imbalance in hadronic collisions in the kinematic region where P⊥ ≫ q⊥ & ΛQCD. A key feature of the factorization is that the perturbatively calculated short-distance hard factors should not be sensitive to details of the factorized long distance physics. We tried, as a test, to derive all short-distance hard factors by using this factorization formula and the Brodsky-Hwang-Schmidt model for SSAs [20]. We were able to recover all hard-scattering factors in this way. We note that the same hard factors HSiversab→cd as above were also found for the weighted (integrated) SSA for di-jet production in [10]. A detailed comparison between the approach of [10] and ours will be presented in Ref. [14]. An all-order proof of the above factorization formula, if correct, is still needed and is beyond the scope of this paper. 4. Summary. We have studied the single-transverse-spin dependent cross section ∆σ(S⊥) for di-jet production momentum imbalance in high-energy hadronic collisions in a kinematic region where P⊥ ≫ q⊥ ≫ ΛQCD, and calculated contributions from both initial- and final-state interactions. At the leading order in q⊥/P⊥, the q⊥ and P⊥ dependences in our calculated results are decoupled and can be factorized into a single-scale hard factor that depends on P⊥, and into perturbatively generated TMD parton distributions. This factor- ization occurs for each partonic channel. Overall, final-state interactions turn out to give the dominant contribution to ∆σ(S⊥) [14]. We therefore expect the SSA in di-jet production to have the same sign as the Sivers asymmetry in SIDIS. We have found that our results are consistent with a more general TMD factorization formula, given in Eq. (11), which we propose to be the correct approach to describing the single-transverse spin asymmetry in di-jet production at hadron colliders when P⊥ ≫ q⊥. We emphasize that obviously the result of our first-order calculation is not able to actually prove this factorization, but should rather be regarded as a “necessary condition” for such a factorization to hold. A full proof remains an important challenge for future work. Also, we recall that we have limited ourselves to the case of collinear initial-state gluon radiation. The effects of large-angle soft-gluon emission have been neglected, even though we have indicated their likely role in Eq. (11). A proof of TMD factorization in this process would naturally incorporate a study of this soft factor. If the proposed factorization formula is valid, the di-jet momentum imbalance at RHIC could be described by the same TMD parton distributions as those used to describe the SIDIS and Drell-Yan processes. Because both initial- and final-state interactions are present, the di-jet momentum imbalance is sensitive to different short distance dynamics, and it will be an excellent process to test QCD factorization and the universality of the TMD parton distributions. In addition it should give valuable information on the partons’ transverse motion in the nucleon. Acknowledgments We thank C.J. Bomhof, S.J. Brodsky, J.C. Collins, X. Ji, A. Metz, P.J. Mulders, and G. Sterman for useful discussions. We are grateful to RIKEN, Brookhaven National Labora- tory and the U.S. Department of Energy (grant number DE-FG02-87ER40371 and contract number DE-AC02-98CH10886) for providing the facilities essential for the completion of this work. J.Q. thanks high energy theory group at Argonne National Laboratory for its hospitality during the writing of this work. [1] G. Bunce et al., Phys. Rev. Lett. 36, 1113 (1976). [2] see, for example: D. L. Adams et al., Phys. Lett. B 261, 201 (1991); Phys. Lett. B 264, 462 (1991); K. Krueger et al., Phys. Lett. B 459, 412 (1999). [3] J. Adams et al., Phys. Rev. Lett. 92, 171801 (2004); S. S. Adler, Phys. Rev. Lett. 95, 202001 (2005); F. Videbaek, AIP Conf. Proc. 792, 993 (2005). [4] A. Airapetian et al., Phys. Rev. Lett. 84, 4047 (2000); ibid., 94, 012002 (2005); V. Y. Alex- akhin et al., Phys. Rev. Lett. 94, 202002 (2005). [5] for reviews, see: M. Anselmino, A. Efremov and E. Leader, Phys. Rept. 261, 1 (1995) [Erratum-ibid. 281, 399 (1997)]; Z. t. Liang and C. Boros, Int. J. Mod. Phys. A 15, 927 (2000); V. Barone, A. Drago and P. G. Ratcliffe, Phys. Rept. 359, 1 (2002). [6] A. V. Efremov and O. V. Teryaev, Sov. J. Nucl. Phys. 36, 140 (1982) [Yad. Fiz. 36, 242 (1982)]; Phys. Lett. B 150, 383 (1985). [7] J. W. Qiu and G. Sterman, Phys. Rev. Lett. 67, 2264 (1991); Nucl. Phys. B 378, 52 (1992); Phys. Rev. D 59, 014004 (1999). [8] X. Ji, J. W. Qiu, W. Vogelsang and F. Yuan, Phys. Rev. Lett. 97, 082002 (2006); Phys. Rev. D 73, 094017 (2006); Phys. Lett. B 638, 178 (2006). [9] D. Boer and W. Vogelsang, Phys. Rev. D 69, 094025 (2004). [10] A. Bacchetta, C. J. Bomhof, P. J. Mulders and F. Pijlman, Phys. Rev. D 72, 034030 (2005); C. J. Bomhof and P. J. Mulders, JHEP 0702, 029 (2007). [11] W. Vogelsang and F. Yuan, Phys. Rev. D 72, 054028 (2005); C. J. Bomhof, P. J. Mulders, W. Vogelsang and F. Yuan, arXiv:hep-ph/0701277. [12] J. Balewski, talk presented at the SPIN 2006 Symposium, Kyoto, Japan, October 2-7, 2006, arXiv:hep-ex/0612036. [13] J. W. Qiu and G. Sterman, AIP Conf. Proc. 223, 249 (1991); Nucl. Phys. B 353, 137 (1991). [14] J. W. Qiu, W. Vogelsang, and F. Yuan, in preparation. [15] C. Kouvaris, J. W. Qiu, W. Vogelsang and F. Yuan, Phys. Rev. D 74, 114013 (2006). [16] J. C. Collins, D. E. Soper and G. Sterman, Adv. Ser. Direct. High Energy Phys. 5, 1 (1988) and references therein. [17] J. C. Collins, D. E. Soper and G. Sterman, Nucl. Phys. B 250, 199 (1985). [18] X. Ji, J. P. Ma and F. Yuan, Phys. Rev. D 71, 034005 (2005); Phys. Lett. B 597, 299 (2004). [19] D. W. Sivers, Phys. Rev. D 41, 83 (1990); Phys. Rev. D 43, 261 (1991). [20] S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B 530, 99 (2002); Nucl. Phys. B 642, 344 (2002). [21] J. C. Collins, Phys. Lett. B 536, 43 (2002). [22] X. Ji and F. Yuan, Phys. Lett. B 543, 66 (2002); A. V. Belitsky, X. Ji and F. Yuan, Nucl. Phys. B 656, 165 (2003). [23] D. Boer, P. J. Mulders and F. Pijlman, Nucl. Phys. B 667, 201 (2003). [24] J. Botts and G. Sterman, Nucl. Phys. B 325, 62 (1989). [25] N. Kidonakis and G. Sterman, Nucl. Phys. B 505, 321 (1997); N. Kidonakis, G. Oderda and G. Sterman, Nucl. Phys. B 531, 365 (1998). http://arxiv.org/abs/hep-ph/0701277 http://arxiv.org/abs/hep-ex/0612036 Acknowledgments References

0704.1154

Information, information processing and gravity

Information, information processing and gravity Stephen D. H. Hsu∗ Institute of Theoretical Science University of Oregon, Eugene, OR 97403 I discuss fundamental limits placed on information and information processing by gravity. Such limits arise because both information and its processing require energy, while gravitational collapse (formation of a horizon or black hole) restricts the amount of energy allowed in a finite region. Specifically, I use a criterion for gravitational collapse called the hoop conjecture. Once the hoop conjecture is assumed a number of results can be obtained directly: the existence of a fundamental uncertainty in spatial distance of order the Planck length, bounds on information (entropy) in a finite region, and a bound on the rate of information processing in a finite region. In the final section I discuss some cosmological issues, related to the total amount of information in the universe, and note that almost all detailed aspects of the late universe are determined by the randomness of quantum outcomes. This paper is based on a talk presented at a 2007 Bellairs Research Institute (McGill University) workshop on black holes and quantum information. I. INTRODUCTION This paper is based on a talk presented at a workshop on black holes and quantum information (Bellairs Re- search Institute of McGill University, Barbados). Most of the participants were quantum information theorists, so I attempted to keep the technical details concerning general relativity or particle physics at a minimum. I tried to summarize, in the most physical and intuitive way, how gravity enforces some surprising constraints on information and information processing. From a practi- cal perspective, due to the feebleness of the gravitational force, all of the limits deduced are incredibly weak. Our technologies are nowhere near saturating them, and they are of much greater interest to theoreticians than exper- imentalists or engineers. Nevertheless, they are funda- mental in nature, and, depending as they do both on quantum mechanics and general relativity, may offer a view into the properties of quantum gravity. In the discussion that follows, gravitational collapse will be our crude but powerful probe of gravitational physics. Complete gravitational collapse leads to the for- mation of black holes and causal horizons. Gravity is a long range force that, as far as we know, cannot be screened. In this respect, it is fundamentally different from gauge forces, such as the strong and electroweak in- teractions, and it is precisely this difference that allows for dramatic phenomena like complete collapse. We use Planck units throughout, in which the speed of light, Planck’s constant and the Planck mass (equiva- lently, Newton’s constant) are unity. In our expressions, any energy or mass is therefore measured in units of 1019 GeV (about 10−5 grams), and any length is measured in units of the Planck length, or about 10−35 meters. ∗Electronic address: [email protected] II. GRAVITATIONAL COLLAPSE Ideally, one would like to know precisely what subset of all possible physical initial data results in gravitational collapse and the formation of a black hole. This is ob- viously a difficult problem and it is currently unsolved. Schoen and Yau [1] proved a celebrated result requiring the existence of a closed trapped surface if the minimum density in a region is sufficiently high. However, this result fails to be useful if the energy of the initial config- uration is distributed in a very nonuniform manner. Note that results for black hole or horizon formation typically require both the assumption of the null or weak energy condition and of cosmic censorship [2]. Under those assumptions a closed trapped surface can be shown to result in a singularity (using the Raychaudhuri equa- tion and assuming the energy conditions hold), and cos- mic censorship requires a horizon to conceal the singu- larity from asymptotic observers. In our analysis we will use the hoop conjecture, due to Kip Thorne [3] as a criterion for gravitational collapse. It states that a system of total energy E, if confined to a sphere of radius R < ηE (η is a coefficient of order one, which we neglect below), must inevitably evolve into a black hole. The condition R < E is readily motivated by the Schwarzschild radius Rs = 2M . This conjecture has been confirmed in astrophysically-motivated numer- ical simulations, and has been placed on even stronger footing by recent results on black hole formation from rel- ativistic particle collisions [4]. These results show that, even in the case when all of the energy E is provided by the kinetic energy of two highly boosted particles, a black hole forms whenever the particles pass within a distance of order E of each other (see Fig. 1). Two par- ticle collisions had seemed the most likely to provide a counterexample to the conjecture, since the considerable kinetic energy of each particle might have allowed escape from gravitational collapse. One can think of the hoop conjecture as requiring that the average energy density of an object of size R be less http://arxiv.org/abs/0704.1154v1 mailto:[email protected] FIG. 1: The hoop conjecture applied to two relativistic par- ticles. than R−2 in order not to collapse to a black hole. Thus, large objects which are not black holes must be less and less dense. For example, a sufficiently large object with only the density of water will eventually form a black hole! III. MINIMAL LENGTH In this section we deduce a fundamental limit on our ability to measure a distance [5, 6, 7, 8]. The results suggest that spacetime may ultimately have a discrete structure. At the end of the section we discuss the impli- cations for quantum information and the ultimate Hilbert space of quantum mechanics. From the hoop conjecture (HC) and the uncertainty principle, we immediately deduce the existence of a min- imum ball of size lP . Consider a particle of energy E which is not already a black hole. Its size r must satisfy r ∼>max [ 1/E , E ] , (1) where λC ∼ 1/E is its Compton wavelength and E arises from the hoop conjecture. Minimization with respect to E results in r of order unity in Planck units, or r ∼ lP [9]. If the particle is a black hole, then its radius grows with mass: r ∼ E ∼ 1/λC . This relationship suggests that an experiment designed (in the absence of gravity) to measure a short distance l << lP will (in the presence of gravity) only be sensitive to distances 1/l. This is the classical counterpart to T-duality in string theory [10]. It is possible that quantum gravitational corrections modify the relation between E and R in the HC. How- ever, if E is much larger than the Planck mass, and R much larger than lP , we expect semiclassical considera- tions to be reliable. (Indeed, in two particle collisions with center of mass energy much larger than the Planck mass the black holes produced are semiclassical.) This means that the existence of a minimum ball of size much smaller than lP does not depend on quantum gravity - the energy required to confine a particle to a region of size much smaller than lP would produce a large, semi- classical black hole. Before proceeding further, we give a concrete model of minimum length that will be useful later. Let the position operator x̂ have discrete eigenvalues {xi}, with the separation between eigenvalues either of order lP or smaller. (For regularly distributed eigenvalues with a constant separation, this would be equivalent to a spa- tial lattice.) We do not mean to imply that nature im- plements minimum length in this particular fashion - most likely, the physical mechanism is more complicated, and may involve, for example, spacetime foam or strings. However, our concrete formulation lends itself to detailed analysis. We show below that this formulation cannot be excluded by any gedanken experiment, which is strong evidence for the existence of a minimum length. Quantization of position does not by itself imply quan- tization of momentum. Conversely, a continuous spec- trum of momentum does not imply a continuous spec- trum of position. In a formulation of quantum mechanics on a regular spatial lattice, with spacing a and size L, the momentum operator has eigenvalues which are spaced by 1/L. In the infinite volume limit the momentum operator can have continuous eigenvalues even if the spatial lattice spacing is kept fixed. This means that the displacement operator x̂(t)− x̂(0) = p̂(0) t does not necessarily have discrete eigenvalues (the right hand side of (2) assumes free evolution; we use the Heisenberg picture throughout). Since the time evolution operator is unitary the eigenvalues of x̂(t) are the same as x̂(0). Importantly though, the spectrum of x̂(0) (or x̂(t)) is completely unrelated to the spectrum of the p̂(0), even though they are related by (2) [11]. Consequently, we stress that a measurement of the displacement is a measurement of the spectrum of p̂(0) (for free evolution) and does not provide information on the spectrum of x̂. A measurement of arbitrarily small displacement (2) does not exclude our model of minimum length. To exclude it, one would have to measure a position eigenvalue x and a nearby eigenvalue x′, with |x− x′| << lP . Many minimum length arguments (involving, e.g., a microscope, scattering experiment or even Wigner’s clock [6]) are obviated by the simple observation of the mini- mum ball. However, the existence of a minimum ball does not by itself preclude the localization of a macroscopic object to very high precision. Hence, one might attempt to measure the spectrum of x̂(0) through a time of flight experiment in which wavepackets of primitive probes are bounced off of well-localised macroscopic objects. Dis- regarding gravitational effects, the discrete spectrum of x̂(0) is in principle obtainable this way. But, detecting the discreteness of x̂(0) requires wavelengths comparable to the eigenvalue spacing. For eigenvalue spacing com- parable or smaller than lP , gravitational effects cannot be ignored, because the process produces minimal balls (black holes) of size lP or larger. This suggests a direct measurement of the position spectrum to accuracy better than lP is not possible. The failure here is due to the use of probes with very short wavelength. A different class of instrument - the interferometer FIG. 2: An interferometer can be sensitive to path length differences much smaller than the wavelength of light used. (Fig. 3) - is capable of measuring distances much smaller than the size of any of its sub-components [12]. An in- terferometer can measure a distance ∆x ∼ λ , (3) where λ = 1/ν is the wavelength of light used, L is the length of each arm, τ the time duration of the measure- ment, and N the number of photons. More precisely, ∆x is the change over the duration of the measurement in the relative path lengths of the two arms of the interfer- ometer. b = τ/L is the number of bounces over which the phase difference builds, so (3) can also be written as , (4) which expresses saturation of the quantum mechanical uncertainty relationship between the phase and number operators of a coherent state. From (3) it appears that ∆x can be made arbitrarily small relative to λ by, e.g., taking the number of bounces to infinity. Were this the case, we would have an exper- iment that, while still using a wavelength λ much larger than lP , could measure a distance less than lP along one direction, albeit at the cost of making the measured ob- ject (e.g., a gravity wave) large in the time direction. This would contradict the existence of a minimum inter- val, though not a minimum ball in spacetime. (Another limit which increases the accuracy of the interferometer is to take the number of photons N to infinity, but this is more directly constrained by gravitational collapse. Ei- ther limit is ultimately bounded by the argument dis- cussed below.) A constraint which prevents an arbitrarily accurate measurement of ∆x by an interferometer arises due to the Standard Quantum Limit (SQL) and gravitational collapse. The SQL [13] is derived from the uncertainty principle (we give the derivation below; it is not specific to interferometers, although see [14]) and requires that , (5) where t is the time over which the measurement occurs and M the mass of the object whose position is mea- sured. In order to push ∆x below lP , we take b and t to be large. But from (5) this requires that M be large as well. In order to avoid gravitational collapse, the size R of our measuring device must also grow such thatR > M . However, by causality R cannot exceed t. Any compo- nent of the device a distance greater than t away cannot affect the measurement, hence we should not consider it part of the device. These considerations can be summa- rized in the inequalities t > R > M . (6) Combined with the SQL (5), they require ∆x > 1 in Planck units, or ∆x > lP . (7) (Again, we neglect factors of order one.) Notice that the considerations leading to (5), (6) and (7) were in no way specific to an interferometer, and hence are device independent. We repeat: no device sub- ject to the SQL, gravity and causality can exclude the quantization of position on distances less than the Planck length. It is important to emphasize that we are deducing a minimum length which is parametrically of order lP , but may be larger or smaller by a numerical factor. This point is relevant to the question of whether an experi- menter might be able to transmit the result of the mea- surement before the formation of a closed trapped sur- face, which prevents the escape of any signal. If we de- crease the minimum length by a numerical factor, the inequality (5) requires M >> R, so we force the experi- menter to work from deep inside an apparatus which has far exceeded the criterion for gravitational collapse (i.e., it is much denser than a black hole of the same size R as the apparatus). For such an apparatus a horizon will already exist before the measurement begins. The ra- dius of the horizon, which is of order M , is very large compared to R, so that no signal can escape. We now give the derivation of the Standard Quantum Limit. Consider the Heisenberg operators for position x̂(t) and momentum p̂(t) and recall the standard inequal- (∆A)2(∆B)2 ≥ − (〈[Â, B̂]〉)2 . (8) Suppose that the position of a free test mass is measured at time t = 0 and again at a later time. The position operator at a later time t is x̂(t) = x̂(0) + p̂(0) . (9) The commutator between the position operators at t = 0 and t is [x̂(0), x̂(t)] = i , (10) so using (8) we have |∆x(0)||∆x(t)| ≥ t . (11) So, at least one of the uncertainties ∆x(0) or ∆x(t) must be larger than of order t/M . As a measurement of the discreteness of x̂(0) requires two position measurements, it is limited by the greater of ∆x(0) or ∆x(t), that is, by t/M . What are the consequences of a minimum length? In a discrete spacetime there need not be any continuous de- grees of freedom, and the number of degrees of freedom in a fixed volume is finite. Further, one can show that dis- cretization of spacetime naturally suggests discretization of Hilbert space itself [15]. Specifically, in a universe with a minimal length (for example, due to quantum gravity), no experiment can exclude the possibility that Hilbert space is discrete. IV. ENTROPY BOUNDS In this section we describe two entropy bounds aris- ing from gravitational collapse. These bounds limit the number of degrees of freedom in a region of size R, or equivalently the amount of information in any system of fixed size. The first bound uses the area–entropy relation for black holes. Black holes radiate [16] and have entropy: S = A/4 [17]. The nature of this entropy is one of the great mysteries of modern physics, especially due to its non-extensive nature: it scales as the area of the black hole (in Planck units), rather than its volume. This pecu- liar property has led to the holographic conjecture [18, 19] proposing that the number of degrees of freedom in any region of our universe grows only as the area of its bound- ary. (See [20] for a review and discussion of covariant gen- eralizations of this idea, and [21] for a general discussion of how area bounds arise in gravitating systems.) The AdS/CFT correspondence [22] is an explicit realization of holography. The entropy of a thermodynamic system is the loga- rithm of the number of the available microstates of the system, subject to some macroscopic constraints such as fixed total energy. In certain string theory black holes, these states have been counted explicitly [23, 24]. Consider a system of size R and total energy E (e.g., the green blob in Fig. 3), which is not a black hole (E < R). Now imagine a spherical shell of energy R−E approaching the system at the speed of light. By causal- ity, the system is unaffected by the shell until the com- bination of the two already satisfy the hoop conjecture. The combined system must evolve into a black hole with entropy A/4. By the second law of thermodynamics, this final entropy is larger than that of the initial sys- tem. Since we made no particular assumptions about the initial system, we deduce that ordinary (non-collapsed) physical systems have entropy less than their surface area FIG. 3: A system collapses to form a black hole. By the second law of thermodynamics, its original entropy is bounded above by that of the black hole. in Planck units. This is quite a counterintuitive result, since in familiar (non-gravitating) systems entropy is typ- ically extensive. The second bound, obtained by ’t Hooft [25], shows that if one excludes states from the Hilbert space whose energies are so large that they would have already caused gravitational collapse, one obtains S = lnN < A3/4, where N is the number of degrees of freedom and A the surface area. To deduce this result, ’t Hooft replaces the system under study with a thermal one. This is justified because, in the large volume limit, the entropy of a sys- tem with constant total energy E (i.e., the logarithm of the phase space volume of a microcanonical ensemble) is given to high accuracy by that of a canonical ensemble whose temperature has been adjusted so that the average energies of the two ensembles are the same. (This is a standard, and central, result in statistical mechanics.) Given a thermal region of radius R and temperature T , we have S ∼ T 3R3 and E ∼ T 4R3. Requiring E < R then implies T ∼ R−1/2 and S < R3/2 ∼ A3/4. We stress that the thermal replacement is just a calculational trick: temperature plays no role in the results, which can also be obtained by direct counting. In [26], it was shown that imposing the condition Tr[ ρH ] < R on a density matrix ρ implies a simi- lar bound SvN < A 3/4 on the von Neumann entropy SvN = Tr ρ ln ρ. For ρ a pure state the result reduces to the previous Hilbert space counting. We note that these bounds are more restrictive than the bound obtained from black hole entropy: S < A/4. One can interpret this discrepancy as a consequence of higher entropy density of gravitational degrees of freedom relative to ordinary matter [27]. The consequences of these bounds are rather striking: they suggest that gravitating systems in d dimensions contain only as much information as analogous, but non- gravitating, systems in d− 1 dimensions. A concrete re- alization of this is the AdS/CFT duality in string theory [22]. V. BOUND ON RATE OF INFORMATION PROCESSING In this section we derive an upper bound on the rate at which a device can process information [28]. We de- fine this rate as the number of logical operations per unit time, denoted as the ops rate R. The operations in ques- tion can be those of either classical or quantum comput- ers. The basis of our result can be stated very simply: information processing requires energy, and general rel- ativity limits the energy density of any object that does not collapse to a black hole. Replacing information pro- cessing by information in the previous sentence leads to holography or black hole entropy bounds, a connection we will explore further below. For related work on funda- mental physical limits to computation, see [29] and [30]. Our result is easily deduced using the Margolus– Levitin (ML) theorem [31] from quantum mechanics, and the hoop conjecture. The Margolus–Levitin theorem states that a quantum system with average energy ǫ requires at least ∆t > ǫ−1 to evolve into an orthogonal (distinguishable) state. It is easy to provide a heuristic justification of this result. For an energy eigenstate of energy E, E−1 is the time required for its phase to change by order one. In a two state system the energy level splitting E is at most of or- der the average energy of the two levels. Then, the usual energy-time uncertainty principle suggests that the sys- tem cannot be made to undergo a controlled quantum jump on timescales much less than E−1, as this would introduce energy larger than the splitting into the sys- Consider a device of size R and volume V ∼ R3, com- prised of n individual components [32] of average energy ǫ. Then, the ML theorem gives an upper bound on the total number of operations per unit time R < nǫ , (12) while the hoop conjecture requires E ∼ nǫ < R. Com- bined, we obtain R < R ∼ V 1/3 . (13) It is interesting to compare this bound to the rate of information processing performed by nature in the evo- lution of physical systems. At first glance, there appears to be a problem since one typically assumes the num- ber of degrees of freedom in a region is proportional to V (is extensive). Then, the amount of information pro- cessing necessary to evolve such a system in time grows much faster than our bound (13) as V increases. Re- call that for n degrees of freedom (for simplicity, qubits), the dimension of Hilbert space H is N = dimH = 2n and the entropy is S = lnN ∼ n. In the extensive case, n ∼ S ∼ V . However, as noted in the previous section, gravity also constrains the maximum information content (entropy S) of a region of space. ’t Hooft [25] showed that if one ex- cludes states from the Hilbert space whose energies are so large that they would have already caused gravitational collapse, one obtains S = lnN < A3/4, where N is the number of degrees of freedom and A the surface area. Given this result we can calculate the maximum rate of information processing necessary to simulate any physical system of volume V which is not a black hole. The rateR FIG. 4: The visible universe is exponentially larger than the initial region from which it evolved. is given by the number of degrees of freedom S ∼ R3/2 times the maximal ML rate T ∼ R−1/2. This yields R ∼ R as in our bound (13). Finally, we note that black holes themselves appear to saturate our bound. If we take the black hole entropy to be S ∼ A ∼ R2, and the typical energy of its modes to be the Hawking temperature TH ∼ R−1, we again obtain R ∼ R. VI. HOW MUCH INFORMATION IN THE UNIVERSE? In this final section we ask how much information is necessary to specify the current state of the universe, and where did it come from? There is convincing observational evidence for the big bang model of cosmology, and specifically for the fact that the universe is and has been expanding. In a radiation-dominated universe, the FRW scale factor grows as R(t) ∼ t1/2, where t is the comoving cosmolog- ical time. From this, it is clear that our universe evolved from a much smaller volume at early times. Indeed, in inflationary cosmology (Fig. 4) the visible universe re- sults from an initial patch which is exponentially smaller than our current horizon volume. The corresponding ra- tio of entropies is similarly gigantic, meaning that there is much more information in the universe today than in the small primordial patch from which it originated. There- fore, the set of possible early universe initial conditions is much, much smaller than the set of possible late time universes. A mapping between all the detailed rearrange- ments or modifications of the universe today and the set of possible initial data is many to one, not one to one [33]. Thus, the richness and variability of the universe we inhabit cannot be attributed to the range of initial con- ditions. The fact that I am typing this on a sunny day, or that our planet has a single moon, or that the books on my office shelves have their current arrangement, was not determined by big bang initial data. How, then, do the richness and variability of our world arise? The answer is quantum randomness – the random- ness inherent in measurements of quantum outcomes. Imagine an ensemble Ψ of n qubits, each prepared in an identical state ψ. Now imagine that each qubit is measured, with a resulting spin up (+) or spin down (−) result. There are 2n possible records, or histories, of this measurement. This is an exponentially large set of outcomes; among them are all possible n-bit strings, including every n-bit work of literature it is possible to write! Although the initial state Ψ contained very little information (essentially, only a single qubit of informa- tion, since each spin is in an identical state), n bits of classical information are required to specify which of the 2n outcomes is observed in a particular universe. For n → ∞ the set of possible records is arbitrarily rich and varied despite the simplicity of initial state Ψ. In the same way, given an initial quantum state Ψ de- scribing the primordial patch of the big bang from which our horizon volume evolved, one must still know the out- comes of a large number of quantum measurements in order to specify the particulars of the universe today. From a many worlds perspective, one must specify all the decoherent outcomes to indicate a particular branch of the wavefunction – a staggering amount of information. Equivalently, from the traditional Copenhagen perspec- tive, each quantum measurement injects a bit (or more) of truly random information into our universe, and this randomness accounts for its variability. The most familiar cosmological quantum randomness comes from fluctuations of the inflaton field, which deter- mine the spectrum of primordial energy density fluctua- tions. It is these density fluctuations that determine the locations of galaxies, stars and planets today. However, from entropic or information theoretic considerations we readily deduce that essentially every detailed aspect of our universe (beyond the fundamental Lagrangian and some general features of our spacetime and its contents) is a consequence of quantum fluctuations! Acknowledgements— The author thanks the partici- pants of the Bellairs workshop, and especially the or- ganizer Patrick Hayden, for a stimulating and pleasant environment. A. Zee provided some important comments on an earlier draft of the manuscript. The author is sup- ported by the Department of Energy under DE-FG02- 96ER40969. [1] R. Schoen and S. T. Yau, Comm. Math. Phys. 90, 575 (1983). [2] See, e.g., R. M. Wald, General Relativity, University of Chicago Press, (1984). [3] K. S. Thorne, Nonspherical gravitational collapse: A short review, in J. R . Klauder, Magic Without Magic, San Francisco 1972, 231–258. [4] D. M. Eardley and S. B. Giddings, Phys. Rev. D 66, 044011 (2002) [gr-qc/0201034]; S. D. H. Hsu, Phys. Lett. B 555, 92 (2003), [hep-ph/0203154]. [5] X. Calmet, M. Graesser and S. D. H. Hsu, Phys. Rev. Lett. 93, 211101 (2004) [arXiv:hep-th/0405033]; Int. J. Mod. Phys. D 14, 2195 (2005) [arXiv:hep-th/0505144]; X. Calmet, arXiv:hep-th/0701073. [6] L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995) [arXiv:gr-qc/9403008]; C. A. Mead, Phys. Rev. 135, B849 (1964); T. Padmanabhan, Class. Quant. Grav. 4, L107 (1987). [7] For earlier work using the quantum and gravitational lim- itations of a clock, see J.C. Baez and S.J. Olson, Class. Quant. Grav. 19, L121 (2002); Y.J. Ng and H. van Dam, Found. Phys. 30, 795 (2000); Phys. Lett. B 477, 429 (2000); G. Amelino-Camelia, Phys. Lett. B 477, 436 (2000) [arXiv:gr-qc/9910023]; R. J. Adler, I. M. Nemen- man, J. M. Overduin and D. I. Santiago, Phys. Lett. B 477, 424 (2000) [arXiv:gr-qc/9909017]. [8] For a discussion of a minimum length in the context on quantum foam see e.g. G. Amelino-Camelia, Nature 398, 216 (1999) [arXiv:gr-qc/9808029]; arXiv:gr-qc/0104005; Y. J. Ng, arXiv:gr-qc/0405078. [9] We focus specifically on spacelike minimum intervals. By Lorentz covariance, we might expect that no spacetime interval ds2 can be measured to accuracy better than l2P , but this is a more subtle issue. [10] J. Polchinski, String Theory, Vol. 1: An Introduction to the Bosonic String, Cambridge Univ. Press, Cambridge, 1998. [11] In general the discrete spectrum of an observable A(0) is unrelated to that of A(t) − A(0). For instance, consider a Hamiltonian H = ασz and A(0) = σx. Then the spec- trum of A(0) is ±1/2, but the spectrum of σx(t)− σx(0) is ±2 sin(αt/2). [12] http://www.ligo.caltech.edu/ [13] V. B. Braginsky and Yu. I. Vorontsov, Usp. Fiz. Nauk 114, 41 (1974) Sov. Phys. Usp. 17, 644 (1975); C. M. Caves, Phys. Rev. Lett. 54, 2465 (1985). [14] C. M. Caves, Phys. Rev. Lett. 45, 75 (1980); K. S. Thorne, R. W. P. Drever, C. M. Caves, M. Zim- mermann and V. D. Sandberg, Phys. Rev. Lett. 40, 667 (1978). [15] R. V. Buniy, S. D. H. Hsu and A. Zee, Phys. Lett. B 630, 68 (2005) [arXiv:hep-th/0508039]. [16] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975). [17] J. D. Bekenstein, Phys. Rev. D7, 2333 (1973). [18] G. ’t Hooft, gr-qc/9310026. [19] L. Susskind, J. Math. Phys. 36, 6377 (1995), [hep-th/9409089]. [20] R. Bousso, Rev. Mod. Phys. 74, 825 (2002), [hep-th/0203101]. [21] J. Oppenheim, Phys. Rev. D 65, 024020 (2002) [arXiv:gr-qc/0105101]. [22] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998), [hep-th/9711200]. [23] A. Strominger and C. Vafa, Phys. Lett. B379, 99 (1996), http://arxiv.org/abs/gr-qc/0201034 http://arxiv.org/abs/hep-ph/0203154 http://arxiv.org/abs/hep-th/0405033 http://arxiv.org/abs/hep-th/0505144 http://arxiv.org/abs/hep-th/0701073 http://arxiv.org/abs/gr-qc/9403008 http://arxiv.org/abs/gr-qc/9910023 http://arxiv.org/abs/gr-qc/9909017 http://arxiv.org/abs/gr-qc/9808029 http://arxiv.org/abs/gr-qc/0104005 http://arxiv.org/abs/gr-qc/0405078 http://www.ligo.caltech.edu/ http://arxiv.org/abs/hep-th/0508039 http://arxiv.org/abs/gr-qc/9310026 http://arxiv.org/abs/hep-th/9409089 http://arxiv.org/abs/hep-th/0203101 http://arxiv.org/abs/gr-qc/0105101 http://arxiv.org/abs/hep-th/9711200 FIG. 5: Each path represents a set of outcomes S from the measurement of n qubits. To specify a particular path re- quires n bits of classical information. [hep-th/9601029]. [24] J. M. Maldacena, A. Strominger and E. Witten, JHEP 12, 002 (1997), [hep-th/9711053]. [25] G. ’t Hooft, [gr-qc/9310026]. [26] R. V. Buniy and S. D. H. Hsu, Phys. Lett. B 644, 72 (2007) [arXiv:hep-th/0510021]. [27] S. D. H. Hsu and B. M. Murray, Phys. Rev. D 73, 044017 (2006), [hep-th/0512033]. [28] S. D. H. Hsu, Phys. Lett. B 641, 99 (2006) [arXiv:hep-th/0607082]. [29] S. Lloyd, Nature 406, 1047 - 1054 (31 August 2000), [quant-ph/9908043]; Phys. Rev. Lett. 88, 237901 (2002), [quant-ph/0110141]. [30] S. Lloyd and Y. J. Ng, Scientific American, 53-61, November 2004; Spektrum Wiss. 2005N1, 32 (2005). [31] N. Margolus and L.B. Levitin, Physica D 120, 188 (1998), [quant-ph/9710043]. [32] We identify n individual components for later compari- son with entropy. The ML theorem could, of course, be applied to the entire device as well. [33] Of course, it is possible that not all seemingly innocu- ous rearrangements of our universe are possible. Perhaps only an infinitesimal subset are possible future evolu- tions of the initial data – this would be the case in a purely classical or deterministic universe. The situation would be analogous to a pseudo-random number gener- ator: from a finite seed of limited entropy, it produces a much larger, seemingly random and aperiodic sequence of output. However, varying over all possible seeds does not produce all possible output strings, since the process is deterministic and the seed entropy is small. http://arxiv.org/abs/hep-th/9601029 http://arxiv.org/abs/hep-th/9711053 http://arxiv.org/abs/gr-qc/9310026 http://arxiv.org/abs/hep-th/0510021 http://arxiv.org/abs/hep-th/0512033 http://arxiv.org/abs/hep-th/0607082 http://arxiv.org/abs/quant-ph/9908043 http://arxiv.org/abs/quant-ph/0110141 http://arxiv.org/abs/quant-ph/9710043

0704.1156

The Mid-Infrared Emission of M87

Draft version October 27, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 THE MID-INFRARED EMISSION OF M87 Eric S. Perlman , R. E. Mason , Christopher Packham , N. A. Levenson , Moshe Elitzur , Justin J. Schaefer , Masatoshi Imanishi , William B. Sparks , James Radomski Draft version October 27, 2018 ABSTRACT We discuss Subaru and Spitzer Space Telescope imaging and spectroscopy of M87 in the mid-infrared from 5-35 µm. These observations allow us to investigate mid-IR emission mechanisms in the core of M87 and to establish that the flaring, variable jet component HST-1 is not a major contributor to the mid-IR flux. The Spitzer data include a high signal-to-noise 15-35 µm spectrum of the knot A/B complex in the jet, which is consistent with synchrotron emission. However, a synchrotron model cannot account for the observed nuclear spectrum, even when contributions from the jet, necessary due to the degrading of resolution with wavelength, are included. The Spitzer data show a clear excess in the spectrum of the nucleus at wavelengths longer than 25 µm, which we model as thermal emission from cool dust at a characteristic temperature of 55 ± 10 K, with an IR luminosity ∼ 1039 erg s−1. Given Spitzer’s few-arcsecond angular resolution, the dust seen in the nuclear spectrum could be located anywhere within ∼ 5′′ (390 pc) of the nucleus. In any case, the ratio of AGN thermal to bolometric luminosity indicates that M87 does not contain the IR-bright torus that classical unified AGN schemes invoke. However, this result is consistent with theoretical predictions for low-luminosity AGNs. Subject headings: 1. INTRODUCTION M87, the dominant galaxy in the Virgo cluster, is one of the nearest (distance=16 Mpc, 1′′= 78 pc) radio galax- ies. Since 1918, when Heber Curtis observed a “curi- ous straight ray” extending from its nucleus, it has been known to host a bright jet that is visible at radio through X-ray wavelengths. The jet, one of the primary hall- marks of M87’s nuclear activity, has a complex, knotty structure (see Perlman et al. 2001b [hereafter P01b] and references therein). M87 also exhibits energetic line emis- sion from a disk that is seen to be in Keplerian mo- tion (Ford et al. 1994, Harms et al. 1994). Longslit spectroscopy of the material within the disk has allowed the mass in the inner 3.5 pc of M87 to be measured at ≈ 3 × 109M⊙ (Marconi et al. 1997), implying the pres- ence of one of the most massive known black holes. M87’s proximity allows us to obtain particularly high spatial resolution. Unfortunately, M87 is quite faint in the infrared, with a core flux of only 16 mJy at 10.8µm (Perlman et al. 2001a, hereafter P01a). This makes Spitzer well-suited to sensitive observations of the in- 1 Joint Center for Astrophysics, Physics Department, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 2 Current Address: Department of Physics & Space Sciences, Florida Institute of Technology, 150 W. University Boulevard, Mel- bourne, FL 32901 3 Gemini Observatory, Northern Operations Center, 670 N. A’Ohoku Place, Hilo, HI 96720, USA 4 Department of Astronomy, University of Florida, 211 BRSC, Gainesville, FL 32611, USA 5 Department of Physics and Astronomy, University of Kentucky, 177 Chem.-Phys. Building, Lexington, KY 40506-0055, USA 6 National Astronomical Observatory, Mitaka, Tokyo 181-8588, Japan 7 Space Telescope Science Institute, 3700 San Martin Drive, Bal- timore,MD 21218, USA 8 Gemini Observatory, Southern Operations Center, c/o AURA, Casilla 603, La Serena, Chile TABLE 1 Known knots in the jet of M87 Knot Dist. from Nuc.1 Approx. Extent2 10.8 µm Flux3 Name (arcsecs) (arcsecs) (mJy) HST-14 0.8 0.5 × 0.1 < 2.3 D 2.7 1.8 × 0.2 1.2± 0.2 E5 6.0 0.9 × 0.3 0.1 F 8.2 1.4 × 0.4 1.3± 0.2 I5 11.2 0.6 × 0.5 0.25 A 12.4 2.2 × 1.3 6.4± 0.6 B 14.3 2.4 × 1.3 6.3± 0.6 C 17.4 1.9 × 1.7 4.0± 0.6 1 Location of the flux maximum, rather than centroid. 2 As measured in optical HST images. 3 Except where noted, as given in P01a. 4 Knot flux is highly variable; the flux maximum is at the upstream end. The 2 σ flux limit given here is derived from these Subaru data. 5 Not detected in Gemini images (P01a); fluxes are extrapolated from the radio-optical spectrum (P01b). frared emission from this object, while at the same time the small spatial scales that can be probed make M87 an appropriate mid-infrared (mid-IR) target for the largest ground-based telescopes – albeit one requiring long inte- grations. The knots in the jet are significantly fainter than the nucleus. Table 1 reviews the locations and extents of major knots in the jet, along with N-band fluxes either directly measured or extrapolated from ra- dio/optical data. Previous ground-based observations showed that most of the mid-IR emission from the core of M87 can be attributed to synchrotron radiation from the innermost regions of the jet (P01a, Whysong & Antonucci 2004), although a minor contribution from dust at T <∼ 150 K could not be ruled out. These results provided strong evidence that the dusty torus of classical unified AGN schemes (Antonucci 1993) may be absent in low- http://arxiv.org/abs/0704.1156v2 luminosity AGNs. Two recent papers have used Spitzer data to address the mid-IR emission processes of M87. Bressan et al. (2006) used 5-20 µm IRS spectra to ex- amine the issue of silicate emission from Virgo Cluster galaxies. They showed that M87, like other Virgo el- lipticals, exhibits a silicate emission feature at 10 µm, but that this feature is not spatially resolved by Spitzer. Shi et al. (2007) used IRAC and MIPS imaging data to re-examine the origin of M87’s mid-IR emission. They found that their photometry of the nucleus plus jet and lobes can be fit by a combination of two synchrotron power laws (usually breaking at optical or higher ener- gies), but that in larger (1′) apertures an excess is present over that model. This excess is ascribed to dust in the host galaxy with a luminosity similar to that observed in other brightest cluster galaxies. In this paper we discuss new mid-IR imaging and spec- troscopy of M87 using the Subaru observatory and Spitzer Space Telescope. We combine data from the IRAC, MIPS and IRS instruments (the latter covering the entire 5- 35µm range) to obtain a more detailed picture of the emission processes in the nucleus and jet. In §2 we present the observations and data reduction procedures, while in §3 we discuss M87’s line and continuum emis- sion. We close in §4 with a summary and discussion. 2. OBSERVATIONS AND DATA 2.1. Ground-Based Observations Mid-IR imaging and spectroscopy of M87 were ob- tained on the nights of UT 2005 April 27 and 28 us- ing the COMICS camera (Kataza et al. 2000) attached to the Cassegrain focus of the Subaru Telescope (Iye et al. 2004). The 320 × 240 SiAs IBC array has a plate scale of 0.129′′/pixel, delivering a 41× 31′′ field of view. The detector was read-out in correlated quadruple sam- pling (CQS) mode (Sako et al. 2003). The observing conditions on both nights were not photometric, and the precipitable water vapor was variable, as high as 3-5 mm on April 27, but reaching 5-7mm on April 28. For the acquisition images, we used the N11.7 filter (central wavelength 11.67µm, bandwidth 1.05µm, 50% cut-on/off). To remove time-variable sky background, telescope thermal noise and so-called “1/f” detector noise, we used the standard chop-nod technique, with a 10′′ chop throw at a position angle of −69.5 degrees (measured north through east), to project the jet along the detector rows and place the reference beam between minima in the jet emission. The chop frequency was 0.45 Hz (standard for Subaru/COMICS observations) and the total on-source time was 700s. We used a similar setup for the spectroscopic ob- servations, with similar chop parameters, but without nodding, as is standard practice with COMICS, and the broad N band order blocking filter. The grating and 0.33′′ slit provided a spectral resolution of ∼ 250 (0.02µm/pix), dispersing the entire N band across the array. The superb design and capabilities of COMICS permit on-slit mid-IR imaging of the polished slit jaws, simultaneous with the spectroscopic observations. Thus we were able to ensure that the emission from the nucleus of M87 passed through the slit and into the spectroscopy arm of COMICS. This proved to be crucial as differen- tial refraction between the telescope’s optical autoguider Fig. 1.— Subaru acquisition image of M87. North is up and east is to the left. The high spatial resolution of the COMICS data is apparent, despite its limited sensitivity. A 3-pixel Gaussian was used to smooth these data. We do not detect emission from knot HST-1, 0.8′′ from the nucleus (location of the circle on this figure). See §§2,3 for details. and the MIR science beam moved the object off the slit in as little as 20 minutes at some airmasses. The varying, suboptimal conditions meant that M87 was not detected in many of the spectroscopy frames, and only those in which the continuum was visible were stacked and used in the final analysis. The data presented in this paper represent 2100 s of on-source time. The data were reduced using IRAF and in-house devel- oped IDL routines. The difference for each chopped pair (and for each nod-set for imaging), was calculated, and the results combined until a single frame was created. During the reduction process, chopped pairs obviously compromised by cirrus, high electronic noise, or other problems were discarded. The PSF was measured from observations of the pho- tometric standard. The measured full-width at half- maximum (FWHM) of the standard was 0.55′′, which agrees well with the FWHM of the spectral trace. Flux calibration was achieved using HD108985 as a flux stan- dard (Cohen et al. 1992, 1999; Tokunaga 1984) and inter- polating the Cohen models to the proper bandpass. Ab- solute errors in flux calibration were estimated from the variations in the counts through the course of the nights. Given the non-photometric conditions under which the data were obtained, we estimate an accuracy of at best 15% for the flux calibration. A 5-pixel extraction aperture was used for the spec- troscopy data, and spectral flux calibration was with ref- erence to the telluric standard star, Vega. No attempt was made to correct for slit losses in the spectrum of M87, but we note that the flux density in the calibrated spec- trum is comparable to that of both the Subaru/COMICS image and the Gemini/OSCIR photometry of P01a. In Figure 1 we show the Subaru image, while in Figure 2 we show the Subaru spectrum. 2.2. Spitzer Observations Spitzer Space Telescope observations of M87 were ob- tained from the Spitzer archives. This includes IRS (In- Fig. 2.— The Subaru/COMICS spectrum of the nucleus, showing the [Ne II] 12.81 µm emission line. See §§2,3 for details. frared Spectrograph, Houck et al. 2004) spectra taken on UT 2004 January 4, as well as MIPS (Multi-band Imag- ing Photometer for Spitzer, Rieke et al. 2004) images taken on UT 2004 December 26. Both datasets were ac- quired as part of PID 82 (PI Rieke). We also obtained data taken on 2005 June 11 with the IRAC (Infrared Array Camera) as part of PID 224 (PID Forman). The advantage of this particular IRS dataset is that it cov- ers the full wavelength range of the short-low and long- low modules. We also comment below on other Spitzer datasets. The IRAC and MIPS data were processed according to the standard pipeline recipes. In both, we used the post- basic calibrated data (post-BCD). Fluxes were extracted using circular apertures appropriate to the diffraction limited resolution at a given wavelength. Background subtraction was done using annular apertures that in- cluded all the flux from the galaxy itself. The Spitzer imaging data (shown in Figure 3) clearly show emission from both the nucleus and the knot A/B complex, as well as extended emission from the galaxy and the south-west hotspot, a feature thought to be associated with the un- seen counterjet (Hines et al. 1989, Sparks et al. 1992, Stiavelli et al. 1992). However, individual knots are not resolved at Spitzer’s angular resolution (≈ 3′′ at 10µm). In this paper, we discuss only the emission from the core and jet knots A and B. We subtracted from the core flux measured on the IRAC data any emission that might be due to galactic emission, using an annulus between 5 − 15′′; we also subtracted flux due to the jet by gaug- ing independently the flux in the four quadrants. For this reason our fluxes differ somewhat from those reported in Shi et al. (2007). The 24µm flux derived in this way agrees fully (within 1σ) with that found in the IRS spec- trum of the core (§3.1). The IRS spectra were acquired using both orders of the low-resolution short-wavelength (“short-low”) and low- resolution long-wavelength (“long-low”) modules of the IRS. Total integration times of 28 s and 60 s were used for the short- and long-wavelength modules, respectively, and spectra were obtained in two nod positions separated by 1/3 of the slit length, translating into nod throws of 19′′ and 56′′ for the short-low and long-low modules respectively. The spacecraft roll angle and relative slit position angles were such that the 3.6− 3.7′′-wide short- 4.5 micron SW Hotspot Nucleus 24 micron Fig. 3.— Spitzer imaging of M87, at 4.5 and 24.5 µm. We have indicated the core, jet and SW hotspot for reference. See §2 for details. low slits were oriented perpendicular to the jet, while the wider (10.5− 10.7′′) long-low slits were aligned along the jet axis and therefore contain emission from the brightest region of the jet (primarily knots A and B, see Table 1). After standard pipeline processing, thermal back- ground emission was removed by subtracting spectra taken at the two nod positions, and 1D spectra then ex- tracted using the SPICE package. The default extraction apertures (7.2′′ at 6 µm, 14.4′′at 12 µm, scaling linearly with wavelength) were employed for the extraction of the short-low data, whereas to better separate the nuclear and jet emission in the long-low data, we used a smaller aperture that varied in size with wavelength, increasing linearly from 5.1” at 17µm to 10.2” at 34 µm, compara- ble to the FWHM of Spitzer’s PSF at those wavelengths. As the long-low slits covered both the nucleus and the knot A/B region, we extracted two 1D spectral datasets. No significant fringing is visible in any of the spectra, and no defringing was attempted. As M87 is an extended source, it is non-trivial to equal- ize the flux calibration of the orders in both the short-low and long-low modules, or between the two modules, due to the different extraction apertures. As a result, the spectra from different orders may contain differing con- tributions from the various extended structures, e.g., the various jet components. The nucleus is essentially unaf- fected by this problem at short wavelengths; however, at longer wavelengths it does have a significant effect. The fact that the nucleus is essentially unresolved at short wavelengths allowed us to use the flux scale from the pipeline for the SL data. We then made use of the over- lap and “bonus segment” data to match up the flux and slope of the spectral segments blueward and redward of each join. No correction was found to be necessary to scale the SL1 data to that from SL2. However, a scaling factor of 1.67 ±0.03 was found to be necessary to scale the LL2 data to the SL1 data, while a scaling factor of 1.38±0.03 was necessary to scale the two long-low orders of the nuclear spectrum. The knot A/B region, by contrast, is highly extended (about 1.3′′×5′′, as measured on the optical and Gemini- N+OSCIR data). This makes the flux calibration process much more complicated, since the SPICE package is not optimized (or intended) to deal with extended sources of this nature. In this case we used the Gemini-N + OSCIR result (P01a) to set the flux scale at 10.8µ m, and then use the best-fit spectral index to extrapolate from the LL spectra down to that wavelength (note that we do not have SL data for the knot A/B region). We then used the overlap and “bonus segment” data to match up the flux and slope of the spectral segments blueward and redward of the LL1/LL2 join (similarly to what we did for the nucleus). Because the structural details of the knot A/B region are rather different from the nuclear region, a smaller scaling factor (1.04± 0.03) was required to scale the LL1 data to the LL2 flux scale once it was fixed using the Gemini-N 10.8µm flux point. The resulting IRS spectra of M87’s nucleus and jet are shown in Fig. 3. RESULTS The reduced Subaru acquisition image is shown in Fig- ure 1. The core is clearly detected, with a flux of 20.8 ± 3.5 mJy, consistent with the findings of P01b to within the 1σ errors The error in the flux calibration is dom- inated by the variable weather conditions. We do not detect any jet features. Even for the knot A/B complex this is not unexpected given the extended nature of the knot, our short integration time, fairly narrow band and the poor sky conditions. Surprisingly, however, we do not detect the flaring jet component HST-1 (Harris et al. 2003, 2005; Perlman et al. 2003), 0.8′′ from the core, despite the fact that at its peak in 2005 March (just a few weeks before the Subaru observations), its 0.8 µm flux was brighter than that of the core itself (Biretta et al., in prep.). We place a limit of 2.3 mJy at 2σ on the flux from knot HST-1. The COMICS nuclear spectrum is shown in Figure 2. The signal-to-noise ratio of the spectrum is low, but we detect continuum emission plus a single emission line at 12.81 µm (rest frame). This spectrum contains only flux from the nucleus; the extraction aperture would have ex- cluded any significant contribution from HST-1 which in any case is faint. In Figure 4, we show the Spitzer spectrum of the knot A/B region (blue) as well as the nucleus (black). The signal-to-noise ratio of the Spitzer data, while still modest (∼ 10 per pixel) is significantly better than that of the Subaru spectrum. We have over- plotted on Figure 4 a binned version of the Subaru nu- clear spectrum, which matches well with that obtained by Spitzer. Despite the difference in angular resolution between the Spitzer (∼ 3′′ at 10 µm) and Subaru (∼ 0.3′′) Fig. 4.— The Spitzer IRS spectrum of the nucleus of M87 (black) as well as knots A and B (blue). For comparison, the binned Subaru COMICS data are shown in red (error bars reflect the standard error on the mean of the points in each bin). To better separate the two IRS spectra, the data for knots A and B have been multipled by 0.75. Several low-ionization emission lines can be seen in the spectrum of the nucleus, as well as hints of silicate emission around 10 and 18 µm. See §§3, 3.1 for details. data, the nuclear spectra are similar in flux density and spectral slope in both datasets. This confirms that HST- 1 (which in the Spitzer data would not be resolved from the nucleus) makes no more than a minimal contribution to the nuclear spectrum. Both the noise in the Subaru data, however, and the known optical variability of the nucleus and HST-1 (measured to be as large as 50% and 500%, respectively; Tsvetanov et al. 1998, Perlman et al. 2003) prevent us from setting stricter limits on the mid-IR flux of HST-1. The overall similarity of the spectral shapes of the nu- cleus and the knot A/B complex, at least below 25 µm, confirms the synchrotron nature of the majority of the nuclear emission, first shown by P01a. A close exami- nation of the nuclear and jet spectra does, however, re- veal differences. The first of these is the presence of five obvious emission lines in the spectrum of the nucleus, whereas only one weak emission line is seen in the knot A/B spectrum. A more subtle difference is that the nu- clear spectrum contains a sharper upturn towards longer wavelengths. We discuss the spectral data in more depth in the following subsections. Note that in the discussion that follows, we use the terms “nuclear region” and “nu- clear spectrum” to refer to the Spitzer spectrum taken at the position of the nucleus (as opposed to that of the knot A/B region), while what we term as the core refers only to the regions within ∼ 0.3′′ of the central black hole (and thus unresolvable in ground-based N-band images). 3.1. IR Continuum Emission The infrared continuum of a galaxy can contain emis- sion from a number of different components. In the case of M87, the most likely sources are thermal emission from warm or cool dust, synchrotron emission from the jet, and (at shorter wavelengths) starlight from K and M stars. In order to determine the nature and origin of this IR emission, it is necessary to correctly account for all possible sources that could fall within the slit and beam of Spitzer at any given wavelength. At Spitzer’s angular Fig. 5.— Spitzer IRS spectrum of jet knots A and B, modeled as discussed in §3.2. In blue, we show the contribution due to knots A and B (summed); in red, the contribution from the core, and in green the contribution from all other knots. The black line represents the sum of the contributions from the core and all knots. The observed spectrum for knots A and B can be satisfactorily reproduced with these synchrotron components. resolution the nuclear spectrum will contain a contribu- tion from emission in the jet (and vice versa; see Table 1), and, because of the decrease in spatial resolution towards longer wavelengths, the magnitude of that contribution will increase with wavelength. We have modeled both the nuclear and knot A/B spectra, and accounted for this effect as follows. We model each component as a 1-dimensional Gaus- sian of width equal to Spitzer’s diffraction limit (which we take to be a Gaussian of FWHM 2.96′′ at 10 µm, located at its flux maximum position (Table 1)). We then integrate each Gaussian across the slit (see §2.2 for details). We use the fluxes of jet knots C, D and F of P01a and take their positions and radio-optical spectral indices αRO from P01b [Fν ∝ ν −α]. For fainter knots, we take 2.1 µm fluxes and positions from P01b and ex- trapolate to 10µm using the αRO values in P01b. For knots A and B, we fix the fluxes to be equal to those observed by P01a but allow their spectral index to vary. We do not include a contribution from HST-1 given that we do not detect it in our 11.7 µm image (but see be- low). The modeling of the knot A/B region spectrum is shown in Figure 5. We show separate curves for the contributions of the core, knot A/B and other knots to the observed spectrum. This modeling procedure can satisfactorily account for the observed spectrum of the knot A/B region. We fit a power-law to the spectrum of the knot A/B emission by holding constant the con- tributions from all other components to the fluxes and spectral indices from P01a and P01b, with the exception of the core (for which we used the power-law fit we detail below). Once this is done, we obtain a best fit spectral index for knots A and B of αIR = 0.75 ± 0.04. This is within 1σ of the radio-optical spectral indices given in P01b for these knots. Thus the spectrum of the knot A/B region is entirely consistent with synchrotron emis- sion, as previously modeled by P01a, P01B and Shi et al. (2007). In Figures 6 and 7, we show the modeling of the nu- Fig. 6.— Spitzer IRS spectrum of the nucleus of M87, modeled as discussed in §3.1. We follow Figure 5 by showing the contributions of the core and the knot A/B region as red and blue respectively, while in green we show the contribution from all other knots. This figure shows the sum of these components as purple, and then op- tically thin and thick thermal dust emission models as the lower black and orange curves respectively. The top set of black curves represents the sum of all synchrotron emission plus the optically thin dust model. The upper and lower black dotted curves repre- sent the result of multiplyin the 55K thermal model by 1.2 and 0.8 respectively; this illustrates the range of normalizations that the data are sensitive to. clear spectrum, with Figure 6 showing the 5 − 35 µm region alone and Figure 7 extending it to longer wave- lengths. To fit the nuclear spectrum, we need to add to the jet components an increase of flux of 15% over that observed at 10.8µm by P01a. Significant nuclear vari- ability in M87 has previously been noted in the optical by Tsvetanov et al. (1998) and Perlman et al. (2003). In addition, our data require a core spectral index that is quite different from that predicted by P01a, specifically αIR = 0.41 ± 0.05 instead of α ∼ 1.0 (as used also in other papers, including Shi et al. 2007). This indicates a synchrotron peak frequency >∼ 5× 10 13 Hz (and likely > 1014 Hz), at least 1-2 decades higher in frequency than that fit by P01a. The spectral index was fit only over the spectral range 7.5-15 µm where no departure is observed from a pure power-law form (see below), but the method was otherwise identical to the one above for the knot A/B spectrum. The different spectral index could be the result of variability, as in blazars (of which M87 is a somewhat misaligned example), where increases in flux are often accompanied by spectral hardenings and large increases in peak frequency (see e.g., Pian et al. 1997). As already noted, the nucleus of M87 is known to be variable in the optical (Tsvetanov et al. 1998, Perlman et al. 2003) on timescales of ∼ 1 month, and on even shorter timescales in the gamma-rays (Aharonian et al. 2006). In contrast to the knot A/B spectrum, we cannot model the observed nuclear emission with the purely power-law emission characteristic of synchrotron radia- tion. The pure power-law model has discrepancies both at the short-wavelength and long-wavelength end of the IRS spectrum. There are two likely causes of the appar- ent short-wavelength excess. The first possibility is that it is due to M and K stars in the nuclear regions of M87, a component that P01a speculated might account for the faint (7% of nuclear flux), extended emission found in the much deeper OSCIR image. A second possibility is that the excess might be the low-frequency tail of the emis- sion from knot HST-1, which Spitzer would be unable to resolve from the nucleus itself. If we were to use a power-law to extrapolate the observed excess at ∼ 5µm down to a wavelength of 0.8 µm, the flux from this ad- ditional component would equal or slightly exceed that of the core, consistent with the observed 0.8 µm flux of knot HST-1 in January 2005 (Biretta et al., in prep.). At longer wavelengths, beyond about 23 µm, we see a much more pronounced excess (Figure 6). By 35 µm the pure synchrotron model – which includes emission from the core plus all jet knots (i.e., all known sources of synchrotron emission in M87) – underestimates the observed nuclear spectrum by about 20%. This excess could be fit by adding an additional power-law compo- nent with spectral index αexcess ∼ 6. However, such a component would be unphysical given known models of synchrotron radiation (e.g., Leahy 1991); and moreover, our model has already accounted for all known sources of synchrotron emission. Also, the addition of such a steep power-law component would dominate over all the other emission sources by ∼ 50 µm, producing a huge excess by 100 µm, something which is ruled out by the longer-wavelength MIPS and IRAS data (Figure 7)9 Al- ternatively, we can fit this excess with thermal emission (Figure 6), which is not unexpected in the nuclear regions of M87 given the dust lanes found in HST and ground- based images (e.g., Sparks et al. 1993, Ford et al. 1994). We believe that thermal emission from cool dust in the nucleus is the most physically realistic explanation for the long-wavelength excess in the few arcsecond-aperture IRS data (distinct from the host galaxy dust detected at large radii by Shi et al. (2007) in 1′-aperture photome- try). In addition, the data and modelling presented here allow us to to place constraints on the mass and temper- ature of dust in the nucleus of M87. A single blackbody fit to this component yields ad- equate fits with temperatures between 45-65K, with higher temperatures being unable to reproduce the rise at 23-35 µm, and lower temperatures resulting in an over- production of the long-wavelength fluxes from IRAS and Spitzer. We are unable to constrain whether the dust is optically thin or thick (both curves are shown in Fig- ure 6, for a characteristic temperature of 55K); however, given that the dust filaments in the nuclear regions of M87 are also typically marked by Hα emission (Sparks et al. 1993, Ford et al. 2004), and also given the low X-ray absorbing column (Perlman & Wilson 2005) the optically thin model may be more likely to be correct. Figure 7 shows only a 55K thermal component, as does Figure 6. Although temperatures as low as 45K are com- patible with the data, and in fact a single 45K thermal emission component can account for all the IRS, MIPS and IRAS data, a single dust temperature this low would mean that the dust in the innermost 5′′ of the galaxy contributes much more flux than dust at radii between 5 − 20′′. As M87 is extended at mid-IR wavelengths 9 Note that the ISO 60µm and 100µm fluxes (Haas et al. 2004) are more than a factor two above all other photometry at similar wavelengths. Fig. 7.— Spitzer IRS spectrum and models of the nucleus of M87, here plotted with a logarithmic y axis and the wavelength axis continued up to 105µm. All curves are as in Figure 6, with the exception of the optically thin thermal emission component, shown in the dash-dot-dot-dot curves (optically thick thermal emission models are not shown). Note that longward of 45µm, thermal emission becomes dominant, a situation that continues to be the case up to much longer wavelengths. Individual points are shown representing data from Spitzer/MIPS (triangle), IRAS (diamonds), and ISO (asterisks). Also shown are (squares) flux points from Perlman et al. (2001a) and Whysong & Antonucci (2004), which most likely lie below our data because of variability. We do not attempt to fit data at longer wavelengths; however, if all the dust were at T < 45K, it would overpredict the IRAS and Spitzer/MIPS 70µm and 160µm fluxes. (e.g., Figure 3), we consider this unlikely. However, sen- sitive, higher angular resolution observations at 30-100 µm would be required to put tighter constraints on dust temperatures and optical thicknesses in the nucleus of The total luminosity of the thermal component re- vealed by the Spitzer spectra is, for T = 55K, 9.7× 1038 erg/s (2.6 × 105L⊙). This figure is highly temperature dependent – for T = 45 K we find a thermal luminos- ity of 3.1 × 1039 erg/s, while for T = 65K, we find a thermal luminosity of 2.9× 1038 erg/s. (By comparison, the error in the normalization of the thermal component, indicated on Figure 6, is significantly smaller, approxi- mately 20%). Thermal radiation from dust is therefore a significant contributor to M87’s IR emission, being 20- 150% as bright as the core’s synchrotron emission be- tween 1-300 µm. It is interesting to consider how much dust could account for this thermal emission, and where the dust would be located. If the dust is heated entirely by the AGN, it must be within a few parsecs of the core. If it is at larger distances it would have to be heated by other mechanisms, for instance within the star-forming disk tentatively detected by Tan et al. (2007). Using standard formulae (Draine 2003 and refs. therein), and assuming an optically thin dust component we derive a dust mass of 83M⊙ for a temperature of T = 55K. This figure is similarly temperature dependent: for T = 45 K we find a dust mass of 180 M⊙, while for T = 65 K we find a dust mass of 19 M⊙. This is a tiny mass compared to that found in the dusty Hα filaments that extend throughout the galaxy (Sparks et al. 1993). It is also tiny compared to the molecular gas mass derived by Tan et al. (2007) from 230 GHz observations. 3.2. Spectral Features The 5-35 µm spectral region covers a range of dust and molecular emission and absorption features, as well as many fine structure lines. As mentioned above, a line at 12.8 µm is detected in the COMICS spectrum of M87. We ascribe this feature to [NeII] rather than the 12.7 µm PAH band because of the lack of other molecular fea- tures (including the stronger 11.2 µm PAH feature) in both the COMICS and IRS spectra. In addition, the IRS spectrum of the nucleus (Figure 4, 6a) also contains lines of [Ne III] 15.6 µm, [S III] 18.7 µm, [O IV]+[Fe II] 25.9 µm and [S III] 33.5µm, while in the knot A/B spectrum (Figures 4, 5), we detect only one emission line at low significance, namely [S III] 33.5µm. (The [Ar II] 7.0 µm line in the spectrum of Bressan et al. 2006 is not clearly detected in our lower S/N spectrum). Table 2 shows the fluxes of the detected lines in the nuclear spectrum, obtained by direct integration under each line and using a continuum level determined by the mean of several points at either side. With the exception of [O IV], which may in principle contribute to the 25.9 µm line, the ionization energies of all of the lines are low enough (< 50 eV) that they can be excited by massive stars and ionizing shocks. High- excitation lines such as [Ne V] 24.3 µm, which can be strong in AGN (e.g. Sturm et al. 2002), are not ob- served in the spectrum of M87. Comparison of line fluxes in different apertures suggest that the line-emitting re- gion may be extended. For instance, the [Ne II] line in the COMICS spectrum is a factor of about three weaker than the same line in the larger-aperture Spitzer spec- trum. Although the continuum level for the [Ne II] line in the COMICS spectrum cannot be determined with great accuracy, such a pronounced difference implies that the line-emitting material extends beyond the compact cen- tral source observed with COMICS. This interpretation is supported by the fact that we observe a factor of about 2.5 lower flux in the [O IV]/[Fe II] line than in the [Fe II] line in a 14′′ x 20′′-aperture ISO spectrum of M87 (Sturm et al. 2002). The ISO spectrum had sufficient spectral resolution to separate the [O IV] 25.89 µm and [Fe II] 25.99 µm lines but no lines other than [Fe II] were detected. As expected, all the lines are unresolved in the R ∼100 IRS spectrum. The lack of high-excitation lines is consistent with the classification of M87’s nucleus as a LINER (a suggestion originally made by Willner et al. 1985). This might be considered surprising given the luminosity of M87’s jet. However, recent radio observations suggest that relativis- tic jets may be quite common among LINERs (Nagar et al. 2000, Falcke et al. 2000, Filho et al. 2004). More- over, due to relativistic beaming, only a very small solid angle of the emission-line regions would be exposed to significant high-energy radiation from the jet. The [S III] 33.5 µm seen weakly in the knot A/B spec- trum (Figures 4, 5) is unlikely to come from knots A and B, since spectra of M87’s jet have been taken many times in other bands, most recently by HST in the UV (Gelderman et al. 2005), and have failed to detect line emission, placing severe limits on the presence of non- stripped nuclei entrained within the jet. It is possible that the [SIII] emission originates in the Hα filaments within a few arcseconds of the knot A/B complex (see TABLE 2 Fluxes of the fine structure lines detected in Line, λrest a Flux Eion µm 10−21 W cm−2 (eV) [Ne II] 12.81 1.9 b 21.6 ........ 5.4 c 21.6 [Ne III] 15.56 4.1 41.0 [S III] 18.71 2.0 23.3 [O IV]/[Fe II] 25.89/25.99 1.9 54.9/7.9 [S III] 33.48 4.5 d 23.3 a From Sturm et al. (2002) b Subaru/COMICS data c Spitzer/IRS data d See text for discussion Sparks et al. 1993). Alternatively, the emission may originate from material close to the nucleus, whose light contributes to the jet spectrum particularly at longer wavelengths where the PSF is largest. This is consis- tent with the relative contribution of the core to the knot A/B spectrum at 33.5µm, as computed in our modeling procedure (§ 3.1). That only [SIII] 33.5 µm, the longest- wavelength line in the nuclear spectrum, is also observed in the jet spectrum suggests that this may be the more likely explanation. In highly obscured objects, such as Seyfert 2 galaxies, the AGN unified model predicts absorption features near 10 and 18 µm due to the Si-O bond stretch in silicate dust grains. Such features are indeed observed quite widely in Seyfert 2 galaxies (e.g. Roche et al. 1991). Such a feature would not necessarily be expected in M87, however, due to the fact that its jet (which would be perpendicular to any dusty torus under unified schemes) is seen at a fairly small angle to the line of sight ( <∼ 15 − 20 ◦, Biretta et al. 1999). Consideration of the difference between means in points at the peak and either side of a typical silicate absorption profile in the COMICS spectrum allows us to put a 95% confidence limit of τ(9.7) . 0.34 on any broad silicate absorption feature in the nucleus of M87. Thus, while a strong absorption feature can be ruled out, this limit is comparable to the silicate optical depth observed in the circumnuclear material of Seyfert 2 galaxies such as NGC 1068 (∼ 0.4; Roche et al. 1984; Jaffe et al. 2004; Mason et al. 2006). It is also consistent with the best available limit (NH < 6 × 10 20 cm−2, much lower than that seen in typical Seyfert 2 galaxies) on nuclear absorption from Chandra X-ray data (Perlman & Wilson 2005), given the recently found correlation between X-ray N(H) figures and silicate absorption (Shi et al. 2006). While the COMICS spectrum sets weak limits on the presence of a 9.7 µm absorption band in the central <0.65′′ of M87, the IRS spectrum can be used to search for dust features which may exist on somewhat larger scales. In fact, there are hints of weak silicate emis- sion features around 10 and 18 µm in the IRS spectrum (the 10µm feature would be well within the noise of the COMICS spectrum). The 10 µm emission feature is also evident in the IRS spectrum of M87 presented by Bressan et al. (2006; their spectrum does not cover wavelengths > 20µm). Previously thought to be rare in AGN, sili- cate emission features have now been seen in the LINER NGC 3998 (Sturm et al. 2005) as well as in several dis- tant quasars (e.g., Hao et al. 2005; Siebenmorgen et al. 2005). Although the AGN unified scheme in its most basic form predicts a silicate emission feature from the hot inner edge of a dusty circumnuclear torus, the aper- tures so far used to detect the silicate feature do not place strong constraints on the location of the silicate-emitting material in these galaxies. Furthermore, the cool (∼ 200 K) temperature implied by the strengths and profiles of the features in the LINER, NGC 3998, suggests an origin in narrow-line-region dust rather than the inner regions of the torus (Sturm et al. 2005). Although Bressan et al. point out that the 10µm feature in their data is not spatially extended, the PSF of Spitzer at 10 µm (3′′) is several times larger than the FWHM of the nucleus itself in higher-resolution ground-based data. Therefore it is entirely possible that the silicate emission could be pro- duced by dust as far as 200 pc from the nucleus. HST and ground-based optical observations show the existence of a significant amount of warm dust in the inner 3” of M87, as evidenced by theHα emission in many of these regions (Sparks et al. 1993, Ford et al. 1994). Therefore, the silicate emission in M87 does not necessarily imply the existence of a nuclear torus. 4. SUMMARY AND CONCLUSIONS We have presented mid-IR imaging and spectroscopy of M87, using the Subaru observatory and the Spitzer Space Telescope. The Spitzer spectroscopy covered both the nucleus and the knot A/B complex in the jet. The knot A/B spectrum can be well modeled by power-law spectra, as expected for synchrotron emission from jet components, with mid-IR fluxes consistent with those published by P01a and spectral indices consistent with the optical-radio spectra of P01b. However, the Spitzer spectrum of the nucleus cannot be modeled by power- law synchrotron emission from the core plus components in the jet. We see clear signs of an infrared excess in the core, which can be well modeled by a thermal spec- trum with a characteristic temperature of 55± 10K. The low temperature and luminosity of this component (§3.2) are consistent with the limits found by Perlman et al. (2001a). Whysong & Antonucci (2004) made a similar argument based on 11.7 µm photometry. The present data sup- port and enhance this result as the thermal contribu- tion is isolated and measured separately from the syn- chrotron contribution. The measured luminosity is an upper limit to the amount of emission from a dusty torus in M87; the thermal contribution measured in the 10.7′′- wide IRS slit could arise anywhere within 420 pc of the nucleus and need not be associated with a torus at all. Moreover, given that the bolometric luminosity of M87 is ∼ 1042 erg s−1 (e.g., Reynolds et al. 1996), classi- cal torus models would predict a mid-IR luminosity at least that high (e.g., Risaliti & Elvis 2005). By com- parison we observe a much smaller thermal IR luminos- ity (∼ 1039 erg s−1), making our observations difficult to reconcile with standard unified models of AGN. What is the overall significance of the weakness of the torus emission in M87, both for this galaxy and for FR I radio galaxies as a whole? One strong possibility is that the weakly-emitting or absent torus in M87 may be a con- sequence of the AGN’s low luminosity. In the disk wind scenario, optically thick regions of an outflowing wind comprise the geometrically and optically thick “torus.” In the model of Elitzur & Shlosman (2006), the torus disappears at low luminosities ( <∼ 10 42 erg s−1), because accretion onto the central black hole can no longer sus- tain the large-scale outflows. That we do not detect the mid-IR signature of the classical torus is entirely consistent with the predictions of the disk wind model. An alternative to the disk-wind scenario was offered by Reynolds et al. (1996) and di Matteo et al. (2003) who modeled the nucleus of M87 with ADAF models. Both of those papers derive accretion rates as much as 4 orders of magnitude below the Eddington rate, which di Matteo et al. speculate may signal that M87’s nuclear activity is at a very late evolutionary stage. It is important to note, however, that ADAF models do not in and of themselves incorporate the notion of a torus. Finally, it is important to point out that the lack of significant torus emission in M87 does not yet constitute strong evidence that FR Is, as a class, lack luminous tori, although Spitzer observa- tions indicate that a significant number may lack this component (Ogle et al. 2006, Birkinshaw et al. in prep). Cen A, for example, shows significant thermal dust emis- sion in the mid-IR, on scales ranging from the unresolved sub-pc scale (likely the torus) to much larger-scale emis- sion (Whysong & Antonucci 2004, Hardcastle et al. 2006 Radomski et al. 2007). Based in part on data collected at Subaru Telescope, which is operated by the National Astronomical Obser- vatory of Japan. We wish to thank the Subaru TAC for their support and the Subaru staff, particularly T. Fujiyoshi, M. Lemmen and M. Letawsky for support- ing our observations. ESP acknowledges support from NASA LTSA grants NAG5-9997 and NNG05-GD63G, as well as HST grant GO-9705. CP acknowledges sup- port from NSF grant no. 0206617. NAL acknowledges work supported by the NSF under grant no. 0237291. ME acknowledges support from NSF grant no. 0507421 and NASA grant NNG05-GC38G. RM and JR were sup- ported by the Gemini Observatory, which is operated by the Association of Universities for Research in Astron- omy, Inc., on behalf of the international Gemini part- nership of Argentina, Australia, Brazil, Canada, Chile, the United Kingdom, and the United States of America. MI is supported by Grants-in-Aid for Scientific Research (16740117). REFERENCES Aharonian, F., et al., 2006, Science, 314, 5804 Antonucci, R. R. J., 1993, ARAA, 31, 473 Biretta, J. A., Sparks, W. B., & Macchetto, F. D., 1999, ApJ, 522, Bressan, A., 2006, ApJL, 639, L55 di Matteo, T., Allen, S. W., Fabian, A. C., Wilson, A. S., & Young, A. J., 2003, ApJ, 582, 133 Elitzur, M., & Shlosman, I., 2006, ApJL, 648, L101 Falcke, H., Nagar, N. M., Wilson, A. S., & Ulvestad, J. S., 200, ApJ, 542, 197 Filho, M. E., Fraternali, F., Markoff, S., Nagar, N. M., Barthel, P. D., Ho, L. C., & Yuan, F., 2004, A& A, 418, 429 Ford, H. C., et al., 1994, ApJ, 435, L27 Harms, R. J., et al., 1994, ApJ, 435, L35 Hao, L., et al., 2005, ApJ, 625, L75 Hines, D. C., Owen, F. N., & Eilek, J. A., 1989, ApJ, 347, 713 Houck, J. R., et al., 2004, ApJS, 154, 18 Iye, M., et al., 2004, PASJ, 56, 381 Kataza, H., Okamoto, Y., Takubo, S., Sako, S., Nakamura, K., Miyata, T., & Yamashita, T., 2000, Proc. SPIE, 4008, 1144 Leahy, J. P., 1991, in Beams and Jets in Astrophysics (Cambridge), p. 100 Marconi, A., Axon, D. J., Macchetto, F. D., Cappetti, A., Sparks, W. B., & Crane, P., 1997, MNRAS, 289, L21 Mason, R. E,. Geballe, T. R., Packham, C., Levenson, N. A., Elitzur, M,. Fisher, R. S., & Perlman, E., 2006, ApJ, 640, 612 Nagar, N. M., Falcke, H., Wilson, A. S., Ho, L. C., 2000, ApJ, 542, 186 Narayan, R., & Yi, I., 1995, ApJ, 452, 710 Narayan, R., Yi, I., & Mahadevan, R., 1996, A & A S, 120, 287 Ogle, P., Whysong, D., & Antonucci, R., ApJ, 647, 161 Perlman, E. S., Sparks, W. B., Radomski, J., Packham, C., Fisher, R. S., Piña, R., & Biretta, J. A.,, 2001a, ApJ, 561, L51 (P01a) Perlman, E. S., et al., 2003, ApJ, 599, L65 Perlman, E. S., Biretta, J. A., Sparks, W. B., Macchetto, F. D., & Leahy, J. P., 2001b, ApJ, 551, 206 (P01b) Perlman, E. S., & Wilson, A. S., 2005, ApJ, 627, 140 Radomski, J., Packham, C., Levenson, E., Perlman, E. S., De Buizer, J., Leeuw, L. L., Mason, R., Matthews, H., Telesco, C., Orduna, M., 2007, ApJ, submitted Reynolds, C. S., di Matteo, T., Fabian, A. C., Hwang, U., & Canizares, C. R., 1996b, MNRAS, 283, L111 Rieke, G. H., et al., 2004, ApJS, 154, 25 Risaliti, G., & Elvis, M., 2004, ASSL Vol. 308: Supermassive Black Holes in the Distant Universe, 187 Sako, S., et al., 2003, PASP, 115, 1407 Shi, Y., et al., 2006, ApJ, 653, 127 Shi, Y., Rieke, G. H., Hines, D. C., Gordon, K. D., Egami, E., 2007, ApJ, 655, 781 Siebenmorgen, R., et al., 2005, A&A, 436, L5 Sparks, W. B., Biretta, J. A., & Macchetto, F., 1996, ApJ, 473, Sparks, W. B., Fraix-Burnet, D., Macchetto, F., & Owen, F. N., 1992, Nature, 355, 804 Stiavelli, M., Biretta, J., Möller, P., & Zeilinger, W. W., 1992, Nature, 355, 802 Sturm, E., Lutz, D., Verma, A., Netzer, H., Sternberg, A., Moorwood, A. F. M., Oliva, E., & Genzel, R., 2002, A&A, 393, Sturm, E., et al., 2005, ApJ, 629, L21 Tan, J. C., Beuther, H., Walter, F., & Blackman, E. G., 2007, ApJ, submitted (astro-ph/0610488). Tonry, J. L., 1991, ApJ, 373, L1 Tsvetanov, Z. I., et al., 1998, ApJ, 493, L83 Urry, C. M., & Padovani, P., 1995, PASP, 107, 803 Willner, S. P., Elvis, M., Fabbiano, G., Lawrence, A., & Ward, M. J., 1985, ApJ, 299, 443 Whysong, D., & Antonucci, R. R. J., 2004, ApJ, 602, 116 http://arxiv.org/abs/astro-ph/0610488

0704.1157

Fermionic construction of tau functions and random processes

arXiv:0704.1157v1 [math-ph] 9 Apr 2007 Fermionic construction of tau functions and random processes 1 J. Harnad†‡2 and A. Yu. Orlov⋆3 † Centre de recherches mathématiques, Université de Montréal C. P. 6128, succ. centre ville, Montréal, Québec, Canada H3C 3J7 ‡ Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montréal, Québec, Canada H4B 1R6 ⋆ Nonlinear Wave Processes Laboratory, Oceanology Institute, 36 Nakhimovskii Prospect Moscow 117997, Russia Abstract Tau functions expressed as fermionic expectation values [14] are shown to provide a natural and straightforward description of a number of random processes and statistical models involving hard core configurations of identical particles on the integer lattice, like a discrete version simple exclusion processes (ASEP), nonintersecting random walkers, lattice Coulomb gas models and others, as well as providing a powerful tool for combina- torial calculations involving paths between pairs of partitions. We study the decay of the initial step function within the discrete ASEP (d-ASEP) model as an example. 1 Introduction Random processes and integrable systems are often regarded as opposites extremes in the study of dynamical systems, since the first, by its very definition involves indeterminacy and prob- abilistic considerations, while the second represents deterministic evolution with the highest degree of coherence in time and space. However if we consider the simplest example of motion of a free quantum particle we find that this simplest model combines integrability with random- ness. The Brownian motion and the motion of quantum particles may be described by similar tools, say, in terms of correlation functions and integrals over paths. If we consider quantum integrable systems, beginning with the earliest studies on spin sys- tems solvable via the Bethe ansatz, it is clear that, like all quantum systems, these involve prob- abilistic considerations, involving averaging and statistical correlations, and therefore such sys- tems combine integrability with randomness. In fact, many are solvable using the same methods as certain related statistical systems since the Hamiltonians may be embedded in the same fam- ily of commuting operators as the transfer matrices governing solvable statistical models. Later, starting with [21], [22], the success was achieved in applying methods of Bethe ansatz to var- ious driven-diffusive and non-equilibrium systems like asymmetric simple exclusion process (ASEP). 1Work of (J.H.) supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds FCAR du Québec; that of (A.O.) by the Russian Academy of Science program “Funda- mental Methods in Nonlinear Dynamics” and RFBR grant No 05-01-00498. [email protected] [email protected] http://arxiv.org/abs/0704.1157v1 We shall focus on the relations of such systems to classical integrable systems rather than to quantum ones. It was first shown in the work of McCoy, Tracy and Wu on the Ising model that certain cor- relation functions in solvable statistical models satisfy Painleve equations, which naturally arise in integrable systems under scaling reductions. In the series of papers by the Kyoto school, a systematic approach to integrable systems based on fermionic constructions of tau functions was developed, in which remarkable links between the quantum and classical interpretations appeared. Classically, a tau function may be seen as a sort of universal “potential” for families of integrable equations like the KP and TL hierarchies. (In the Hamiltonian setting, it may be interpreted as Hamilton’s principle function evaluated along the integral manifolds mapped out by a complete set of commuting flows.) At the same time, it has an interpretation as a certain correlation function for models of free fermions. Although in the classical integrable systems context this formulation at first appeared to simply be an interesting reinterpretation of the previous approaches to classical integrable systems, based on isospectral flows of linear oper- ators, inverse spectral theory and the Zakharov-Shabat’s dressing method [66], it subsequently turned out to be a powerful tool linking soliton theory with many other fields of physics and mathematics. Tau functions have long been known to play a central role in the remarkable links found be- tween random matrix models and integrable systems [19], [16], [3]. Subsequent applications of tau functions were also found in combinatorics, probability theory, statistical physics and quan- tum chaos. (See, e.g., the series of papers by C.Tracy, G. Widom, A.Okounkov, A. Borodin, K.Johansson, J. Baik, P. Forrester, P. Zinn-Justin, M. Adler, P. van Moerbeke. In particular see [63], [46], [48], [9], [10], [11], [12], [33], [6], [15], [16], [17], [18], [4], [5]). Some of these links are quite subtle, while others are still not completely understood. They appear however to be quite central to the seemingly endless new applications of ideas and methods originating in the modern theory of integrable systems to these other domains. In this work, we present some direct links of the fermionic construction of tau functions as developed in [14], [32] with certain types of random processes and combinatorial problems that arise in relation to partitions. As in [14], [32], we use the language of free fermions, Maya diagrams and Young diagrams. This “quantum-like” formulation turns out to be very fruitful in providing natural links between many stochastic systems and classical integrability, as well as suggesting new methods for approaching the combinatorial aspects common to these problems. Suitably interpreted, it will be shown to provide a natural and straightforward description of a number of random processes and statistical models. This approach is based on relating a natural combinatorially defined basis for the fermionic Fock space to states of the random system. To each basis Fock vector is associated, both a 1D configuration of a lattice gas (via its “Maya diagram”) and a Young diagram, which may be viewed as a region of 2D hard core particles. This provides a fermionic setting for well-known models like asymmetric simple exclusion processes,1D- lattice log gases subject non-intersecting lattice walkers and models of fluctuating interfaces. Other work along similar or related lines may be found, e.g. in the work of Okounkov, Olshansky, Borodin, Reshetihin, and others. The specific random process or equilibrium statistical model is determined by specification of the tau function and the interpretation of time variables. A central tool consists of the well- known [60], [61, 62] Schur function expansions of tau functions, τ(t, t̄) = τλ,µsλ(t)sµ(̄t), (1) where the sum is over all pairs of partitions λ, µ, which may be deduced very simply from the fermionic representation. We show that the tau function τλ,µ may be interpreted as a generating function for the (unnormalized) transition probabilities between states of certain models of stochastic systems. After a suitable normalization, the procedure also leads to discrete analogs of models of random orthogonal matrices. In the theory of integrable systems the variables t and t̄ are known as “higher times” de- scribing the simultaneous evolution under commuting flows, and providing solutions to inte- grable “hierarchies”, such as the Toda lattice TL and KP systems. In the present work, these variables are either chosen to have specific values, thereby defining the statistical weight of a given configuration, or just play the role of formal expansion parameters which, through the Schur function series allow the tau function to be interpreted as a generating function for transition probabilities. We use the generating tau function for constructing models of random walks of non-intersecting (”hard core”) particles along one dimensional lattice. We obtain two types of models intro- duced by M. Fisher [15]: (a) random turn models and (b) vicious walkers (whose particular case M.Fisher called lock step models). In models (a) at each tick of the clock a randomly cho- sen walker takes a random step, while in models (b) a certain number of walkers take random step in each time instant. In both types of models each site may be occupied by only one walker at the same time. The alternative approach to the same models is to consider the related Young diagrams, which are 2D figures which are in one-to-one correspondence with 1D configurations of hard core particles. This yields models of oscillating domains. Models (a) will be related to the repeating actions of elements of the so-called ĝl(∞) Lie algebra to Fock vectors and tau functions. The number of these actions is identified with the discrete time T parameter of the stochastic process. Models (b) will be related to the repeating action of elements of the related ĜL(∞) group. The important role of group ĜL(∞) and its algebra in theory of integrable systems was discovered in [14], [32]. The structure of the paper is as follows. Second part of the introductory section is devoted to the general conception of our approach. Here we introduce a time-dependent normalization function Zν(T) which plays a central role in the approach. In section 2 we consider models of type (a). After necessary preliminaries we come to subsection 2.3 where we relate a graph to each Toda lattice tau function and consider a certain random turn walk along this graph. Here notions of path and of permutation factor of a path are introduced. In subsection 2.4 we show that each tau function is a generating functions which counts differences of weights for two types of random motion along the related graph, which are notable by a value of a permutation factor of their paths. In subsection 2.5 we consider known models of Coulomb lattice gas in thermodynamical equilibrium, see [17] for a review. The Coulomb potential is modified in a way that the vacuum configuration (the step function presenting the Fermi surface) has minimal energy. We point out a two fold relation of these models to classical integrable systems. In subsection 2.6 we introduce a model of random turn walkers in external potential which may be viewed as a discrete-time version of ASEP (d- ASEP) with variable site-depending hopping rates and compare related normalization function with the partition function of a model of Coulomb gas in thermodynamical equilibrium. We show that Z0(T) may be viewed as a certain correlation function in a discrete model of a random ensemble of orthogonal T × T matrices. For constant hopping rate we find the asymptotic configuration in the large T limit. We found that the limiting shape coincides with one obtained by Vershik and Kerov in [38] in the study of a limiting shape of random partitions with fixed large area which are distributed according to the so-called Plancherel measure. In our case the size of asymptotic configuration depends on the hopping rate as it is given by formula (92). In subsection 2.7 we introduce additional dependence of the hopping rates which now depend on a mutual position of particles via the modified Coulomb potential with an arbitrary charge and show how it corrects the size of the asymptotic configuration. In section 3 we consider models (b). In subsection 3.1 we show how the so-called determinantal ensembles [13] appear as a result of actions of elements of ĜL(∞) group on tau functions. In subsection 3.2 we mark the convenience of usage of multi-component fermions which allows in a simple way to construct fermionic representations for models of vicious walkers in an external potential and via Wick’s theorem yields answers for correlation functions in form of determinants. In subsection 3.3 we consider models of ”random layering” and models of ”rocks” (figures which generalize diagrams of the so-called 3D partitions) which we believe were basically known in certain versions, for instance, as the so-called polynuclear growth model. Here again we introduce the external potential which results in a variable site-dependent rate (this is the hopping rate in 1D picture and a gluing rate in 2D one). The normalization function of some of these models coincides with the tau function of Pfaff lattice [4], [35]. Except the consideration in subsection 2.6 we restrict ourselves to the presentation of var- ious models of random processes arising from integrable systems more than to the analysis of random models themselves which is mainly an analysis of the large time behavior of dif- ferent correlation functions. As it was very successfully demonstrated in series of papers by A.Okounkov the usage of fermionic language may be quite helpful for such purposes. This part of work will be published separately. 1.1 Action on Fock vectors. Time-dependent normalization function Zν(T) We refer the reader to Appendices A.1- A.3 for a compendium useful facts and conventions regarding fermions, partitions and Schur functions that are used repeatedly in what follows. For present purposes, this involves only charged, one-dimensional free fermions satisfying the standard canonical anti-commutation relations fif̄j + f̄jfi = δij, fifj + fjfi = 0, f̄if̄j + f̄j f̄i = 0, i, j ∈ Z (2) These act on a fermionic Fock space whose “vacuum state” |0〉 is, like the Dirac sea, annihilated by all negative component creation operators fi and positive component annihilation operators f−i−1|0〉 = f̄i|0〉 = 0, i ≥ 0 (3) More generally, we have the “charge n” vacuum states |n〉, |n〉 = fn−1 . . . f0|0〉 for n ≥ 0, |n〉 = f̄n . . . f̄−1|0〉 for n < 0, (4) where the integer n denotes the Dirac “sea level”. We may visualize the level - n Dirac sea as a diagram consisting of a set of integer sites on a vertical axis, with all sites below n occupied, while the n-th site and all those above it are empty, as in Fig.1. A complete set of basis vectors for the Fock space may be associated with Maya diagrams, which we shall denote by Greek letters λ, ν, . . . , to be more precise, by two letters, Greek and Latin, the first denotes the configuration of particles, the second denotes the level of the Dirac 1. Dirac sea of level n 2. One particle one hole Figure 1: Maya diagrams sea, which we perturb by placing a finite number of particles of the Dirac sea to the upper positions. In this way all basis vectors of Fock space may be obtained. Since no two particles can occupy the same site, each Fock basis vector may be encoded by a partition, these are the ordered sets of weakly decreasing numbers, say, ν = (ν1, ν2, . . . ) where ν1 ≥ ν2, · · · ≥ 0, and by a level, say n, related to the coordinates of the particles, h1 > h2 > · · · , as follows: νi = hi + i− n (5) We number (identical) particles in downward direction, the highest one be the first. Provided we view each Fock vector as being created by a perturbation of vacuum (’initial’) vector, one may say, that νi measures the deviation of a current (’Lagrangian’) coordinate, hi, of the i-th particle from its initial value, n− i, in the vacuum state. The partitions are very useful and conventional notations, that is why we need them. There are two suitable way to figure partitions: via Maya diagram, which is just configuration of the particles h = (h1, h2, . . . ) placed on the vertical line, and via Young diagram which is a set of ν = (ν1, ν2, . . . ) related to h = (h1, h2, . . . ) by (5). They also yields links with the topic of growing interfaces, since the diagram of each partition (Young diagram) may be viewed as a discrete model of a two-dimensional region. A set of partitions placed one above another in three-dimensional space may be a discrete model of 3D region, and so on. A particular case of such 3D figure, called plane partition, was considered in the papers of Okounkov and Reshetikhin [48], [49] as a certain statistical model. In the present paper we consider a sequence of Fock vectors obtained by application of operators oi,i−1 to an initial basis Fock vector, say, |ν〉, as follows |ν, n〉 → o1,0|ν, n〉 → o2,1o1,0|ν, n〉 → · · · → oT,T−1 · · · o1,0|ν, n〉 (6) This sequence may be viewed as a T-step process on the space of configurations of hard core particles (1D lattice gas) where each configuration is related to some basis Fock vector and may be visualized via the so-called Maya diagram, see the picture. The number T will be related as discrete time of the process. Alternatively, (6) may be viewed as a T-step process on the Young diagrams (a type of 2D regions). In what follows everywhere, where it will not be not confusing, we shall omit the label related to the level of configuration (or, the same, the level of Dirac sea), keeping only Greek letters, say, λ, µ ,ν for configurations of the hard core particles, having in mind the pairs (λ, l), (µ,m) ,(ν, n). The result of the application of o(T) := oT,T−1 · · · o1,0 (7) to the initial state ν (corresponding to an initial configuration of hard core particles), is a Fock vector which is a linear combination of basis Fock vectors (=configurations of hard core par- ticles). One of the objects of an interest is the relative weight of a given configuration, say, Wν→λ(T) = 〈λ|o(T)|ν〉 (8) with respect to the weight of all possible configurations, Pν→λ(T) = Wν→λ(T) Zν(T) where the normalization function in the denominator is Zν(T) = 〈λ|o(T)|ν〉 (10) the sum ranges over all possible configurations. Matrix element j,j−1 λ,µ = 〈λ, l|o j,j−1|µ,m〉 (11) will be referred as (one-step) transition weight between a configuration |µ,m〉 and a configura- tion |λ, l〉 at moment j. If we want to keep in mind the probabilistic interpretation of the process we ask transition weights to be positive. The completeness of the basis Fock vectors implies µ,m |µ,m〉〈µ,m| = 1. We have 〈λ, l|oT,T−1 · · · o1,0|ν, n〉 = {µ(i),mi} T,T−1 λ,µ(T−1) · · · o1,0 µ(1),ν where one sums over all possible intermediate configurations µ(i), m(i), i = 1, . . . , T − 1. Each monomial non-vanishing term in the sum is related to a certain set of configurations, ν, n → µ(1), m(1) → · · · → µ(T−1), m(T−1) → λ, l , (13) which will be refereed as a path from the configuration ν, n to the configuration λ, l, having length (or, the same, duration) T. The value of the monomial is referred as the weight of the path, which is equal to a product of all one-step transition weights (11) along the path. Let us notice that it may have sense to evaluate Zν(T) also for the case where the positivity condition, j,j−1 λ,µ ≥ 0, (14) is violated and therefore we have no a probabilistic interpretation of (12) as a random process on the space of configurations of the hard core particles. In certain examples considered at the present paper transition weights (11) take the form 〈λ, l|oj,j−1|µ,m〉 = e−Uλ,l(j)+Uµ,m(j−1) (15) The case where the weight of each path depends only on the end points of the path, i.e. on initial and on final configurations shall be referred as potential weights. In what follows we shall consider only potential transition weights. To compare with, let us assign to each configuration of hard-core particles, λ, an energy Uλ. What people are interested in statistical physics is the partition function of the system e−Uλ (16) (everywhere we shall put temperature times the Boltzmann constant to be one). In case of random process the normalization function (10) - which yields the sum of weights of all pathes of durations T which start at the configuration ν - plays the role similar to the role of partition function in thermodynamics. For potential transition weights the normalization function takes the form Zν(T) = e−Uλ(T)+Uν(0) Nλ,ν(T) (17) where Nλ,ν(T) counts the number of pathes of duration T. Given T, this factor may be also included into the exponent to define a partition function of a T-dependent statistical model. At the present paper we shall focus to figure out the links of random processes (6) to clas- sical integrable systems, random matrices and growing surfaces. Benefits of these links will be presented in our forthcoming papers [29], [30]. Our main example is a discrete version of asymmetric simple exclusion process (d-ASEP), where we present the answer for the decay of the step function. In short we shall consider random walk of N non-intersecting particles and other examples. The mentioned links of random processes with classical integrable systems are two-fold. The first one is the usage of series (1) as generating functions for transition weights between different configurations of the hard-core particles. Then higher times t and t̄ are just formal parameters. There are different tau functions, related to A, B, D and C root systems; the way of construction of these may involve different type of fermionic (or, bosonic) operators. At the present paper we deal with A type, which may be realized in terms of fermions (2). We show that A-type tau function may generate certain classes of transition weights (11). We consider two ways of generating weights, see below. This is one link between random processes and integrable systems. The second link is the fact that the normalization functions Z0(T) may be related to a tau function of some other classical integrable systems. It occurs if we consider deformations of transition weights with a set of deformation parameters t̃ = (t̃1, t̃2, . . . ) 〈λ, l|oj,j−1|µ,m〉 → 〈λ, l|oj,j−1(t̃)|µ,m〉 = e−Uλ,l(j,t̃)+Uµ,m(j−1,t̃)〈λ, l|oj,j−1|µ,m〉 (18) specified later by (58) and caused by transforms of operators oj,j−1 of form oj,j−1 → oj,j−1(t̃) = eH0(t̃)oj,j−1e−H0(t̃), j = 1, . . . , T (19) (operators H0 are defined in Appendix A.3). These deformation parameters may be recognized as higher times of different underlying integrable equations. These are dual integrable hierar- chies in sense of papers [50] and [54] but not exactly4. In many cases, under flows (19), the normalization function Z0(T, t̃) of (10) is a tau function of B-type, namely, the B type in the realization of Kac-van de Leur, see [34] (also known as Pfaff lattice independently introduced as a tool for study of random matrices see [4], [40]). Thus, what we mean is a sort of duality between A and B type of equations. The last remark is that we may consider a sort of coupling of random processes with statis- tical models on the space of configuration, 2Eλ , (20) where q is some constant (charge) as follows. We use the energy q2Eλ of the model to modify each transition weight: 〈λ|oj,j−1|µ〉 → e−q2Eλ+q2Eµ 〈λ|oj,j−1|µ〉, j = 1, 2, . . . , T (21) Zν(T) = 〈λ|o(T)|ν〉 → 2Eλ+q 2Eν 〈λ|o(T)|ν〉 (22) As we shall show in subsection 2.7 this procedure may keep relations with integrable sys- tems if q2Eλ is chosen in a special way. 2 Action on Fock vectors via ĝl(∞) algebra Maya diagrams Let us consider an infinite set of vertexes labelled by integers. Each vertex will be sketched as either a white or black disk. Following Sato, we shall call it Maya diagram, (after a Maya game in which small stones are moved to occupy free sites according to a certain rule). In what follows we shall describe certain rules for motion of “stones”- hard core particles. The Maya diagrams for the vacuum state are presented in Figure 1.2. We may consider each black ball as a particle and each white ball as the lack of a particle at this site (empty site). The Maya diagram is a way of visualization of basis Fock vectors for fermions on the circle, sites correspond to Fourier components of the fermions, no two fermions can occupy the same site; therefore, there is one site for at most one particle (these are what is called hard core particles). Sites with positive large enough numbers are supposed to be free, while sites with large enough negative numbers are supposed to be occupied, that is the related Maya diagrams possesses this property: all sites to the up of a certain number are all white, while sites to the down of a certain number are all black. The Maya diagram where all sites below n-th one are occupied, and all sites above n− 1 are free, is called the vacuum Maya diagram of level n, see the Figure 1. It represent the so-called Dirac sea of level n. 4This duality might be related in a general notion of duality of integrable systems developed in [2], [24], [25], Each Maya diagram may be obtained from the vacuum diagram of a certain level, say, n, via placing of certain number, say, k, of black balls (particles) to higher empty sites. We say that each Maya diagram has level n. Basis Fock vectors, partitions and Maya diagrams By its origin Maya diagrams is a visualization of vectors in the fermionic Fock space, see Appendix A.1. Each black ball (a particle) located at a site i indicates that the site is occupied. Let us introduce the following basis vectors in the fermionic Fock space |λ, n〉 = (−1) i=1 βifn+αk f̄n−1−βk · · · fn+α1 f̄n−1−β1|n〉, α1 > · · · > αk ≥ 0, β1 > · · · > βk ≥ 0 where λ = (α1, . . . , αk|β1, . . . , βk) is the Frobenius notation for the partition λ, and n is the level of Dirac sea, see Introduction and Appendix A.1 for more details. This vector is one-to-one related to the Maya diagram of level n, where particle are situated as follows. A particle number i (black ball) counted from the top has a coordinate n + αi, i = 1, . . . , k. Thus, the upmost particle has the coordinate α1. Underneath of the sea level all sites are occupied except free sites (holes) which we count from the bottom. Hole number i has a coordinate n− 1− βi, thus the down-most hole has the coordinate n− 1− β1. Notice that β1 + 1 is the length of partition lambda, ℓ(λ). One may give an alternative description of coordinates of particles on the Maya diagram related to the vector |λ, n〉. Take any integer N which is not less than ℓ(λ). Then all site with coordinates n−N − 1, n−N − 1, . . . are occupied. On the sites n−N, n−N + 1, . . . there are N particles which occupy sites with coordinates hi = λi − i+N, i = 1, . . . , N, N ≥ ℓ(λ) (24) All other sites including and above n−N are empty. Dual vector is 〈λ, n| := (−1) i=1 βi〈n|f−1−β1 f̄α1 · · · fn−1−βk f̄n+αk · · · (25) One can verify the orthonormality condition 〈λ, l|ν, n〉 = δl,nδλ,ν (26) Along these notations we shall identify partitions, Maya diagrams and basis Fock vectors. Being identical to basis Fock vectors, Maya diagrams may be multiplied by numbers and may form (formal) linear combinations. Action of ĝl(∞) on Maya diagrams It was old observation that quadratic operators of type fif̄k, i, k = 1, . . . , N , form gl(N) Lie algebra. In the papers of Kyoto school it was studied within symmetry analysis of integrable equations like Toda lattice equation. One of the observations made in the wonderful series devoted to hierarchies of integrable equations (in particular, see [14], [32]) is that N = ∞ case needs a special consideration. When there are an infinite number of fermionic modes one needs normal ordering. Then, natural combination are Ei,k := fif̄k − 〈0|fif̄k|0〉 −∞ < i, k < +∞ (27) which form an algebra called ĝl(∞) (the algebra of infinite matrices with central extension). We do not need properties of this algebra. Let us only note that such algebras play essential role in the construction of hierarchies of integrable classical (not quantum) equations, where the algebra of infinite matrices appears as an underlying symmetry algebra of integrable equations. This algebra naturally acts on the fermionic Fock space. For our purposes we are interested in the action of Ei,k on the basis Fock vectors, which may be viewed as an action on Maya diagrams. The point is that this action transforms one Maya diagram to the other one multiplied by a sign factor. Then any product of operators (27) do the same. Keeping in mind definition of basis Fock vector and applying (27) one can verify that the action of Ei,k on Maya diagram is as follows. First, let us notice that the second term in the definition of operator Ei,k is vanishing for i 6= k. Let us consider this case first. Each Ei,k acts on a Maya diagram trivially (as multiplication by zero of the related basis Fock vector) in two cases. They occur either if k-th site is empty, or/and if i-th site is already occupied by a particle as on Figure 2.1. Otherwise Ei,k action places black ball from its k-th site to the i site, multiplying the new Maya diagram by (−1)cik , where cik is a number of black balls between i-th and k-th sites, in other words, the number of particles which the particle originally located at k-th site ”jumps over” when moving to its new, i-th, site, see Figure 2.2. An arrow on this figure shows that the particle hope from the site k to the site i. .Ei,k 1. Elimination of Maya diagrams 2. Nontrivial action Figure 2: Elementary gl(∞) actions on Maya diagrams The action of operators Ei,i on Maya diagrams may be called testifying action. They acts trivially on the vacuum Maya diagram of level 0. For i < 0 the operator Ei,i eliminates any Maya diagram which (similar to the vacuum diagram) contains a particle at the site i < 0. If the site i is empty then Ei,i acts as the multiplication by unity. For i ≥ 0 the operator Ei,i eliminates any Maya diagram which (as the vacuum diagram of the level 0) contains the white ball at the site i. If there is a particle at the site i then Ei,i acts as multiplication by unity. Because Maya diagrams are in one-to-one correspondence with basis Fock vectors we can act on them by sums and products of operators (27). The element Ei,i (28) plays a special role. One can easily check that operator Q acts a Maya diagram of a level n by the multiplication by n. It is called the charge operator. Graphs Let us take vertices of a Maya diagram as vertices of a graph, where vertices are labelled by integers and each pair of vertices may be connected by a pair of opposite directed arrows. Given i 6=k ai,kEi,k ∈ ĝl(∞), (29) we assign the weight ai,k to the arrow which starts at the vertex k and ends at the vertex i. For the sake of simplicity, we shall not consider diagonal terms, Ei,i. Also let us not draw arrows of zero weight. In this way we obtain some weighted graph, related to given A =∈ ĝl(∞). For instance, for i=n−2 (Ei+1,i + Ei,i+1) (30) we obtain the graph on Figure 3.1. 1. Graph Figure 3: Maya diagrams projected on graphs In the present paper we are interested in motion of particles along arrows of the graph whose vertices coincide with vertices of Maya diagram. Having in mind this picture one may forget about all vertices of Maya diagrams which are not linked by vertices. For instance, the Maya diagrams depicted on Figures 3.2 and 3.3, projected on the graph depicted on Figure 3.1, respectively, are as on Figures 3.4 and 3.5. If the number of arrows is finite we shall ignore the rest part of a Maya diagram. 2.1 Young diagram versus Maya diagram. Interface between 2D area versus configuration of particles on the 1D lattice For discrete systems the difference between 1D and 2D systems of hard core particles is rather conventional in the following sense: any 1D lattice gas configuration may be converted into a Young diagram viewed as an example of a condensate of hard core 2D particles. Let us figure out an example: These two figures show the relation between the motion of particles on Maya diagrams and a motion of the interface. One may say, that each hop of a 1D particle upward is related to an increasing of the area of the corresponding Young diagram (a condensation of 2D particles to the Young diagram), while 1. Upward/downward hop of a particle on a Maya diagram 2. Adding/removing a box to a Young diagram Figure 4: Equivalent representations of elementary steps a hop downward results in the inverse process of ’evaporation of particles’ and the decreasing of the area the Young diagram. (A figure similar to Fig. 4.2 may be found in [14] without relations to the topic of the present paper: to stochastic motion.) 2.2 Generating vector for Maya diagrams There are vectors, coherent states, 〈n|eH(t) and eH̄(t)|n〉, which depend on parameters t = (t1, t2, . . . ), see Appendix A.3, and which generate all Fock space with fixed given sea level n, 〈n|eH(t) = 〈λ, n|sλ(t), eH̄(t)|n〉 = |λ, n〉sλ(t) (31) where sλ(t) is the Schur polynomial [42]. The Schur polynomial is a polynomial in many variables t1, t2, . . . , which are labelled by partitions. The appearance of the Schur polynomial is typical for integrable systems. 2.3 Random turn motion on graphs. Permutation factor and weight of Consider the graph related to i 6=k ai,kEi,k, Ei,k = fif̄k , i 6= k (32) which is a set of vertices, some of which may be linked by arrows: an arrow (i, k) starts on a vertex k and ends on a vertex i. We assign a weight ai,k to each arrow (i, k). In case ai,k = 0 there is no arrow. The matrix a with entrances ai,k is called connecting matrix for a graph. For simplicity of consideration we omit diagonal terms. We shall consider a discrete dynamical motion along a Maya diagram with an initial dia- gram ν given by |ν〉 → A|ν〉 → · · · → AT|ν〉 (33) Here T is viewed as discrete time variable. As we have explained in the Introduction we have different paths connecting two different configurations, see (13); each path is related to a sequence of intermediate partitions. Consider AT which is a sum of monomials, each one is a product of T terms aiα,kαEiα,kα, α = 1, . . . , T. Each path of process (33) is generated by a unique monomial applied to a initial Maya diagram. The number T may be referred as duration of path. It may be described as set of con- sequent events, single hops. Each time instant, say, α, a particle hops along an arrow (iα, kα): one hop at each time instant. Say, monomial aiT,kTEiT,kT · · · ai1,k1Ei1,k1 , provided it acts non- trivially on ν, at first time instant describes the hop of a particle along arrow (i1, k1) and at last time instant, T, describes the hop of particle along an arrow (iT, kT). The product weights of T arrows which were crossed by particles, α=1 aiα,kα, is called the weight of path. Each pair of configurations may be connected by a number of paths each of which has its own weight. The fact that AT is a sum of weighted monomials means randomness of the motion de- scribed by (33). Let us call it random turn motion along a graph, given by A. The random motion where in each time instant only one chosen at random particle hops is called random turn motion. To each path we shall assign permutation factor as follows. Given acting nontrivially monomial, we can follow each particle involved into the motion along graph. Let us enumerate particles located on the initial Maya diagram the from the top of diagram, and keep these (personal) numbers of particles on the target Maya diagram (where each particle ’remembers’ it’s given number). Let us re-enumerate particles of the target Maya diagram from the top: this may be also obtained by a certain permutation of personal numbers which we assigned to the particles in the beginning of the motion. If the sign of the permutation is even then the permutational factor of the path is equal to +1. If the sign is odd the permutation factor of the path is equal to −1. Thus, to a each path we assign a weight, a permutation factor, and a duration. 2.4 Graphs and tau functions. Tau function as counting function Consider the random turn motion described in the previous subsection. Consider all paths of duration T starting with a configuration described by a partition λ′ and ending on a configuration λ. The sum of weights of all such paths with positive permutation factor we denote by λ′→λ(T) (34) The sum of weights of all such paths with negative permutation factor we denote by λ′→λ(T) (35) Let us note that permutation sign appears due to the fermionic approach to the problem. In problems of random motion of hard-core particles along graphs which know nothing about fermions, one may wonder about a total weight, which is Wλ′→λ(T) = W λ′→λ(T) +W λ′→λ(T) (36) From the previous consideration we have 〈λ, n|AT|λ′, n〉 = W (+)λ′→λ(T)−W λ′→λ(T) (37) By (31) we relate these numbers to the so-called TL tau function [14], [64] in the following τA(n, t, t̄) := 〈n|eH(t)ezAeH̄(t̄)|n〉 = λ′→λ(T)−W λ′→λ(T) sλ(t)sλ′(t̄) (38) where A is i 6=j ai,jfif̄j (39) and z, t = (t1, t2, . . . ), t̄ = (t̄1, t̄2, . . . ) are formal parameters, and fi, f̄i are free fermions. H(t) = Hmtm, H̄(t̄) = H−mt̄m, Hm = fif̄i+m (40) Thus, we say, that the power of z counts the number of steps. T-step process is |ν〉 → A|ν〉 → · · · → AT|ν〉 (41) The variable T is treated as the (discrete) time of the random process described by A. The most simple case is random processes where λ′→λ(T) = 0 (42) for all pairs λ′ λ. Then tau function may be interpreted as counting function for weights Wλ′→λ(T). Then it follows that Wλ′→λ(T) is a subject to certain discrete bilinear equations (Hirota equations), which we shall write down not here. However we can weaken condition (42) keeping the interpretation of tau function as count- ing function. Consider two examples where particles may hop only to the nearest neighboring sties. One example is presented on the figures 5. Figures 5.1 and 5.3 depict graphs where ini- tial step function, ν = 0, configuration will decay. In figure 5.1 at each time instant a particle chosen at random will hop upward along an arrow if the neighboring place is not occupied. In figure 5.2, motion is impossible: in this configuration particles are locked. Each finite per- turbation of ν = 0 will come to ν = 0 in a final number of steps. In Figure 5.3 not only particles are chosen at random but also arrows are to be chosen at random: if the neighboring vertices are free, chosen at random particle may hop either upward or downward. It is called non-intersecting random turn walkers, (on this topic see [6], [17], [8] and [3]). These examples of random motion keep the condition (42) because it is ordered motion and the signs of each path is positive. For the choice of A as in Figures 5.1-5.3, namely, A = H1, H−1, H1 + H−1 the weight of any path is either zero, or one. It means that tau function is a generating function for number of ways to get configuration λ starting with a configuration λ′ in T steps. Second example is a walk on a ring, where particles are allowed to hop to the nearest neighboring site. How to obtain such graphs is depicted on figure 6: figures 6.3 and 6.4 are obtained respectively from figure 6.1 and 6.2. On figures 6.2 (or, the same 6.4) condition (42) is violated because when a particle hops along the arrow linking vertices n and n+5 permutation of particles is odd in case the total number of particles on the ring is even. Thus each path containing odd number of hops along arrows linking sites n and n+5 has negative permutation factor. Nevertheless if we assign positive weight to all arrows except arrows linking n and n+5 1. H−1 2. H1 3. H−1 +H1 Figure 5: Random turn motion along graphs ✟✟✙✟✟✯ ❍❍❨❍❍❥ t n ✟✟✙✟✟✯ ❍❍❨❍❍❥ t n Figure 6: For the case depicted in 2 and 4, in order to keep the probabilistic interpretation, one has to assign negative weights to the arrows that link n + 5 and n sites, tau function will count the transition weights between any pair of configurations. Say, if arrows linking n and n + 5 have weight −1, while all other arrows have weight 1 tau function counts number of ways linking any pair of configuration on the ring. Processes like those depicted in Figures 5 and 6 are also known as discrete versions of simple exclusion processes (totally asymmetric (d-TASEP) for figures 5.1, 5.2, and symmetric for figure 5.3), which may be considered either on the line as on figures 1-3, or on a ring as in the Figure 6. This type of models was introduced by M.E.Fisher. It may be called discrete time asymmetric simple exclusion process (d-ASEP) in case the probability rate for hops in the opposite directions are different and does not depend upon site numbers. The solvability of ASEP was proven by H. Spohn in [21, 22] by method of Bethe anzats and a huge literature is devoted to the study of different aspects of this model which, in non-equilibrium physics, plays a role comparable to the role of Ising model in statistical physics. The problem of evaluation of correlation functions, asymptotic behavior we hope to con- sider separately using fermionic approach. The dependence of transition probabilities on parameters (weights of arrows) will be con- sidered in subsection 2.6. Before this consideration of non-equilibrium system of particles we find that a consideration of fermions in thermal equilibrium will be helpful. 2.5 Fermions in thermal equilibrium Consider the following statistical ensemble: there is an infinite set of levels labelled by inte- gers. Each level i has energy Ui and may be occupied by a fermion. In thermodynamics the probability to occupy a site i is proportional to exp(−Ui), where Ui is energy level numbered by i (we put the Boltzmann constant times temperature to be one). In case Ui is a monotonous function of the site number we can use Maya diagrams of some level, say n. At zero tempera- ture all lower site are occupied up to this level (in solid state physics called Fermi level). The bottom limit of Maya diagram is fully packed by particles, while top limit is free of particles. A fragment of the picture is sketched out in the following figure i-2 exp (-Ui-2) i-1 exp (-Ui-1) i exp (-Ui) i+1 exp (-Ui+1) Fig.1. Fermionic levels where some sites are occupied Each configuration of fermions distributed among levels has its weight given by the product of the Gibbs factors related to the occupied sites. The contribution of the fragment depicted on the figure is e−Ui−2−Ui−1−Ui+1 . These are non-interacting fermions. The equilibrium ensemble is described by its partition function. First, let us introduce the notation which we shall use throughout the paper Uλ(n) = (Uλi−i+n − U−i+n) (43) Notice that for the vacuum configuration of any level n we have U0(n) = 0. The normalized partition function is then a sum over partitions e−F0 = e−Uλ(n) (44) where F0 is the free energy of the system of non-interacting fermions living on a Maya diagram of a level n. Remark. Let notice, that we can cut Maya diagram from the bottom via the following procedure. We send all energies of sites located bellow, say, site N , to minus infinity. Without the loss of generality we take N = 0. Then, we have in particular, eU−1−U0 = 0 (45) In this way we restrict our consideration by finite number of fermions, n. It is available in case our system is related to one-dimensional solid state system where there are finite number of particles. In such cases n will be the number of particles equal to Fermi level. If only n fermions are involved we have Uλ(n) = (Uλi−i+n − U−i+n) , (46) where λi − i is the coordinate of the particle numbered by i (i = 1, . . . , n) on a Maya diagram of a level n which is cut at zero site. We count particles from the top and their coordinates from the bottom. We shall consider both cases, N = 0 and N → −∞, the first will be described with the help of semi-infinite Maya diagram and the second by usual Maya diagram. Modified Coulomb potential Eλ. Now, suppose that fermions pair-wise interact, that is, now, the energy of the system is Uλ(n) plus the energy of pair-wise interaction. We choose this interaction as Coulomb interaction (if sites are interpreted as level under Fermi surface then, this is Coulomb interaction in the momentum space) q2 log(i− j), (47) which describes the repulsion of fermions of a charge q at sites i and j. The problem that Coulomb energy goes to +∞ for semi-infinite lattice. In order to avoid the divergency of the energy for semi-infinite lattice we shall modify it taking the energy of particles in a configura- tion λ in form Eλ = − log sλ(t∞) := log i<j(hi − hj)∏L i=1 hi! , hi = λi − i+ L, i = 1, . . . , L, L ≥ ℓ(λ) (48) Notation sλ(t∞) is taken from the Appendix A.1, see (177). Remark 2.1. This expression contains pair-wise Coulomb interaction (47) in the enumerator and an external electric potential in the denominator. As one may check this expression vanishes on the vacuum configuration for any value of L ≥ 0. Let us mark that L is not a parameter: the energy does not depend on the choice of L provided it exceeds the length of a partition. It is reasonable to recall the meaning of numbers i, λi and hi. The number i numerate particles counted in the downward direction. The number λi measure the shift of the particle i in the upward direction from it’s homesite in the vacuum configuration. Given L the number hi is the coordinate of the particle numbered by i which is counted from the origin located on the site n− L on the Maya diagram of level n. The point that the interaction (48) does not depend on the choice of this origin which one can send to −∞. It is perfectly adopted for the lattice gas with fully packed ”bottom” limit. Moreover, one can show that the vacuum configuration is the configuration with minimal energy equal to is zero. The related free energy will be denoted by Fq. In this case the partition function of the system is e−Fq(n) = e−Uλ−q 2 Eλ+q 2|λ| log(t1 t̄1) = e−Uλ (sλ(t∞)) (t1t̄1) |λ| (49) where we introduce dependence on axillary parameters t1t̄1 for further convenience. There are special cases, q2 = 1 and q2 = 2, where (49) may be identified with tau functions of quite different integrable systems. (The case q2 = 4 is also related to integrable systems, however we will not need it at present paper). We shall consider these cases separately. For q2 = 2 the slightly modified series (49) is equal to e−F2(n,t1,t̄1) := e−Uλ−q 2 Eλ+|λ| log(t1 t̄1) = e−Uλ(t1t̄1) |λ| (sλ(t∞)) = c−1n 〈n|et1H1e i≥0 Uifif̄i− i<0 Uif̄ifiet̄1H−1 |n〉 (50) which is a Toda lattice (TL) tau function. (Here cn is a normalization constant chosen in a way that the right hand side is equal to unity for t1 = t̄1 = 0, see (211)). The equality directly follows from (31) and (26) if we take into account (210) and finely use (173) and (177). This formula is true for both cases N = −∞ and N = 0. To obtain the last case we need keep in mind relation (45). Let us refer to the paper [41] where the Coulomb gas on the lattice was related to the tau functions of different type. In general, the following vacuum expectation value which is known to be an example of the Toda lattice tau function, namely τ(n, t, U, t̄) = 〈n|eH(t)e i≥0 Uifif̄i− i<0 Uif̄ifieH̄(t̄)|n〉 = cn e−Uλsλ(t)sλ(t̄) (51) yields the partition function for the system of fermions in a configuration |λ, n〉 whose pair-wise interaction (47) is replaced by interaction of all particles via the potential given by log sλ(t) + log sλ(t̄) (52) which, in general, is not of a pair-wise type. The second equality of (51) is derived from relations (31),(26) and (210), exactly in the same way as previous formula (50) but without specification of Schur functions. The pair-wise type interactions one may obtain via specifi- cation of parameters t and t̄ which enter the tau function (51), via formulas (179)-(176) of Appendix A.2. (Such specifications were considered in [52], [53], [54] and [50] in quite differ- ent contexts). Remark 2.2. To complete the talk about links between the Coulomb lattice gas and tau functions we note, that if we choose a special parametrization of the set {Ui, i ∈ Z}: via sets of variables t̃ := (t̃1, t̃2, . . . ) and t̃ ∗ := (t̃−1, t̃−2, . . . ) as follows: t̃m, i ∈ Z (53) we obtain (see [50]) that e−F2 is a TL tau function in variables of a certain dual hierarchy. Similar parametrization of the parameters U which enter tau function (50) was used in the context of completely different problems in [47] and in [1]. In [1] relation (53) has a meaning of the dispersion law for fermions in solid state physics, where i plays the role of momentum and Ui plays the role of energy of fermions. Authors thanks P. Wiegmann for explaining [1] before it was published. The case q2 = 1 is most important in view of its application in the next section, and is less trivial from the point of view of what it may be in integrable hierarchies. Given level of vacuum configuration n consider ̺n(U) := ℓ(λ)≤n 〈λ, n|e i≥0 Uifif̄i− i<0 Uif̄ifieH−1|0, n〉 (54) ℓ(λ)≤n e−Uλ(n)sλ(t∞) = cn h1>···>hn≥0 e−Uhi+U−i+n (hi − hj) where hi = λi − i + n. The second equality is obtained from formulas (210) and (206) and the last equality from (177) of Appendix A.2, and where cn is given by (211) (We shall omit this constant cn below). Then replacing summation over the cones h1 > · · · > hn ≥ 0 by sum- mation over non-ordered non-negative h1, . . . , hn which may be done due to the permutational symmetry of the terms (and getting the factor (n!)−1) we arrive at the partition function for n fermions with the electric charge 1: ̺n(U) = h1,...,hn≥0 e−Uhi+U−i+n |hi − hj | = e−F1(n) (55) Notice that we do not need condition (45) because the reduction to the n particle partition function is achieved by imposing the condition ℓ(λ) ≤ n. Remark 2.3. The interesting fact is the following. Let us write the grand partition function ̺(µ,U) = ̺n(U) (56) where µ is a chemical potential. Then one can show that this is an example of the infinite-soliton tau function of a version of the BKP hierarchy, suggested by V.Kac and J. van de Leur [35] (called them ”charged” BKP hierarchy). This BKP hierarchy is different from the BKP hierarchy presented in [14], [32]. The fermionic representation is a specification of the fermionic representation found in [40] in the context of the study of Pfaff lattice [4] and its generalizations. For the sake of simplicity we shall write down the fermionic representation only for ̺n(U), where n is an even number. It is ̺n(U) = n!〈n|eH(t̃)e i,j≥0 f(i)f(j)sign(j−i) −H̄(t̃∗)|0〉 (57) where f(i) = and whereH(t̃) and H̄(t̃∗) are given by (40) and sets of variables t̃ := (t̃1, t̃2, . . . ) and t̃ ∗ := (t̃−1, t̃−2, . . . ) are related to the set U := (U0, U1, . . . ) via Ui = 2 imt̃m (58) where two semi-infinite sets of parameters t̃m are called higher times of the coupled charged BKP hierarchy. Let us figure out the equidistant energy levels case: Ui−1 − Ui = −2t̃1 (59) which is described by all t̃m vanish except t̃1. Notice that (58) is not one-to-one correspondence, given sets t̃ and t̃ uniquely define the variables U but not vice versa. One need to make some comments on (57). We obtain it by developing the exponential term in the middle of the vacuum expectation value into the Taylor series. Then, as one can see, only 1 n-th term contributes. After one uses certain tricks invented in [40] he gets (57). In the same way, using results of [40] we obtain that e−Fq is also a tau function. We shall not write precise formulas for this case. Thus, for special values of electric charges, q = 1, 2 (and also for the case q = 2 which we omit in the present paper) we relate Coulomb particles to tau functions. 2.6 Random turn walk in an external potential. Decay of the step func- Let us describe a certain model. Consider the random turn walk related to the weighted graph given by A(U) = A+(U) + A−(U), (60) and where A+(U) = eUi−1−Uifif̄i−1, A−(U) = eUi−Ui−1fi−1f̄i, (61) We use the same letters U = {Ui} as in the previous subsection in order to pay attention to a certain similarity of the answers for probabilities to find particles in given configurations ob- tained in rather different problems: particles in the thermal equilibrium and particles subjected to the chosen random motion. According to the previous consideration our model is the following. There is the infinite graph, which is the one-dimensional lattice Z (which we view as a set of integer points on a vertical line), with all neighboring sites linked by pairs of oppositely directed arrows. Each site may be either empty, or filled by at most one particle. The ’bottom’ limit of this graph is fully packed by particles, while the ’top’ limit is free of particles. At each time step, in the ’middle’ of the graph we have some configuration of particles which are subject to random motion. In each unit time interval a particle chosen at random hops to the neighboring position provided it is free. It may hop either upward, or downward with different probabilities. The rate of the hop of a particle from the site i− 1 to the site i is given by up : r(i) = e−Ui+Ui−1 (62) while the rate of the hops from the site i to the site i− 1 is given by down : r∗(i) = e−Ui−1+Ui (63) which is the number inverse to (62). One may say that we consider random turn walking particles in an external field, which affects the rates of the hops. Notice that this model should be considered as a generalization of the so-called random turn model which was introduced in [15]. (See e.g., [8], [18], for other approaches.) A simplifying auxiliary picture may be the following. Each particle moves in the potential field given by U where Ui is referred as the potential of a site i. The rate of a hop depends on the difference of the potentials between final and target sites as given by (62) and (63). The alternative picture is obtained via the Young diagram of a Maya diagram. The process under consideration is a random changing of the shape of a Young diagram, viewed as random gluing and evaporating of boxes to it in such a way that only one box is added (or removed) at each time step. Each box has a ’gluing energy’ UR − UR−1, given by the distance, R, of the box to the main diagonal of the Young diagram. (This distance is called a content of a box of a Young diagram [42]). An example is given below, where a Maya diagram of level 0 and the related Young diagram are drawn (the Young diagram is drawn by bold lines). The distances (the ”contents”) of boxes marked by x on the figure are R = −1 for the lower box and R = 2 for the upper box (these numbers coincide with the height coordinates of black balls on the first figure). For boxes marked by *, as going up and right we respectively obtain R = −2, 0, 3. 1. Random turn walk of particles on a Maya diagram. 2. Random adding/removing a box to a Young diagram related to the up/downward hops of particles on Maya diagram. At unit time instant either a box has to be added at any of vacant places marked by star, or a box marked by x has to be removed. Figure 7: Two realizations of random turn walk Let us note that the one directed process of only adding of boxes (obtained from the general case by the imposing of condition that all Ui ≪ Ui−1) is well studied, then the problem is equivalent to the enumeration of the so-called standard Young tableau [42], see below. Let us evaluate transition probabilities. For the sake of simplicity we take Dirac level n to be zero and denote Uλ(n) by Uλ where we use notation (43). First of all let us note that the weight of the transition from an initial configuration |λ′〉 to the final configuration |λ〉 in T steps is given by the formula Wλ′→λ(T) = e Uλ′−UλNλ,λ′(T) (64) where Nλ,λ′(T) is the number of ways to come from the position λ′ to the position λ in T steps. This follows from the choice of the weight of a single hop in form (62)-(63) and from the definition of the weight. For the fermions in the state of the thermal equilibrium we have the partition function e−Uλ (65) Each Z−1e−Uλ yields the probability to find fermions in a state |λ〉. In our case of random turn walk, the hard core particles which start from a given initial configuration |λ′〉 are described via Zλ′(T) = eUλ′−UλNλ,λ′(T) (66) (One may call Nλ,λ′(T) kinematic entropy factor.) Then Pλ′→λ(T) = Zλ′(T) −1eUλ′−UλNλ,λ′(T) (67) yields the probability to find the hard core particles in a state |λ〉 as a result of T-step random turn walk of an initial configuration |λ′〉 in an external potential {Ui, i ∈ Z}. In particular, in case the random turn walk starts with the vacuum configuration, then Z0(T) = e−UλNλ,0(T) (68) where Nλ,0(T) is the number of ways to gain the configuration |λ〉 in T steps if the initial configuration is the vacuum one. The probability to find the hard-core particles (subject to random turn walk in a potential {Ui, i ∈ Z}) in a configuration |λ〉 is P0→λ(T) = Z0(T) −1e−UλNλ,0(T) (69) yields the probability to find the hard core particles in a state |λ〉 as a result of T-step random turn walk of an initial configuration |λ′〉. As we shall show below Nλ,0(T) = 2 |λ|−T T−|λ| · sλ(t∞) (70) where sλ(t∞) is the Schur function evaluated in a special point, t∞=(1,0,0,. . . ), see Appendix A.3 (177). Let us obtain (68)-(70) via the fermionic representation. Notice that we take [A−(U), A+(U)] = 1 (71) which simplifies calculations. We are interested in the evaluation of Wλ′→λ(T) = τλ,λ′(T) = 〈λ|(A+(U) + A−(U))T|λ′〉 (72) which gives the weight of the random process where an initial configuration of hard core parti- cles |λ′〉 comes to the final configuration 〈λ| in T steps. By (71) we have ez(A++A−) = e 2 ezA+ezA− (73) At the present paper we shall study the simplest case: λ′ = 0. Then, our model describes a decay of the step configuration as a function of the discrete time T. Let us take into account that A−(U)|0〉 = 0. By (73), by Appendix A.5 and also by (177) we obtain 〈λ|ez(A++A−)|0〉 = e 2 z|λ| · e−Uλsλ(t∞) (74) Now let us decompose both sides in Taylor series in z to find the right hand side of (72) where λ′ = 0. For time duration T = 2m+ |λ| we obtain W0→λ(T) = T! e−Uλsλ(t∞) = T!2 |λ|−T T−|λ| · e−Uλsλ(t∞) (75) which, for U = 0, gives the number of ways to get a configuration |λ〉 starting from vacuum configuration |0〉 in T-steps (for T − |λ| = 2m is even and non-negative). The index m counts how many times particles, which are involved in the random process, move downward. The left hand side vanishes if T − |λ| is odd. Also, notice that the length of partition λ can not exceed the duration of motion T. The equality ℓ(λ) = T is related to the motion (or, the same, to the ”path” as was explained in the subsection 2.3) where a single (the upmost on the vacuum Maya diagram) particle moves, and each hop is upward one. Thus in this very case the distance it has pass (which is equal to the first part of the partition λ, or, the same, equal to the length of the first row of Young diagram) is equal to T. Let us consider different cases described by relation (75). (1) For λ = 0 the Boltzmann factor eU0 = 1 and the weight is equal to the number of ways to return back W0→0(T) = N0,0(T) = (T − 1)!! = e T log T+··· (76) where the last relation describes the large T limit. (2) The case T = |λ| corresponds to the non-stop forward motion: all jumps of the par- ticles are in upward direction, and described by W0→λ(|λ|) = 〈λ|A|λ|+ |0〉. If one turns to the description of the motion via Young diagrams, one may see that the number Nλ,0(|λ|) = |λ|!sλ(t∞) =: d(λ) (77) describes the number of ways to create a Young diagram of given shape λ by gluing at random box by box in such a way that each time we have a Young diagram, see fig.7. (In other words it is the number of the so-called standard tableau of the shape λ, see Appendix A.1 and for details see [42]). Thus we have W0→λ(|λ|) = e−Uλ |λ|! sλ(t∞) = e−Uλd(λ), (78) (3) The case ℓ(λ) = 1 means that the final state is a single particle configuration (however, in the middle of the process more particles may be involved, thus, the problem is different from the random walk of a single particle restricted to the half-line, which in T → ∞ limit yields Brownian motion on the half-line). Thus, in this case λ = (λ1) where λ1 is equal to the shift of the upmost particle from its position in the initial vacuum configuration. Suppose the hopping e−Ui+Ui−1 = r(i) = r (79) does not depend on i (which was figured out in (59), r = exp(−2t̃1)). As a result of sim- ple evaluation one shows that in T → ∞ limit the dominant term in the subset of ℓ(λ) = 1 configurations λ is related to the partition λ̃ = (λ̃1) given by λ̃1 = r 1− r2 (1− r2)2 The weight for one-particles configurations in large T limit near λ̃ is given by the formula which resembles formula for the Brownian motion W0→λ ≃ T! exp −(λ− r As we see the variance is given by T 4 . At last we note, that in case the rate (62) depends on site, then in a wide class of rates in large T limit λ̃1 may be evaluated as the solution of λ̃1 = r(λ̃1) T. For instance, for Gauss potential, Ui = i2, one obtains λ̃1 ∼ log T. (4) The case where T − |λ| is large enough. Via Stirling’s formula we obtain W0→λ(T) ≃ )e−Uλsλ(t∞), |λ| ≪ T (80) Remark 2.4. Let us notice that the sum in the right hand side of (68) may be considered as a certain correlation function of a discrete version of the orthogonal ensemble of random matrices. To see it we use the same trick as in [54]. First, according to (177) we write sλ(t∞) = i<j(hi − hj)∏ i=1 hi! , hi = λi − i+ T ≥ 0, i = 1, . . . , T (81) Then we use the fact that we summarize a symmetric in the variables hi function multiplied by the right hand side of (81). In this case we can use permutations of h1, . . . , hT to replace the sum over the cones h1 > · · · > hT ≥ 0 by sum over all hi ≥ 0 getting a factor 1 over T!. At last we obtain Z0(T) = c h1,··· ,hT≥0 T2 + T i=1 V (hi) i,j=1 |hi − hj | (82) h1,··· ,hT≥0 T2 + T −E(h,1) (83) τλ,0(T) (84) where c = 2− 2 and where V (hi) = −Uhi + UT−i − log Γ(hi + 1) + log 2 hi (85) and E(h, q), h = (h1, . . . , hT) is the electrostatic energy of T charges q located on the one-dimensional lattice and having coordinates h1, . . . , hT: E(h, q) = (Uhi − UT−i)− log 2 hi − q2 log |hi − hj |+ q2 log Γ(hi + 1) (86) The number of terms in sums (82) and (83) is finite thanks to the Gamma function insertion. However in T → ∞ limit, the sum ranges over all positive integers hi. Notice that E(h, q) vanishes on the vacuum configuration λ = 0. Terms in the right hand side of (86) are to be interpreted as follows: the probability rates given by U are related to the external electric potential (which may depend on a site coordinate), gives rise to the first term. The third term is the Coulomb interaction of particles with the unit charge. The last term has a meaning of an external electric field which provides the vanishing of the Coulomb energy of the vacuum configuration of the particles. Two last terms appeared due to the Schur function in the right hand side of (75) and descended from the hard core interaction of the particles. The (discrete) time of the random turn walk is equated with the total number of Coulomb particles, or, the same, with the ’size’ of orthogonal matrix whose eigenvalues are presented by the positive integers h1, . . . , hT. The τλ,0(T) gives the weight of the random process where the vacuum initial configuration of hard core particles decays to a final configuration given by coordinates h1, . . . , hT in T steps. Now let us turn to the problem of finding large T limit. In a usual way, we variate positions of particles in order to obtain the dominant term in sum (82). First let us remind that the length, say R, of the partition we are looking for does not exceed T. Introducing the density of particles σ ≤ 1 (which the number of particles for one site) as a function of the variable hi = λi − i+ R (the origin of Maya diagram is related to h = R) in a way ∫ ∞ σ(h)dh = R (87) we come to the equation for σ which defines the dominant configuration of particles at time T >> 1 in the continues limit r(h− T) σ(x)dx xσ(x)dx = 0 (88) where P stands for the principal value. Let us note that this equation may have no solutions. This equation one can solve in case the hopping rate (62) does not depend on site: r(i) = r, see (79). With the help of the formula arcsin dh = log (which may be extracted from [39], our proof the reader will find in the Appendix) we obtain the following solution to (88) σ(h) = arcsin , h ∈ [0, 2R] (90) σ(h) = 1, h < 0; σ(h) = 0, h > 2R, (91) where the length R of the partition is found by substitution of (90)-(91) into (88) and is equal R = 2 1 + r−2 , (92) see fig.8 which describes the decay of initial vacuum Maya diagram of level 0 in large T limit. Let us consider the partition λ = λ(T) and it’s Young diagram related to the asymptotic density function σ(h). Then solution (90)-(92) correspond to a Young diagram symmetrical under the reflection with respect to the main diagonal (this follows from the symmetry between particles and holes which exists in case of constant rate). The length of the partition ℓ (λ(T)), it’s area |λ(T)| and the number of boxes on the main diagonal k (λ(T)) are given by the following formulae ℓ (λ(T)) = R = 2 1 + r−2 |λ(T)| = hσ(h)dh− R 1 + r−2 1 + r−2 k (λ(T)) = 1 + r−2 We obtain (95) using the fact that the last number is equal to the number of particles which passed the origin (the origin is related to h = R) after duration T. This number is obtained by evaluating the following integral: k (λ(T)) = σ(h)dh = The derivative dk(λ(T)) (T(1 + r−2))− 2 yields current of particles through the origin related to asymptotic configuration. The number of downward steps m(T) of particles (see (75)) related to the asymptotic con- figuration is m(T) = T − |λ| 2(1 + r2) 1 + r−2 These formulae give simple answers for particular cases: (a) completely symmetric simple exclusion process, r = 1, (b) locking potential r → 0 resulting to λ(T) = 0 (c) totally asym- metric exclusion process (d-TASEP) where particles hop only upward, r → ∞. Two last cases were considered in the list of examples which follows (75). Let us write down a formula for the Schur function evaluated for the asymptotic partition λ = λ(T) sλ(t∞) = e σ̃(u)(u−log uR)du+ 1 σ̃(u)σ̃(u′) log(uR−u′R)dudu′)+··· = e− R2 logR+R2O(1) (98) 2(1+r−2) log T+··· where by dots we denote minor terms in T → ∞ limit and where σ̃(u) := σ(uR). Together with (70) and (97) it yields the number of ways to get the asymptotic configuration in T steps via random turn walk in large T limit as Nλ(T),0(T) = 2m(T)m(T)! sλ(T)(t∞) = T !e 2(1+r2) log T− T 2(1+r−2) log T+··· log T+··· (99) Notice that in the large T limit the leading term of logNλ(T),0(T) does not depend on the rate r, though the asymptotic configuration λ(T) depends. In the large T limit the leading term of logNλ(T),0(T) is the same for the locking case r → 0 where answer is given by (76) and in the d-TASEP case r → ∞ where answer is given by (77). It will be interesting to compare considered numbers (length, weight, current,...) evaluated on the asymptotic configuration with their average values. Let us mark that the shape of the asymptotic Young diagram given by (90)-(91) coincides with the shape found by Kerov and Vershik [38] in the problem of study of limiting shape of 1 0 1 0 Figure 8: Decay of step function for the case of a constant hopping rate in T → ∞ limit random partitions with fixed weight |λ| = N distributed according to the so-called Plancherel measure. We obtain the identification if we put N = R2. As soon as we arrive at (75) this fact is not so striking because the Plancherel measure is equal to N !sλ(t∞) 2. Thus, it is similar to the variational problem for d-TASEP where (75) is replaced by (78). Then, the variational problem for d-ASEP may be split into two parts: given weight of partition and taking it into account via Lagrangian multiplier to variate the shape of the partition and obtain (90)-(91) (this is quite similar to the d-TASEP problem where the weight is fixed as |λ| = T), then to variate with respect to the weight. This might be the other way to get formulae (90)-(92). The problem of evaluation of correlation functions for various versions of random turn walk is studied in the forthcoming paper [29]. 2.7 Weights which depend on mutual configuration: coupling with Coulomb- type weight As we have seen the case U = 0 may be interpreted as a free random turn walk of the non- intersecting particles on the lattice. Here ”free” means that each particle hops either upward or downward with the same probability (provided, the target site is empty). Then, as we have seen, in the answer for the weight of transition between initial vacuum configuration to a con- figuration given by (integer) coordinates h, hard-core interaction gives rise to the factor ∆(h). Then (83) can be interpreted as a partition function for the Coulomb gas of particles (each one has the unit charge) living on the lattice. For non-vanishing U the weights of hops also depend on the location of the hop. In Coulomb gas picture Ui is interpreted as an electric potential at a point i. Now let us consider a model of random motion where the weight of each step is additionally dependent on the particle configuration as follows. Consider the ’coupling’ of the model studied in the previous subsection 2.6 to the modified Coulomb interaction (48) where particles posses an arbitrary charge q. Namely, suppose that the weight of the hop is additionally multiplied by a factor which depends on the configuration of the particles according to formula hi + 1 j=1,2,... j 6=i hi + 1− hj hi − hj (100) for the hop of i-th particle from the site hi to the site hi + 1 (provided this site is empty), and according to  hi − 1 j=1,2,... j 6=i hi − 1− hj hi − hj (101) for the hop of i-th particle from the site hi to the site hi − 1 (provided this site is empty). Then instead of (75) we obtain W0→λ(T; q) = T! e−Uλsλ(t∞) 1+q2 = T!2 |λ|−T T−|λ| · e−Uλsλ(t∞)1+q (102) Normalization function is a sum of the weights over all transitions of the vacuum configura- tion of particles which are described by the non-intersecting random turn walk in the Coulomb potential of time duration T. After the corresponding changing of the weights of steps ac- cording to the appearance of the Coulomb interaction with a charge q, we get the resulting normalization function as Z(T, q) := (sλ(t∞)) q2 〈λ|(A− + A+)T|0〉 (103) h1,...,hT=1 T−|λ(h)| ) T−|λ(h)| e−V (hi) |hi − hj|β (104) In (104) h is (non-ordered) set of positive numbers, h1, . . . , hT, |λ(h)| := h1+· · ·+hT+ 12T(T− 1), and β = 1 + q2 (105) V (h) = Uh + β log h! (106) Thus, by choosing the rates of hops of the random turn walk we arrive to effective electric charges model in thermal equilibrium with particles forced by (1) external field given by U (2) hard-core interaction between particles (3) Coulomb interaction between particles with coupling constant q Note that if q 6= 0 we have no anymore need to mention that our particles are hard-core ones, since the Coulomb repulsion force does not allow to occupy the same place by more than one particle. One can see that the special cases, q = 0, 1, 3, result in links with discrete analogues of orthogonal (β = 1), unitary (β = 2) and symplectic (β = 4) matrix ensembles [43] which as it is known may be related to the integrable hierarchies. Then, for constant rate the asymptotic configuration will be the same (90)-(91), see fig.(8), where now 1 + r−2 (107) As we see the size of the asymptotic domain shrinks because of the adding of the modified Coulomb interaction which tries to confine particles near the vacuum configuration, see Remark 2.1. For asymptotic configuration λ(q, T) we straightforwardly obtain ℓ(λ(q, T)) = 1 + r−2 2(1+q2) , |λ(q, T)| = 1 1 + r−2 1 + r−2 2(1+q2) k (λ(q, T)) = 1 + r−2 2(1+q2) , m(q, T) = 1 + r−2 1 + r−2 2(1+q2) which gives answers for the asymptotic partition length, weight, for the number of particles which passed the origin and for the number of backward steps. For q2 > 0 in the large T limit and we also obtain log sλ(q,T)(t∞) = − 2(1 + q2) 1 + r−2 log T + · · · logNλ(q,T),0(T) = log T + · · · At last let us mark that the introduction of the free parameter q is similar to the introduction of a parameter s in [37], where a notion of an entropy of stochastic dynamical systems was studied. We hope to consider this problem in future. 2.8 Determinantal formulae: Wick theorem and Gessel-Viennot formu- Consider A+(U) of (61). Then 〈λ′|A+(U)N |λ〉 = 〈λ′|eA+(U)|λ〉, N = |λ′| − |λ| (108) The form of right hand side allows to apply the Wick theorem: 〈λ′|eA+(U)|λ〉 = det 〈m|f̄m+h′ eA+(U)fm+hi|m〉 i,j=1,...,m = det Um+hi−Um+h′j (h′j − hi)! i,j=1,...,m (109) where hi = λi − i+m, h′i = λ′i − i+m, m ≥ ℓ(λ′) ≥ ℓ(λ) (110) where we use 〈0|f̄i+NeA+(U)fi|0〉 = eUi−Ui+N (111) The right hand side also allows to consider tau functions τ(t, U, t̄) = 〈0|eH(t)eA+(U)eH̄(t̄)|0〉 (112) as a generating function for the transition weights 〈λ′|A+(U)N |λ〉. Remark 2.5. Let us mark the formula 〈λ′|eA+(U)|λ〉 = sλ′/λ(t∞) ℓ(λ′)∏ −Uhi+Uh′i (113) where t∞ = (1, 0, 0, . . . ) and λi = hi − i+m, λ′i = h′i − i+m, m = max(ℓ(λ), ℓ(λ′)) = ℓ(λ′) (114) The left hand side of (108) is interpreted as a N-step random turn process, described above. Let us consider an example. Gessel-Viennot formula for binomial determinants Consider two sets of positive integers ak > · · · > a1 ≥ 0 and bk > · · · > b1 ≥ 0, where ai ≥ bi, i = 1, . . . , k. Following [20] let us study the following determinant a1 . . . ak b1 . . . bk := det i,j=1,...,k (115) called binomial determinant. Let us identify sets of bk−i and ak−i with sets of hi and h i related by (110) to partitions λ and λ′ that is λi = bk−i + i− k, λ′i = ak−i + i− k, i = 1, . . . , k (116) [x]k := x2i , . . . [y]k := y2, . . . (117) It is known [42] that Schur functions sλ(t), where t = [x]k, vanish on partitions whose length exceed k: ℓ(λ) > k. Then, as a specification of (109) and by using formulas (31), we get that the following tau function τ(k, [x]k, [y]k) = 〈k|eH([x]k)eAeH̄([y]k)|k〉 (118) λ,λ′∈P ℓ(λ),ℓ(λ′)≤k 〈k|eH([x]k)|λ′, k〉〈λ, k|eH̄([y]k)|k〉〈λ′, k|eA|λ, k〉 λ,λ′∈P ℓ(λ),ℓ(λ′)≤k sλ′([x]k)sλ([y]k)det a1 . . . ak b1 . . . bk , (119) where ifif̄i−1, (120) is a generating function for the binomial determinants. This A is related to the motion along the following graph There are (i+1)(i+2)(i+3) ways toi+2 come from a point i to the point i+3 Figure 9: The numbers to the left are the weights of the arrows Remark 2.6. A tau function similar to (118) was used in [27] for the study of the so-called two matrix model. The graph related to (120) is sketched out above. The weight of an arrow may be considered as the number of identical arrows of unit weight. Thus, the product of weights of successive arrows connecting two vertices yields the number of ways connecting these vertices. The number of ways connecting points k and i on the Maya diagram is counted by i! 〈0|f̄iAi−jfj |0〉. Therefore, 〈0|f̄ieAfj |0〉 = (i− j)! 〈0|f̄iAi−jfj|0〉 = (121) From (113) we have the relation obtained in [20]: i,j=1,...,k (n)λ′ sλ′/λ(t∞) (122) where is the so-called skew Schur function (170). Gessel and Viennot suggested to interpret the binomial determinant as the number of non- intersecting ways connecting points ai with points bi, where i = 1, . . . , k (in our approach connecting a pair of Maya diagrams) as it is depicted at the right hand side figure [20]. Each step is either down or left directed: Notice that if we introduce the space-time axis (x, t) as it is figured on the picture Fig.3 we observe non-intersecting one-dimensional random walk of k vicious walkers, moving with speed ±1, changing at random the direction at each time instant ∆t = 1 . The particles start at t = 0 their motion from positions a1, . . . , ak. Later a particle by particle meets a ’light’ signal spreading with the speed −1 which starts motion in the origin at t = 0. A particle which started at ai meets the signal at point Considering random path to be an interface one can relate the right hand figure to the ran- domly growing area below the path. By stars we mark the last portion of area (ice covering of a step scala). Then the parameter k in the tau function (118) may be viewed as discrete time. These problems will be considered in the forthcoming paper. 1. The transition weight is given by the number of paths to the next figure ❞❅❅ t 2. Paths connecting a pair of Maya diagrams may be viewed as the interface of growing area Figure 10: Counting connecting paths If we replace eA by eA · · · eA = eTA in the expression for tau function (118) we obtain T-ple mapping of partitions described in the next section. In certain sense the model considered in this section interpolates between random turn and vicious walkers models, which we consider in the next section. 3 Action of ĜL(∞) group elements on Fock vectors and ran- dom processes Below we shall construct certain versions of stochastic motion on the space of partitions, which may be also called random slices (”rocks”) and random layering. Namely, one may view stochastic dynamics on partitions as models of random slices, which may be visualized as a set of partitions placed one above another according to its time evolution. Each three-dimensional figure of height T obtained in this way present a path (13) and may be compared with a rock inscribed into a corner between two walls. In this section we present only descriptions of the models in terms of fermions without study of these models. 3.1 Determinantal processes Now consider the case where each oj,j−1 in (6) has a special form oj,j−1 = gj,j−1 := e i,k a j,j−1 fif̄k , o(T) = g(T) := gT,T−1 · · · g1,0 (123) These exponentials are known to be elements of ĜL(∞) group, see [14], [32], where the im- portance of ĜL(∞) for constructing of integrable equations together with their solutions was established. τT(x, y) = 〈0|f̄(x−1N ) · · · f̄(x 1 ) g(T) f(y1) · · · f(yN)|0〉∆(x)−1∆(y)−1 is known [14], [32] to be a TL tau function. In our case we want tau functions to be dependent on the axillary discrete variable T via (123). In what follows we shall assume the following simplifying condition (j,j−1) ik = 0, i, k < 0, j = 1, 2, . . . (124) Let us remind definitions of the Schur function sν(x) = det ∆(x)−1, hi = νi − i+N and basis Fock vectors |ν,N〉 = fhN · · · fh1|0〉, 〈ν,N | = 〈0|f̄h1 · · · f̄hN Denoting the matrix elements of g(T): τλ,ν(T) = 〈λ,N |g(T)|ν,N〉, (125) we may rewrite tau function in form τT(x, y) = τλ,ν(T)sλ(x)sν(y) If we have (124), then Wick’s theorem has especially simple form: 〈µ(j)|gj,j−1|µ(j−1)〉 = det gj,j−1(h(j)a , h (j−1) b ), g j,j−1(a, b) = 〈0|fagj,j−1f̄b|0〉 (126) Remark 3.1. Thanks to the Wick theorem, we write (12) as τT(x, y) = {h(j)a } ∆(x)∆(y) (127) where the sum ranges over T sets of all admissible configurations h a , j = 0, . . . , T {h(j)a } = det ) T−1∏ j+1,j(h(j+1)a , h (128) {h(j)a } {h(j)a } τT(x, y) {h(j)a } λ,ν τλ,ν(T)sλ(x)sν(y) (129) almost coincides with the formula (1.10) of [23] (under some specification of functions ψ and φ in this paper, and under certain notational replacements: say, h a of (129) are x a of [23] and τT(x, y) of (129) is ZM,m of [23]), where it describes the so-called reduced probability density . The only difference between formula (129) and formula (1.10) of [23] is that, in our case, variables h a take discrete values as they are related to the parts of the intermediate partitions, while the main example of [23] is the multimatrix model where these variables coincide with eigenvalues of random matrices (then, a serves for the number of an eigenvalue, and j serves for a number of a matrix in the chain of coupled matrices). More generally (129) describes a discrete analogue of the so-called multi-level determinantal en- sembles where the ”levels” are numbered by j in (128). Any oj,j−1 in form (123) gives rise to a multi-level determinantal ensemble [13]. In our approach to random processes the tau function τT(x, y) plays the role of the generat- ing function for τλ,ν(T). The normalization function is Zν(T) = τλ,ν(T) In the present paper we are interested not in the ratio P TM {h(j)a } mentioned in the remark (see (129)) but in: Pν→λ(T) = τλ,ν(T) Zν(T) (130) yielding the normalized weight (which may be interpreted as the probability in case all weights are positive) to arrive to a configuration λ in T steps if an initial configuration is ν. Let us note that the most natural way to consider (128) is to use the so-called multi- component fermions and multi-component integrable hierarchies. This will be done below. 3.2 Multi-component fermions For certain problems it is suitable to re-write (128) with the help of multi-component fermions, [f (j)n , f m ]+ = [f̄ n , f̄ m ]+ = 0, [f n , f̄ m ]+ = δj,iδn,m (see Appendix A.2 for details). The multicomponent fermions were used in [14], [32] to con- struct hierarchies of multicomponent integrable equations. We shall use the following notations. Given set of partitions, ν(j), j = 0, . . . , T, we intro- duce a basis Fock vector |ν(0), n(0); . . . ; ν(T), n(T)〉 := Π(T) · · ·Π(1)|0, . . . , 0〉 where Π(j) = f · · · f (j) i = ν i − i+ n(j), n(j) = ℓ(ν The dual Fock vector we denote by 〈ν(0), n(0); · · · ; ν(T), n(T)|: 〈λ(0), l(0); . . . ;λ(T), l(T)|ν(0), n(0); . . . ; ν(T), n(T)〉 = δλ(0),ν(0) · · · δλ(T),ν(T) Now, we replace (6) by |ν,N ; 0, 0; . . . ; 0, 0〉 → o1,0|ν,N ; 0, 0; . . . ; 0, 0〉 → · · · → oT,T−1 · · · o1,0|ν,N ; 0, 0; . . . ; 0, 0〉 (131) where N = ℓ(ν(0)) (132) and where each operator oj+1,j is an intertwining operator between different Fock spaces F (j+1) and F (j) : oj+1,j = Gj+1,j := e i,k≥0 g j+1,j (j+1) i (133) Then (131) describes the same process on partitions as (6) conditioned by (123) and by (124) provided that gj+1,jk,i coincides with g j+1,j(k, i) of (126). Denoting G(T) = GT,T−1 · · ·G1,0 (134) we re-write tau function (127) as τT(x, y) = (N !) T〈0, . . . , 0|f (T)(x1) · · ·f (T)(xN)G(T)f̄ (1)(yN) · · · f̄ (1)(y1)|0, . . . , 0〉∆(x)−1∆(y)−1 (135) τλ,ν(T)sλ(x)sν(y) where τλ,ν(T) of (125) is reexpressed as τλ,ν(T) = 〈0, 0; . . . ; 0, 0;λ,N |G(T)|ν,N ; 0, 0; . . . ; 0, 0〉 (136) Tau function (135) is a specification of the multi-component tau function introduced in [14] (see also [34]), and almost coincides with the fermionic representation of the milti-matrix model presented in [31]. Regardless to the choice of representation for τλ,ν(T) which is obtained either from the one-component fermionic expectation value (125), or from the multi-component one, (136), the formulae (130) yields the transition probability to come to a configuration λ if the initial configuration is ν. Example: N vicious walkers (discrete Brownian motion of N hard core particles) Let us apply it to the model of N vicious walkers introduced by M.Fisher in [15] (actually he mainly considered the so-called lock step model where all walkers start on odd (or even) numbered coordinates on the lattice. At each tick of the clock each walker moves either up or downward subject to no two walkers occupying the same site at the same time. In various contexts this model was studied in [6], [17], [18], [63]. In this case for (133) we take Gj,j−1 = exp i+1+U (j−1) i+1f̄ (j−1) i + e −U (j) i−1+U (j−1) i−1f̄ (j−1) (137) The number j plays the role of physical time, the parameter T coincides with the duration of the process. Application of operator Gj,j−1 to a Fock vector (=Maya diagram) describes the unit-time transition of N particles, each of which hops to the nearest site conditioned that in both initial and final configurations there are no sites occupied by more than one particle. In (137) U plays the role of an external field: a particle located at a site i hops upward with the rate e−U i+1+U (j−1) i and downward with the rate e−U i−1+U (j−1) Then (131) describes the motion of hard-core particles which start with a configuration ν. As it follows from (184) the particles are constrained to move only along the positive half-line. Particles are not influenced by this restriction in case their initial configuration (given by ν) is far enough from the origin. The exponent in (137) should be compared with expression (61) which was used earlier to get the random turn motion. Vicious walkers on the circle We need to replace (137) by Gj,j−1 = exp gj,j−1, gj,j−1 = e−U (j−1) (j−1) n + e −U (j)n +U (j−1) 0 f (j)n f̄ (j−1) 0 (138) i+1+U (j−1) i+1f̄ (j−1) i + e −U (j) i−1+U (j−1) i−1f̄ (j−1) On correlation functions. An advantage of the multicomponent approach is the simple representation for the correlation functions in form of determinants. Given set of partitions ρ(j), j = 1, . . . , T − 1, whose lengths, ℓ(ρ(j)), do not exceed N , one may address the following questions: (a) what is the weight of the process where all paths (13) are constrained by the condition that each number ρ i , i = 1, . . . , ℓ(ρ (j)) ≤ N is a part of j-th intermediate partition µ(j) belonging to the path. Let us consider ’characteristic operators’ χ(j) := χ(ρ(j)) = f · · · f (j) · · · f̄ (j) (139) where h(j)i = ρ i − i+N . Introduce oχ(T) = o T,T−1χ(T−1)oT−1,T−2 · · · o2,1χ(1)o1,0 (140) Then, the weight we are looking for is the following correlation function Kν→λ(ρ (1), . . . , ρ(T−1)) = 〈0, 0; . . . ; 0, 0;λ,N |oχ(T)|ν,N ; 0, 0; . . . ; 0, 0〉 Zν(T) 〈0|f̄ (T)hi oχ(T)f Zν(T) (141) where hi = λi − i + N and h′i = νi − i + N (i = 1, . . . , N), and where we use the notation |0〉 := |0, 0; . . . ; 0, 0〉. The last equality is due to the Wick theorem. (b) what is the weight of the process where all paths (13) constrained by the condition that no one part of an j-th intermediate partition µ(j) belonging to the path coincides with any of parts of the ρ(j), where j = 1, . . . , T − 1. Let us consider ’characteristic operators’ χ̄(j) := χ̄(ρ(j)) = f̄ · · · f̄ (j) · · · f (j) (142) where h i = ρ i − i+N . Introduce oχ̄(T) = o T,T−1χ̄(T−1)oT−1,T−2 · · · o2,1χ̄(1)o1,0 (143) The weight of this process is the following correlation function Kν→λ(ρ̄ (1), . . . , ρ̄(T−1)) = 〈0, . . . , 0, λ|oχ̄(T)|ν, 0, . . . , 0〉 Zν(T) 〈0|f̄ (T)hi oχ̄(T)f Zν(T) (144) where hi = λi − i + N and h′i = νi − i + N (i = 1, . . . , N). The last equality is due to the Wick theorem. Equivalent representation is given by Kν→λ(ρ̄ (1), . . . , ρ̄(T−1)) = 〈0, 0; ρ(T−1), n(T−1); . . . ; ρ(1), n(1);λ,N |o(T)|ν,N ; ρ(T−1), n(T−1); . . . ; ρ(1), n(1); 0, 0〉 Zν(T) (145) where n(T−1) = ℓ(ρ(T−1)) This form allow to write it via the Wick theorem as Kν→λ(ρ̄ (1), . . . , ρ̄(T−1)) = 〈0|f̄hi o(T) fhk |0〉 Zν(T) By an analogy with the so-called spectral correlation functions in random matrices, we call the weights Kν→λ(ρ (1), . . . , ρ(T−1)) and Kν→λ(ρ̄ (1), . . . , ρ̄(T−1)) correlation functions. 3.3 Random layering. Chains of Darboux transformations Below we consider simple application of certain GL∞ transformations, sometimes called Dar- boux transformations. For U = 0 below and for the case of growing partitions these examples are mainly reformulation of different known random models considered in literature. (I) Random lay Let us consider four basic examples of ĜL(∞) operators. They are basic in the sense that their non-vanishing matrix elements oλ,λ′ are related to pairs of Young diagrams different by adding/removing layers (strips) to Young diagrams: vertical or horizontal ones. Thus, we pick up four different types of o, say, o(i) = expA(i), i = 1, 2, 3, 4. Let us consider these four cases separately. (1) The exponential of A(1)(x, U) := −A(1)(−x, U) = − k=1,2,... e−Ui+Ui−k(−x)kfif̄i−k (146) yields laying of a given Young diagram by a vertical strip, see fig. 11, as it is given by o(1)(x, U)|λ′〉 = |λ〉 o(1)λ,λ′(x, U) where λ,λ′(x, U) := 〈λ|o (1)|λ′〉 = eUλ′−Uλx|λ|−|λ′| is non-vanishing only if the difference between λ and λ′ is the so-called vertical strip [42]. Example: Now turn to the related Maya diagrams which describes 1D configuration of the related lattice gas. Let us compare it with the random turn walk, see Fig.7 in the subsection ”Decay of the step function”. In the random turn walk model in each time instant only one particle hops either one step upward or one step downward to the nearest neighboring site. The Fig.11 above describes the one step of a group of randomly chosen particles upward to the nearest neighboring sites which occurs at one time instant, this hop being conditioned that after the hop each site is occupied by no more than one particle. One may say that Fig.11b describes ’trains’ of particles which hop one step upward. The potential U gives rise to the gluing rate in 2D picture and the hopping rate in 1D picture which site-depending. The weight to absorb n boxes is proportional to xn. (2) The exponential of A(2)(x, U) := − k=1,2,... e−Ui+Ui+k(−x)kfif̄i+k (147) x Fig.11a. Condensation of a lay onto a Young diagram λ′. Random ’rain’ of boxes from rightward. These boxes may be fitted only to admissible places. Fig.11b. Examples of diagrams of λ where o λ,λ′ is nontrivial. One layer (one strip) is added at random to the Young diagram λ′. Boxes marked by x depict the difference between λ′ and λ, called the vertical strip Figure 11: A layering by a vertical strip yields getting away of a given Young diagram a vertical strip, that is o(2)(x, U)|λ′〉 = |λ〉 o(2)λ,λ′(x, U) where λ,λ′(x, U) := 〈λ|o (2)|λ′〉 = eUλ′−Uλx|λ′|−|λ| is non-vanishing only if the difference between λ′ and λ is the vertical strip [42]. Presenting this process as a random motion of 1D particles on Maya diagram we see that it describes a hop downward to the nearest site of randomly chosen group of particles (they may form ’trains’) at each time instant. (3) The exponential of A(3)(x, U) = k=1,2,... e−Ui+Ui−kxkfif̄i−k (148) yields laying of a given Young diagram by a horizontal strip, as it is given by o(3)(x, U)|λ′〉 = |λ〉 o(3)λ,λ′(x, U) where the transition weight λ,λ′(x, U) := 〈λ|o (3)|λ′〉 = eUλ′−Uλx|λ|−|λ′| is non-vanishing only if the diagram λ includes the diagram λ′, and the difference between these diagrams is the so-called horizontal strip [42]. Example is given by fig. 12: ✻ ✻ ✻ ✻ ✻ ❵ ❵ ❵x x x x x Fig.12a. Condensation of a lay onto a Young diagram λ′. Random ’rain’ of boxes from downward. These boxes may be fitted only to admissible places, which means that new figure should be a Young diagram again ❵ ❵ ❵ ❵ ❵ ❵ Fig.12b. Examples of diagrams of λ where o λ,λ′ is nontrivial. One layer (one strip) is added at random to the Young diagram λ′. Boxes marked by x depict the difference between λ′ and λ, called the horizontal strip Figure 12: A layering by a horizontal strip Turning to a 1D configuration of the related lattice gas we see that it is suitable to inter- change roles of particle (pictured as black balls) and free sites (white balls). In this dual picture trains of holes hop one step downward at each time instant. The rate of this process depends on the difference of potential in initial and final positions and is proportional to xn, where n is the number of absorbed boxes. (4) The exponential of A(4)(x, U) := k=1,2,... e−Ui+Ui+kxkfif̄i+k (149) yields the random process of getting away of a given Young diagram a horizontal strip, that is o(4)(x, U)|λ′〉 = |λ〉 o(4)λ,λ′(x, U) where λ,λ′(x, U) := 〈λ|o (4)|λ′〉 = eUλ′−Uλx|λ′|−|λ| is non-vanishing only if the difference between λ′ and λ is the horizontal strip [42]. These are four cases. Now one can consider a chain of transformations, each transformation is given by a set of corresponding U and x which define gluing rates. Examples of these chains were considered in the reviewing paper [65] from the point of view of discrete Hirota equations. (II) Chains of Darboux transformations. Random layering and ”rocks” The chain of transformations of initial Fock vector is as follows: |λ′〉 → eAσ1 (x1)|λ′〉 → eA T (xT) · · · eA 1 (x1)|λ′〉 (150) where σj = 1, 2, 3, 4, and where T is a discrete time. Given j = 1, 2, 3, . . . , T, each A j (xj) is characterized by a given set of {U (j)i , i ∈ Z} which enters definitions (146),(147),(148) and (149). Given set {σj , j = 1, 2, 3, . . . , T} and {U (j)i , j = 1, 2, 3, . . . , T, i ∈ Z} gives rise to a chain of Darboux transformations. Such chains may be viewed as words of lengths T formed by four types of characters (a ’DNA’ coding a set of random processes of adding or eliminating of layers). Tau function is constructed as τ(T, t, t̄) = 〈0|eH(t)o(T)eH̄(t̄)|0〉 = τλ,λ′(T)sλ(t)sλ′(t̄) o(T) = eA T (xT) · · · eA 1 (x1) then, each word is related to a certain generalized Darboux transformation of the tau func- tion. This tau function may be evaluated because we know all matrix elements of each ’letter’ (xj). Let us note that tau functions of this type were considered in [52], [51] as series in skew Schur functions which in special cases provided examples of Gelfand-Graev hypergeo- metric series. This is a certain random process on partitions (and their Young diagrams) describing ran- dom adding and elimination of lays to their Young diagrams, each time step gives rise to an act of either creation, or elimination of lays, as it is shown in figures 11-14. For the process of time duration T, the relative weight of a configuration λ is given by the ratio Pλ′→λ(T) = τλ,λ′(T) Zλ′(T) where Zλ′(T) = τλ,λ′(T) is the sum of weights of all processes of duration T started with an initial configuration λ′. One may call the random process of creating and eliminating of lays as model of random slices, or, a ”rock” inscribed into a corner between two walls, which may be viewed as a set of Young diagrams placed one above another as sections of 3D figure according to its time evolution. Each three-dimensional figure of height T obtained in this way presents a path (13). Examples. Choose λ′ = 0, and let σj = 1 (which means that we shall consider growing Young diagrams) and U i = Ui , for all j = 1, . . . , T. Let us note that for U = 0 models considered below may be connected to various models related to the so-called semi-standard tableau (see Appendix A.1 and for details [42]), like a model of polynuclear growth (for instance, see [15] for a review). j (xj , U) = A (1)(xj, U) = − k=1,2,... e−Ui+Ui−k(−xj)kfif̄i−k (151) In this model, xj , j = 1, . . . , T and Ui, i ≥ 0 are sets of parameters which define transition probabilities. Then the process may be called a version of random process of growing Young diagram which is as follows. The initial configuration is the zero diagram. Each time step one lay (a set of strips) is added at random to the previous Young diagram (in a way that new figure is again a Young diagram). This is a Markov process. The transition weight for each step, say, a step number j, where the initial configuration (related to the moment j − 1) is, say, µ(j−1), and the next configuration (related to the moment j) is µ(j), is equal to Wµ(j−1)→µ(j)(1) = e µ(j−1)x |µ(j)−µ(j−1) | j (152) This yields a un-normalized probability to add a lay (a strip) of a weight (i.e. number of boxes) equal to |µ(j) − µ(j−1)| to a Young diagram. One can see that for this random process W0→λ(T) = e −Uλsλ(x1, . . . , xT), Z0(T) = e−Uλsλ(x1, . . . , xT) (153) Since U0 = 0 and s0 = 1 one can notice notice that 1/Z0(T) yields the probability to create zero Young diagram, λ = 0, i.e. the probability to create no any nonvanishing diagram). Let us consider these formulae and find links to a few known random topics. (1) First, consider the Gauss potential Ui = e 2 and chose xi = e i , where c is a constant and h′1, . . . , h T are ordered positive integers h 1 > · · · > h′T. Then the weight W0→λ(T) = 2+···+c(h′T)2 i<j(e i − ech′j ) e−c(hi−h i,j=1,...,T , hi = λi − i+ T, (154) unexpectedly coincides with the (un-normalized) transition probability for T non-intersecting (”ordered”) Brownian particles on the line whose initial and final coordinates are positive in- tegers h′1, . . . , h T and h1, . . . , hT. In large T limit it may be also related to the Itsykson-Zuber integral (relation between this integral and the Brownian motion is quite known topic, for in- stance see [36]). Then there are few limiting cases for (153), which are simple. (2) The first is x1 = · · · = xT = x < 1. Then using (174) we obtain Z0(T) = e T2− 1 eUT−i (T − i)! h1,...,hT≥0 e−Vhi i,j=1 |hi − hj |β, β = 1 where Vhi = Uhi − hi log x Thus Z0(T) is the partition function of a discrete version of ensemble of random orthogonal matrices, which is also a certain tau function: the tau function of the Pfaff lattice introduced in [3], [34]. Let us note that the ensemble of random orthogonal matrices is well studied object, see [63], [6], [33]. For Ui = 0 it is a version of the well-known Laguerre ensemble. (3) A similar result we obtain in case we take xj = e −1(ϑ+(j−1)φ), where φ, ϑ are parame- ters. Then with the help of (176) we obtain unitary orthogonal ensemble Z0(T) = T2− 1 eUT−i (q; q)T−i h1,...,hT≥0 e−Vhi i,j=1 |eφhi − eφhj |β, β = 1 (155) where Vhi = Uhi − hiϑ where q-factorials (q; q)n are defined in (172), and Ui may be chosen as series analogues to −1t̃m, i ∈ Z (156) (Such parametrization was used in [50] for different purposes. These three cases can be also compared with [54]). At last let us write down a group of examples (found in [42]) where Z0(T) (and, therefore, P0→0(T) = Z0(T) −1) may be explicitly evaluated : (a) Take all Ui = 0. Then, there is a simple formula Z0(T) = sλ(x1, . . . , xT) = (1− xj)−1 (1− xixj)−1 (b) Take all eUi = 1, i ≤ m and eUi = 0, i > m. Then Z0(T) = sλ(x1, . . . , xT) = Dm/D0, Dm = det x2T+m−ij − xi−1j where sum is going over all partitions whose Young diagrams are contained in the rectangle T ×m. (c) Take all eUi = 1, i ≤ m and eUi = 0, i > m, as before, and put xj = e(2T−2j+1)φ. Then Z0(T) = (2T−1)φ, . . . , eφ) = eφ(m+2j−1) − 1 eφ(2j−1) − 1 e2φ(m+i+j−1) − 1 e2φ(m+i+j−1) − 1 N(Θ)e|Θ|φ where first sum is going over all partitions whose Young diagrams are contained in the rectangle T ×m. The second sum ranges over all symmetric plane partitions Θ whose diagram are contained in the box T × T ×m. (Diagram of a plane partition is a 3D figure constructed from identical cubes, whose sections by horizontal planes are Young diagrams, see [42]). Here N(Θ) is the number of symmetric plane partitions of weight |Θ|. Detailed explanations of the right hand sides of formulae in the last three examples (together with some other different examples of particular cases of summation formulae for λ sλ) may be found in [42] from where they were borrowed by the authors. Let us also note that the model of growing partition via layering (151) is directly related to the problem of enumerating of the so-called semi-standard tableau. 4 Discussion In the present paper we want to figure out links between classical integrable systems and ran- dom system. These links in a natural way suggest to introduce site-dependent weights for hops of the particles and provide certain analogies between hard-core particles in equilibrium and non-equilibrium states. Along this line we introduced a version of discrete ASEP with site de- pending hopping rates. We show that the normalization function for probabilities of this model is related to classical integrable hierarchy of type B rather than to the hierarchy of the type A (the Toda lattice hierarchy) we started with. The normalization function plays the role similar to the role of partition function for statistical ensembles. We present a compact formula which describes the decay of the step function which also converts this problem to the evaluation of a certain correlation function for a discrete version of orthogonal matrix ensemble, where potential is defined by the site-dependent hopping rates of our model. The further project aims the following problems: (1) to study a model which is a modification of the model considered in the subsection 2.6. Namely, we replace the operator A+ + A− by a linear combination of the so-called Vira- soro generators L1, L0 and L−1 which form SL(2) algebra. This model is also quite solvable, however we will show that it presents different behavior (2) to present a description of models of stochastic motion which may be obtained from various integrable hierarchies (various realizations of hierarchies of type A,B,C and D and their multi-component versions) [30] (3) to study the asymptotic behavior of transition probabilities between different configura- tions and of certain correlation functions [29] (4) In particular for random turn decay of step function to evaluate the time dependence of : (a) the mean number of involved particles, < ℓ(λ) > (b) the mean height < |λ| > (c) the mean flow through the origin < k(λ) > (k(λ) is the number of hooks, or, the same, the number of Frobenius coordinates (5) to consider d-ASEP with various boundary conditions. To study the phenomenon of shock waves and phase transition of the first order known for the problem with open boundaries [55], [58] (6) to study the phenomenon of thermalization which should occur for finite graphs (7) to understand links with other approaches to the 1D non-equilibrium phenomena, in particular with these of papers [21], [22], [59], [57]. (8) to understand relations with the Bethe anzats method for the ASEP model Acknowledgements The starting point for the ideas developed in this work were discussions between A.O. and T. Shiota during a stay at Kyoto university in 1999-2001. He wishes to thank A. Odzijevicz for discussions and kind hospitality during his stay in Bialystok in June 2005, where a large part of this work was done. The work was completed and written in Montreal (July 2005 and October 2006). Results contained in this paper were presented by A.O. at the workshop “Mathematics and Physics of Growing Interfaces“ in Santa Fe, January 2006. He also wishes to thank P. Zinn- Justin, J. van de Leur, M. Mineev and V. B. Priezzhev for helpful discussions and giving [20], and to O. Zaboronski and P. Wiegmann who called attention to papers on ASEP and chemical reactions [57] and S.I.Badulin for technical support. J. H. would like to thank Herbert Spohn for helpful discussions in Nov. - Dec. 2006, at the Kavli Institute for Theoretical Physics, U.C. Santa Barbara program on ”Spectral geometry and field theory: from growth phenomena to disordered systems” and also to thank the organizers of this program for the invitation to take part and the kind hospitality extended throughout his stay by the Kavli Institute. A Appendices A.1 Partitions and Schur functions Partitions. Polynomial functions in many variables are parameterized by partitions. A parti- tion is any (finite or infinite) sequence of non-negative integers in decreasing order: λ = (λ1, λ2, . . . , λr, . . . ) , λ1 ≥ λ2 ≥ · · · ≥ λr ≥ · · · > 0 (157) The numbers λi are called the parts of the λ. The number of the parts is the length of the λ, denoted ℓ(λ). The sum of the parts denoted |λ|, is called the weight of λ. If |λ| = n, we say that λ is a partition of n. It is often convenient to extend the λi’s to an infinite sequence {λi}i∈N where λi := 0 if i > ℓ(λ) (158) The zero partition, with ℓ(λ) = 0 is denoted by 0. The set of all partitions, including 0, is denoted by P . The Young diagram of a partition is defined as the set of points (or nodes) (i, j) ∈ Z2 such that 1 ≤ j ≤ λi. The Young diagram is viewed as a subset of entries in a matrix with l(λ) rows and the λ1 columns, with the nodes denoted by squares aligned adjacently to form rows of length λ1, λ2, . . . , stacked so that successive rows downwards have equal or diminishing lengths, as in the example below (159) which is the diagram of the partition λ = (4, 3, 1), with weight |λ| = 8 and length ℓ(〈λ|) = 3. The partition whose diagram is obtained by transposition of the diagram λ with respect to the main diagonal is called the conjugate partition and denoted by ν. Another notation is due to Frobenius. Suppose that the main diagonal of the diagram of λ consists of r nodes (i, i) (1 ≤ i ≤ r). For 1 ≤ i ≤ r, let αi = λi − i be the number of nodes in the ith row to the right of (i, i),, and βi = νi− i the number of nodes in the ith column below (i, i). We then have α1 > α2 > · · · > αr ≥ 0 β1 > β2 > · · · > βr ≥ 0 (160) The Frobenius notation for the partition λ is then λ = (α1, . . . , αr|β1, . . . , βr) = α|β). (161) The Frobenius notation may be viewed as a decomposition of a diagram λ into hooks with corners situated along the main diagonal, the largest hook being (α1|β1), the next one (α2|β2), and so on down to the smallest (αr|βr). For example, the partition (4, 3, 1) consists of two hooks (3, 2) and (1, 0): and (162) and in Frobenius notation is written (3, 1|2, 0). If λ = (α|β), then λt = (β|α) If we insert the increasing sequence of integers 1, 2, . . . |λ| into a Young diagram such that 1 is in the first box, and the numbers are increasing to the right within each row and downward within each column, the result is called a (standard) Young Tableau. If we insert a sequence 1, 2, . . . n with n ≤ |λ such that the numbers are nondecreasing to the right in each row, and decreasing downward in each column, this is called a semi-standard Young Tableau. Schur functions. The Schur functions Sλ([x]) associated with the partition λ may be viewed either as a sym- metric homogeneous polynomial in N variables (x1, . . . , xN ), where N may be any integer ≥ |ℓ(λ)| or, equivalently a weighted homogeneous polynomial in the infinite sequence of vari- ables (t1, t2, . . . ) defined by: sλ([x]) := ( x2a, . . . , xia, . . . ) := (t1, t2, . . . , ti, . . . ) (163) where each ti has weight i, having total weight |λ|. Because of the homogeneity condition, Sλ can only depend on the finite set of variables (t1, . . . , t|λ|). Viewed as functions of the xa’s, they may be defined as the following ratio of determinants (Jacobi-Trudi formula) Sλ([x]) = det(x λj−j+N ∆(x1, . . . , xN) 1 ≤ i, j ≤ N (164) where ∆(x1, . . . , xN) = (xi − xj) = det(x i ) (165) is the Vandermonde determinant. For N ≤ |λ|, this is equivalent to the following combinatorial definition: Sλ([x]) = 1 . . . x N (166) where the sum is over all semi-standard Young Tableau of shape λ and µi is the number of times i appears in the Tableau. Representation theoretically, the significance of this is that Sλ([x]) is the character of the irreducible, rank λ| tensor representation of U(N) or GL(N) whose symmetry properties are given by the Young diagram of λ. (i.e. the irreducible representation consisting of ten- sors obtained by first symmetrizing all components labelled by the rows of the Young di- agram, and then antisymmetrizing those labelled by the columns. The xa’a are viewed as eigenvalues of the U(N) or GL(N) group element or one simply chooses diagonal elements X = diag(x1, . . . , xN) and Sλ([x]) is the trace of the representation evaluated at X .) Polynomial functions in many variables, like the Schur functions, are parameterized by partitions. Consider a semi-infinite set of variables t = (t1, t2, t3, . . . ). Given partition λ, the Schur function sλ(t) is defined by sλ(t) = det hλi−i+j(t) 1≤i,j≤ℓ(λ) , where zkhk(t) = exp zmtm , (167) and, for k < 0, we put hk = 0 . The hk(t) is called the elementary Schur function. t = t(x(n)) = (t1(x (n)), t2(x (n)), . . . ), tm(x (n)) = xmi , (168) then definitions (167) and (164) are equivalent [42]: sλ(t(x (n))) = sλ(x (n)). (169) From definition (167) it follows that sλ(t(x (n))) = 0 if ℓ(λ) > n. We use the underline in sλ only to distinguish the two definitions. If an n×n matrix X has eigenvalues x1, . . . , xn, we may denote sλ(x1, . . . , xn) by sλ(X), without underline, since in this paper the Schur function with uppercase argument is used only in this sense. Skew Schur function is defined as follows sλ/µ = det hλi−µj−i+j 1≤i,j≤n , (170) Properties of these polynomials are described in details in [42]. Schur functions evaluated at special points. We need notations: (a)λ := (a)λ1(a− 1)λ2 · · · (a− k + 1)λk , (a)m := Γ(a+m) , (171) (qa; q)λ := (q a; q)λ1(q a−1; q)λ2 · · · (qa−k+1; q)λk , (q a; q)m := (1− qa) · · · (1− qa+m−1) , (172) where k = ℓ(λ). We set (a)0 = (q a; q)0 = 1 and (a)−k = (q a; q)−k = 0 for k > 0. We introduce the following notations [54]: t∞ = (1, 0, 0, 0, . . . ) , (173) t(a, 1) = , . . . , (174) t(∞, q) = (t1(∞, q), t2(∞, q), . . . ), tm(∞, q) = m(1− qm) , m = 1, 2, . . . , (175) t(a, q) = (t1(a, q), t2(a, q), . . . ) , tm(a, q) = 1− (qa)m m(1− qm) , m = 1, 2, . . . (176) Note that t(a, q) tends to t(∞, q) (resp. t(a, 1)) as a → ∞ (resp. q → 1). As for t∞, if f satisfies f(ct1, c 2t2, c 3t3, . . . ) = c df(t1, t2, t3, . . . ) for some d ∈ Z, we have ~df(t(∞, q)) → f(t∞) as ~ := lnq → 0. Below ∆(h) := i<j(hi − hj). Lemma 1. For a partition λ = (λ1, λ2, . . . ), let hi := n+λi−i (1 ≤ i ≤ n), where n ≥ ℓ(λ)). sλ(t∞) = ∆(h)∏n i=1 hi! , (177) sλ(t(a, 1)) = ∆(h)∏n i=1 hi! Γ(a− n+ hi + 1) Γ(a− i+ 1) , (178) sλ(t(∞, q)) = ∆(qh)∏n i=1(q; q)hi , (179) sλ(t(a, q)) = ∆(qh)∏n i=1(q; q)hi (qa−i+1; q)hi−n+i , (180) Note that those quantities (177)–(180) are independent of the choice of n ≥ ℓ(λ). We also mark that for integer positive a the Schur functions (178) and (180) vanish if ℓ(λ) > a. A.2 Fermionic Fock space, gl(∞) and GL(∞) The following is a summary regarding the one and two-component free fermion algebra based on the introductory section of [28]. The reader may refer to [32], [14] for further details. In the following, A denotes the complex Clifford algebra over C generated by charged free fermions {fi, f̄i}i∈Z, satisfying the anticommutation relations [fi, fj]+ = [f̄i, f̄j]+ = 0, [fi, f̄j]+ = δij . (181) where [, ]+ denotes the anticommutator. Elements of the linear part W := (⊕m∈ZCfm)⊕ ⊕m∈ZCf̄m (182) will be referred to as a free fermions. The fermionic free fields f(x) := k, f̄(y) := −k−1, (183) may be viewed as generating functions for the fj , f̄j’s. This Clifford algebra has a standard Fock space representation F and dual space F̄ (see e.g. [26, 28]) which contain unique vacuum states |0〉 and 〈0| respectively satisfying the properties fm|0〉 = 0 (m < 0), f̄m|0〉 = 0 (m ≥ 0), 〈0|fm = 0 (m ≥ 0), 〈0|f̄m = 0 (m < 0). (184) The Fock spaces F and F̄ are mutually dual, with the hermitian pairing defined via the linear form 〈0||0〉 on A called the vacuum expectation value. This satisfies 〈0|1|0〉 = 1; 〈0|fmf̄m|0〉 = 1, m < 0; 〈0|f̄mfm|0〉 = 1, m ≥ 0, (185) 〈0|fn|0〉 = 〈0|f̄n|0〉 = 〈0|fmfn|0〉 = 〈0|f̄mf̄n|0〉 = 0; 〈0|fmf̄n|0〉 = 0, m 6= n, .(186) Wick’s theorem implies that for any finite set of elements {wk ∈ W}, we have 〈0|w1 · · ·w2n+1|0〉 = 0, 〈0|w1 · · ·w2n|0〉 = σ∈S2n sgnσ〈0|wσ(1)wσ(2)|0〉 · · · 〈0|wσ(2n−1)wσ(2n)|0〉. (187) Here σ runs over permutations for which σ(1) < σ(2), . . . , σ(2n − 1) < σ(2n) and σ(1) < σ(3) < · · · < σ(2n− 1). If {wi}i=1,...,N , are linear combinations of the fj’s only, j ∈ Z, and {w̄i}i=1,...,N linear combinations of the f̄j’s, j ∈ Z, then(187) implies 〈0|w1 · · ·wN w̄N · · · w̄1|0〉 = det (〈0|wiw̄j|0〉) |i,j=1,...,N (188) Following [14], [32], for all N ∈ Z, we also introduce the states 〈N | := 〈0|CN (189) where CN := f̄0 · · · f̄N−1 if N > 0 (190) CN := f−1 · · · fN if N < 0 (191) CN := 1 if N = 0 (192) |N〉 := C̄N |0〉 (193) where C̄N := fN−1 · · · f0 if N > 0 (194) C̄N := f̄N · · · f̄−1 if N < 0 (195) C̄N := 1 if N = 0 (196) The states (189) and (193) are referred to as the left and right charged vacuum vectors, respec- tively, with charge N . From the relations 〈0|f̄N−kf(xi)|0〉 = xN−ki , 〈0|f−N+k−1f̄(yi)|0〉 = yN−ki , k = 1, 2, . . .N, (197) and (188), it follows that 〈N |f(x1) · · ·f(xn)|0〉 = δn,N∆N (x), N ∈ Z, (198) 〈−N |f̄(y1) · · · f̄(yn)|0〉 = δn,N∆N (y), N ∈ Z. (199) For free fermion generators with |x| 6= |y|, 〈0|f(x)f̄(y)|0〉 = 1 (200) Note that the expression on the right hand side is actually defined, by (185), as the infinite series∑∞ n=0 y nx−n−1 which converges only inside |x| < |y|. However one can consider expression (200) for the whole region of x and y (when |x| 6= |y|) in the sense of analytical continuation. From Wick’s theorem it follows that 〈n−m|f(x1) · · ·f(xn)f̄(y1) · · · f̄(ym)|0〉 = ∆n(x)∆m(y)∏ i=1,...,n j=1,...,m (xi − yj) (201) A.3 Commuting flows, τ functions and Schur functions In the theory of integrable systems the following ĝl(∞) operators are important: fif̄i+m, m = ±1,±2, . . . (202) These operators form Heisenberg algebra relations [Hm, Hn] = mδm+n,0 (203) These oscillator algebra properties together with Hm|n〉 = 0; 〈n|H−m = 0, m > 0, n ∈ Z (204) allow to refer vectors 〈0|eH(t) and H̄(t̄) as generalized coherent states which depend on param- eters t = (t1, t2, . . . ) and t̄ = (t̄1, t̄2, . . . ). Here H(t) = Hmtm, H̄(t̄) = H−mt̄m (205) We have the following fermionic representation of Schur functions (which follows, as usual from Wick’s theorem) Lemma 1 [14] For α1 > · · · > αk ≥ 0, β1 > · · · > βk ≥ 0 the next formula is valid: 〈0|eH(t)f̄−β1−1 · · · f̄−βk−1fαs · · · fα1 |0〉 = (−1) β1+···+βk+ksλ(t) , (206) where in the Frobenius notation λ = (α1, . . . , αk|β1, . . . , βk). Defining, as in eq. (61) A+(U) := eUi−1−Uifif̄i−1, (207) we have the following expression for the matrix elements of its exponential in terms of the Schur function evaluate at the special value t∞ := (1, 0 . . . ) Lemma 2 [52] 〈λ, n|eA+(U)|n〉 = e−Uλ(n)sλ(t∞) (208) This relation follows from A+(U) = e H0(U)H−1e −H0(U), H0(U) := Uif̄ifi − Uifif̄i (209) Indeed, by definitions of H0(U) and 〈λ, n| one evaluates: 〈λ, n|eH0(U) = cne−Uλ(n)〈λ, n| (210) where Uλ(n) is given by (43), where cn is defined by 〈n|eH0(U)|n〉 = cn, cn = e−U0···−Un−1, n > 0; cn = eU−1+···+Un, n < 0 (211) Then, (208) follows (210) and (206). Different detailed proof was written down in [26], where the notation r of (62) was used. In papers [50], [26] instead of e−Uλ(n) of the present paper the notation rλ(n) was used. Skew Schur function is defined as follows sλ/µ = det hλi−µj−i+j 1≤i,j≤n , (212) The following is a generalization of (208) in terms of skew Schur functions Lemma 3 [52] 〈λ, n|eA+(U)|µ, n〉 = eUµ(n)−Uλ(n)sλ/µ(t∞) (213) A.4 Derivation of (90) and some other formulae of subsection 2.6 Formula (90) results from arcsin (y − 1) dy = log 2u, u ∈ [0, 2] (214) where we put u = h . We have obtained formula (214) as follows. First, it is widely used in random matrix theory (the proof may be found, say, in [43] or in [17]) the relation called the Wigner semi-circle law, which is the first relation of 1− v2 dv − x = 0, x ∈ [−1, 1], P 1− v2 = 0 (215) while the last relation is obtained from the first one via taking the derivative with respect to x and via integrating by parts. Now, keeping in mind (arcsin v)′ = (1 − v2)− 12 and taking the derivative of the l. h. s. of (214) with respect to u, using ((u− y)−1)u = − ((u− y)−1)y and integrating by parts after a shift of the integration variable (y − 1 = v) we obtain 1− v2 u− 1− v where as u − 1 ∈ [−1, 1] the second term vanishes due to (215), while the first term (which originates from the lower boundary term; upper boundary term vanishes) coincides with the derivative of the r.h.s. of (214) with respect to u. Thus we have proven (214) up to a term, say C, independent of u. Let us find it. Consider arcsin (y − 1) dy = P arcsin (y − 1) log |u− y|dy = log u− 1 log |u− y|dy√ 1− (y − 1)2 =: log u− C, u ∈ [0, 2], where we integrated by parts. We should find the last term, C, which (we know) is the constant we are looking for. We have log |u− y|dy√ 1− (y − 1)2 log(x− sinφ)dφ, x ∈ [−1, 1], which we evaluate at point x = 1 (since it does not depends on x we are free to chose any point of the interval [−1, 1]): log(1− sin φ)dφ = 1 dφ sin2n φ (1− z2)− 2 − 1 1− cosφ sin φ dφ = 2 log cos 0 = − log 2 which completes the proof of (214). Now let us obtain (94). |λ| = (hi + i− R) = hσ(h)dh− R Then ∫ 2R hσ(h)dh = arcsin = − 1 h2dh√ (1 + sin2 φ)dφ = Now let us derive (92). We have log(T − |λ|) = T − R which we equate to log 2 and obtain (92). At last let us obtain (96): σ(h)dh = dh = −R References [1] A. G. Abanov, E. Bettelheim and P. Wiegmann, ”Nonlinear Dynamics of Quantum Sys- tems and Soliton Theory”, nlin.SI/0605006 [2] M. Adams, J. Harnad and J. Hurtubise, “Dual Moment Maps into Loop Algebras” , Lett. Math. Phys . 20, 299-308 (1990) [3] M.Adler and P. van Moerbeke, ”A matrix integral solution to two-dimensional Wp grav- ity” Commun. Math. Phys. 147 (1992) 25-56 [4] M.Adler, P. van Moerbeke and T. Shiota, “Pfaff τ -functions”, solv-int/9909010 [5] M. Adler, P. van Moerbeke, ”Virasoro action on Schur function expansions, skew Young tableaux and random walks”, math.PR/0309202 [6] J. Baik, “Random vicious walk and random matrices”, Comm. Pure Appl. Math., 53(11) 11 85-1410 (2000.) [7] M. Bertola, B. Eynard, J. Harnad Differential systems for biorthogonal polynomials ap- pearing in 2-matrix models, and the associated Riemann-Hilbert problem , Commun. Math. Phys. 243, 193-240 (2003). [8] T.H. Baker and P.J. Forrester, ”Random walks and random fixed point free involutions”, arXiv: math.CO/0107128 [9] A. Borodin, “Periodic Schur process and cylindric partitions”, arXiv:math.CO/0601019 [10] A. Borodin and G. Olshanski, “Asymptotics of Plancherel-type random partitions”, math.PR/0610240 [11] A. Borodin and G. Olshanski, ”Meixner polynomials and random partitions“ math.PR/0609806 [12] A. Borodin and G. Olshanski, “Random partitions and Gamma kernel”, math- ph/0305043. [13] A. Borodin, A. Soshnikov, ”Janossy Densities I. Determinantal Ensembles” math- ph/0212063 [14] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, “Transformation groups for soliton equa- tions”, in Nonlinear integrable systems—classical theory and quantum theory eds. M. Jimbo, and T. Miwa, World Scientific, pp. 39–120, (1983). [15] M. Fisher, “Walks, walls, wetting and melting”, J. Stat. Phys. 34 667-729 (1984) [16] A.S. Fokas, A.R. Its and A.V. Kitaev “The isomonodromy approach to matrix models in 2D quantum gravity”, Commun. Math. Phys. 147 395-430 (1992) [17] P.J. Forrester, “Log Gases and Random Matrices”, book in preparation http://www.ms.unimelb.edu.au/ matpjf.matpjf.html [18] P.J.Forrester, ”Random walks and random permutations”, arXiv: math.CO/9907037 [19] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A.Yu.Orlov, “Matrix Models of 2D Gravity and Toda Theory”, Nucl. Phys. B 357, 565–618 (1991). [20] I. Gessel and G. Viennot, ”Binomial Determinants, Paths and Hook Length Formulae” 1985 Academic Press, Inc [21] K-H Gwa and H. Spohn, “Six-vertex model, roughened surfaces and an asymmetric spin hamiltonian?”, Phys. Rev. Let. 68, 725 - 728 (1992); [22] , K-H Gwa and H. Spohn, “Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation”, Phys. Rev. A 46, 844-854 (1992) [23] J. Harnad, ”Janossy Densities, Multimatrix Spacing Distributions and Fredholm Resol- vents”, International Mathematical Reasearch Notices 48 (2004) 2599-2609 [24] J. Harnad, Dual Isomonodromic Deformations and Moment Maps into Loop Algebras , Commun. Math. Phys. 166, 337-365 (1994). [25] J. Harnad and Alexander R. Its Integrable Fredholm Operators and Dual Isomonodromic Deformations , Commun. Math. Phys. 226, 497-530 (2002). [26] J. Harnad and A.Yu. Orlov, “Scalar products of symmetric functions and matrix integrals”, Theor. Math. Phys. 137, 1676–1690 (2003). [27] J. P. Harnad and A.Yu. Orlov, “Scalar products of symmetric functions and matrix inte- grals”, Theor. Math. Phys. 137, 1676–1690 (2003); nlin.SI/0211051. [28] J. Harnad and A.Yu. Orlov, “Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions”, J. Phys. A 8783-8810 (2006) [29] J.P. Harnad and A.Yu.Orlov, “Evaluation of asymptotics of correlation functions for ran- dom turn Brownian motion”, in progress [30] J.P. Harnad, J.W. van de Leur and A.Yu.Orlov, “Fermionic construction of BKP tau func- tions for random processes”, in progress [31] J.P.Harnad and A.Yu. Orlov, “Fermionic construction of partition function for multi- matrix models and multi-component TL hierarchy”, preprint 2005 [32] M. Jimbo and T. Miwa, “Solitons and Infinite Dimensional Lie Algebras”, Publ. RIMS Kyoto Univ. 19, 943–1001 (1983). [33] K. Johansson, “Shape fluctuations and random matrices“ Comm. Math. Phys. 209 437- 476 (2000) [34] V.G. Kac and J.W. van de Leur. The n-component KP hierarchy and representation the- ory, in Important developments in soliton theory, eds. A.S. Fokas and V. E. Zakharov, Springer Series in Nonlinear Dynamics (1993), 302 – 343 [35] V.G. Kac and J.W. van de Leur. “The geometry of spinors and multicomponent BKP and DKP hierarchies”, CRM Proceedings and Lecture Notes 14, 1998, 159-202 [36] M.Katori, H.Tanemura, ”Noncolliding Brownian motions and Harish-Chandra formula” , math.PR/0306386 [37] V. Lecomte, C. Appert-Rolland, F. van Wijland, “Chaotic properties of systems with Markov dynamics”, cond-mat/0505483 [38] A.M.Vershik and S.V.Kerov, ”Asymptotics of the Plancherel measure of the symmetric group and the limitinf form of Young tableaux”, Soviet Mathematics Doklady 18 (1977) 527-531 [39] A.M.Vershik and S.V.Kerov, ”Asymptotics of the largest and the typical dimensions of the irreducible representations of a symmetric group”, Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 19, No. 1, pp. 25-36 (1985) [40] J.W. van de Leur, “Matrix Integrals and Geometry of Spinors”, J. of Nonlinear Math. Phys. 8, 288-311 (2001) [41] I.M. Loutsenko and V. Spiridonov, “Soliton Solutions of Integrable Hierarchies and Coulomb Plasmas”, J. Stat. Phys, 99, 751, (2000). [42] I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, (1995). [43] Mehta, M. L., Random Matrices, 2nd edition (Academic, San Diego, 1991). [44] A. Okounkov, “Symmetric functions and random partitions“, math.CO/0309074 [45] A. Okounkov, “Toda equations for Hurwitz numbers”, AG/00044128 v1. [46] A. Okounkov, “Symmetric functions and random partitions“, math.CO/0309074 [47] N. Nekrasov and A. Okounkov, ”Seiberg-Witten Theory and Random Partitions”, hep- th/0306238 [48] A. Okounkov, N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram J. Amer. Math. Soc. 16 581-603 (2003); mathCO/0503508 [49] A. Okounkov, N. Reshetikhin, “Random skew plane partitions and the Pearcey process”, mathCO/0503508 [50] A.Yu. Orlov, “Hypergeometric Functions as Infinite-Soliton Tau function” Theoreti- cal and Mathematical Physics 146(2): 183-206 (2006); “Hypergeometric τ -functions τ(t, T, t∗) as ∞-soliton τ -function in T variables”,nonlin.SI/0305001 [51] A. Yu. Orlov ”Tau Functions and Matrix Integrals”, math-ph/0210012 [52] A.Yu. Orlov and D.M.Scherbin “Fermionic representation for basic hypergeometric func- tions related to Schur polynomials” nlin.SI/0001001 [53] A.Yu. Orlov and D.M. Scherbin, “Multivariate hypergeometric functions as tau func- tions of Toda lattice and Kadomtsev-Petviashvili equation”, Physics D 152-153 pp 51-56 (2001) [54] A. Yu. Orlov and T. Shiota, “Schur function expansion for normal matrix model and associated discrete matrix models”, Phys. Lett. A 343, 384-396 (2005); (more completed version in math-ph/0501017) [55] K.Krebs, F.H. Jaforpour and G. M. Schu”tz, ”Microscopic structure of travelling wave solutions in a class of stochastic interacting particle systems”, cond-mat/0307680 [56] V.B. Priezzhev, ”Exact Nonstationary Probabilities in the Asymmetric Exclusion Process on a Ring”, Phys Rev Let. 19 050601-1 - 050601-4 (2003) [57] V. Rittenberg, ”Reaction-diffusion processes as physical realizations of Hecke algebras”, cond-mat/9402020 [58] F. Roshani, M. Khorrami, ”Exclusion Processes and boundary conditions”, cond- mat/0211167 [59] G. M. Schu”tz, ”Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles”, cond-mat/0308450 [60] Takasaki,K.: Initial value problem for the Toda lattice hierarchy, Adv. Stud. Pure Math. 4 (1984) 139-163 [61] Takebe, T.: Representation Theoretical Meaning of Initial Value Problem for the Toda Lattice Hierarchy I, Lett. Math. Phys. 21 (1991), 77-84. [62] Takebe, T.: Representation Theoretical Meaning of Initial Value Problem for the Toda Lattice Hierarchy II, Publ. RIMS, Kyoto Univ. 27 (1991), 491-503. [63] C. A. Tracy, H. Widom, “Nonintersecting Brownian Excursions”, math.PR/0607321 [64] K. Ueno and K. Takasaki, Adv. Stud. Pure Math.4, 1-95, (1984). [65] A.Zabrodin, “A survey of Hirota’s difference equations”, solv-int/9704001 [66] V.E. Zakharov and A.B. Shabat, J. Funct. Anal. Appl. 8 226, (1974)

0704.1158

Novelty and Collective Attention

Novelty and Collective Attention Fang Wu and Bernardo A. Huberman Information Dynamics Laboratory HP Labs Palo Alto, CA 94304 October 28, 2018 Abstract The subject of collective attention is central to an information age where millions of people are inundated with daily messages. It is thus of interest to understand how attention to novel items propagates and eventually fades among large populations. We have analyzed the dy- namics of collective attention among one million users of an interactive website — digg.com — devoted to thousands of novel news stories. The observations can be described by a dynamical model character- ized by a single novelty factor. Our measurements indicate that nov- elty within groups decays with a stretched-exponential law, suggesting the existence of a natural time scale over which attention fades. http://arxiv.org/abs/0704.1158v1 The problem of collective attention is at the heart of decision making and the spread of ideas, and as such it has been studied at the individual and small group level by a number of psychologists [1, 2], economists [3], and researchers in the area of marketing and advertising [4, 5, 6]. Attention also affects the propagation of information in social networks, determining the effectiveness of advertising and viral marketing [7]. And while progress on this problem has been made in small laboratory studies and in the theoretical literature of attention economics [8], it is still lacking empirical results from very large groups in a natural, non-laboratory, setting. To understand the process underlying attention in large groups, consider as an example how a news story spreads among a group of people. When it first comes out, the story catches the attention of a few ones, who may further pass it on to others if they find it interesting enough. If a lot of people start to pay attention to this story, its exposure in the media will continue to increase. In other words, a positive reinforcement effect sets in such that the more popular the story becomes, the faster it spreads. This growth is counterbalanced by the fact that the novelty of a story tends to fade with time and thus the attention that people pay to it. There- fore, in considering the dynamics of collective attention two competing ef- fects are present: the growth in the number of people that attend to a given story and the habituation that makes the same story less likely to be attrac- tive as time goes on. This process becomes more complex in the realistic case of multiple items or stories appearing at the same time, for now people also have the choice of which stories to attend with their limited attention. In order to study the dynamics of collective attention and its relation to novel inputs in a natural setting, we analyzed the behavioral patterns of one million people interacting with a news website whose content is solely determined by its own users. Because people using this website assign each news story an explicit measure of popularity, we were able to determine the growth and decay of attention for thousands of news stories and to validate a theoretical model which predicts both the dynamics and the statistical distribution of story lifetimes. The website under study, digg.com, is a digital media democracy which allows its users to submit news stories they discover from the Internet [9]. A new submission immediately appears on a repository webpage called “Up- coming Stories”, where other members can find the story and, if they like it, add a digg to it. A so-called digg number is shown next to each story’s headline, which simply counts how many users have digged the story in the digg.com past.1 If a submission fails to receive enough diggs within a certain time limit, it eventually falls out of the “Upcoming” section, but if it does earn a critical mass of diggs quickly enough, it becomes popular and jumps to the digg.com front page.2 Because the front page can only display a limited number of stories, old stories eventually get replaced by newer stories as the page gets constantly updated. If a story however, becomes very popular it qualifies as a “Top 10” and stays on the right side of the front page for a very long time. When a story first appears on the front page it attracts much attention, causing its digg number, Nt, to build up quickly. After a couple of hours its digg rate slows down because of both its lack of novelty and its lack of prominent visibility (reflected in the fact that it moves away from the front page). Thus the digg number of each story eventually saturates to a value, N , that depends on both its popularity growth and its novelty decay. In order to determine the statistical distribution of this saturation number, which corresponds to the number of diggs it has accumulated throughout its evolution, we measured the histogram of the final diggs of all 29,864 popular stories in the year 2006. As can be seen from Fig. 1, the distribution appears to be quite skewed, with the normal Q-Q plot of log(N ) a straight line. A Kolmogorov-Smirnov normality test of log(N ) with mean 6.546 and standard deviation 0.6626 yields a p-value of 0.0939, suggesting that N follows a log-normal distribution. It is then natural to ask whether Nt, the number of diggs of a popular story after finite time t, also follows a log-normal distribution. To answer this question, we tracked the digg numbers of 1,110 stories in January 2006 minute by minute. The distribution of log(Nt) again obeys a bell shape curve. As an example, a Kolmogorov-Smirnov normality test of log(N2 hours) with mean 5.925 and standard deviation 0.5451 yields a p-value as high as 0.5605, supporting the hypothesis that Nt also follows a log-normal distri- bution. The log-normal distribution can be explained by a simple stochastic dy- namical model which we now describe. If Nt represents the number of people who know the story at time t, in the absence of any habituation, 1In fact, digg users are given the option to “bury” a story, which will decrease the story’s digg number. Because this rarely happens, we ignore this possibility and simply assume that a story’s digg number can only grow with time. 2The actual machine-learning algorithm used to determine whether a story qualifies to appear on the front page is very complex and will not be discussed in this paper [10]. digg.com diggs 0 1000 2000 3000 4000 5000 −4 −2 0 2 4 Normal Q−Q Plot Theoretical Quantiles Figure 1: (a) The histogram of the 29,684 diggs in 2006, as on January 9, 2007. (b) The normal Q-Q plot of log(N ). The straight line shows that log(N ) follows a normal distribution with a slightly longer tail. This is due to digg.com’s built-in reinforcement mechanism that favors those “top stories”, which can stay on the front page and can be found at many other places (e.g. “popular stories in 30 days” and “popular stories in 365 days”). on average a fraction µ of those people will further spread the story to some of their friends. Mathematically this assumption can be expressed as Nt = (1 + Xt)Nt−1, where X1,X2, . . . are positive i.i.d. random variables with mean µ and variance σ2. The requirement that Xi must be positive ensures that Nt can only grow with time. As we have discussed above, this growth in time is eventually curtailed by a decay in novelty, which we pa- rameterize by a time dependent factor rt consisting of a series of decreasing positive numbers with the property that r1 = 1 and rt ↓ 0 as t ↑ ∞. With this additional parameter, the full stochastic dynamics of story propagation is governed by Nt = (1 + rtXt)Nt−1, where the factor rtXt acts as a dis- counted random multiplicative factor. When Xt is small (which is the case for small time steps) we have the following approximate solution: (1 + rsXs)N0 ≈ ersXsN0 = e rsXsN0, (1) where N0 is the initial population that is aware of the story. Taking loga- rithm of both sides, we obtain logNt − logN0 = rsXs. (2) The right hand side is a discounted sum of random variables, which for rt near one (small time steps) can be shown to be described by a normal distribution [11]. It then follows that for large t the probability distribution of Nt will be approximately log-normal. Our dynamic model can be further tested by taking the mean and vari- ance of both sides of Eq. (2): E(logNt − logN0) var(logNt − logN0) . (3) Hence if our model is correct, a plot of the sample mean of log(Nt)− log(N0) versus the sample variance for each time t, should yield a straight line passing through the origin with slope µ/σ2. One such plot for 1,110 stories collected in January 2007 is shown in Fig. 2. As can be seen, the points indeed lie on a line with slope 6.947. The decay factor rt can now be computed explicitly from Nt up to a constant scale. Since we have normalized r1 to 1, we have E(logNt)−E(logNt−1) E(logN1)− E(logN0) . (4) 0.0 0.1 0.2 0.3 0.4 sample variance Figure 2: Sample mean of logNt − logN0 versus sample variance, for 1,110 stories in January 2007. Time unit is one minute. The points are plotted as follows. For each story we calculate the quantity logNt−logN0, which is the logarithm of its digg number measured t minutes after its first appearance on the front page, minus the logarithm of its initial digg number. We collect 1,110 such quantities for 1,110 stories. Then we compute their sample mean y and sample variance x, and mark the point (x, y). This is for one t. We repeat the process for t = 1, 2, . . . , 1440 and plot 1440 points in total (i.e. 24 hours). They lie roughly on a straight line passing through the origin with slope 6.947. The curve of rt estimated from the 1,110 stories in January 2007 is shown in Fig. 3(a). As can be seen, rt decays very fast in the first two to three hours, and its value becomes less than 0.03 after three hours. Fig. 3(b,c) show that rt decays slower than exponential and faster than power law. Fig. 3(d) shows that rt can be fit empirically to a stretched exponential relaxation or Kohlrausch-Williams-Watts law [12]: rt ∼ e −0.4t . The halflife τ of rt can then be determined by solving the equation e−0.4t e−0.4t dt. (5) A numerical calculation gives τ = 69 minutes, or about one hour. This characteristic time is consistent with the fact that a story usually lives on the front page for a period between one and two hours. The stretched exponential relaxation often occurs as the result of multi- ple characteristic relaxation time scales [12, 13]. This is consistent with the fact that the decay rate of a story on digg.com depends on many factors, such as the story’s topic category, the time of a day when it appears on the front page. The measured decay factor rt is thus an average of these various rates and describes the collective decay of attention. These measurements, comprising the dynamics of one million users at- tending to thousands of novel stories, allowed us to determine the effect of novelty on the collective attention of very large groups of individuals, while nicely isolating both the speed of propagation of new stories and their de- cay. We also showed that the growth and decay of collective attention can be described by a dynamical model characterized by a single novelty factor which determines the natural time scale over which attention fades. The exact value of the decay constant depends on the nature of the medium but its functional form is universal. These experiments, which complement large social network studies of viral marketing [7] are facilitated by the availability of websites that attract millions of users, a fact that turns the internet into an interesting natural laboratory for testing and discovering the behavioral patterns of large populations on a very large scale [14]. References [1] Kahneman, D. Attention and effort. Englewood Cliffs, N.J.: Prentice Hall (1973). digg.com 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 0 1 2 3 4 5 6 log(t) 2 4 6 8 10 12 14 Figure 3: (a) The decay factor rt as a function of time. Time t is measured in minutes. (b) log(rt) versus t. rt decays slower than exponential. (c) log(rt) versus t. rt decays faster than power law. (d) log(rt) versus t 0.4. The slope is approximately −0.4. [2] Pashler, H. E. The psychology of attention. MIT Press (1998). [3] Camerer, C. The behavioral challenge to economics: understanding normal people. Paper presented at Federal Reserve of Boston meeting (2003). [4] Pieters, F. G. M., Rosbergen, E. and Wedel, M. Visual attention to repeated print advertising: A test of scanpath theory. Journal of Mar- keting Research 36(4), 424–438 (1999). [5] Dukas, R. Causes and consequences of limited attention. Brain Behavior and Evolution 63, 197–210 (2004). [6] Reis, R. Inattentive Consumers. Journal of Monetary Economics Vol. 53, 1761–1800 (2006). [7] Lefkovic, J., Adamic, L. and Huberman, B. A. The dynamics of vi- ral marketing. Proceedings of the ACM Conference on Electronic Com- merce (2006). [8] Falkinger, J. Attention Economies. Forthcoming in Journal of Eco- nomic Theory (2003). [9] How Digg Works. http://www.digg.com/how [10] Private communication with the digg.com support team. [11] Embrechts, P. and Maejima, M. The central limit theorem for summa- bility methods of i.i.d. random variables. Probability Theory and Related Fields, Volume 68, Number 2 (1984). [12] Lindsey, C. P. and Patterson, G. D. Detailed comparison of the William- Watts and Cole-Davidson functions. J. Chem. Phys. 73(7) (1980). [13] Frisch, U. and Sornette, D. Extreme deviations and applications. J. Phys. I France 7 1155–1171 (1997). [14] Watts, D. A twenty first century science. Nature 445, pp. 489 (2007). http://www.digg.com/how digg.com

0704.1161

The Fincher-Burke spin excitations and omega/T scaling in the insulating 5% Sr-doped La2CuO4

Fincher-Burke spin excitations and ω/T scaling in insulating La1.95Sr0.05CuO4 Wei Bao,1 Y. Chen,2, 3 K. Yamada,4 A. T. Savici,5 P. L. Russo,6 J. E. Lorenzo,7 and J.-H. Chung2, 3 Condensed Matter and Thermal Physics, Los Alamos National Laboratory, Los Alamos, NM 87545 NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899 Dept. of Materials Science and Engineering, University of Maryland, College Park, MD 20742 Institute of Materials Research, Tohoku University, Sendai 980-8577, Japan Brookhaven National Laboratory, Upton, NY 11973 TRIUMF, Vancouver, BC V6T 2A3, Canada Institut Néel, CNRS, BP 166X, F-38043, Grenoble, France (Dated: November 27, 2018) Insulating La1.95Sr0.05CuO4 shares with superconducting cuprates the same Fincher-Burke spin excitations, which usually are observed in itinerant antiferromagnets. The local spectral function satisfies ω/T scaling above ∼16 K for this incommensurate insulating cuprate. Together with previous results in commensurate insulating and incommensurate superconducting cuprates, these results further support the general scaling prediction for square-lattice quantum spin S = 1/2 systems. The width of incommensurate peaks in La1.95Sr0.05CuO4 scales to a similar finite value as at optimal doping, strongly suggesting that they are similarly distant from a quantum critical point. They might both be limited to a finite correlation length by partial spin-glass freezing. It is well known that the (π, π) peak of the Néel anti- ferromagnetic order of the parent compounds is replaced by a quartet of incommensurate peaks when cuprates are sufficiently doped to have become superconducting1. The very recent excitement is that spin excitations measured in superconducting La2−xSrxCuO4 (LSCO) (x = 0.10 and 0.16)2, YBa2Cu3O6.6 (YBCO) 3, as well as in the “stripe-ordered” La1.875Ba0.125CuO4 with a suppressed 4 share a common spectral feature: broad excitation continua, originating from the incommensurate quartet, disperse towards the (π, π) point with increasing en- ergy at the rate of the spin-wave velocity of the par- ent compounds. The spectrum is distinct from the spin-waves in two important ways: 1) the excitations are not resolution-limited; 2) there are no the outward branches. Such a spin excitation spectrum previously has been observed in itinerant spin-density-wave anti- ferromagnets such as elemental metal Cr5 and strongly correlated metal V2−yO3 6, and the single-lobed disper- sive continuum is referred as the Fincher-Burke mode. A self-consistent theory has been developed by Moriya and others to describe the mode7, and quantitative agree- ment has been achieved for three-dimensional itinerant antiferromagnets6. Itinerant theories have also been de- veloped to account for the Fincher-Burke-like modes in superconducting cuprates8. Sandwiched between the parent antiferromagnetic in- sulator and the high-TC superconductor in the phase dia- gram of LSCO is a distinct doping regime from x = 0.02 to 0.0559. Cuprates in the regime are insulators with- out the long-range Néel order, and a spin-glass transi- tion occurs at Tf . 10 K 10,11. This doping range is often referred to as the spin-glass regime, although the spin-glass phase extends to both lower and higher dop- ings in the Néel and superconducting states. Magnetic correlations were extensively investigated and regarded as being commensurate, as in the parent compound12,13. However, with improved single-crystal samples, mag- netic correlations show a novel incommensurate dou- blet, which also differs distinctly from the quartet in the superconductors9,14. In this paper, we report that spin excitations of La1.95Sr0.05CuO4 in the insulating spin- glass regime are also composed of the Fincher-Burke modes, originating from the incommensurate doublet, with a velocity the same as in superconducting LSCO. Although cuprates in the spin-glass regime are insula- tors, they are not the usual band insulators, and part of the Fermi surface may have survived15,16. It would be interesting to investigate whether the Fincher-Burke modes reported here can be accounted for by extending theories for similar spin excitations of superconducting cuprates. Meanwhile, for quasi-two-dimensional (2D) spin S=1/2 cuprates, the temperature range where spin fluc- tuations are investigated is within T << J/kB ∼ 1000 K17, where classical statistical mechanics has to be replaced by quantum statistical mechanics18. The gen- eral scaling argument of 2D quantum statistical systems leads to a prediction for samples which are not exactly at a quantum critical point (QCP) that for T ≪ J/kB but above a low-temperature limit TX , the energy scale for 2D spin fluctuations at long wavelengths is kBT 18,19. Hence, for TX < T ≪ J/kB, the spin excitation spectrum follows the ω/T scaling19, which is not a robust feature of the Moriya theory7. For T < TX , a constant energy gap is predicted19. We test these predictions against the distinct incommensurate spin excitations from the dou- blet in La1.95Sr0.05CuO4. We also compare our results with previous investigations of incommensurate spin fluc- tuations from the quartet in superconducting cuprate20 and commensurate ones at the (π, π) point in insulating cuprate21. Our data support the scaling above TX but no gap is observed below TX . Surprisingly, scaling analysis of the q and ω dependent spin excitation spectra indi- cates that La1.95Sr0.05CuO4 and optimally doped LSCO are similarly distant from the QCP. http://arxiv.org/abs/0704.1161v3 A single piece of La1.95Sr0.05CuO4 crystal of 5.2 g was used in this work. It was grown using a traveling- solvent floating-zone method as described previously9,14. We use the orthorhombic Cmca unit cell (a = 5.338Å, b = 13.16Å, c = 5.404Å at 1.5 K) to describe the q- space for measurements at NIST using the cold neutron triple-axis spectrometer SPINS. The sample temperature was controlled by a pumped He4 cryostat. The horizon- tal collimations before and after the sample were both 80′, and a Be filter cooled by liquid nitrogen was used after the sample to reduce higher order neutrons pass- ing through the pyrolytic graphite (002) used for both monochromator and analyzer. The intensity of magnetic neutron scattering was counted against a flux monitor placed before the sample in a fixed Ef = 5 meV configu- ration and normalized to yield S(q, ω) in absolute units. Such a cold neutron spectrometer readily resolves the in- commensurate doublet near (100), i.e., the (π, π) point (see Fig. 1), while it is difficult to resolve the doublet us- ing a thermal neutron triple-axis spectrometer due to its coarser resolution. Hence, we will present only the cold neutron scattering results here. We first present the nominal elastic signal at various temperatures. Fig. 1(a) shows constant-energy ~ω=0 scans through the incommensurate doublet from 1.2 to 80 K. The sharp peak at (100) is due to higher-order diffrac- tion of (200). Its width indicates the instrument resolu- tion. The magnetic doublets at qIC=(1,0,±0.058(2)) are obviously broader than the resolution. The deconvoluted peak width yields the in-plane correlation length for the nominally elastic spin correlations, ξ� = 34(2)Å at 1.2 K, about 9 nearest-neighbor Cu spacings. With increas- ing temperature, the doublet monotonically decreases in intensity without appreciable change in either the peak width or position. Above ∼ 20 K, the doublet disappears, consistent with previous studies9. At finite energies, however, the temperature depen- dence of the doublet is entirely different from that at ~ω = 0. Fig. 1(b) shows constant-energy ~ω=0.5 meV scans measured in the same temperature range. Instead of monotonically decreasing, the intensity first increases, reaches a maximum between 15 and 20 K, and then de- creases with further rising temperature. The intensity at the peak shoulder, e.g., at q = (1, 0, 0.2), in Fig. 1(a)-(b) measures a temperature-independent background, which has been subtracted in Fig. 1(c). Note that at 1.2 K, S(q, ω) at 0.5 meV is more than one order of magni- tude weaker than at ~ω = 0. This reflects the fact that the energy spectrum of S(q, ω) shows a prominent sharp “central peak” at ω = 0 at low temperatures, see Fig. 1(c). The “central peak” is energy-resolution-limited at the SPINS spectrometer, with the full-width-at-half- maximum (FWHM) of 0.3 meV. However, the nominal elastic signal from La1.95Sr0.05CuO4 is not truly static at T > Tg ≈ 5 K as determined by our µSR measure- ments, which has an energy resolution of ∼10−6 meV. Details of the µSR study will be published elsewhere. Similar “central peak” phenomenon has been reported FIG. 1: (color online) The generalized spin correlation func- tion S(q, ω) as a function of q along the c-axis measured at (a) ~ω = 0 and (b) 0.5 meV, respectively, at various temper- atures. (c) S(q, ω) as a function of energy at q = (1, 0, 0.05) and at 1.2 and 20 K, respectively. The “central peak” at 1.2 K is also shown on a 1/3 scale with open circles. Background has been subtracted in (c). for Li-doped La2CuO4 and YBa2Cu3O6+x, and the very slow spin dynamics is attributed to a partial spin-glass transition22,23. What is the energy dependence of the doublet? Fig. 2(a) shows scans at various energies at 20 K. Below 3 meV, the two incommensurate peaks are clearly distin- guishable. As energy increases, the doublet merges into a flat-top peak. The scans in Fig. 2(a) can be fitted using two gaussians of the same width. The peak positions are shown as the black circles in Fig. 2(b). The dispersion is consistent with the inner branches of spin-waves (solid lines)17. The same dispersion rate has been reported for superconducting La2−xSrxCuO4 (x = 0.10 and 0.16) The shaded area in Fig. 2(b) covers the FWHM, which grows slowly with energy from 0.089(3) Å−1 at 0.5 meV. FIG. 2: (color online) (a) Constant-~ω scans at 20 K. The solid lines are two equal-width gaussians. The horizontal bars indicate instrument resolution (FWHM). (b) The double peak positions. The shaded area represents the FWHM. The spec- tral shape strongly resembles that for La1.84Sr0.16CuO4 2 and V1.97O3 6. The lines denote the spin-wave branches with a velocity of 850 meVÅ of La2CuO4 17. The FWHM/2 mea- sured at those energies listed in (a) is shown as a function of temperature in (c) and reduced temperature in (d). The width is comparable to that for the x = 0.16 sample. Because of the smaller doublet separation, the merge of the peaks occurs at ∼4 meV, much lower than at ∼40 meV for La1.84Sr0.16CuO4 2. Thus, despite very differ- ent ground states and spatial spin correlations, insulating and superconducting LSCO share the common Fincher- Burke spectral shape. Now we turn to examination of scaling behavior of spin excitations. Historically, it was done in the spin- glass regime through the local spin correlation function S(ω) = dqS(q, ω) using cuprate samples showing com- mensurate spin correlations12,13. It is re-examined here with the very different incommensurate spin correlations of our improved sample. The imaginary part of the local dynamic magnetic susceptibility relates to S(ω) by the fluctuation-dissipation theorem χ′′(ω) = π(1 − e−~ω/kBT )S(ω). (1) Fig. 3 shows χ′′(ω) from 1.2 to 80 K, which is well de- scribed by the Debye relaxor model χ′′(ω) = χ0(~ω/Γ) 1 + (~ω/Γ)2 , (2) where χ0 is the local staggered static magnetic suscep- tibility, and Γ the spin relaxation constant. In previous studies12,13, the local χ′′(ω) was modeled by χ′′(ω) = I(ω) arctan[a1(~ω/kBT ) + a2(~ω/kBT ) 2]. (3) The arctan function is stipulated by the marginal Fermi liquid theory12,24, but I(ω) has no determined analytic form13. Hence, we opt for the well-known Debye relaxor FIG. 3: (color online) The local dynamic magnetic suscep- tibility χ′′(ω) measured at various temperatures. The solid lines are the least-square fit to Eq. (2). The relaxation con- stant Γ is shown on the ~ω-T plane as a function of T , and departs from the dashed line Γ = 0.73kBT below TX ≈ 16 K. model, Eq. (2), to fit our data. The Debye relaxor model has also successfully described measured χ′′(ω) of insu- lating La2Cu0.94Li0.06O4, which has commensurate mag- netic correlations21. On the base plane of Fig. 3, the spin relaxation con- stant Γ obtained from the least-squares fit is shown as a function of temperature. The good instrument reso- lution, 0.3 meV (FWHM), has a negligible effect during fitting. One interesting result is that Γ = 0.73(2)kBT for temperatures above TX ≈ 16 K. Hence, when χ ′′(ω), normalized by its maximum χ0/2 at ~ω = Γ, is plot- ted as a function of ~ω/kBT , Eq. (2) dictates that all data collected above TX collapse onto a single universal curve y = 2/[1 + (x/0.73)2] and Fig. 4(a) bears this out. The result is commonly referred to as the ω/T scaling, and Γ/kBT = O(1) is a hallmark of quantum magnetic theory18,19. For T < TX , Fig. 3 shows that Γ departs from the proportionality to temperature. Consequently, the low temperature data would not follow the scaling curve, as demonstrated by Fig. 4(b). Note that the spec- tral function Eq. (2) does not become gapped below TX , contrary to non-random quantum theory19, but can be explained by dopant scattering25,26. The ω/T scaling and its departure below TX shown in Fig. 4 for La1.95Sr0.05CuO4 bears a striking similar- ity to what reported for La2Cu0.94Li0.06O4 21. The two cuprates have similar hole concentration and develop spin-glass at similar Tg. However, they differ in several important ways: i) The dopants are out of the CuO2 plane in the Sr compound, but directly replace Cu2+ in the Li compound. ii) The former becomes a super- conductor with additional 0.5% more holes, but the lat- ter always remains an insulator. iii) Magnetic correla- tions are incommensurate in the former, but commen- surate in the latter. iv) The in-plane correlation length ξ� ≃ 34(2)Å for the glassy spin component in the former, but ξ� ≫ 274Å in the latter, and the κ(ω, T ) shown in Fig. 2(c) is more than double that in the latter22. v) FIG. 4: (color online) The normalized local dynamic magnetic susceptibility χ′′(ω)2/χ0 as a function of ~ω/kBT . (a) The ω/T scaling is followed for data taken above TX ≃ 16 K. The solid curve is the scaling function y = 2/[1 + (x/0.73)2]. (b) Data taken below TX does not follow the scaling function. Γ/kBT = 0.73 for the former, and 0.18 for the latter In spite of these differences, Γ saturates at Γ0 ∼ 1 meV and χ′′(ω) becomes essentially T -independent (see Fig. 3) for both cuprates below TX . As a consequence, Eq. (1) requires a reduced S(ω) at low energies when the temper- ature decreases below TX , as observed in Fig. 1(b) and (c), in sharp contrast to a magnet at the QCP. Scaling of spin excitations has also been examined near the optimal doping for La1.86Sr0.14CuO4 20. The mate- rial is concluded to be near a QCP, namely, TX→0, with some caveat27. The χ′′P /ω in [ 20], equaling to χ0/Γ of Eq. (2), would saturate below TX , but TC = 35 K sets the upper limit for measurable TX in La1.86Sr0.14CuO4. Hence it cannot be determined whether La1.86Sr0.14CuO4 or La1.95Sr0.05CuO4 is closer to a QCP with a lower TX . Another method to assess the distance from the QCP is to examine the width of constant-~ω scans, see Fig. 2(c). Adapting the ansatz in [20], the κ(ω, T ) plotted as a func- tion of a reduced temperature better collapses our data in Fig. 2(d), and the solid line is κ(ω, T )2 = κ20 + (kBT/c) 2[1 + (~ω/0.73kBT ) 2], (4) where κ0 = 0.044(1)Å −1 and c = 4(1) × 102 meVÅ. At the QCP, κ0 is expected to be zero, and its value for La1.95Sr0.05CuO4 is comparable to κ0 in the supercon- ducting state, but narrower than κ0 at 40 K in the normal state for La1.84Sr0.16CuO4 2. Hence, La1.95Sr0.05CuO4 and the optimally doped LSCO with a short 1/κ0, about 6 Cu-Cu spacings, seem equally distant from the QCP. In conclusion, the Fincher-Burke modes, the broad and dispersive spin excitations of itinerant antiferromagnets, are observed in the spin-glass regime of La2−xSrxCuO4. Theoretical understanding of similar excitation modes in superconducting cuprates now has an added task in the insulators. Befitting to the generality of its theoretical argument, the ω/T scaling is shown to be valid for a new type of cuprates above TX . Spin excitations below TX remain gapless contrary to the prediction of non-random quantum theory. The spin-glass transition at finite dop- ing introduces an extra component of slow spin fluctu- ations. It would be interesting to explore whether the glassy state22,23,28, limiting the correlation length of the rest of spins, is responsible for the equal distance from the QCP for La1.95Sr0.05CuO4 and La1.84Sr0.16CuO4. Work at LANL is supported by U.S. DOE, and SPINS partially by NSF under Agreement No. DMR-0454672. 1 J. M. Tranquada, in Handbook of High-Temperature Super- conductivity, edited by J. R. Schrieffer (Springer, Berlin, 2007), p. 257. 2 N. B. Christensen, D. F. McMorrow, H. M. Rønnow, B. Lake, S. M. Hayden, G. Aeppli, T. G. Perring, M. Mangkorntong, M. Nohara, and H. Takagi, Phys. Rev. Lett. 93, 147002 (2004). 3 S. M. Hayden, H. A. Mook, P. Dai, T. G. Perring, and F. Doğan, Nature (London) 429, 531 (2004). 4 J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, Nature (London) 429, 534 (2004). 5 C. R. Fincher, G. Shirani, and S. A. Werner, Phys. Rev. B 24, 1312 (1981). 6 W. Bao, C. Broholm, S. A. Carter, T. F. Rosenbaum, G. Aeppli, S. F. Trevino, P. Metcalf, J. M. Honig, and J. Spalek, Phys. Rev. Lett. 71, 766 (1993); W. Bao, C. Bro- holm, J. M. Honig, P. Metcalf, and S. F. Trevino, Phys. Rev. B 54, R3726 (1996). 7 T. Moriya, Spin Fluctuations in Itinerant Electron Mag- netism (Springer-Verlag and Berlin, 1985). 8 A. Abanov and A. V. Chubukov, Phys. Rev. Lett. 83, 1652 (1999); U. Chatterjee et al., Phys. Rev. B 75, 172504 (2007). 9 M. Fujita, K. Yamada, H. Hiraka, P. M. Gehring, S. H. Lee, S. Wakimoto, and G. Shirane, Phys. Rev. B 65, 64505 (2002). 10 B. J. Sternlieb, G. M. Luke, Y. J. Uemura, T. M. Rise- man, J. H. Brewer, P. M. Gehring, K. Yamada, Y. Hidaka, T. Murakami, T. R. Thurston, et al., Phys. Rev. B 41, 8866 (1990). 11 F. C. Chou, F. Borsa, J. H. Cho, D. C. Johnston, A. Las- cialfari, D. R. Torgeson, and J. Ziolo, Phys. Rev. Lett. 71, 2323 (1993). 12 B. Keimer, N. Belk, R. J. Birgeneau, A. Cassanho, C. Y. Chen, M. Greven, M. A. Kastner, A. Aharony, Y. Endoh, R. W. Erwin, et al., Phys. Rev. B 46, 14034 (1992). 13 M. Matsuda, R. J. Birgeneau, Y. Endoh, Y. Hidaka, M. A. Kastner, K. Nakajima, G. Shirane, T. R. Thurston, and K. Yamada, J. Phys. Soc. Japn 62, 1702 (1993). 14 S. Wakimoto, G. Shirane, Y. Endoh, K. Hirota, S. Ueki, K. Yamada, R. J. Birgeneau, M. A. Kastner, Y. S. Lee, P. M. Gehring, et al., Phys. Rev. B 60, R769 (1999). 15 M. L. Tacon, A. Sacuto, A. Georges, G. Kotliar, Y. Gallais, D. Colson, and A. Forget, Nature Physics 2, 537 (2006). 16 T. Yoshida, X. J. Zhou, T. Sasagawa, W. L. Yang, P. V. Bogdanov, A. Lanzara, Z. Hussain, T. Mizokawa, A. Fuji- mori, H. Eisaki, et al., Phys. Rev. Lett. 91, 027001 (2003). 17 S. M. Hayden, G. Aeppli, R. Osborn, A. D. Taylor, T. G. Perring, S.-W. Cheong, and Z. Fisk, Phys. Rev. Lett. 67, 3622 (1991). 18 D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod. Phys. 77, 579 (2005). 19 S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993). 20 G. Aeppli, T. E. Mason, S. M. Hayden, H. A. Mook, and J. Kulda, Science 278, 1432 (1997). 21 W. Bao, Y. Chen, Y. Qiu, and J. L. Sarrao, Phys. Rev. Lett. 91, 127005 (2003). 22 Y. Chen, W. Bao, Y. Qiu, J. E. Lorenzo, J. L. Sarrao, D. L. Ho, and M. Y. Lin, Phys. Rev. B 72, 184401 (2005). 23 C. Stock, W. J. L. Buyers, Z. Yamani, C. L. Broholm, J.- H. Chung, Z. Tun, R. Liang, D. Bonn, W. N. Hardy, and R. J. Birgeneau, Phys. Rev. B 73, 100504(R) (2006). 24 C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abra- hams, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 1996 (1989). 25 S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). 26 Y. L. Liu and Z. B. Su, Phys. Lett. A 200, 393 (1995). 27 S. Chakravarty, Science 278, 1412 (1997). 28 C. Panagopoulos, A. P. Petrovic, Hillier, J. L. Tallon, C. A. Scott, and A. P. Rainford, Phys. Rev. B 69, 144510 (2004).

0704.1162

Binary Capture Rates for Massive Protostars

Draft version November 24, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 BINARY CAPTURE RATES FOR MASSIVE PROTOSTARS Nickolas Moeckel, John Bally Center for Astrophysics and Space Astronomy, and Department of Astrophysical and Planetary Sciences University of Colorado, Boulder, CO Draft version November 24, 2018 ABSTRACT The high multiplicity of massive stars in dense, young clusters is established early in their evolu- tion. The mechanism behind this remains unresolved. Recent results suggest that massive protostars may capture companions through disk interactions with much higher efficiency than their solar mass counterparts. However, this conclusion is based on analytic determinations of capture rates and esti- mates of the robustness of the resulting binaries. We present the results of coupled n-body and SPH simulations of star-disk encounters to further test the idea that disk-captured binaries contribute to the observed multiplicity of massive stars. Subject headings: binaries: general — circumstellar matter — stars: formation 1. INTRODUCTION Massive stars such as those of the Trapezium in the Orion Nebula cluster (ONC) have a high mul- tiplicity compared to low mass stars (Mason et al. 1998; Preibisch et al. 1999; Stahler et al. 2000; Garćıa & Mermilliod 2001). The formation of a multiple system in the early life of a cluster can occur through fragmentation of the prestellar material, or by capture. Capture can occur via multi-body interactions with other cluster members, or through disk interactions in the protostellar phase. There is growing observational evidence for massive, embedded disks surrounding massive protostars (Cesaroni et al. 2007, for a recent review). Simulations of massive star formation suggest that the masses of the disks may build up to values ∼ 30% of the central star’s mass before global instabilities trigger a sudden accretion event (Krumholz et al. 2007). Fragmentation of such massive disks into companions is possible (Kratter & Matzner 2006), but in this work we consider capture of a lower mass cluster member by a massive star-disk system, continuing the analysis pre- sented in Moeckel & Bally (2006) and Moeckel & Bally (2007, hereafter MB07). Disk assisted capture in low mass systems has been studied using analytic methods (Clarke & Pringle 1991) and smoothed particle hydrodynamics (SPH) codes (Heller 1995; Boffin et al. 1998). The capture rates from these studies are too low to be a significant contributer to the binarity of roughly solar mass stars; for 20 M⊙ stars with a 2 M⊙ disk in a Trapezium-like environment, MB07 show that the capture rate is high enough to account for 50% binarity after 1 Myr. MB07 estimated analyti- cally the likelihood of survival for these binaries, and con- cluded that higher mass captured companions are more likely to survive, while lower mass companions would be preferentially ionized by encounters with other cluster members. However, in calculating capture rates and es- timating survival fractions, many averages and integrals are taken. It is desirable to further test these results in an unaveraged, more ‘realistic’ setting. Electronic address: [email protected] In this paper we describe the results of n-body simula- tions of a cluster similar to the ONC, in which we include the dissipative effects of passages through a circumstellar disk surrounding a central 20 M⊙ star. This test demon- strates that the rates derived by MB07 are reasonable in a cluster setting, and that the most likely outcome of en- counters between the captured binary and other cluster members leaves the central star in a binary. 2. SIMULATIONS In this study we combine the SPH results of MB07 with n-body simulations of a cluster similar to the ONC. This is similar in spirit to the work of Scally & Clarke (2001) and Pfalzner et al. (Pfalzner et al. 2006; Pfalzner 2006; Pfalzner & Olczak 2007), in which n-body simula- tions of a cluster are used to determine encounter fre- quency and parameters, which are then used to study the effect of encounters on disks. This work is differ- ent in that the results of the close encounters are in- cluded in the simulation as it is running, similar to McDonald & Clarke (1995), who used an analytic pre- scription for disk-mediated binary formation in simula- tions of a cluster with 10 stars. The simulations are performed using Aarseth’s code NBODY6 (Aarseth 2000). The cluster is set up as a King model (e.g. Binney & Tremaine 1987) with W0 = 9, core radius rc = 0.2 pc, central density n0 = 2.0× 104 pc−3, and central velocity dispersion σ0 = 2.19 km s−1. These parameters are similar to those of the ONC (Hillenbrand & Hartmann 1998), and with a cut-off ra- dius of 2.0 pc yield 4725 stars in the simulation. The IMF used is that of Kroupa (2002) in the range 0.3 - 9.0 M⊙ . Each star is randomly placed according to the King model density distribution, with a random veloc- ity appropriate to its radius. In addition, we place a 20 M⊙ star at the center of the simulation with a random velocity. 1000 sets of initial conditions were generated; each is run with and without the effects of disk dissipa- tion. In the dissipation runs, the only star with a disk is the central star. This is an artificial situation; while the most massive disk will dominate an interaction, the presence of disks around all stars in the cluster would http://arxiv.org/abs/0704.1162v1 mailto:[email protected] Fig. 1.— Energy change as a function of remnant disk mass for three of the binaries simulated in Moeckel & Bally (2006). The outlying point for i = 180◦ shows that the true effect of disk dis- ruption on repeated encounters is more complicated than a simple mass scaling. However, for most encounters, a scaling between the two bold lines is reasonable. increase the effect of disk encounters. MB07 used a modified version of the publicly released SPH code GADGET-2 (Springel 2005) to simulate en- counters between the massive star-disk system and im- pactors with masses in the range 0.3 - 9.0 M⊙, perias- tra in the range 50 - 550 AU, and inclinations from 0◦ - 180◦. The disk radius is 500 AU. Interpolation of the data from these simulations gives, for any impactor mass, periastron, and inclination angle, a change in orbital en- ergy and change in disk mass associated with the en- counter. We have modified NBODY6 to detect encoun- ters between the central star and other cluster members. At the encounter periastron, we determine the orbital pa- rameters, and then modify the velocity of both encounter partners in a momentum conserving fashion so that the change in orbital energy is equal to that found via in- terpolation of the data from MB07. The change in disk mass due to the encounter is also tracked. Because we only have data for periastra ≥ 50 AU, encounters with smaller periastra are assumed to occur at 50 AU. When the orbit of a relatively massive impactor is coplanar to the disk, MB07 find that accretion and disk capture can increase the impactor’s mass by up to 10%. However, the change in orbital energy during these pas- sages is an order of magnitude greater than can be ac- counted for by accretion drag; the dominant contribution to the energy change is the change in velocity due to disk interactions. We make the simplification that the change in orbital energy is due entirely to a change in the rel- ative velocity of the two stars. Under this assumption, at periastron the velocity kick δv on the impactor for a given orbital energy change ∆E is given by −2v + 4v2 + m(1 +m/M) v̂, (1) where v is the pre-kick velocity, m is the impactor mass, and M is the primary mass. The change in velocity for Fig. 2.— The distribution of encounter frequency for runs with no disk dissipation. Also plotted are the best fit Poisson distribution to the realizations in which the central star remains in the cluster core (solid), and the expected value from equation 3 (dotted). The error bars are single sided 1-σ confidence limits (Gehrels 1986) the primary is δV = −(m/M)δv. The change in the total energy due to these kicks is tracked and included in energy checks. In order to account for the change in disk mass, we scale the change in energy for an encounter according to ∆E(md, i,mi, rp) = ∆E0(i,mi, rp) . (2) Here ∆E is the orbital energy change from an encounter with disk massmd, impactor massmi, inclination angle i, and periastron rp. ∆E0 is the energy change for the same parameters with the disk at its original mass md0, and is found from interpolation of the data in MB07. The index n we take to be 1 or 2, and scales the proportionality of the the energy change with the remaining disk mass fraction. The scaling of energy change with disk mass is not completely straightforward. Heller (1995), simulating en- counters with disks of different masses, found that the change in energy scales roughly linearly with the disk mass, in which case n = 1. Moeckel & Bally (2006) sim- ulated repeated encounters between a captured impactor and the same remnant disk. Analysis of that data (Fig. 1) shows that the scaling of energy change and disk mass is more like n = 2. However, this also takes into ac- count the changing radius of the disk, an effect which we do not include in this study. Since the proper scaling depends on details of the encounters that would require full SPH simulations of each close passage, we instead run all simulations with both scalings and compare the results. Our scheme preserves the binary periastron separation, which is shown by Moeckel & Bally (2006) to decrease during repeated retrograde encounters, and increase with prograde encounters. In the most extreme (and rarest) case, an in-plane retrograde passage, the periastron sepa- ration decreases by ∼ 25% after the first encounter. The data in MB07 show that the change in energy scales com- parably to the periastron radius; thus we would expect an error . 25% due to our artifically fixed periastron. This is similar in magnitude to the uncertainty in our mass scaling, which as shown below has a negligible im- pact on our results. Each case is run for 0.5 Myr; we limit the analysis to this time for three reasons. After this time the disks are mostly destroyed. In our simulations this destruction is by encounters, while in reality the additional effect of photo-evaporation will contribute. Mass segregation also begins to take effect after this time, which is an added complication in the analysis of the results. Finally, the multiplicity of stars is established early in their evolution (Mathieu 1994), and a mechanism for binary formation should work on short enough timescales to reflect that. 3. RESULTS 3.1. Encounter Rates The calculation of binary capture rates (Clarke & Pringle 1991; Heller 1995; Boffin et al. 1998; MB07) is largely similar to the estimation of collision or encounter rates. We begin by comparing the standard encounter timescale calculation to the simulations. The encounter rate for a star of mass M in a cluster with number density n, velocity dispersion σ, and en- counter radius r is given by γ = 4 πnσr2 . (3) Here M is M + m̄, with m̄ the average stellar mass in the cluster (Binney & Tremaine 1987; MB07 for general masses). For the parameters of our simulated clusters, the encounter rate is γ = 1.02 × 10−5 yr−1, with 5.1 encounters expected over 0.5 Myr. Plotted in figure 2 is the distribution of the number of unique cluster members encountered by the central star, without disk dissipation. Multiple encounters with the same star, for instance in a binary, are counted only once. Because some of the central stars have a high ini- tial velocity and escape to the cluster outskirts, there is an excess of low encounter-number runs. Therefore the distribution for all runs is plotted, as well as only those in which the central star remained within the cluster core. One would expect that the distribution for stars in the same environment would be Poisson; the data shows oth- erwise. Because the number density and velocity disper- sion are radially dependent, central stars that move to larger radii are exposed to a much different environment than those that remain near the cluster center. By limit- ing our analysis to the cluster core, where the properties are closest to those used in equation 3, the best fit (with mean value 5.78) is in reasonable agreement with the theoretical value of 5.1. Equation 3 is averaged over the mass function. Because of the dependence of the encounter rate on the mass of the stars, encounters with more massive cluster members are more frequent. The mass function of encounter part- ners is well fit by a single power-law mass function of the form ξ(m) ∝ m−α with α = 0.92, while a Salpeter mass function has α = 2.35. 3.2. Binary Fraction and Mass Function Of greatest interest is the fraction of the massive stars at a given time that are in a multiple system. MB07 Fig. 3.— The number of runs with the central star in a binary, NB(t), as a fraction of the total number of runs NT for the three simulation series. Cases with both n = 1 and n = 2 in equation 2 are at ∼ 30% after 0.5 Myr. calculate that for a cluster with the parameters of our models, the binary formation rate is Γ = 0.6 Myr−1 (Figure 3 in MB07). We compare this to our simula- tions as follows. For each simulation, a list of stars that encounter the central star is generated, with the times of their encounters. If the same star is involved in consec- utive encounters, a binary has formed, and we track the following events. Ionization: The next two encounters are with differ- ent stars. Even if the intruder and the original binary partner form a binary, the central star is no longer in a multiple system. Exchange: The next two encounters are with the same star, which is different from the initial bi- nary. Flyby: The next encounter is with a different star, but the one after that is with the initial binary partner. In the latter two cases, the central star is considered to be in a binary throughout. In this scheme the possibility ex- ists that random, consecutive encounters with the same star could contaminate the binary statistics. In practice, the number of binaries with semi-major axes larger than the average inter-stellar distance in the cluster core is on the order of 1%, and we consider the detected binaries to be true binaries. Plotted in figure 3 are the number of central stars in binaries as a function of time, for each of the three series of simulations. Considering first the series with no disks, we see an initial rise in the binary fraction, followed by a leveling off to ∼ 12% at 0.5 Myr, as binary creation through random dynamical processes is balanced by ion- ization. For the two series with disk dissipation, there is a steady rise up to a value of approximately 30%, in good agreement with the calculations in MB07. The binary fraction does not appear to depend on the specifics of the energy-change disk-mass scaling; the early passages, when the disk still has nearly its original mass, account for the increased binarity. Since massive impactors are preferentially captured by the disk and more likely to remain bound during ex- changes and flybys, the mass function of binary partners at 0.5 Myr is flatter than that of the encounter partners. TABLE 1 Binary fate statistics at 0.5 Myr Series NB NE NI NF NS (NE +NF )/NI no disk 387 44 224 431 119 2.12 n = 1 708 186 217 1619 305 8.32 n = 2 698 169 238 1482 291 6.94 With no disks, the best fit single power-law mass func- tion has α = 0.67. The disk case with n = 1 has α = 0.16, and for n = 2 we have α = 0.12. 3.3. Binary Robustness MB07 estimated the survival probabilities of capture- formed binaries, finding that massive companions are more likely to survive encounters with other cluster mem- bers, and that lower mass companions are likely to be ion- ized. The binary fractions found here are in agreement with calculations that don’t include ionization effects; in order to explain this we turn to the question of ionization versus exchange and flybys. Shown in table 1 are NB, the number of binaries formed, NE , the number of exchanges, NI , the number of ionizations, and NF , the number of flybys that occur in each series of 1000 simulations. NB includes all unique binary pairings, so that an exchange contributes twice. Thus the number of surviving binaries NS is given by NB − NE − NI . The ratio of exchanges and flybys to ionizations, (NE +NF )/NI , is indicative of the survival chances of a binary in each series. For the diskless case, n = 1, and n = 2 this ratio is 2.12, 8.32, and 6.94 respec- tively. Encounters between binaries and other cluster members are much more likely to end with the central star in a binary for the simulations with dissipation com- pared to the diskless case, and suggest that the binaries formed via this capture mechanism are more robust than indicated by the simple estimations of MB07. The in- creased disk dissipation in the n = 1 series yields slightly harder binaries, but the total binary fraction is not sig- nificantly affected by the scaling. 4. DISCUSSION The simulations presented here are intended to test and verify the capture rates calculated in MB07. The conclu- sions of that work are largely upheld; encounters occur at approximately the expected frequency, and binaries are captured at a rate consistent with the analytical es- timates. In addition, once a binary is formed it is less likely to be destroyed by ionization than the estimates in MB07 suggest. Encounters between the captured bi- naries and intruding cluster stars are far more likely to result in a flyby or exchange than in an ionization, leav- ing the central star in a binary. The capture rates for the central star-disk system are not high enough to fully account for the high multiplicity observed in massive, cluster-bound stars. Our simula- tions produce ∼ 30% binarity at 0.5 Myr, the time when the disks are mostly destroyed by encounters or photo- evaporation, an effect not modeled here. Recent work (Krumholz et al. 2007) shows that during the formation of a massive star, material moves through the protostel- lar disk in sporadic, massive accretion events, between which the disk builds up to large masses. A disk that is ∼ 30−50% of the central star’s mass, instead of the 10% used here, could increase the capture rates by a factor of several. Additionally, continued accretion onto the sys- tem or the presence of disks around all the stars could increase the capture rates. It is worth noting that our simulations here are tailored to ONC-like systems. Since the capture rate is linear with stellar density and drops with higher velocity dispersion, changing the cluster pa- rameters will affect the results. The effect of encounters on accretion processes in mas- sive star formation is unclear. It is possible that the de- struction of the disk by repeated passages could truncate accretion at the time of binary formation. Alternatively, accretion could resume onto the binary and tighten the orbit, leading to a massive, close binary system. The frequency of the encounters modeled here is high enough that such a situation warrants further investigation. As concluded in MB07, this rate can not be ignored. However, additional binary formation mechanisms must be employed to explain the observed multiplicity of mas- sive stars. In the Trapezium the massive stars have, on average, 1.5 companions (Zinnecker & Bate 2002). Since disks are effectively destroyed during binary capture, this is a process that can only account for a single compan- ion, unless further accretion creates a new circumstellar or circumbinary disk. This work was supported by NASA grant NNA04CC11A to the CU Center for Astrobiology. REFERENCES Aarseth, S. J. 2000, in The Chaotic Universe, Proceedings of the Second ICRA Network Workshop, Advanced Series in Astrophysics and Cosmology, vol.10, Edited by V. G. Gurzadyan and R. Ruffini, World Scientific, 2000, p.286, ed. V. G. Gurzadyan & R. Ruffini Binney, J., & Tremaine, S. 1987, Galactic Dynamics (Princeton, NJ, Princeton University Press, 1987, 747 p.) Boffin, H. M. J., Watkins, S. J., Bhattal, A. S., Francis, N., & Whitworth, A. P. 1998, MNRAS, 300, 1189 Cesaroni, R., Galli, D., Lodato, G., Walmsley, C. M., & Zhang, Q. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, 197–212 Clarke, C. J., & Pringle, J. E. 1991, MNRAS, 249, 584 Garćıa, B., & Mermilliod, J. C. 2001, A&A, 368, 122 Gehrels, N. 1986, ApJ, 303, 336 Heller, C. H. 1995, ApJ, 455, 252 Hillenbrand, L. A., & Hartmann, L. W. 1998, ApJ, 492, 540 Kratter, K. M., & Matzner, C. D. 2006, MNRAS, 373, 1563 Kroupa, P. 2002, Science, 295, 82 Krumholz, M. R., Klein, R. I., & McKee, C. F. 2007, ApJ, 656, Mason, B. D., Gies, D. R., Hartkopf, W. I., Bagnuolo, W. G., Brummelaar, T. T., & McAlister, H. A. 1998, AJ, 115, 821 Mathieu, R. D. 1994, ARA&A, 32, 465 McDonald, J. M., & Clarke, C. J. 1995, MNRAS, 275, 671 Moeckel, N., & Bally, J. 2006, ApJ, 653, 437 —. 2007, ApJ, 656, 275 Pfalzner, S. 2006, ApJ, 652, L129 Pfalzner, S., & Olczak, C. 2007, A&A, 462, 193 Pfalzner, S., Olczak, C., & Eckart, A. 2006, A&A, 454, 811 Preibisch, T., Balega, Y., Hofmann, K.-H., Weigelt, G., & Zinnecker, H. 1999, New Astronomy, 4, 531 Scally, A., & Clarke, C. 2001, MNRAS, 325, 449 Springel, V. 2005, MNRAS, 364, 1105 Stahler, S. W., Palla, F., & Ho, P. T. P. 2000, in Protostars and Planets IV, ed. V. Mannings, A. Boss, & S. Russell (Tucson: Univ. Arizona Press), 327 Zinnecker, H., & Bate, M. R. 2002, in ASP Conf. Ser. 267: Hot Star Workshop III: The Earliest Phases of Massive Star Birth, ed. P. Crowther, 209

0704.1163

Sharp Asymptotics for KPP Pulsating Front Speed-up and Diffusion Enhancement by Flows

SHARP ASYMPTOTICS FOR KPP PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT BY FLOWS ANDREJ ZLATOŠ Abstract. We study KPP pulsating front speed-up and effective diffusivity enhancement by general periodic incompressible flows. We prove the existence of and determine the limits c∗(A)/A and D(A)/A2 as A → ∞, where c∗(A) is the minimal front speed and D(A) the effective diffusivity. 1. Introduction We study reaction-diffusion fronts in the presence of strong incompressible flows. We consider the PDE Tt + Au · ∇T = ∆T + f(T ) (1.1) on Rn, with T (t, x) ∈ [0, 1] the normalized temperature of a premixed combustible gas. The non-linear reaction rate f is of Kolmogorov-Petrovskii-Piskunov (KPP) type [11]: f ∈ C1,ε([0, 1]), f(0) = f(1) = 0 and f is non-increasing on (1− ε, 1) for some ε > 0, (1.2) 0 < f(s) ≤ sf ′(0) for s ∈ (0, 1). The 1-periodic flow u : Tn → Rn satisfies u ∈ C1,ε(Tn), ∇ · u ≡ 0, u dx = 0. (1.3) That is, u is incompressible and mean-zero. The number A ∈ R is the flow amplitude. We will consider the case of strong flows (i.e., large A) and their influence on the speed of propagation of pulsating fronts for (1.1). This problem has recently seen increased activity and has been addressed by various authors — see, e.g., [1, 2, 5, 7, 9, 10, 13, 14]. A pulsating front in the direction e ∈ Rn, |e| = 1, is a solution of (1.1) of the form T (t, x) = U(x · e− ct, x), with c the front speed, and U 1-periodic in x and such that U(s, x) = 1, U(s, x) = 0, uniformly in x. It is well known [4] that in the KPP case there is c∗e(A), called the minimal pulsating front speed, such that pulsating fronts exist precisely for c ≥ c∗e(A) (we suppress the Department of Mathematics, University of Chicago, Chicago, IL 60637; email: [email protected]. The author acknowledges partial support by the NSF through the grant DMS-0632442. http://arxiv.org/abs/0704.1163v2 2 ANDREJ ZLATOŠ u and f dependence in our notation). We note that c∗e(A) also determines the propagation speed of solutions to the Cauchy problem with general compactly supported initial data [4, 15]. Mixing by flows (coupled to diffusion) typically increases the speed of pulsating fronts for (1.1). The minimal front speed c∗e(A) can grow at most linearly with A [5] and does so for shear (unidirectional) flows [1, 2, 7, 9] u(x) = (α(x′), 0, . . . , 0) (x′ = (x2, . . . , xn)) and e = (1, 0, . . . , 0). (1.4) The same is true for so-called percolating flows which possess infinite channels [7], contrasting with the case of cellular flows when, at least in two dimensions, c∗e(A) = O(A 1/4) [1, 7, 9, 13] (see also [14] for a three-dimensional example). We are interested here in all flows which maximally (i.e., linearly) enhance the minimal front speed for (1.1) and our goal is to determine the asymptotic rate of this front speed-up — to prove the existence and evaluate the limit of c∗e(A)/A as A→ ∞. For shear flows, this limit has been known to exist [2] and has been determined in [9], but both problems have been open in general. We thus consider general periodic flows (1.3) and let w ∈ H1(Tn) ∣ Imw = 0 and u · ∇w = 0 (1.5) be the set of real-valued first integrals of the flow u. We then have the following main result. Theorem 1.1. If u and f satisfy (1.2) and (1.3) and |e| = 1, then c∗e(A) = sup ‖∇w‖2 ≤f ′(0)‖w‖2 (u · e)w2 dx ‖w‖22 . (1.6) In particular, the limit exists. Moreover, f ′(0)→0 c∗e(A) f ′(0)A = sup (u · e)w dx ‖∇w‖2 , (1.7) f ′(0)→∞ c∗e(A) = sup (u · e)w2 dx ‖w‖22 ≤ max {u(x) · e}. (1.8) Remarks. 1. Inequality “≥” in (1.6) (with lim infA→∞ in place of limA→∞) has been proved in [5], and [9] showed equality in the case of shear flows (1.4). 2. (1.8) already appeared in [5], with either lim infA→∞ or lim supA→∞ in place of limA→∞. For shear flows (1.4) the inequality becomes an equality [9] due to (1.6) and continuity of u. 3. Notice that (1.6) (for any f ′(0)) is positive precisely when there exists w ∈ I such that (u · e)w dx 6= 0 (take 1± εw in (1.6)). This is also the condition for positivity of (1.7) and (1.12) below. 4. The result extends directly to the more general case of x-dependent and 1-periodic reaction and second-order term (see Theorem 3.2). We perform the proof in the simpler setting above for the sake of transparency. PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 3 It has been shown in [13, 14] that, at least in two dimensions, there is a close relationship between the minimal front speeds for (1.1) and the effective diffusivity in the homogenization theory for the related advection-diffusion problem Φt + Au · ∇Φ = ∆Φ. (1.9) As is well known, the long-time behavior of solutions to (1.9) is governed by the effective diffusion equation i,j=1 σij(A) ∂xi∂xj Here σ(A) is a constant effective diffusivity matrix. If e ∈ Rn and we let χe,A be the mean-zero solution of −∆χe,A + Au · ∇χe,A = Au · e (1.10) on Tn, then σ(A) is given by e · σ(A)e′ = (∇χe,A + e) · (∇χe′,A + e′)dx = e · e′ + ∇χe,A · ∇χe′,Adx. The effective diffusivity for (1.9) in the direction e ∈ Rn, |e| = 1, is now De(A) ≡ e · σ(A)e = 1 + ‖∇χe,A‖22. (1.11) Again, mixing by flows enhances the effective diffusivity. It is easy to show that De(A) can grow at most quadratically with A, and flows that achieve this are said to maximally enhance diffusion (see [6, 8, 12] and references therein). It turns out that our method applies to the problem of determining the asymptotic rate of this enhancement as well, and we find the limit De(A)/A 2 as A→ ∞ for general periodic flows. To the best of the author’s knowledge, existence of this limit has not been known before. Theorem 1.2. If u satisfies (1.3) and |e| = 1, then De(A) = sup (u · e)w dx ‖∇w‖2 . (1.12) In particular, the limit exists. Moreover, there is w0 ∈ I which is a maximizer of (1.12) and χe,A/A→ w0 in H1(Tn). Remarks. 1. It follows that the left hand side of (1.12) is the square of the left hand side of (1.7). This has been established in two dimensions by Ryzhik and the author [14], even without the A→ ∞ limit (see also [13]). 2. We show that if (1.12) is positive, then the maximizers are precisely w = aw0 + b with a, b ∈ R, a 6= 0. 3. If one considers the small diffusion problem φt = ε∆φ+u ·∇φ instead of (1.9), then the corresponding effective diffusivity satisfies D̃e(ε) = εDe(ε −1). Hence the limit limε→0 εD̃e(ε) also equals (1.12). 4. Again, there is a straightforward extension to the case of x-dependent second order term and even non-mean-zero flows (see Theorem 2.1). 4 ANDREJ ZLATOŠ We prove Theorem 1.2 in Section 2 and Theorem 1.1 in Section 3. The generalizations to the case of x-dependent second-order and reaction terms are Theorems 2.1 and 3.2 below. 2. Effective Diffusivity Enhancement Proof of Theorem 1.2. Let ψA ≡ χe,A/A, so that −∆ψA + Au · ∇ψA = u · e (2.1) Multiplying this by ψA and integrating over T n we obtain using incompressibility of the flow, ‖∇ψA‖22 = (u · e)ψAdx ≤ ‖u · e‖2‖ψA‖2. (2.2) Poincaré inequality ‖w‖2 ≤ C‖∇w‖2 (2.3) for some C <∞ and any mean-zero w then yields ‖ψA‖H1 ≤ C‖u · e‖2. (2.4) It also follows from (2.2) that De(A) + ‖∇ψA‖22 = (u · e)ψAdx. (2.5) Since ‖ψA‖H1 is uniformly bounded, there is a sequence Ak → ∞ such that ψAk converges to some w0 ∈ H1(Tn), weakly in H1(Tn) and strongly in L2(Tn). Then ∆ψAk → ∆w0 and ∇ψAk → ∇w0 in the sense of distributions and (2.1) divided by Ak implies u · ∇w0 = 0 (2.6) in the sense of distributions. Since w0 ∈ H1(Tn), this equality holds almost everywhere and w0 ∈ I. We also have ‖∇w0‖22 ≤ lim sup ‖∇ψAk‖ (u · e)w0dx = ∇ψA∇w0dx ≤ ‖∇ψA‖2‖∇w0‖2 where we used (2.2) in the second step, and (2.1) multiplied by w0 and integrated over T (together with (2.6)) in the third step. Thus ‖∇w0‖2 ≤ ‖∇ψA‖2 (2.7) as well as lim sup ‖∇ψAk‖2 ≤ ‖∇w0‖2. These give ‖∇ψAk‖2 = ‖∇w0‖2, which turns the weak H1-convergence into a strong one: ψAk → w0 in H 1(Tn). (2.8) PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 5 Let us assume w0 6≡ 0. Then ∇w0 6≡ 0 because each ψA is mean-zero. From (2.5) and (2.8), De(Ak) = ‖∇w0‖22 = (u · e)w0 dx = (u · e)w0 dx ‖∇w0‖2 . (2.9) Pick an arbitrary non-constant w ∈ I. If we multiply (2.1) by w and integrate, we obtain (u · e)w dx ∇ψAk∇w dx ∇w0∇w dx ≤ ‖∇w0‖2‖∇w‖2. (2.10) Hence (u · e)w dx ‖∇w‖2 ≤ ‖∇w0‖22 = lim De(Ak) with equality precisely when ∇w is a multiple of ∇w0 (and so w = aw0+ b). This also means that w0 is a maximizer for (1.12). If now Bk → ∞ is any sequence, then as above we can find a subsequence (which we again call Bk) such that ψBk → w1 ∈ I. But then w1 must also maximize (1.12), thus w1 = aw0+b. Moreover, b = 0 because ψA are mean-zero, and (2.9) with Bk in place of Ak forces a = 1. Hence ψA → w0 in H1(Tn) and (1.12) follows. Finally, if w0 ≡ 0 is the only limit point of ψA, then ψA → 0 in H1(Tn), and (1.12) follows from (2.5) and (2.10). � Notice that (2.5), (2.7), and (2.9) show that De(A) ≥ 1 + δA2, where δ is the limit in (1.12). We also note that in the special case of shear flows u(x) = (α(x′), 0, . . . , 0) equation (1.10) becomes −∆x′χe,A = Ae1α(x′) with χe,A(x) = χe,A(x ′). Hence χe,A = Ae1∇x′(−∆x′)−1α and the limit in (1.12) equals |e1|‖∇x′(−∆x′)−1α‖22. This can be found, e.g., in [8, Lemma 7.3]. As mentioned above, the result easily extends to the case of x-dependent second order term and a non-mean-zero flow. We consider Φt + Au · ∇Φ = ∇ · (a∇Φ) (2.11) instead of (1.9) with 1-periodic and real symmetric uniformly elliptic matrix a and 1-periodic flow u such that a ∈ C2(Tn), u ∈ C1,α(Tn), ∇ · u ≡ 0, ū ≡ u dx. (2.12) Then (1.10) and (1.11) are replaced by −∇ · (a∇χe,A) + Au · ∇χe,A = A(u− ū) · e, De(A) ≡ |||∇χe,A + e|||22, with |||w|||22 ≡ ∇w · (a∇w) dx. If we define w ∈ I w dx = 0 6 ANDREJ ZLATOŠ then we have Theorem 2.1. If a and u satisfy (2.12) and |e| = 1, then De(A) = sup (u · e)w dx |||∇w|||2 . (2.13) In particular, the limit exists. Moreover, there is w0 ∈ I0 which is a maximizer of (2.13) and χe,A/A→ w0 in H1(Tn). 3. KPP Front Speed-up In this section we prove Theorem 1.1. We start with an auxiliary lemma. Let us define κe(λ) ≡ sup (u · e)w2 dx− ‖∇w‖22 ‖w‖22 . (3.1) Note that κe(λ) must be convex as it is a supremum of linear functions. Also, κe(λ) ≥ 0 because w ≡ 1 ∈ I. Lemma 3.1. Assume the setting of Theorem 1.1. Then for each λ > 0, the supremum in (3.1) is attained, the maximizer is unique up to multiplication, and c∗e(A) = inf f ′(0) + κe(λ) . (3.2) Proof. It has been shown in [4] that the minimal front speed c∗e(A) can be computed using the variational principle c∗e(A) = inf f ′(0) + λ2 + κ(λ;A) . (3.3) Here κ(λ;A) is the unique eigenvalue of the problem ∆ϕ− Au · ∇ϕ− 2λe · ∇ϕ+ λAu · eϕ = κ(λ;A)ϕ, ϕ > 0 (3.4) on Tn, with a unique normalized eigenfunction ϕA(x;λ). Moreover, the function µ(λ;A) ≡ λ2 + κ(λ;A) is monotonically increasing and convex in λ ≥ 0, with µ(0;A) = 0 (see [3, 13]). We now rewrite (3.3) and (3.4) as c∗e(A) = inf f ′(0) + (λ/A)2 + κ(λ/A;A) . (3.5) ∆ϕA − Au · ∇ϕA − e · ∇ϕA + λu · eϕA = κ(λ/A;A)ϕA, ϕA > 0. (3.6) We multiply (3.6) by ϕ−1A and integrate to obtain (using incompressibility of u) 0 ≤ ‖∇ lnϕA‖22 = κ(λ/A;A). (3.7) PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 7 Similarly, multiplication by ϕA yields κ(λ/A;A) + ‖∇ϕA‖22 = λ (u · e)ϕ2Adx ≤ λ‖u · e‖∞. (3.8) since ‖ϕA‖2 = 1. This again means that there is a sequence Ak → ∞ such that ϕAk converges to some w0 ∈ H1(Tn), weakly in H1(Tn) and strongly in L2(Tn). The convergence ∆ϕAk → ∆w0 and ∇ϕAk → ∇w0 in the sense of distributions, boundedness of κ(λ/A;A) in A, and (3.6) divided by A then imply (2.6) and so w0 ∈ I (note that ‖w0‖2 = ‖ϕAk‖2 = 1). Now we multiply (3.6) by w0 and integrate to obtain (with o(1) = o(k 0) and using (3.8)) ∇ϕAk∇w0dx+ λ (u · e)w20dx+ o(1) = κ(λ/Ak;Ak) + o(1) (u · e)w20dx− ‖∇ϕAk‖ 2 + o(1). Once again it follows that ‖∇w0‖22 ≤ lim sup ‖∇ϕAk‖ 2 ≤ ‖∇w0‖2 lim sup ‖∇ϕAk‖2 and so as in Section 2, ϕAk → w0 in H 1(Tn). (3.9) (3.8) then yields κ0 ≡ lim κ(λ/Ak;Ak) = λ (u · e)w20dx− |∇w0|2 dx. Let w ∈ I ∩ L∞(Tn), multiply (3.6) for A = Ak by w2/ϕAk and integrate to obtain (using that ∇ϕA/ϕA = ∇ lnϕA are uniformly bounded in L2(Tn) by (3.7) and (3.8)) κ0‖w‖22 = λ (u · e)w2dx+ lim w2 − 2∇ϕAk w∇w − 2λ e · ∇ϕAk w2 dx (u · e)w2dx− |∇w|2 dx. (3.10) Since each w ∈ I is the H1-limit of wN(x) ≡ sgn(w(x))min{|w(x)|, N} ∈ I ∩ L∞(Tn), this inequality extends to all w ∈ I. Hence κ0 = κe(λ) from (3.1), and w0 is a maximizer for (3.1) (because ‖w0‖2 = 1). Moreover, if Bk → ∞ is any sequence with κ(λ/Bk;Bk) ≡ κ1, then repeating the above argument we find that there must be a subsequence (which we again call Bk) such that ϕBk → w1 ∈ I in H1(Tn), ‖w1‖2 = 1. But then as before, (u · e)w21dx− |∇w1|2 dx = κ1 ≥ (u · e)w2dx− |∇w|2 dx ‖w‖22 for any w ∈ I. Taking w = w0 we obtain κ1 = κe(λ), and so κe(λ) = lim κ(λ/A;A) = λ (u · e)w20dx− |∇w0|2 dx. (3.11) 8 ANDREJ ZLATOŠ The function κe(λ) is convex, monotonically increasing, and non-negative, as it is the pointwise limit of functions µ(λ/A;A) = (λ/A)2+κ(λ/A;A) which have the same properties. This also implies that the convergence in (3.11) is uniform on each bounded interval of λ. We then have f ′(0) + µ(λ/A;A) = inf f ′(0) + κe(λ) (≤ is immediate, whereas ≥ uses convexity of µ(λ/A;A) once more). This proves (3.2). We are left with showing that any maximizer of (3.1) is a multiple of w0. Denote ϕk ≡ ϕAk and notice that (3.9) shows that (after passing to a subsequence — we will repeat this without mentioning it below), ∇ϕk(x) → ∇w0(x) and ϕk(x) → w0(x) for a.e. x. Next (3.7) and (2.3) imply that if ck is the average of lnϕk, then lnϕk − ck → ω strongly in L2 and weakly in H1. But then lnϕk(x)− ck → ω(x) for a.e. x. Since lnϕk(x) → lnw0(x) for a.e. x, it follows that ck → c and ω = lnw0 − c. We thus obtain lnw0 ∈ H1 which means w0(x) 6= 0 for a.e. x, and so for a.e. x, ∇ϕk(x) ϕk(x) → ∇w0(x) w0(x) . (3.12) Let now w 6≡ 0 be a maximizer of (3.1) and let us first assume w ≥ 0 almost everywhere. Then (3.10) for wN and wN → w in H1 show lim sup wN −∇wN But then (3.12) and pointwise convergence of wN and ∇wN to w and ∇w, respectively, give for a.e. x, ∇w0(x) w0(x) w(x) = ∇w(x). We now let wε(x) ≡ max{w(x), ε} so that lnwε ∈ H1 and ∇ lnwε(x) = ∇ lnw0(x) wε(x) > ε, 0 wε(x) = ε. This and lnw0 ∈ H1 means that ‖∇ lnwε‖2 is bounded, and again we must have lnwεk−cεk → ω strongly in L2, weakly in H1, and pointwise almost everywhere. But lnwεk(x) → lnw(x), so again cεk → c and lnw ∈ H1. Hence w(x) > 0 for a.e. x, and so ∇ lnw(x) = ∇ lnw0(x) for a.e. x. This means lnw − lnw0 is constant, that is, w is a multiple of w0. If w is an arbitrary maximizer of (3.1), then both w±(x) ≡ max{±w(x), 0} ∈ I must be maximizers of (3.1) (or ≡ 0). But then w±(x) > 0 for a.e. x, meaning that one of them is zero while the other is a multiple of w0. � Proof of Theorem 1.1. Inequality “≥” in (1.6) is immediate from (3.1) and (3.2). To prove the opposite inequality it is sufficient to find λ such that the unique normalized non-negative maximizer w0(λ) of (3.1) satisfies γ(λ) ≡ ‖∇w0(λ)‖22 = f ′(0). To this end notice that if λ = 0, then w0(λ) ≡ 1 and so γ(0) = 0. Also, γ must be continuous. Indeed — let λk → λ∞ < ∞ and denote wk ≡ w0(λk). Then (3.8) and (3.9) imply that wk are uniformly bounded in H 1. Thus a subsequence (again called wk) converges PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 9 strongly in L2 and weakly in H1 to some ω. Obviously ω ∈ I as well as λk (u · e)w2k dx → (u · e)ω2 dx and ‖∇ω‖2 ≤ lim inf ‖∇wk‖2. But ‖∇ω‖2 < lim inf ‖∇wk‖2 is impossible (otherwise wk would not maximize (3.1) for large k) and so for a subsequence, κe(λk) = λk (u · e)w2k dx− ‖∇wk‖22 → λ∞ (u · e)ω2 dx− ‖∇ω‖22. Since κe is continuous, this means ω = w∞. We have thus proved that every sequence λk → λ∞ has a subsequence with γ(λkj) → γ(λ∞), that is, γ is continuous with γ(0) = 0. Let now Γ ≡ supλ≥0 γ(λ). If f ′(0) ∈ (0,Γ), then there is λ > 0 with γ(λ) = f ′(0) and (1.6) is proved. If, on the other hand, Γ <∞ and f ′(0) ≥ Γ, then (3.2) is bounded above by lim inf (u · e)w0(λ)2 dx− ‖∇w0(λ)‖22 + f ′(0) = lim inf (u · e)w0(λ)2 dx, which does not exceed the right hand side of (1.6) due to ‖∇w0(λ)‖22 ≤ Γ ≤ f ′(0)‖w0(λ)‖22. Since (1.8) is immediate from (1.6), we are left with proving (1.7). Let us consider any w ∈ I with ‖w‖2 = 1 and ‖∇w‖22 ≤ f ′(0). Let w̄ ≡ w dx ∈ [−1, 1] and ω ≡ w − w̄. Then ‖ω‖22 ≤ C‖∇ω‖22 ≤ Cf ′(0), and so w̄2 + 2w̄ω + ω2 dx = w̄2 +O( f ′(0)) as f ′(0) → 0. Hence w̄ = 1 +O( f ′(0)) and we have (u · e)w2 dx = 2 (u · e)ω dx+O(f ′(0)) ≤ 2 f ′(0) (u · e)ω dx ‖∇ω‖2 +O(f ′(0)) with equality when ‖∇ω‖22 = f ′(0). Picking first w that maximizes (1.6) and then ω that maximizes (1.7) with ‖∇ω‖22 = f ′(0) (and adjusting w̄ accordingly) finishes the proof. � Note that in the case of shear flows (1.4) equation (3.6) becomes ∆x′ϕ+ λαϕ = κ(λ/A;A)ϕ, ϕ > 0. (3.13) with ϕ(x) = ϕ(x′). As a result κ(λ/A;A) = κ(λ; 1) = κe(λ) and w0(λ) = ϕ, and (3.5) shows that c∗e(A)/A is non-increasing. This has been proved in [2]. If the limit is γ, then (3.5) gives c∗e(A) − γ ≤ f ′(0) (which has been already observed in [9]). Here one uses convexity of κe and κe(0) = 0 to show that the infimum in (3.5) is achieved at some λ ≤ λA ≡ f ′(0)A, as well as f ′(0) + κe(λ) ≥ min λ∈(0,λA) f ′(0) + κe(λ) κe(λA) Moreover, if the infimum in (3.2) is achieved at a finite λ, then (3.5) gives that c∗e(A) − γ = O(A2). 10 ANDREJ ZLATOŠ This condition is satisfied for all f ′(0) < Γ, where Γ is from the proof of Lemma 3.1, that is, it is the supremum over λ > 0 of the Ḣ1 norms of the principal eigenfunctions of (3.13). This is because of (3.2), the definition of κe, and the fact that κe(λ) = sup (u · e)w2 dx ‖w‖22 Finally, we note that Γ < ∞ is possible — in the shear flow case it holds when there is an open set U ⊆ Tn−1 such that α(x′) = maxTn−1 α for all x′ ∈ U . Then any w ∈ H1(Tn) supported on T × U and independent of x1 belongs to I and maximizes (1.6) whenever f ′(0) ≥ ‖∇w‖22/‖w‖22. Thus the limit in (1.6) need not be strictly increasing with f ′(0) (which happens precisely when Γ <∞). In the more general case when the second order term and the non-linearity depend on x, we consider Tt + Au · ∇T = ∇ · (a∇T ) + f(x, T ) (3.14) with a 1-periodic real symmetric uniformly elliptic matrix and u 1-periodic such that a ∈ C2(Tn), u ∈ C1,ε(Tn), ∇ · u ≡ 0, u dx = 0. (3.15) The non-linearity f is 1-periodic in x and satisfies for some ε > 0 f ∈ C1,δ(Tn × [0, 1]), f(x, 0) = f(x, 1) = 0 and f(x, ·) is non-increasing on (1− ε, 1) for each x ∈ Tn, (3.16) 0 < f(x, s) ≤ sf ′s(x, 0) for (s, x) ∈ (0, 1)× Tn. We let ζ(x) ≡ f ′s(x, 0) > 0 and ζ0 ≡ ζ(x) dx. Equations (3.3) and (3.4) are then replaced by (see [4]) c∗e(A) = inf κ(λ, f ;A) ∇ · (a∇ϕ)− Au · ∇ϕ− 2λe · a∇ϕ+ [λAu · e+ ζ + λ2e · ae− λ∇ · (ae)]ϕ = κ(λ, f ;A)ϕ. If we now define κe(λ, f) ≡ sup (λu · e+ ζ)w2 dx− |||∇w|||22 ‖w‖22 then (3.2) becomes c∗e(A, f) = inf κe(λ, f) and mimicking the above proofs one obtains the following extension of Theorem 1.1. Theorem 3.2. If a, u, and f satisfy (3.15) and (3.16) and |e| = 1, then c∗e(A) = sup |||∇w|||2 ζw2 dx (u · e)w2 dx ‖w‖22 PULSATING FRONT SPEED-UP AND DIFFUSION ENHANCEMENT 11 In particular, the limit exists. Moreover, c∗e(A, αf) = sup (u · e)w dx |||∇w|||2 c∗e(A, αf) = sup (u · e)w2 dx ‖w‖22 ≤ max {u(x) · e}. References [1] B. Audoly, H. Berestycki and Y. Pomeau, Réaction diffusion en écoulement stationnaire rapide, C. R. Acad. Sci. Paris 328, Série IIb, 2000, 255–262. [2] H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, in Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, H. Berestycki and Y. Pomeau eds, Kluwer, Doordrecht, 2003. [3] H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure and Appl. Math., 55, 2002, 949–1032. [4] H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, I - Periodic framework, J. European Math. Soc. 7, 2005, 173–213. [5] H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applica- tions to nonlinear propagation phenomena, Comm. Math. Phys. 253, 2005, 451–480. [6] R.N. Bhattacharya, V.K. Gupta and H.F. Walker, Asymptotics of solute dispersion in periodic porous media, SIAM J. Appl. Math. 49, 1989, 86–98. [7] P. Constantin, A. Kiselev, A. Oberman and L. Ryzhik, Bulk burning rate in passive-reactive diffusion, Arch. Ration. Mech. Anal. 154, 2000, 53–91. [8] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM Jour. Appl. Math., 54, 1994, 333–408. [9] S. Heinze, Large convection limits for KPP fronts, Max Planck Institute for Mathematics Preprint Nr. 21/2005, 2005. [10] A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 2001, 309–358. [11] A.N. Kolmogorov, I.G. Petrovskii and N.S. Piskunov, Étude de l’équation de la chaleur de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh. 1 (1937), 1–25. [12] P. Kramer and A. Majda, Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena, Phys. Rep. 314, 1999, 237–574. [13] A. Novikov and L. Ryzhik, Bounds on the speed of propagation of the KPP fronts in a cellular flow, Arch. Rat. Mech. Anal., to appear. [14] L. Ryzhik and A. Zlatoš, KPP Pulsating front speed-up by flows, preprint. [15] H. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, Jour. Math. Biol., 45, 2002, 511–548. 1. Introduction 2. Effective Diffusivity Enhancement 3. KPP Front Speed-up References

0704.1164

Pulsating Front Speed-up and Quenching of Reaction by Fast Advection

PULSATING FRONT SPEED-UP AND QUENCHING OF REACTION BY FAST ADVECTION ANDREJ ZLATOŠ Abstract. We consider reaction-diffusion equations with combustion-type non-linearities in two dimensions and study speed-up of their pulsating fronts by general periodic incom- pressible flows with a cellular structure. We show that the occurence of front speed-up in the sense limA→∞ c∗(A) = ∞, with A the amplitude of the flow and c∗(A) the (minimal) front speed, only depends on the geometry of the flow and not on the reaction function. In particular, front speed-up occurs for KPP reactions if and only if it does for ignition reac- tions. We provide a sharp characterization of the periodic symmetric flows which achieve this speed-up and also show that these are precisely those which, when scaled properly, are able to quench any ignition reaction. 1. Introduction and Examples In this paper we study the effects of strong incompressible advection on combustion. We consider the reaction-advection-diffusion equation Tt + Au(x) · ∇T = ∆T + f(T ), T (0, x) = T0(x) ∈ [0, 1] (1.1) on D ≡ R× Td−1, with u a prescribed flow profile and A ≫ 1 its amplitude. Here T (t, x) ∈ [0, 1] is the normalized temperature of a premixed combustible gas and f is the burning rate. We assume that u ∈ C1,ε(D) is a periodic incompressible (i.e., ∇·u ≡ 0) vector field which is symmetric across the hyperplane x1 = 0. That is, u(Rx) = Ru(x) where R(x1, . . . , xd) = (−x1, x2, x3, . . . , xd) is the reflection across x1 = 0. If the period of u in x1 is p, then this implies that u is symmetric across each hyperplane x1 = kp, k ∈ Z. Hence u is a periodic symmetric flow of cellular type (since u1(x) = 0 when x1 ∈ pZ) with [0, p] × Td−1 a cell of periodicity. The reaction function f ∈ C1,ε([0, 1]) is of combustion type. That is, there is θ0 ∈ [0, 1) such that f(s) = 0 for s ∈ [0, θ0] ∪ {1} and f(s) > 0 for s ∈ (θ0, 1), and f is non-increasing on (1−ε, 1) for some ε > 0. This includes the ignition reaction term with θ0 > 0 and positive reaction term with θ0 = 0. In the latter case we single out the Kolmogorov-Petrovskii- Piskunov (KPP) reaction [13] with 0 < f(s) ≤ sf ′(0) for all s ∈ (0, 1). We will be interested in two effects of the strong flow Au on combustion: pulsating front speed enhancement and quenching of reaction. This problem has recently seen a flurry of activity — see [1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18, 22, 23, 24]. A pulsating front is a solution of (1.1) of the form T (t, x) = U(x1 − ct, x), with c the front speed and U(s, x) periodic in x1 (with period p) such that U(s, x) = 1 and lim U(s, x) = 0, http://arxiv.org/abs/0704.1164v2 2 ANDREJ ZLATOŠ uniformly in x. It is well known [4] that in the case of positive reaction there is c∗(A), called the minimal pulsating front speed, such that pulsating fronts exist precisely for speeds c ≥ c∗(A). In the ignition reaction case the front speed is unique and we again denote it c∗(A). In the present paper we will be interested in the enhancement of this (minimal) front speed by strong flows. We say that the flow Au quenches (extinguishes) the initial “flame” T0 if the solution of (1.1) satisfies ‖T (t, ·)‖∞ → 0 as t → ∞. Here one usually considers compactly supported initial data. The flow profile u is said to be quenching for the reaction f if for any compactly supported initial datum T0 there is an amplitude A0 such that T0 is quenched by the flow Au whenever A ≥ A0. We note that quenching never happens for KPP reactions — the solutions of (1.1) for compactly supported non-zero T0 always propagate and the speed of their spreading equals c∗(A) [4, 20]. In this paper we characterize those periodic symmetric incompressible flows in two di- mensions which achieve speed-up of fronts and, if scaled properly, quenching of any ignition reaction. For l > 0 we denote by lT the interval [0, l] with its ends identified, and we let u(l)(x) ≡ u(x/l) be the scaled flow on R× lT (with cells of size lp× l). Theorem 1.1. Let u be a C1,ε incompressible p-periodic flow on D = R × T which is sym- metric across x1 = 0, and let f be any combustion-type reaction. (i) If the equation u · ∇ψ = u1 (1.2) on pT× T has a solution ψ ∈ H1(pT× T), then lim sup c∗(A) <∞ (1.3) and no u(l) is quenching for f . (ii) If (1.2) has no H1(pT× T)-solutions, then c∗(A) = ∞ (1.4) and if f is of ignition type, then there is l0 ∈ (0,∞) such that the flow u(l) on R× lT is quenching for f when l < l0 and not quenching when l > l0. Remarks. 1. The proof shows that in (ii), l0 ≥ c‖f(s)/s‖−1/2∞ for some u-independent c > 0. It can also be showed that the claim l0 > 0 in (ii) extends to some positive reactions that are weak at low temperatures (more precisely, f(s) ≤ αsβ for some α > 0 and β > 3 — see Corollary 4.4), in particular, the Arrhenius reaction f(s) = e−C/s(1 − s), C > 0. On the other hand, if f(s) ≥ αsβ for some α > 0, β < 3, and all small s, then l0 = 0 for any u [22]. 2. We note that l0 = ∞ is impossible for cellular flows in two dimensions — see [23] which studies strongly quenching flows u, that is, quenching for any ignition reaction and any l. 3. Although we only consider periodic boundary conditions here, it is easy to see that Theorem 1.1 remains valid for (1.1) on R×[0, 1] with Neumann boundary conditions, provided u2(x) = 0 when x2 ∈ {0, 1}. FRONT SPEED-UP AND QUENCHING 3 4. Although a part of our analysis — Sections 2 and 3 — is valid in any dimension, it remains an open quenstion whether Theorem 1.1 also extends beyond two dimensions. Theorem 1.1 has the following corollary: Corollary 1.2. Let u be a C1,ε incompressible p-periodic flow on D = R × T which is symmetric across x1 = 0. Then speed-up of pulsating fronts by u in the sense of (1.4) occurs for ignition reactions if and only if it occurs for KPP reactions. Remark. Although speed-up of KPP fronts has been studied extensively (see, e.g., [3, 5, 6, 10, 11, 15, 18, 24]), rigorous results on ignition front speed-up have so far been established only in two dimensions for percolating flows and special cellular flows [11] (see below). It is not surprising that the flows which achieve speed-up of fronts are precisely those which quench large initial data. Fast fronts are long, the latter being due to short time–long distance mixing by the underlying flow. Such mixing yields quenching, although possibly only away from regions where the flow is relatively still (e.g., the centers of the cells in Figure 1 below). If these regions are sufficiently small, for instance when the flow is scaled, then reaction cannot survive inside them and global quenching follows. This relation of front speed to flow mixing properties also illuminates Corollary 1.2. Note that the above assumptions on u exclude the class of percolating flows (in particular, shear flows u(x) = (α(x2, . . . , xd), 0, . . . , 0)) which possess streamlines connecting x1 = −∞ and x1 = +∞. In two dimensions, the conclusions of Theorem 1.1(ii) for these flows have been established in [6, 7, 11, 12, 18]. Moreover, results from [5, 24] can be used to prove linear pulsating front speed-up (namely, limA→∞ c∗(A)/A > 0) by percolating flows in the presence of KPP reactions in any dimension. As for cellular flows in two dimensions (the kind we consider here), the claims about the front speed c∗(A) in Theorem 1.1 have been proved for KPP reactions in [18]. The special case of the flow u(x) = ∇⊥H(x) ≡ (−Hx2 , Hx1) with the stream function H(x1, x2) = sin 2πx1 sin 2πx2 has been addressed in [9, 11, 15], which proved (1.4) for any reaction and quenching by u(l) for small enough l and ignition reactions. The streamlines of this flow are depicted in Figure 1. Figure 1. A cellular flow. 4 ANDREJ ZLATOŠ We note that it is easy to show that (1.2) has no H1(T2)-solutions in this case [18], and so one can recover these results from Theorem 1.1(ii). Our general method does not yield the more precise asymptotics c∗(A) ∼ A1/4 in the KPP case [15] and A1/5 . c∗(A) . A1/4 in the ignition case [11] for this particular flow. We conclude this introduction with two more examples of types of flows to which Theo- rem 1.1 applies. Example 1.3. Checkerboard flows. Consider the cellular flow above vanishing in every other cell as depicted in Figure 2, thus forming a checkerboard-like pattern. This flow is both periodic (with period 2) and symmetric but it is not C1,ε. Let us remedy this problem by letting the stream function be H(x1, x2) = (sin 2πx1 sin 2πx2) α with α > 2 in the cells where u does not vanish. Again, (1.2) has no H1(2T×T)-solutions [18], and so Theorem 1.1(ii) — speed-up of fronts and quenching by u(l) — holds. Moreover, the same conclusion is valid for other flows with this type of structure, even if the angle of contact of the “active” cells is π. Figure 2. A checkerboard cellular flow. Example 1.4. Flows with gaps. Consider again the cellular flow above but with a vertical “gap” of width δ > 0, in which the flow vanishes, inserted in place of each vertical segment {k}×T, k ∈ Z, such as shown in Figure 3. We again need to alter the stream function as we did in the previous example in order to make the flow C1,ε. This time it is easy to see that (1.2) has H1((1+ δ)T×T)-solutions [18], and so Theorem 1.1(i) — no speed-up of fronts and no quenching by u(l) — holds in this case. The same conclusion is valid for other flows with similar structures of streamlines, even when the gaps are replaced by channels in which the flow moves “along” the channel only (see [18] for more details). We also note that Sections 2 and 3 below yield the conclusions of Theorem 1.1(i) for cellular flows with gaps in any dimension (using that gaps force Lemma 2.2(ii) to hold). The rest of the paper consists of Section 2 where we prove a few preliminary lemmas, and Sections 3 and 4 which contain the proof of Theorem 1.1. The author would like to thank Sasha Kiselev, Tom Kurtz, and Greg Lawler for useful dis- cussions. Partial support by the NSF through the grant DMS-0632442 is also acknowledged. FRONT SPEED-UP AND QUENCHING 5 Figure 3. A cellular flow with gaps. 2. Some Preliminaries In this and the next two sections we will assume the hypotheses of Theorem 1.1 with the period p = 1 — the general case is handled identically. This implies that u is symmetric across each hyperplane x1 = k, k ∈ Z. The analysis in this section and the next applies to (1.1) on D = R× Td−1 for any d ∈ N. Let us consider the stochastic process X t starting at x ∈ D and satisfying the stochastic differential equation 2 dBt −Au(XA,xt )dt, X 0 = x, (2.1) where Bt is a normalized Brownian motion on D. We note that by Lemma 7.8 in [16], we have that if φt + Au(x) · ∇φ = ∆φ, φ(0, x) = φ0(x), (2.2) φ(t, x) = E . (2.3) In particular, φ0(x) = χ[−L,L](x) gives φ(t, x) = P |XA,xt | ≤ L , (2.4) where we define |x| ≡ |x1| for x ∈ D. Also notice that if φ0 = T0 ∈ [0, 1], then by comparison theorems [19] for any t, x, 0 ≤ T0(t, x) ≤ et‖f(s)/s‖∞φ(t, x) ≤ et‖f ′‖∞φ(t, x). (2.5) Lemma 2.1. (i) If k ∈ Z and y1 = k then the distribution of XA,yt is symmetric across the hyperplane x1 = k, that is, t ∈ V ) = P t ∈ R(V − (k, 0)) + (k, 0) for each V ⊆ D. (ii) If k ∈ Z and y1 ≥ k, then for any I ⊆ R+, t )1 ∈ k + I t )1 ∈ k − I . (2.6) When y1 ≤ k, the inequality in (2.6) is reversed. 6 ANDREJ ZLATOŠ (iii) If L ∈ N, then |XA,yt | ≤ L . (2.7) Proof. (i) and (ii) are obvious from the symmetry of u across x1 = k and from almost sure continuity of X t in t. To show (iii), it is sufficient to consider y1 > L. Applying (ii) with k = jL for j = 1, . . . , ⌈y1/L⌉ − 1, we see that t )1 ∈ [−L, L] t )1 ∈ [(2j − 1)L, (2j + 1)L] The claim follows. � Next we prove the following key dichotomy. Lemma 2.2. For any sequence {An}∞n=1 one of the following holds. (i) For any t, ε > 0 and L <∞ there are x, n such that |XAn,xt − x| ≤ L < ε. (2.8) (ii) For any t, ε > 0 there is L <∞ such that for any x, n, |XAn,xt − x| ≤ L > 1− ε. (2.9) Proof. Let us first assume that there is t′ > 0 such that for any ε′ > 0 and L′ <∞ there are x, n such that |XAn,xt′ − x| ≤ L < ε′. (2.10) Given any ε > 0, L ∈ N, let m > 2/ε be an integer and let x, n be as in (2.10) with ε′ = 1/m, L′ = (2m+ 1)L. Notice that by periodicity of u we can assume |x1| ≤ 1. For any t ≥ t′ we |XAn,xt − x| ≤ L |XAn,xt | ≤ 2L |XAn,xt′ | ≤ 2mL + sup |y|≥2mL |XAn,yt−t′ | ≤ 2L The first term is smaller than ε′ < ε/2 by (2.10) and the second is at most 1/m < ε/2 by (2.7). This yields (i) for t ≥ t′. On the other hand, if (i) does not hold for some t ∈ (0, t′), then there are ε, L such that for all x, n, |XAn,xt − x| ≤ L Choose m ∈ N so that mt ≥ t′. It follows that |XAn,xmt − x| ≤ mL for all x, n. But this contradicts (i) for mt, which has just been proven. Therefore (i) holds for all t > 0 under the hypothesis above. Now assume the opposite case to the one above. Namely, that for each t′ > 0 there are ε′ > 0 and L′ <∞ such that for all x, n, |XAn,xt′ − x| ≤ L ≥ ε′. (2.11) We will show that then (ii) holds, thus finishing the proof. FRONT SPEED-UP AND QUENCHING 7 For each t > 0 let ε0(t) ≡ sup |XAn,xt − x| ≤ L Periodicity of u guaranties that ε0(t) = sup |x|≤1,n |XAn,xt | ≤ L ≡ sup ε1(t, L) Notice that ε0(t) is non-increasing. Indeed, for L,m ∈ N and t ≥ t′, ε1(t, L) ≤ ε1(t′, mL) + (2.12) by (2.7), and so ε0(t) ≤ ε0(t′) + 1/m for any m. We will now show that ε0(t) = 1 for all t. To this end assume ε0(t) < 1 for some t. Let m be large (to be chosen later), and let L be such that ε1(t, L) > ε0(t)− (2.13) Consider any |x| ≤ 1, n such that |XAn,xt | ≤ (2m+ 1)L ≤ ε0(t) + . (2.14) Such x, n do exists because of ε0(t) ≥ ε1(t, (2m+ 1)L). Then the set of Brownian paths for which there is t′ ∈ [0, t] such that |XAn,xt−t′ | = (m+ 1)L has measure at least 1− ε0(t)− 1/m. Since |XAn,xt | ∈ [L, (2m+ 1)L] ∣ |XAn,xt−t′ | = (m+ 1)L for some t ′ ∈ [0, t] ≥ inf t′∈[0,t] ′, mL) > ε0(t)− by (2.12) and (2.13), this means |XAn,xt | ≤ (2m+ 1)L |XAn,xt | ≤ L |XAn,xt | ∈ [L, (2m+ 1)L] ≥ ε1(t, L) + 1− ε0(t)− ε0(t)− 2− ε0(t)− ε0(t)− Since ε0(t) < 1, this is larger than ε0(t) + 1/m when m is large enough. This, however, contradicts (2.14). Therefore we must have ε0(t) = 1 for all t, which is (ii). � We will also need the following result which is essentially from [8]. Lemma 2.3. For any d ∈ N, there is c > 0 such that for any Lipschitz incompressible flow u, any A, and any t ≥ 0, the solution φ of (2.2) on Ω ≡ [0, 1]×Td−1 with Dirichlet boundary conditions on ∂Ω satisfies ‖φ(t, ·)‖∞ ≤ 2e−ct‖φ0‖∞. (2.15) 8 ANDREJ ZLATOŠ Proof. The maximum principle implies that it is sufficient to show that there is τ > 0 such ‖φ(τ, ·)‖∞ ≤ ‖φ0‖∞. uniformly in u and A. For incompressible flows on Td and mean-zero φ0 this follows from Lemma 5.6 in [8]. The proof extends without change to our case, the Dirichlet boundary condition replacing the mean-zero assumption when the Poincaré inequality is used. � 3. Proof of Theorem 1.1: Part I Let us now assume that u and f are as in Theorem 1.1 and An → ∞ is such that Lemma 2.2(ii) holds. We will then show that the minimal front speeds c∗(An) are uniformly bounded and the flows Anu do not quench large enough compactly supported initial data T0 for (1.1). The analysis in this section applies to D = R× Td−1 for any d ∈ N. Lemma 3.1. Consider the setting of Theorem 1.1 with D = R× Td−1, and let An → ∞ be such that Lemma 2.2(ii) holds. Then c∗(An) are uniformly bounded above. Proof. Choose L ∈ N that satisfies Lemma 2.2(ii) for t = 1 and ε = 1 . Let x be such that x1 ∈ Z and consider XAn,xt from (2.1). Take τ0 = 0 and let τj be the first time such that |XAn,xτj −X | = 3L (recall that |x| = |x1|). We then have from (2.9) and (2.7), P(τj − τj−1 ≤ 1) ≤ because 1 p+ (1− p) ≥ 3 implies p ≤ 1 . This means that for any large enough C, t ∈ N, P(|XAn,xt − x| ≥ 3LCt) ≤ P(τCt ≤ t) ≤ )Ct−j 2(C−1)t (5/4)C−1CC 2C−1(C − 1)C−1 ≤ κ(C)t with κ(C) ≡ 2Ce(2/3)C → 0 as C → ∞. We used here the fact that fewer than t of the differences τj−τj−1 can exceed 1 in the second inequality, and Stirling’s formula in the fourth. Let now T be the solution of (1.1) with A = An and T0 ≡ χR−×Td−1 . If φ solves (2.2) with A = An and φ0 ≡ T0, then we have by (2.5) for x(s) ≡ (s, 0, . . . , 0), T (t, x(3LCt)) ≤ et‖f ′‖∞φ(t, x(3LCt)) ≤ et‖f ′‖∞P(|XAn,x(3LCt)t − x(3LCt)| ≥ 3LCt) → 0 as t→ ∞, provided C is large enough. On the other hand, it is well known that T (t, x(ct)) → 1 as t→ ∞ when c < c∗(An) [4, 20, 21]. This means c∗(An) ≤ 3LC and we are done. � Lemma 3.2. Consider the setting of Theorem 1.1 with D = R× Td−1, and let An → ∞ be such that Lemma 2.2(ii) holds. Then there is compactly supported T0(x) ∈ [0, 1] such that the solution T of (1.1) with A = An does not quench for any n. FRONT SPEED-UP AND QUENCHING 9 Proof. By comparison theorems, we only need to consider f of ignition type — with θ0 > 0. We again choose L ∈ N that satisfies Lemma 2.2(ii) for t = 1 and ε = 1 . We next note that there is δ > 0 such that P(|XAn,xt − x| ≥ t8/15) ≤ e−t (3.1) for all large enough t and all x ∈ D and n. Indeed, assume x1 ∈ Z and t ∈ Z (the general case follows immediately from this), and let j(t) = inf{j | τj > t}, with τj from the proof of Lemma 3.1. Then that proof shows that for C ∈ Z we have P(j(t) > Ct) = P(τCt ≤ t) ≤ κ(C)t (3.2) with κ(C) < 1 if C is large. On the other hand, symmetry of u across each hyperplane x = k ∈ Z shows that Yj ≡ (XAn,xτj −X )1 are iids with P Yj = ±L . This gives P(|XAn,x − x| ≥ L(Ct)9/17 ∣ j(t) ≤ Ct) ≤ e−(Ct)δ for some δ > 0 by ≈ (1 + jδ− 2 )(1− jδ− (1− j2δ−1)−j1−2δ(1 + jδ− 2 )−j (1− jδ− ]j2δ/2 ≈ e−j2δ/2, where we used Stirling’s formula again. This, the fact that |XAn,xτj(t) − X t | ≤ L (by the definition of τj and j(t)), and (3.2) yield (3.1) for large enough t (with a different δ > 0). We will also need the conclusion of Lemma 3.1 in [9] which says that there is c̃ > 0 such that for any x ∈ D, m ∈ Z, A ∈ R, incompressible u, and t ≥ 1 we have ∈ [m,m+ 1] ≤ c̃t−1/2. (3.3) We note that [9] only considers d = 2, but the general case is identical. Let us now take non-negative ψ0 ∈ C(R) ∩ C3([−2, 2]) such that suppψ0 = [−2, 2], ψ0(s) = ψ0(−s) and ψ0(0) = 2+θ03 , ψ0(s) = (3− |s|)2 − 1 for |s| ∈ [1, 2], ψ′0 is decreasing on [−1, 1]. Note that this means that ψ0 is non-negative, symmetric, non-increasing on R +, and convex where f(ψ0(s)) = 0. We then let T0(x) ≡ ψ0 with a large M ∈ Z to be determined later. We will show using the properties of ψ0 that if T solves (1.1) with A = An, then for τ ≡M3/2 we have T (τ, x) ≥ T0(x) (3.4) (which gives the desired result by comparison theorems). 10 ANDREJ ZLATOŠ Let ε be such that ψ0(1 + ε) = 1+2θ0 and M such that εM +M4/5 ≤M − 2. Let φ be the solution of (2.2) with φ0 ≡ T0 and assume first that x1 ∈ [(1 + ε)M, 2M −M4/5] ∩ Z. Let x′ ≡ (x2, . . . , xd). Then by (2.3), monotonicity of ψ0 on R+, and symmetry of u, φ(τ, x) ≥ m=−M4/5−1 P((XAn,xτ )1 ∈ [x1 +m, x1 +m+ 1])φ0(x1 +m+ 1, x′) P((XAn,xτ )1 ∈ [x1 +m, x1 +m+ 1])(φ0(x1 +m+ 1, x′) + φ0(x1 −m, x′)). (3.5) We have x1 +m+ 1 x1 −m = 2ψ0 +ψ′′0 and τ = M3/2 together with (3.1) implies that the sum of the P(·) terms in (3.5) is larger than 1 (1− e−τδ) = 1 (1− e−M3δ/2). This and ψ′′0(s) = 1+θ03 for s ∈ (1, 2) yields φ(τ, x) ≥ (1− e−M3δ/2)φ0 1 + θ0 (4c̃M1/4)−2 +O(M−3/5), where we also used that (3.3) gives |XAn,xτ − x| ≥ Since φ0(x)− φ0(x1 + 12 , x ′) = O(M−1), this means φ(τ, x) ≥ φ0(x) + c′M−1/2 (3.6) for some c′ > 0 and any large enough M . The same argument applies for any τ ′ ∈ [τ/2, τ ] (with a uniform c′) in place of τ . This, Lemma 2.3, and the fact that φ0 varies on a scale O(M −1) on [⌊x⌋, ⌊x⌋+1]×Td−1 yield (3.6) for any x1 ∈ [(1 + ε)M, 2M −M4/5], provided M is large enough. If x1 ∈ [2M −M4/5, 2M ], then (3.6) follows in the same way because ψ0(s) > [(3− |s|)2 − 1] for s ∈ (2, 3). And if x1 > 2M , then (3.6) is immediate from φ(τ, x) ≥ 0. Symmetry and T ≥ φ give (3.4) whenever |x| ≥ (1 + ε)M , so let us now consider |x| ≤ (1 + ε)M . As above we obtain for large M , φ(τ, x) ≥ φ0(x)− c′M−1/2, (3.7) where c′ only depends on ‖ψ′′0‖∞. We now choose a convex g : R+ → R+ with g(s) ≤ f(s) for s ≤ 3+θ0 and g(s) ≥ α for some α > 0 and all s ≥ 1+3θ0 . Define β > 0 so that if γ(0) = 2+θ0 and γ′(s) = g(γ(s)), then γ(β) = 3+θ0 . Next let f̃ ≡ β g ≤ g when τ = M3/2 ≥ β and let w : (R+)2 → R+ satisfy w(0, s) = s and wt(t, s) = f̃(w(t, s)). FRONT SPEED-UP AND QUENCHING 11 Notice that w(τ, 2+θ0 ) = 3+θ0 and w(τ, s) ≥ s+ αβ for s ≥ 1+3θ0 . (3.8) It is easy to show using f̃ ′, f̃ ′′ ≥ 0 that ws, wss ≥ 0. It then follows that T̃ (t, x) ≡ w(t, φ(t, x)) is a sub-solution of (1.1) with A = An and T̃0 = T0 as long as ‖T̃ (t, ·)‖∞ ≤ 3+θ04 (so that f̃(T̃ ) ≤ f(T̃ )). Since ‖φ‖∞ ≤ ψ0(0) = 2+θ03 , this is true for all t ≤ τ by (3.8) and wt, ws ≥ 0. But then T (τ, x) ≥ T̃ (τ, x), while large enough M guarantees for |x| ≤ (1 + ε)M , φ(τ, x) ≥ φ0(x)− c′M−1/2 ≥ 1+2θ03 − c ′M−1/2 ≥ 1+3θ0 So for these x by (3.8), T (τ, x) ≥ T̃ (τ, x) ≥ φ(τ, x) + αβ ≥ φ0(x)− c′M−1/2 + αβ ≥ φ0(x) = T0(x) when M is large. This is (3.4) and thus concludes the proof. � 4. Proof of Theorem 1.1: Part II We now assume that u and f are as in Theorem 1.1 and An → ∞ is such that Lemma 2.2(i) holds. We will then show that lim supn→∞ c∗(An) = ∞, and that there is c > 0 such that if f is of ignition type with ‖f(s)/s‖∞ ≤ c, then any compactly supported initial datum T0 for (1.1) is quenched by some flow Anu. The analysis in this section applies in two dimensions only, so we will consider d = 2 and D = R× T. Lemma 4.1. Consider the setting of Theorem 1.1 with D = R×T and let An → ∞ be such that Lemma 2.2(i) holds. Then lim supn→∞ c∗(An) = ∞. Proof. Assume that c∗(An) ≤ c0 <∞ for all n and let T be a pulsating front solution of (1.1) with A = An and speed c∗(An), that is, T (t+ c∗(An) −1, x1 + 1, x2) = T (t, x1, x2), T (t,±∞, x2) = 12 ∓ uniformly in x2 (4.1) (recall that u has period 1 in x1). We note that [2] shows Tt(t, x) ≥ 0. (4.2) Integrating (1.1) over [0, c∗(An) −1]×D and using (4.1) and incompressibility of u, we obtain ∫ c∗(An) f(T (t, x)) dxdt. Next we multiply (1.1) by T and again integrate as above to get ∫ c∗(An) T (t, x)f(T (t, x))− |∇T (t, x)|2 dxdt ≤ 1− ∫ c∗(An) |∇T (t, x)|2 dxdt. This means that for some t ∈ [0, c∗(An)−1] (which we take to be 0 by translating T in time), f(T (0, x)) dx ≤ 2c0, (4.3) 12 ANDREJ ZLATOŠ |∇T (0, x)|2 dx ≤ c0. (4.4) We will now show that (4.1)–(4.4) force the reaction zone (front width) to be bounded in the following sense. LetD−ε be the rightmost cell [m ε , m ε +1]×T such that infx∈D−ε T (0, x) ≥ 1−ε (i.e., m−ε is the largest integer for which this condition holds). We also let D ε be the leftmost cell [m+ε , m ε + 1]× T such that supx∈D+ε T (0, x) ≤ 1− ε. Obviously m ε < m ε . We will now show that for each small ε > 0 there is Lε <∞ such that for each n we have m+10ε −m−ε ≤ Lε. (4.5) Assume for a moment that (4.5) holds. Periodicity and (2.8) tell us that there are n and x ∈ D−ε such that |XAn,xτ − x| ≥ Lε for τ ≡ ε‖f ′‖−1∞ > 0. Since x1 ≥ m−ε ≥ m+10ε − Lε, symmetry of u implies (XAn,xτ )1 ≥ m+10ε Using (2.5) and (2.3) we have T (τ, x) ≤ eτ‖f ′‖∞ 1− 10ε < 1− ε ≤ T (0, x) if ε > 0 is small. This contradicts (4.2), so our assumption c∗(An) ≤ c0 <∞ must be invalid. Thus the proof will be finished if we establish (4.5) for all small ε > 0. Let us consider an arbitrary small ε > 0 such that f is bounded away from zero on [1− 13ε, 1− ε ] and assume, towards contradiction, that for each L ∈ N there is n such that m+10ε −m−ε ≥ 10L. (4.6) Let T0(x) ≡ T (0, x), T̄0(x) ≡ [⌊x1⌋,⌊x1⌋+1]×T T0(x) dx, and denote Dj ≡ [m−ε + j,m−ε + j + 1] × T. Then (4.4) and Poincaré inequality (with constant C) imply that for each small δ > 0 and L ≡ ⌈Cc0/δ⌉, at least 7L of the cells Dj , j = L, . . . , 9L, satisfy ‖T0 − T̄0‖2L2(Dj) ≤ C‖∇T0‖ L2(Dj) ≤ δ. (4.7) Hence there are at least ⌊3L ⌋ disjoint 5-tuples of consecutive cells satisfying (4.7). Then (4.3), f bounded away from zero on [1−13ε, 1− ε ], and T̄0(Dj) decreasing in j (by (4.2)) imply that for some j0 ∈ [L, 9L] we must have either (4.7) and T̄0(Dj) ≤ 1−12ε for j = j0−2, . . . , j0+2, or (4.7) and T̄0(Dj) ≥ 1 − ε2 for j = j0 − 2, . . . , j0 + 2 (provided δ is small enough and L large). Let us assume the case T̄0(Dj) ≤ 1− 12ε for j = j0− 2, . . . , j0+2, j0 ∈ [L, 9L]. Then (4.2) and (4.6) say that there must be y ∈ Dj0 such that for t ≥ 0, T (t, y) ≥ T0(y) ≥ 1− 10ε. (4.8) FRONT SPEED-UP AND QUENCHING 13 Let S−2γ ⊂ Dj0−2∪Dj0−1∪Dj0 be the square of a small side 2γ > 0 (to be chosen later) centered at y− ≡ y − (1, 0) and denote by Γ− the intersection of S−2γ with the connected component Ω− of the set {x | T0(x) ≥ 1−11ε} containing y− (recall that that T0(y−) ≥ T0(y) ≥ 1−10ε). If Γ− has diameter less than γ (in particular, Γ− = Ω− ⊆ S−2γ), then for Γ ≡ Γ− + (1, 0), all x ∈ ∂Γ, and all t ≤ c∗(An)−1, T (t, x) ≤ T (0, x− (1, 0)) ≤ 1− 11ε by (4.1) and (4.2). It follows by comparison that T (t, x) ≤ et‖f ′‖∞(R(t, x) + 1 − 11ε) where R(t, x) solves (2.2) on S2γ ≡ S−2γ + (1, 0) with Dirichlet boundary conditions and R(0, x) = 11εχΓ(x). But then the uniform bound in Lemma 2.3 and parabolic scaling in (t, x) gives that for any t > 0 there is small enough γ > 0 such that ‖R(t, x)‖∞ ≤ ε2 , and if t is chosen small enough (and γ accordingly), then T (t, y) < 1 − 10ε follows. This clearly contradicts (4.8). If instead (for the chosen γ) the set Γ− ⊂ Dj0−2 ∪ Dj0−1 ∪ Dj0 has diameter at least γ, then T̄0(Dj) ≤ 1 − 12ε and inf T0(Γ−) ≥ 1 − 11ε imply that the second inequality in (4.7) must be violated for at least one of j = j0 − 2, j0 − 1, j0, provided δ > 0 is chosen small enough (depending on γ, ε). Indeed — if ‖∇T0‖2L2(Dj) is small enough, then T must be close to 1− 11ε on some vertical line passing through Γ−, and then T must be close to 1− 11ε on most horizontal lines inside Dj by the same argument. This contradicts T̄0(Dj) ≤ 1− 12ε. Finally, if we instead assume T̄0(Dj) ≥ 1 − ε2 for j = j0 − 2, . . . , j0 + 2 and T (t, y) ≤ T0(y − (1, 0)) ≤ 1− ε for small t ≥ 0, a similar argument again leads to contradiction. This means that (4.6) cannot hold for small ε > 0 and (4.5) follows. The proof is finished. � Lemma 4.2. Consider the setting of Theorem 1.1 with D = R×T. There is c > 0 such that if f is of ignition type with ‖f(s)/s‖∞ ≤ c and An → ∞ is such that Lemma 2.2(i) holds, then for any compactly supported T0(x) ∈ [0, 1] there is n such that the solution T of (1.1) with A = An quenches. Remark. We note that c is from Lemma 2.3 and can be easily evaluated from its proof. Proof. By comparison theorems, it is sufficient to consider initial data T0(x) ≡ χ[−L,L](x1) for all L ∈ N. Let φ be the solution of (2.2) with A = An and initial datum φ0 ≡ T0. We first claim that for each τ, δ > 0 there is n and a continuous curve h : [0, 1] → [0, 1]× T such that (h(0))1 = 0 and (h(1))1 = 1 , and for all s ∈ [0, 1] and t ≥ τ , φ(t, h(s)) ≤ δ. (4.9) To this end we let ψ be the solution of (2.2) with initial condition ψ0 ≡ χ[−K−2,K](x1) where K ≥ 3Lδ−1. By periodicity of u and (2.8), there must be n (which will be kept constant from now on) and y ∈ [−1, 0]× T such that ψ(τ, y) = P (XAn,yτ )1 ∈ [−K − 2, K] The maximum principle for (2.2) implies that the connected component of the set {(t, x) ∈ [0, τ ]×D |ψ(t, x) ≤ δ 14 ANDREJ ZLATOŠ containing (τ, y) must intersect {x ∈ D |ψ(0, x) ≤ δ } = (R \ [−K − 2, K])× T. Since by symmetry ψ(t, x1, x2) = ψ(t,−2−x1, x2) for x1 ≥ 0, this means that there is a curve h(s) joining {0} × T and {K} × T such that for each s there is τs ≤ τ with ψ(τs, h(s)) = P (XAn,h(s)τs )1 ∈ [−K − 2, K] Lemma 2.1(iii) and the definition of K then mean that for all t ≥ τ , φ(t, h(s)) = P |XAn,h(s)t | ≤ L] which is (4.9) (after reparametrization of h and restriction to s ∈ [0, 1]). Symmetry of u and φ0 implies that (4.9) holds for h(s) extended to s ∈ [−1, 1] by h(−s) = (−(h(s))1, (h(s))2). Finally, (4.9) applies to h(s) extended periodically (with period 2) onto R. This last claim holds because φ(t, x) ≥ φ(t, x + (2, 0)) when x1 ≥ −1 (and φ(t, x) ≥ φ(t, x − (2, 0)) when x1 ≤ 1), which in turn follows because φ(t, x) − φ(t, x + (2, 0)) solves (2.2) with initial datum that is symmetric across x1 = −1 and non-negative on [−1,∞)× T (and hence stays such by the symmetry of u). This means that ‖φ(t+ τ, ·)‖∞ ≤ ‖ψ(t, ·)‖∞+ δ where ψ is the solution of (2.2) on 2T×T with ψ0 ≡ 1 and ψ(t, h(s)) = 0 for all t > 0 and s ∈ [0, 2]. Since the Poincaré inequality and the proof of Lemma 2.3 extend to this setting with the same universal constant c > 0, we obtain that ‖φ(t, ·)‖∞ ≤ δ + 2e−c(t−τ). If now ‖f(s)/s‖∞ = c′ < c and τ, δ > 0 are chosen small enough depending on c − c′ (and n accordingly), we obtain ‖T (t0, ·)‖∞ ≤ ′t0(δ+2ecτe−ct0) ≤ θ0 for some t0. The maximum principle then implies ‖T (t, ·)‖∞ ≤ θ0 for any t ≥ t0 and quenching follows. � The proof of Theorem 1.1 is now based on the last four lemmas and this result from [18]: Lemma 4.3. Assume the setting of Theorem 1.1 with f a KPP nonlinearity and D = R×T. (i) If (1.2) on 2T× T has a solution ψ ∈ H1(2T× T), then (1.3) holds. (ii) If (1.2) has no H1(2T× T)-solutions, then (1.4) holds. Proof of Theorem 1.1. If (1.2) has a solution ψ ∈ H1(2T × T), then c∗(An) is bounded for any KPP f and any An → ∞, and so Lemma 4.1 gives Lemma 2.2(ii). Lemmas 3.1 and 3.2 now give (i) for any f . Note that if each sequence An does not quench some compactly supported initial datum T0 for (1.1) with A = An, then there is T0 that is not quenched by any A. This holds because if each T0(x) ≡ χ[−n,n](x1) is quenched by some An, then this sequence would yield a contradiction. If, on the other hand, (1.2) has no H1(2T×T)-solutions, then c∗(An) → ∞ for any KPP f and any An → ∞, and so Lemma 3.1 gives Lemma 2.2(i). Lemma 4.1 now gives (1.4) for any f . The claim about the existence of l0 follows from the fact that T solves Tt − Au(l) · ∇T = ∆T + f(T ) on R× lT if and only if S(t, x) ≡ T (l2t, lx) solves St − Alu · ∇S = ∆S + l2f(S) on R × T. Comparison theorems and f ≥ 0 then show that if u(l) is quenching for f , then so is u(l̃) for any l̃ < l. This only guarantees l0 ∈ [0,∞], but l0 < ∞ follows from Theorem FRONT SPEED-UP AND QUENCHING 15 8.2 in [23] and the fact that the flow u leaves the bounded domain [0, p] × T invariant. For ignition reactions Lemma 4.2 shows l0 > 0 — if each T0 is quenched by at least one Anu for any sequence An → ∞, then each T0 is quenched by Au for all large A. � Finally, we provide the following extension of Theorem 1.1(ii) to some positive reactions. Corollary 4.4. The claim l0 > 0 in Theorem 1.1(ii) holds for any combustion-type reaction satisfying f(s) ≤ αsβ for some α > 0, β > 3, and all s ∈ [0, 1]. Proof. By the proof of Theorem 1.1, it is sufficient to show that there is l > 0 such that u is quenching for l2f(s). The proof is essentially identical to that of Theorem 8.3 in [23]. We let IA ≡ ‖φ(t, ·)‖β−1∞ dt where φ is the solution of (2.2) and φ0(x) ≡ T0(x). It follows from [14] (see also [22, Lemma 2.1]) that u is quenching for l2f(s) when for each compactly supported T0 there is A0 such that l 2α(β − 1)IA < 1 whenever A ≥ A0. So fix T0 and notice that the bound ‖φ(t, ·)‖∞ ≤ c̃|suppT0|t−1/2 for t ≥ 1, which follows from (3.3), gives ‖φ(t, ·)‖β−1∞ dt ≤ 1 if t0 is chosen appropriately (depending on c̃|suppT0|). For t ≤ t0 we use the bound ‖φ(t, ·)‖∞ ≤ 5e−ct, which follows from the proof of Lemma 4.2 (with the same c) provided A0 is chosen large enough so that δ in that proof is smaller than e −ct0 for each A ≥ A0 (and τ is such that ecτ ≤ 2). This choice is possible because each sequence An → ∞ has a term An guaranteeing δ < e −ct0 . Hence for A ≥ A0 we have ‖φ(t, ·)‖β−1∞ dt ≤ (5e−ct)β−1 dt ≡ C <∞. Now let l > 0 be such that l2α(β − 1)(1 + C) < 1, and we are done. � References [1] B. Audoly, H. Berestycki and Y. Pomeau, Réaction diffusion en écoulement stationnaire rapide, C. R. Acad. Sci. Paris 328, Série IIb (2000), 255–262. [2] H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure and Appl. Math. 55 (2002), 949–1032. [3] H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, H. Berestycki and Y. Pomeau eds, Kluwer, Doordrecht, 2003. [4] H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, I - Periodic framework, J. European Math. Soc. 7 (2005), 173–213. [5] H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys. 253 (2005), 451–480. [6] P. Constantin, A. Kiselev, A. Oberman and L. Ryzhik, Bulk burning rate in passive-reactive diffusion, Arch. Ration. Mech. Anal. 154 (2000), 53–91. [7] P. Constantin, A. Kiselev, L. Ryzhik, Quenching of flames by fluid advection, Comm. Pure Appl. Math. 54 (2001), 1320–1342. [8] P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoš, Diffusion and Mixing in Fluid Flow, Ann. of Math. (2), to appear. [9] A. Fannjiang, A. Kiselev and L. Ryzhik, Quenching of reaction by cellular flows, Geom. Funct. Anal. 16 (2006), 40–69. [10] S. Heinze, Large convection limits for KPP fronts, Max Planck Institute for Mathematics Preprint Nr. 21/2005, 2005. 16 ANDREJ ZLATOŠ [11] A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 309–358. [12] A. Kiselev and A. Zlatoš, Quenching of combustion by shear flows, Duke Math. J. 132 (2006), 49–72. [13] A.N. Kolmogorov, I.G. Petrovskii and N.S. Piskunov, Étude de l’équation de la chaleur de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh. 1 (1937), 1–25. [14] P. Meier, On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal. 109 (1990), 63–71. [15] A. Novikov and L. Ryzhik, Bounds on the speed of propagation of the KPP fronts in a cellular flow, Arch. Rat. Mech. Anal. 184 (2007), 23–48. [16] B. Øksendal, Stochastic Differential Equations, Springer-Verlag, Berlin, 1995. [17] J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of para- bolic equations in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 499–552. [18] L. Ryzhik and A. Zlatoš, KPP pulsating front speed-up by flows, Commun. Math. Sci. 5 (2007), 575–593. [19] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. [20] H. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, Jour. Math. Biol. 45 (2002), 511–548. [21] J. Xin, Existence and nonexistence of travelling waves and reaction-diffusion front propagation in periodic media, J. Stat. Phys. 73 (1993), 893–926. [22] A. Zlatoš, Quenching and propagation of combustion without ignition temperature cutoff, Nonlinearity 18 (2005), 1463–1475. [23] A. Zlatoš, Diffusion in fluid flow: Dissipation enhancement by flows in 2D, preprint. [24] A. Zlatoš, Sharp Asymptotics for KPP Pulsating Front Speed-up and Diffusion Enhancement by Flows, preprint. Department of Mathematics, University of Chicago, Chicago, IL 60637, USA Email: [email protected] 1. Introduction and Examples 2. Some Preliminaries 3. Proof of Theorem ??: Part I 4. Proof of Theorem ??: Part II References

0704.1165

Electroweak Higgs as a pseudo-Goldstone boson of broken scale invariance

arXiv:0704.1165v3 [hep-ph] 23 Sep 2007 April 2007 Electroweak Higgs as a pseudo-Goldstone boson of broken scale invariance Robert Foot, Archil Kobakhidze and Raymond R. Volkas 1 School of Physics, Research Centre for High Energy Physics, The University of Melbourne, Victoria 3010, Australia Abstract We point out that it is possible to associate the electroweak Higgs boson with the pseudo- Goldstone boson of broken scale invariance, thus resolving the hierarchy problem in a technically natural way. We illustrate this idea with two specific gauge models. Besides being consistent with all currently available experimental data, both models maintain the predictive power of the standard model, since the first model has only one additional parameter beyond the standard model, and the second has the same number of free parameters as the standard model. 1E-mail: [email protected], [email protected], [email protected] http://arxiv.org/abs/0704.1165v3 1 Introduction Understanding the origin of mass is one of the key problems within the standard model (SM). The chiral nature of the gauge symmetry of the Standard Model forbids masses for quarks and leptons, apart from right-handed neutrino Majorana masses. Taking neutrinos to be Dirac, the only mass parameter in the theory is the electroweak µ2 mass parameter in the Higgs potential. Setting it to zero renders the SM scale invariant at the classical level. However, the scale invariance is known to be anomalous; it is broken at the quantum level. This manifests through the important effect called dimensional transmutation. In particular, in a scale invariant theory a scalar field can develop a vacuum expectation value (VEV) radiatively as a result of the quantum conformal anomaly [1]. On general grounds one can argue that the existence of a light scalar field, with its mass generated entirely at loop level, is inevitable within scale invariant models that involve also multiple scalar fields [2]. This light scalar field is nothing but a pseudo-Goldstone boson (PGB) accompanying the breaking of the anomalous scale invariance. It is tempting to associate this PGB with the electroweak Higgs boson. The reasons are obvious. Firstly, a massless theory is more predictive than the corresponding theory with a priori unknown mass parameters. Secondly, and perhaps more importantly, the electroweak scale generated through the dimensional transmutation will be radiatively stable, thus resolving the technical aspect of the hierarchy problem. Unfortunately, this very appealing theoretical framework applied within the SM predicts a very light Higgs boson with mh < 10 GeV and also requires the top quark to be light: mt <∼ 40 GeV . Both of these predictions are in sharp contradiction with observations. A quick inspection of the Coleman-Weinberg effective potential reveals the way to circum- vent the above problem in weakly-coupled (perturbative) theories. One needs to ensure the overall dominance of the bosonic contributions to the effective potential over the fermionic ones, such that the experimental lower bound on the Higgs boson mass is satisfied while keeping the coupling constant in the perturbative domain. The simplest way to achieve this is by simply adding one (or more) scalar fields to the theory.2 The minimal model of this type involves a Higgs potential with just three parameters – which is only one more parameter than the SM case. We study this model in section II, where we show that it does provide a phenomenological consistent theoretical framework to realise the Higgs boson as a PGB of broken scale invariance. Extended scalar sectors are a feature of many theories beyond the standard model. One 2Scale invariant models with additional scalars have been considered in Refs. [4] and [5] (see also the discussion of the scale-invariant case in Ref. [6]). The philosophy of those models was somewhat different though, with the additional scalar gaining a large VEV in both models which means that the electroweak Higgs could not be interpreted as the PGB of broken scale invariance. Also in Ref. [4] there was an additional U(1)X gauge boson. The models which we consider are simpler, with less parameters than these alternative models. such theory, with a particularly simple Higgs sector, is the mirror model [3]. In that theory, one essentially has two isomorphic sectors of particles, the standard particles and a ‘mirror’ sector. The mirror particles are governed by a Lagrangian of exactly the same form as the standard model, so that a discrete Z2 mirror symmetry can be defined interchanging the ordinary and the mirror particles. If we make the theory scale invariant, by eliminating the µ2 mass parameter in the Higgs potential, then we can generate electroweak symmetry breaking radiatively, via the Coleman-Weinberg mechanism. The Higgs potential then has just two parameters (the same as in the SM) since the Z2 mirror symmetry in the model fixes the quartic coupling constant in the mirror sector to be the same as the corresponding coupling constant in the ordinary particle sector. We show in section III that this classically scale invariant model is phenomenologically consistent, and also realises the Higgs boson as a PGB of spontaneously broken scale invariance. 2 The next-to-minimal scale-invariant Standard Model Consider a minimal extension of the scale-invariant SM with a single extra real scalar field. The most general renormalisable potential is V0(φ, S) = (φ†φ)2 + (φ†φ)S2 , (1) where φ is the electroweak Higgs doublet and S is the real singlet field. observe that the above potential (as well as total Lagrangian) has an accidental discrete Z2 symmetry: S → −S. We parameterise the fields (in the unitary gauge) as: , S = r sinω (2) In this parameterisation, the potential (1) takes the form V0(r, ω) = r cos4 ω + sin4 ω + sin2 ω cos2 ω . (3) Note that the radial component r of the Higgs fields (2) factors out in the absence of the tree-level mass parameter. The renormalised quantum-corrected potential, in addition to (3), includes a sum of δVk−loop contributions generated at k-loop level (k = 1, 2, . . .). Each k-loop contribution is a kth order polynomial in log(φ/Λ), where Λ is some renormalisation scale. Perturbation theory is valid if the logarithms are not too large, so that V0 > δV1−loop > δV2−loop > . . . is satisfied. In general, the minimisation of even a 1-loop corrected effective potential involving multiple scalars cannot be done analytically, but one may resort to a numerical analysis. Here we instead follow the approximate analytic method suggested in [2] which is suitable in weakly coupled scale invariant theories. Following [2], we first ignore perturbatively small radiative corrections and concentrate on the tree-level potential (3). Minimising the potential, assuming that r 6= 0 but is otherwise at this stage arbitrary, leads to two possible cases (we ignore a third case with unbroken electroweak symmetry because it is phenomenologically not viable). If λ3 > 0 then 〈sinω〉 = 0 , 〈r〉 = 2〈φ〉 ≡ v ≈ 246 GeV , < S >= 0 , (4) λ1(Λ) = 0 . (5) In this case only the electroweak Higgs develops a nonzero VEV. If λ3 < 0 then 〈tan2 ω〉 = ǫ , 2〈φ〉 = 〈r〉 1 + ǫ ≡ v ≈ 246 GeV , 〈S〉 = v〈tanω〉 , (6) λ3(Λ) + λ1(Λ)λ2(Λ) = 0 (7) where ǫ ≡ λ1(Λ) λ2(Λ) . In this case both scalar fields develop VEVs and the discrete Z2 symmetry is also broken spontaneously.3 Relations such as Eqs.(5) and (7) can be satisfied by an appropriate choice of the renor- malizsation point µ = Λ [2], where the running coupling constants depend on µ, and Λ is the specific value where the required relations hold. In each case, the tree-level potential then has a flat direction along the vacuum solution, and the relation Eq.(5) [or Eq.(7)] removes a dimensionless parameter in favour of the renormalisation point Λ which has the dimension of mass (dimensional transmutation). Because the tree-level potential vanishes along a specific direction, the 1-loop correction to it will dominate along that direction. Next we calculate the tree-level masses by expanding the Higgs potential, Eq.(1), around the vacuum: φ = 〈φ〉+φ′, S = 〈S〉+S ′. In each case there are two physical scalars, but only one of these gains mass at tree-level, since there is a flat direction in the Higgs potential. Let us call H the state that gets mass at tree-level, and h the state which is massless at tree-level (the PGB of broken scale invariance). For the first case, where λ3 > 0 ⇒ 〈S〉 = 0 [Eq.(4)], we find m2H = , H = S. (8) 3This might cause a problem with cosmological domain walls unless 〈S〉 is sufficiently small. If it is very small but nonzero, then a network of domain walls would form with an equation of state (p = wρ) parameter w = −2/3 in the non-relativistic limit. This is somewhat interesting for dark energy reasons, though present indications are that a w value closer to −1 is preferred by the data. The PGB in this case is h = φ′0. In the second case, where λ3 < 0 and both S and φ gain VEVs, we find m2H = λ1v 2 − λ3v2 , H = − sinωφ′0 + cosωS ′. (9) In this case the PGB is h = cosωφ′0 + sinωS Let us now calculate the mass of the PGB boson for each pattern of symmetry breaking. The 1-loop correction to the tree-level potential (3) (along the radial direction) has the form δV1−loop = Ar 4 + Br4 log , (10) where 64π2〈r〉4 M4V log M4S log − 4Tr M4F log 64π2〈r〉4 3TrM4V + TrM S − 4TrM4F . (12) The traces in the above equations go over all internal degrees of freedom, and MV,S,F are the tree-level masses respectively for vectors, scalars and fermions evaluated for the given VEV pattern. The stationary condition ∂δV1−loop |r=〈r〉 = 0 implies the relation . (13) The PGB mass can be calculated directly from δV1−loop. Using (13) one finds [2]: m2h = ∂2δV1−loop |r=〈r〉 = 8B〈r〉2 8π2〈r〉2 3TrM4V + TrM S − 4TrM4F . (14) Applying this equation to determine the mass of the PGB, we find: m2h = 8π2〈r〉2 6m4W + 3m H − 12m4t m4H cos 8π2v2 , (15) since we need mH to dominate over the other terms if the PGB mass is to be larger than the experimental lower limit of about ∼ 115 GeV. Precision electroweak tests put an upper bound on the Higgs boson mass. The current upper limit for the standard model Higgs is mhiggs < MEW , with MEW ≈ 186 GeV at 95% C.L. [7]. Since the Higgs boson mass gives a radiative correction at l-loop level via a log term, we can find the corresponding limit in this model by the replacement λ2SM logm higgs → λ2SM cos 2 ω logm2h + λ SM sin 2 ω logm2H . Thus, the limit on the scale invariant model from precision electroweak tests is mH < MEW , (16) where cω ≡ cosω. Clearly, in the first symmetry breaking scenario, where cosω = 1, this bound only constrains the mass of the PGB h to be less than MEW which can easily be satisfied (as is also the case in the SM). In the second case, where cosω is essentially a free parameter, the above bound gives a contraint on ω (given the relation, Eq.15). Using the current experimental bound, MEW ≈ 186 GeV, we find that tanω < 0.65. That is to say, the PGB mainly “resides” in the electroweak doublet. Note that the above analysis with one real scalar field can be simply extended to N real scalar fields. Taking for simplicity an O(N) symmetric potential, we simply need to replace S2 → ∑Ni=1 S2i in the potential, Eq.(1). In this case only the λ3 > 0 region, where 〈Si〉 = 0, is phenomenologically viable, since having 〈Si〉 6= 0 would lead to massless Goldstone bosons from the spontaneous breaking of O(N) symmetry. Also, if 〈Si〉 = 0 we can also give S gauge quantum numbers. For example, having S complex and transforming as an SU(3)c colour triplet would be equivalent to having N = 6 real scalar fields with an O(6) symmetric potential. Having N scalar fields, will give a factor N in the right-hand side of Eq.(15) and thus reduce the mass of the heavy scalar (for a fixed mh). To find the domain of validity of perturbation theory we need to look at the renormali- sation group equations [8]: = 3λ21 + N + 8 λ22 + λ 3λ1λ3 (N + 2) λ2λ3 + , (17) where t = log µ. Contributions from gauge and Yukawa coupling constants can be approxi- mately neglected relative to the Higgs potential couplings constants. These equations must be supplemented by the constraint Eq.(5) or (7) for the two sym- metry breaking scenarios of interest. Due to the relatively large mH mass, the position of the Landau pole, µL, is typically only a few orders of magnitude above the weak scale in both symmetry breaking scenarios. For example, for mh at the experimental limit of 115 GeV, taking N = 1 (N = 6) and the case where λ3 > 0 so that 〈S〉 = 0, we find, numerically solving the equations, that µL ≈ 20Λ ≈ 104 GeV (µL ≈ 4× 102Λ ≈ 105 GeV). [Note that Λ is determined from Eq.(13)]. 3 Electroweak Higgs as a PGB of broken scale invari- ance in mirror models In the previous section we showed that extending the Higgs sector in a scale invariant theory allows consistent models giving a naturally light and radiatively stable Higgs boson. Many theories beyond the SM actually require an extended Higgs sector. One theory with a particularly simple Higgs potential is the mirror matter model [3]. In the simplest version of that theory each type of ordinary particle (other than the graviton) has a distinct mirror partner. The ordinary and mirror particles form parallel sectors, each with gauge symmetry GSM = SU(3)⊗SU(2)⊗U(1), so that the overall gauge group is GSM⊗GSM. The interactions within each sector are governed by Lagrangians of exactly the same form, except that mirror weak interactions are right-handed.4 In other words, the full Lagrangian has the form L = L1 + L2 + Lmix , (18) where L1 is the usual Lagrangian of the SM, while L2 is the Lagrangian for the mirror SM. They are related by a parity symmetry P, such that PL1P−1 = L2, which is imposed as an exact symmetry of the whole Lagrangian: PLP−1 = L. The third term Lmix contains only two parity invariant renormalisable interactions that couple the ordinary sector with the mirror one: Lmix = ǫF 1µνF 2 µν + 2λ(φ 1φ1)(φ 2φ2) , (19) where F 1µν and F µν are the U(1) field strengths for the ordinary and mirror sectors and φ1 and φ2 are ordinary and mirror electroweak Higgs doublets, respectively. Thus we have only two extra parameters, ǫ and λ, in addition to those of the ordinary SM. The U(1) kinetic mixing term will not be of interest to us here. The most general tree-level Higgs potential of the scale invariant mirror model can be expressed as V0(φi) = λ 1φ1 + φ 1φ1 + φ . (20) This potential is bounded from below if λ + δ ≥ 0 and λ + δ ≥ 0. We take the Higgs field φi of each sector in unitary gauge and express them as , φ2 = . (21) 4There is a related model where the interchange symmetry is just an internal Z2, unrelated to parity. In that version, the “shadow” weak interactions are left-handed. Then the potential (20) takes the form, V0(r, ω) = λ+ δ(cos4 ω + sin4 ω) . (22) As in the previous section, we can minimise the potential, assuming r 6= 0, which leads to two cases (depending on the sign of λ): Broken − P case for λ > 0 : sinω = 0 or cosω = 0 requiring λ+ δ = 0; Unbroken − P case for λ < 0 : sinω = cosω = 1√ requiring λ+ = 0. (23) As before, the relations between λ and δ will be satisfied by an appropriate choice of the renormalisation point µ = Λ. For the broken-P case, we have either 〈φ1〉 or 〈φ2〉 being zero. Since we are identifying sector 1 as the ordinary sector, we shall concentrate on the sinω = 0 configuration. Next we calculate tree-level mass-squared matrix at µ = Λ: Broken− P Case : M2ij = λ〈r〉2 Unbroken − P case : M2ij = λ〈r〉2 The above mass matrices each have vanishing determinant, so one of the physical scalar fields, the PGB, is massless at tree level. As usual, the conformal anomaly means it obtains a relatively small mass from the 1-loop correction to V0. Also, it can be easily seen that the matrices in Eq.(24) have non-negative eigenvalues provided that λ > 0 (λ < 0) for the broken- (unbroken-) P cases. Thus, there are only two physically distinct minima of the potential (22): 〈r〉 ≡ v ≈ 246 GeV, 〈cosω〉 = 1, if λ > 0 , (25) 〈r〉 ≡ 2 · 246 GeV, 〈cosω〉 = , if λ < 0 . (26) In case (25) the electroweak symmetry is broken in the SM sector, while the electroweak symmetry in the mirror sectors is intact5, and hence the Z2 parity symmetry is spontaneously broken. In this model, the scalar sector consists of the standard Higgs boson, which is massless at tree level, and a complex mirror-doublet of bosons with mass squared: m2H = λv2 ≈ λ(246)2 GeV2. 5SU(2) ⊗ U(1) electroweak symmetry in the mirror sector is eventually broken through mirror SU(3) quark condensation. For details, see Ref. [9]. In case (26), the SU(2)×U(1) symmetry is broken in each sector so the Z2 mirror sym- metry remains exact. Consequently, the gauge bosons and fermions in each sector obtain the same masses as the corresponding particles in the SM. At tree level there is one mas- sive scalar, with m2H = −4λv2 ≈ −4λ(246) GeV2, and one massless state. This massless state corresponds to the PGB of broken scale invariance. The mass eigenstates are parity eigenstates, maximal superpositions of the ordinary and mirror physical Higgs bosons. Let us now calculate the mass of the PGB boson for each pattern of symmetry breaking. As briefly reviewed in section II, the mass of the PGB is, in general, given by Eq.(14). Applying this to the spontaneously broken mirror symmetry case (25) we obtain, m2h ≃ 8π2v2 6m4W + 3m Z + 4m H − 12m4t 2π2v2 for mh >∼ 115 GeV, (27) while for the unbroken mirror symmetry case (26) we find: m2h ≃ 16π2v2 12m4W + 6m H − 24m4t 16π2v2 for mh >∼ 115 GeV. (28) Thus, we effectively have a mass relation between the light and heavy Higgs bosons in the models. Numerically, we obtain the approximate relations: 115 GeV 360 GeV Broken case (Eq.25); 115 GeV 600 GeV Unbroken case (Eq.26). (29) Phenomenologically, the broken mirror symmetry case, Eq.(25), mimics the SM. The light Higgs, h, couples in exactly the same way as does the SM Higgs field, while the heavier states, H , couple to the mirror sector. In this case, the experimental lower limit on mh is the same as the SM limit of approximately 115 GeV. Also, the upper bound on mh inferred from precision electroweak measurements is mh < MEW ≈ 186 [7]. There is no difficulty in satisfying these constraints. In the case of the unbroken mirror symmetry, Eq.(26), the light Higgs field h, couples to both of the sectors, with coupling strength 1/ 2 compared with the SM Higgs. Thus the limit from precision electroweak measurements in this model is given by Eq.(16) with c2ω = 1/2. However, for MEW ≈ 186 GeV, this bound is inconsistent with the relation, Eq.(28). We conclude that the Coleman-Weinberg mechanism is consistent with existing phenomenological bounds only for the broken-P situation6. Below, we therefore discuss the broken-P case only. The experimental lower bound on the mass of the SM Higgs boson, mh > 115 GeV, can be translated into a lower bound on the coupling λ2(Λ) ≡ 2λ(Λ): λ2(Λ) > 4.2 115 GeV , (30) for the broken P case of Eq.(25). To find the domain of validity of perturbation theory we look, as before, at renormalisation group equations: = 3λ21 + λ 2 , (31) = 3λ1λ2 + λ22 , (32) where 1 λ1 = λ + δ. As λ2(Λ) is significantly larger than any other coupling in the SM we can approximately neglect top-quark and gauge boson contributions. These equations must be supplemented by the constraint equation (23), which reads λ1(Λ) = 0 , (33) for the case (25). Numerically solving these equations, we find that the position, µL, of the Landau pole is µL ≈ 3 × 102Λ ≈ 105 GeV (for mh at the experimental limit of 115 GeV). The perturbative domain thus extends confortably above the electroweak scale. 4 Conclusion We have examined the idea that the standard model Higgs boson might be the pseudo- Goldstone boson of broken scale invariance. The simplest version of this idea is the original Coleman-Weinberg model. The problem there is that the mass of the Higgs is too small, and the spontaneous symmetry breaking can only occur if the top quark is also very light. However, if there is an extended Higgs sector, then the additional scalar degrees of freedom can compensate for the heavy top quark, and give a Higgs mass in excess of the experimental lower limit, currently around 115 GeV. Specifically, we have considered two phenomenologically consistent models. The first involves the addition of one (or more) real scalar fields. The additional bosonic degrees of freedom can lead to phenomenologically successful electroweak symmetry breaking. An 6Of course this conclusion is only for the minimal mirror model with two sectors. In this context it will be interesting to study the case of generalized mirror models with N-sectors [10]. However, we will leave this study for the future. even more constrained symmetry breaking sector arises in the mirror model, which has a discrete symmetry interchanging the standard model Higgs with a mirror partner. The discrete symmetry eliminates one parameter in the Higgs potential, and leads to a consistent electroweak symmetry breaking with the same number of parameters as in the standard model case. The proposed scale-invariant models (and their generalizations) can be testable in up- coming LHC experiments in the case of non-zero cosω. The light PGB Higgs interacts with Standard Model particles with couplings reduced by factor cosω relative to the correspond- ing couplings of the Standard Model Higgs boson. In addition heavier Higgs-like boson with a mass correlated with the mass of PGB Higgs (see Eq.(15) can be observed. The couplings of this heavy boson with Standard model particles is suppressed by sinω and its total decay width will be dominated by the decay width into two PGB Higgses. In the case of cosω = 0, the difference between the PGB Higgs and the Standard Model Higgs is rooted in the scalar potential, and, most probably, LHC will not be capable to distinguish among them. Future linear collider (e.g. the proposed ILC) should be able to study this case. 5 Acknowledgements This work was supported by the Australian Research Council. References [1] S. R. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888. [2] E. Gildener and S. Weinberg, Phys. Rev. D 13 (1976) 3333. [3] R. Foot, H. Lew and R. R. Volkas, Phys. Lett. B 272 (1991) 67; The concept of mirror matter was discussed prior to the advent of the standard model in T. D. Lee and C. N. Yang, Phys. Rev. 104 (1956) 256; I. Kobzarev, L. Okun and I. Pomeranchuk, Sov. J. Nucl. Phys. 3 (1996) 837; M. Pavsic, Int. J. Theor. Phys. 9 (1974) 229. [4] R. Hempfling, Phys. Lett. B 379 (1996) 153 [arXiv:hep-ph/9604278]; W. F. Chang, J. N. Ng and J. M. S. Wu, arXiv:hep-ph/0701254. [5] K. A. Meissner and H. Nicolai, arXiv:hep-th/0612165. [6] J. R. Espinosa and M. Quiros, arXiv:hep-ph/0701145. [7] W. M. Yao et al. [Particle Data Group], J. Phys. G 33 (2006) 1. [8] T. P. Cheng, E. Eichten and L. F. Li, Phys. Rev. D 9 (1974) 2259. [9] R. Foot and H. Lew, arXiv:hep-ph/9411390. R. Foot, H. Lew and R. R. Volkas, JHEP 0007 (2000) 032 [arXiv:hep-ph/0006027]. [10] R. Foot, Phys. Lett. B 632 (2006) 467 [arXiv:hep-ph/0507294]; R. Foot and R. R. Volkas, Phys. Lett. B 645 (2007) 75 [arXiv:hep-ph/0610013].

0704.1168

Electronic Properties of Carbon Nanotubes Calculated from Density Functional Theory and the Empirical pi-Bond Model

Electronic Properties of Carbon Nanotubes Calculated from Density Functional Theory and the Empirical π-Bond Model Deep Shah, Nicolas A. Bruque,∗ Khairul Alam, Roger K. Lake,† and Rajeev R. Pandey Department of Electrical Engineering, University of California, Riverside, CA 92521-0204 J. Computational Electronics (accepted, 20 February 2007) Abstract The validity of the DFT models implemented by FIREBALL for CNT electronic device modeling is assessed. The effective masses, band gaps, and transmission coefficients of semi-conducting, zigzag, (n, 0) carbon nanotubes (CNTs) resulting from the ab-initio tight-binding density functional theory (DFT) code FIREBALL and the empirical, nearest-neighbor π-bond model are compared for all semiconducting n values 5 ≤ n ≤ 35. The DFT values for the effective masses differ from the π-bond values by ±9% over the range of n values, 17 ≤ n ≤ 29, most important for electronic device applications. Over the range 13 ≤ n ≤ 35, the DFT bandgaps are less than the empirical bandgaps by 20-180 meV depending on the functional and the n value. The π-bond model gives results that differ significantly from the DFT results when the CNT diameter goes below 1 nm due to the large curvature of the CNT. The π-bond model quickly becomes inaccurate away from the bandedges for a (10, 0) CNT, and it is completely inaccurate for n ≤ 8. Keywords: FIREBALL, CNT, DFT, NEGF, Bandstructure. ∗Electronic address: [email protected] †Electronic address: [email protected] mailto:[email protected] mailto:[email protected] I. INTRODUCTION Carbon nanotube (CNT) systems are of high research interest for use in sensing and nanoelectronics. Electronic device and circuit architecture concepts for bio-assembled CNTs have been described and demonstrated [1, 2, 3, 4, 5]. One system that we have found particularly interesting is a molecule joining two semiconducting CNTs. Such a system can display the electrical response of a resonant tunnel diode (RTD) [6, 7, 8, 9], and we refer to it as a CNT-Mol-RTD. The current-voltage response of a CNT-Mol-RTD depends on the alignment of the CNT bandedges with molecular states of the organic group [6, 7, 8]. Therefore, quantitative simulations of such a structure require models which are suitable for calculating the electronic states of both semiconductors, molecules, and the chemical bonds between the molecule and semiconductor. The models should be general enough to treat local distortions in the semiconductor lattice and distortion of the molecule. A CNT-Mol-RTD structure tends to be large compared to the benzene dithiol type of molecules that have been so heavily studied for molecular electronics [10, 11, 12, 13, 14, 15, 16]. Therefore, the model and its implementation must have the ability to handle large systems. One model widely used for computational molecular electronics is density functional theory (DFT). DFT is a general, flexible theory with a proven record of calculating the electronic states of both semiconductors and molecules. The well known weakness of DFT implemented with widely used density functionals is its underestimation of the bandgap of semiconductors [17]. For electronic device modeling, this is not a trivial issue. The most famous example is that of Ge which is predicted within the local density approximation (LDA) or generalized gradient approximation (GGA) to be a metal. Another example is Zener tunneling which is exponentially dependent on the bandgap. It is this process that limits the maximum on-off current ratio in CNT field effect transistors. Linear errors in the bandgap result in exponentially large errors in the simulated minimum off-current [18]. The final example is the electron current of a CNT-Mol-RTD. It depends on the alignment of a molecular state with the conduction band edge of the CNT. If the conduction band edge is too low, the predicted current - voltage response will be inaccurate. While hybrid functionals that include exact exchange greatly improve the situation, they also significantly increase the computational burden [19]. Another option for the simulation of large structures is to use empirical models [20, 21]. Such models can reproduce the bandgaps and energy levels of semiconductors with high accuracy. However, simulations of heterogeneous CNT, metal, organic, biological systems are difficult with empirical models due to issues of transferability of parameters. Conversely, electronic device simulations of semiconductors are difficult with DFT models due to the underestimation of the bandgap. One is forced to make a choice, and we have chosen the DFT model as implemented by the code FIREBALL [22, 23] since it has demonstrated the ability to model large biological molecules [24]. In this paper, we compare predictions of the properties relevant to electronic device modeling of semiconducting (n,0) CNTs calculated from both DFT theory and an empirical, tight-binding, π-bond model. Two important band-edge properties are the bandgap and the effective masses. For larger diameter CNTs, the empirical, tight-binding model is expected to predict the bandgap and band-edge effective masses with good accuracy since the empirical parameters have been chosen by fitting to those quantities. Therefore, for the larger diameter CNTs, we use the values of the bandgap and effective masses obtained from the π-bond model to assess the values obtained from the DFT model. For smaller diameter CNTs where the curvature becomes significant, the π-bond model breaks down. Bandedge quantities alone are not sufficient for electronic device modeling. Since elec- tronic devices are operated at biases on the order of a volt, accurate modeling of the higher energy states away from the bandedges is necessary for device modeling. The quantity that best characterizes the higher energy spectra for device simulations is the transmission coef- ficient. Therefore, we compare transmission coefficients calculated from the DFT and the empirical, tight-binding models. Below, we calculate and compare the bandgaps, effective masses, and transmission spectra of semiconducting zigzag CNTs, ranging from (5, 0) to (35, 0) corresponding to diameters ranging from 0.39 nm to 2.8 nm. The bandgaps and effective masses are calculated, plotted, and compared for every non-metallic (n, 0) CNT with 5 ≤ n ≤ 35. Selected transmission coefficients are plotted and compared for n = 10, 20, 31, and 35. II. METHOD OF CALCULATION The FIREBALL calculations are performed using the local density approximation (LDA) (the Ceperley-Alder [25] form as parameterized by Perdew and Zunger [26]) and the BLYP exchange-correlation functional [27, 28]. A self-consistent calculation is performed using a generalization of the Harris-Foulkes [29, 30] energy functional referred to as DOGS after the authors of the original paper [31, 32]. A separable non-local pseudopotential [33] and a minimal sp3 FIREBALL basis are used. The localized pseudoatomic orbitals are slightly excited due to hard wall boundary conditions imposed at certain cutoffs, r2sc and r c [32]. An excitation energy of approximately 2.0 eV is used to preserve the chemical bonding trends of carbon which results in r2sc = 4.0 Å and r c = 4.5 Å. Further details are given in [23]. After the FIREBALL DFT calculation finishes, the device Hamiltonian matrix elements, the overlap matrix, and the device-to-contact coupling matrices are extracted. The spatial extent of the non-zero matrix elements (the sparsity of the matrices) is determined by the pseudopotential cutoff limits and the FIREBALL orbital radii. Each CNT system, pictured in Fig. 1, consists of one CNT unit cell composed of 4 atomic layers periodically repeated in the axial direction. Non-zero matrix elements of a given atomic layer extend to the left and right 4 atomic layers, or one unit cell of the zigzag CNT. In terms of the 4-atomic layer unit cells, there is only nearest neighbor unit-cell coupling. Transmission is calculated using the non-equilibrium Green function formalism (NEGF). The CNT is partitioned into a ‘device’ consisting of one unit cell and a left and right ‘contact.’ The left and right ‘contacts’ are taken into account exactly by self-energies Σ` and Σr, respectively, as illustrated in Fig. 1. Details of the NEGF algorithm are described in [6, 7, 8]. For the empirical π-bond model, we use a nearest-neighbor model with matrix element Vppπ = −2.77 eV and �p = 0.0 eV [34]. The NEGF algorithm is the same as that used with the FIREBALL matrix elements. The effective mass for both models is calculated from the 1-D dispersion using 1/m∗ = 1~2∂ 2E/∂k2. The energy band gap is determined from an E − k calculation by reading the difference between the highest occupied band energy and the lowest unoccupied band energy at the Γ point. III. RESULT AND DISCUSSION Figure 2(a) compares the calculated band gaps for (n, 0) CNTs with 5 ≤ n ≤ 35 leaving out the metallic CNTs with n values that are integer multiples of 3. The n values are shown on the bottom horizontal axis and the corresponding CNT diameters are shown on the top horizontal axis. On the plot itself, the data points indicate the n values for which calculations were performed. All other n values in that range correspond to metallic CNTs. At first glance, the bandgaps that result from the FIREBALL calculations closely track the bandgaps determined by the π-bond model for n ≥ 10. CNTs in this size range are the ones that are most commonly synthesized, and they are the ones that are most important for electronic devices [35]. Below n = 10, the π-bond model breaks down due to the large curvature of the CNT. For the smallest CNT with n = 5, the FIREBALL calculations show zero bandgap. Upon closer inspection of Fig. 2(a), one notices a sawtooth shape to the plot of bandgap versus n for the DFT calculations. We observe that the bandgap resulting from the DFT calculations corresponds closely to the bandgap resulting from the π-bond model for (n, 0) CNTs when n = 3p+ 1 where p is an integer ≥ 3. This is shown clearly in Fig. 2(b). in which we plot the differences between the bandgaps calculated from the LDA and BLYP functionals and those from the π-bond model for n ≥ 10. For n = 3p+1 ≥ 13, the LDA model underestimates the bandgap by 24 - 28 meV. For these n values, the BLYP model underestimates the bandgap by 46 - 84 meV. For n = 3p−1 with p an integer, the discrepancies between the bandgaps resulting from the DFT models and the π-bond model are larger. For values of n = 3p− 1 ≥ 14, the LDA model underestimates the bandgap by 52 - 147 meV. For these n values, the BLYP model underestimates the bandgap by 66 - 176 meV. Figs. 3 shows calculations and comparisons of the electron and hole effective masses. The plots of the left column show calculations and comparisons of the electron effective mass, and the plots of the right column show calculations and comparisons of the hole effective mass. In each row consisting of two plots, the scale and range of values is identical to facilitate easy comparisons between the values for electrons and holes. The calculated electron and hole effective masses (normalized to the bare electron mass) are plotted in Figs. 3(a) and (d), respectively, versus n for all semiconducting values in the range 7 ≤ n ≤ 35. One immediately notices that all models result in very similar values of effective masses for both the electrons and holes. To discern the differences, we plot the difference of the mass values calculated from the DFT models and the pi-bond model in Figs. 3(b) and (e) for electrons and holes, respectively. In other words, Figs. 3(b) and (e) show the normalized difference values (m∗DFT − m π−bond)/m0. Note that on the mass difference plots, (b) and (e), the n values range from 10 - 35 whereas on the mass plots, (a) and (d), the n values range from 7 - 35. The range of n is reduced on the difference plots to keep the plotted range of differences small. The difference plots show that the effective mass determined from the BLYP model tends to be larger than the effective mass determined by the LDA model. Over the range of n values 14 ≤ n ≤ 35, the mass determined by the BLYP model has a maximum difference from the mass determined from the LDA model of 0.008 at n = 14 and a minimum difference of 0.002 at n = 34. Also, the sawtooth peaks in the difference curves are out of phase with the sawtooth peaks in the mass curves above. The maximum differences occur at the minimums of the mass curves. Overall, the differences in the mass values predicted by the DFT models and the π-bond model are relatively small. To quantify the differences, the percent differences, 100 ∗ (mDFT −mπ−bond)/mπ−bond, are plotted in (c) and (f). Over a wide range of the most useful n values for device applications, 17 ≤ n ≤ 29, the values for the effective masses from the DFT models fall to within +8% to −9% of the values from the π-bond model. So far, we have considered bandedge properties of bandgaps and effective masses. As we noted above, the higher energy electronic spectra is also important for electronic device modeling, and the best way to characterize it is to calculate the transmission coefficients. Figure 4 shows the transmission spectra for (10, 0), (20, 0), (31, 0) and (35, 0) CNTs calcu- lated using DFT/BLYP, DFT/LDA, and the empirical, π-bond model. In all cases, the energy axis of the transmission curves has been shifted such that the center of the bandgap lies at 0 eV. The energy region in Fig. 4 for which the transmission is zero is the band gap for the CNTs. For the 2 largest CNTs, n = 31 and 35, the transmission resulting from the DFT and π-bond models all have similar, symmetric forms. There is some compression of the energy scale for the transmission coefficients calculated from the DFT models compared to the transmission coefficient calculated from the π-bond model. The energy separation between higher modes is smaller in the DFT models then in the π-bond model. For the smallest CNT, n = 10, there is a noticeable, qualitative difference between the transmission calculated from the DFT models and the π-bond model. The transmission resulting from the π-bond model is always symmetric around the center of the bandgap. For the DFT models, the transmission is noticeably asymmetric. Approximately 0.5 eV above the conduction band edge, the DFT models predict 3 bands closely spaced and 2 bands doubly degenerate. These 7 bands, multiplied by 2 for spin, give rise to the large step of 14 in the transmission coefficient 0.5 eV above the conduction band edge in the transmission coefficient of the (10, 0) CNT. This large increase in the transmission and density of states 0.5 eV in the conduction band is significant for device modeling. A similar large step is also observed in a transmission calculation based on the SIESTA code for a (7, 0) CNT [36]. A similar large increase in the transmission also occurs in the (20,0) CNT at 1.3 eV above the conduction band edge. The differences in the (10,0) valence band transmission resulting from the DFT and π- bond models, while not as dramatic as those in the conduction band, are still significant from a device modeling perspective. The 0.5 eV gap between the valence band edge and the next lower pair of bands found from the π-bond model is reduced to approximately 0.3 eV in the DFT models. These energies lie within the applied voltage window, (VDD), of any of the most optimistically scaled CNT field effect transistors, and will, thus, affect the physics of the carrier transport. While our main focus is on assessing the validity of the DFT models for device modeling, these results also provide an assessment of the π-bond model and show that the π-bond model should be used with care and scepticism for (10,0) CNTs. IV. SUMMARY The goal of this work is to assess the validity of the DFT models implemented by FIRE- BALL for CNT electronic device modeling. Our approach is to compare the electronic properties resulting from the DFT models with those resulting from the π-bond model since the parameters of the π-bond model have been empirically chosen to give a good fit to the bandgap and effective mass for CNTs with diameters that are ‘not too small.’ We have com- pared the bandgaps, effective masses, and transmission coefficients of (n, 0) CNTs calculated from the empirical π-bond model, DFT/LDA, and DFT/BLYP models. For values of n in the range 17 ≤ n ≤ 29, the calculated effective masses from the DFT models are within ±9% of those calculated from the π-bond model. For n ≥ 10, the difference between the bandgap calculated from the π-bond model and the bandgaps calculated from the DFT models oscillates as a function of n. The differences are smallest for n = 3p + 1 where p is an integer, and the differences are largest for n = 3p − 1. For n = 3p + 1 ≥ 13, the LDA model underestimates the bandgap by 24 - 58 meV and the BLYP model underestimates the bandgap by 46 - 84 meV. For n = 3p − 1 ≥ 14, the LDA model underestimates the bandgap by 52 - 147 meV and the BLYP model underestimates the bandgap by 66 - 176 meV. Overall, in the important range of n values most relevant for CNT devices, 17 ≤ n ≤ 29, the bandgaps, effective masses, and transmission coefficients calculated from the DFT models implemented by FIREBALL are sufficiently accurate for electronic device simulations. These simulations also quantify what is meant by ‘not too small’ when applying the π-bond model. For n = 10, the bandedge properties resulting from the π-bond and DFT models agree to within 10%, however, the π-bond model quickly becomes inaccurate away from the bandedges. The transmission from the higher energy modes resulting from the π- bond model has differences with those resulting from the DFT models which are significant for device modeling. For n ≤ 8, the π-bond model is completely inaccurate. Acknowledgments This work was supported by the Microelectronics Advanced Research Corporation Fo- cus Center on Nano Materials (FENA), SRC/SRCEA, the NSF (ECS-0524501), and DARPA/DMEA-CNID (H94003-04-2-0404). [1] K. Keren, R. S. Berman, E. Buchstab, U. Sivan, and E. Braun, Science 302, 1380 (2003). [2] E. Braun and K. Keren, Adv. Phys. 53, 441 (2004). [3] K. A. Williams, P. T. M. Veenhuizen, B. G. de la Torre, R. Eritja, and C. Dekker, Nature 420, 761 (2002). [4] R. den Dulk, K. A. Williams, P. T. M. Veenhuizen, M. C. de Konig, M. Overhand, and C. Dekker, in DNA-Based Molecular Electronics, edited by W. Fritzsche (AIP, New York, 2004), Vol. 725, pp. 25 – 31. [5] C. Dwyer, V. Johri, M. Cheung, J. Patwardhan, A. Lebeck, and D. Sorin, Nanotechnology 15, 1240 (2004). [6] N. Bruque, R. R. Pandey, R. Lake, H. Wang, and J. Lewis, Molecular Simulation 31, 859 (2005). [7] R. R. Pandey, N. Bruque, K. Alam, and R. Lake, Phys. Stat. Sol. (a) 203, R5 (2006). [8] N. A. Bruque, K. Alam, R. R. Pandey, R. K. Lake, J. P. Lewis, X. Wang, F. Liu, C. S. Ozkan, M. Ozkan, and K. L. Wang, J. Nanoelectronics Optoelectronics 1, 74 (2006). [9] X. Wang, F. Liu, G. T. S. Andavan, X. Jing, N. Bruque, R. R. Pandey, R. Lake, K. Singh, M. Ozkan, K. L. Wang, and C. S. Ozkan, Small 2, 1356 (2006). [10] P. Damle, A. Ghosh, and S. Datta, Phys. Rev. B 64, 201403 (2001). [11] P. Damle, A. Ghosh, and S. Datta, Chem. Phys. 281, 171 (2002). [12] T. Rakshit, G. Liang, A. Ghosh, and S. Datta, Nano Lett. 4, 1083 (2004). [13] Y. Xue, S. Datta, and M. A. Ratner, Chem. Phys. 281, 151 (2002). [14] Y. Xue, S. Datta, and M. A. Ratner, J. Chem. Phys. 115, 4292 (2001). [15] S. N. Yaliraki and M. A. Ratner, J. Chem. Phys. 109, 5036 (1998). [16] J. M. Seminario, L. E. Cordova, and P. A. Derosa, Proc. IEEE 91, 1958 (2003). [17] R. M. Martin, Electronic Structure Basic Theory and Practical Methods (Cambridge University Press, Cambridge, 2004). [18] K. Alam and R. K. Lake, Appl. Phys. Lett. 87, 073104 (2005). [19] J. Muscat, A. Wander, and N. M. Harrison, Chem. Phys. Lett. 342, 397 (2001). [20] A. Canning, L. W. Wang, A. Williamson, and A. Zunger, J. Comp. Phys. 160, 29 (2000). [21] G. Klimeck, F. Oyafuso, T. B. Boykin, R. C. Bowen, and P. von Allmen, CMES-Computer Modeling in Engineering and Sciences 3, 601 (2002). [22] O. F. Sankey and D. J. Niklewski, Phys. Rev. B 40, 3979 (1989). [23] J. P. Lewis, K. R. Glaesemann, G. A. Voth, J. Fritsch, A. A. Demkov, J. Ortega, and O. F. Sankey, Phys. Rev. B 64, 195103 (2001). [24] J. P. Lewis, T. E. C. III, E. B. Starikov, H. Wang, and O. F. Sankey, J. Phys. Chem. B 107, 2581 (2003). [25] D. M. Ceperley and G. J. Alder, Physical Review Letters 45, 566 (1980). [26] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [27] A. D. Becke, Phys. Rev. A 38, 3098 (1988). [28] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988). [29] J. Harris, Phys. Rev. B 31, 1770 (1985). [30] W. M. C. Foulkes and R. Haydock, Phys. Rev. B 39, 12520 (1989). [31] A. A. Demkov, J. Ortega, O. F. Sankey, and M. P. Grumbach, Phys. Rev. B 52, 1618 (1995). [32] O. F. Sankey, A. A. Demkov, W. Windl, J. H. Fritsch, J. P. Lewis, and M. Fuentes-Cabrera, Int. J. Quantum Chem. 69, 327 (1998). [33] D. R. Hamann, Phys. Rev. B 40, 2980 (1989). [34] J. W. Mintmire, D. H. Robertson, and C. T. White, J. Phys. Chem. Solids 54, 1835 (1993). [35] Z. Chen, J. Appenzeller, J. Knoch, Y.-M. Lin, and P. Avouris, Nano Lett. 5, 1497 (2005). [36] C. Adessi, S. Roche, and X. Blase, Phys. Rev. B 73, 125414 (2006). Figure Captions FIG. 1: One unit cell (10,0) zigzag CNT with 4 atomic layers. Self-energies take into account the semi-infinite leads for transmission calculations. FIG. 2: (a) (n, 0) CNT band gaps as a function of n and diameter calculated from the π-bond model and DFT with LDA and BLYP functionals. (b) Difference between the band gap calculated from the DFT models and the π-bond model. FIG. 3. Electron, (a) - (c), and hole, (d) - (f), effective mass comparisons. Top: Normalized effective mass (m∗/m0), calculated from the π-bond and DFT models. Middle: Difference between the effective mass calculated from the DFT models and the π-bond model (mDFT −mπ−bond)/m0. Bottom: Percent difference, 100 ∗ (mDFT −mπ−bond)/mπ−bond. FIG. 4. Transmission calculated from π-bond and DFT models for (n, 0) CNTs with n values of (a) 10, (b) 20, (c) 31, and (d) 35. 1 2 3 4 UNIT CELL FIG. 1: � �� �� �� �� �� �� π����� � �� � � �� � �������� ���� �� �� �� �� �� �� �� �� �� �� �� �� ��� � π����� ���� � π����� FIG. 2: ����� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� �∆ ��� �∆ �� �� �� �� �� �� ���� �∆ ��� �∆ �� ��� � ��� � ��� �� �� �� �� �� �� ���� � π�� ��� � π�� �� ��� � ��� � ��� �� �� �� �� �� �� ���� � π�� ��� � π�� �� ����� �� �� FIG. 3: �� ���� �� ���� � ��� � ��� � ������ ���� π���� ��� ���!�� ������ ���� �� ���� �� ���� � ��� � ��� � π���� ���!�� �� ���� �� ���� � ��� � ��� � ������ ���� π���� �"�!�� ������ ���� �� ���� �� ���� � ��� � ��� � �"�!�� π���� FIG. 4: INTRODUCTION METHOD OF CALCULATION RESULT AND DISCUSSION SUMMARY Acknowledgments References

0704.1169

Holographic bound and protein linguistics

Holographic bound and protein linguistics Dirson Jian Li∗ and Shengli Zhang Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, PR China (Dated: November 16, 2018) The holographic bound in physics constrains the complexity of life. The finite storage capability of information in the observable universe requires the protein linguistics in the evolution of life. We find that the evolution of genetic code determines the variance of amino acid frequencies and genomic GC content among species. The elegant linguistic mechanism is confirmed by the experimental observations based on all known entire proteomes. PACS numbers: 87.10.+e, 87.15.Aa, 87.14.-g, 04.70.Dy The phenomenon of life is governed by the general prin- ciples in physics, so the progress in understanding the physical world may provide new insight into the origin and evolution of life. The analogy between biology and linguistics at the level of sequences hints that the bio- information is processed by underlying linguistic rules. Several attempts have been made to combine linguistic theory with biology [1]. But the existence of linguistics in the biomacromolecular sequences needs a physical ex- planation. The holographic bound, intimately related to the holographic principle, came from the deep insights of Bekenstein and Hawking in 70’s [2][3]. Its validity is insured by the second law of thermodynamics. In- terestingly, the problem on the existence of linguistics in the biomacromolecular sequences can be explained by the holographic bound. In the past decades, the biology has been changed greatly. Wada advocated “... to deter- mine the ‘first principles’ of bio-sciences and link them with the first principles of non-bio-sciences in order to understand the complex systems.” and Gilbert also em- phasized the importance of the theoretical methods in biology [4]. Nowadays, the intimacy between biology and physics is unprecedented. Considering the significant role of information either in physics or in biology [5][6], the gap between physics and biology may be bridged from the viewpoint of information. In the post-genomic era, the number of entire pro- teomes increases rapidly. We can take all known entire proteomes as samples to study the global properties of life on our planet. The variance of GC content [7] is a global property, which varies greatly among species. We also found another global property of the evolution of amino acid frequencies though they vary slightly. The mecha- nism of the variance of genomic GC content and amino acid frequencies was a long-standing and far-reaching problem [8]. The genetic code evolved in the context of four-letter alphabet when the 20 amino acids joined pro- tein sequences chronologically [9]. The nature of prime biased AT/GC pressure and the reason for the correla- tion of GC content between total genomic DNA and the 1st, 2nd and 3rd codon positions were unknown. The profound mechanism behind the variance of amino acid frequencies has not been studied; it is worse that the amino acid frequencies are routinely assumed to be con- stant. All of these basic problems in biology are solved in our theoretical framework based on the formal linguistics and the evolution of genetic code. In this paper, firstly we explain the existence of pro- tein linguistics and the limited complexity of life in the universe in terms of the holographic bound. Secondly, a linguistic model is proposed to reveal the mechanism of the evolution of amino acid frequencies and genomic GC content as well as the protein length distribution. The ex- cellent fit between our simulations and the experimental observations strongly supports the linguistic mechanism, where the experimental observations are based on the data of 106 entire proteomes (85 eubacteria, 12 archae- bacteria, 7 eukaryotes and 2 viruses) in database PEP [10] and the data of GC content in database Genome Properties system [11]. The “information” is the thread of the paper, which connects traditionally irrelative prob- lems in physics and biology with each other. According to the holographic bound, which states that the information storage capacity of a spatially finite sys- tem is limited by a quarter of its boundary area measured in Plank area unless the second law of thermodynamics is untrue, the entropy S in a volume of radius R satisfies S 6 Smax ≈ ( )2, (1) where lp is the Plank length. There is much astronom- ical evidence that our universe may be headed for an infinite deSitter space. The holographic bound can be applied to the observable universe with finite event hori- zon. Therefore we can estimate the upper limit of the information storage capability of the observable universe as Iuniv ≈ 10 122 bits [3]. In the point of view in physics, the entropy in our universe is given primarily by the num- ber of black body cosmic background photons, ∼ 1090, which is definitely less than Iuniv. However, the upper limit of information Iuniv is not so large when considering the information of a system of life. Firstly, let us give a reasonable definition of a “liv- ing” system from the viewpoint of information. Many restless functional proteins, composed of 20 amino acids (a.a.), distinguish the life from the lifeless matter. Thus, http://arxiv.org/abs/0704.1169v1 a general living system L(n) is defined as a set of all pos- sible proteins with length no more than n a.a., each of which is in either the folded state or the unfolded state. The maximum length n indicates the complexity of the system. Then, let’s calculate the information of the sys- tem. The number of states of L(n) is Ω(n) = k=2 2 hence its information is I(n) = log2Ω(n) ≈ 20 n bits. (2) The exquisite single-chain structure of proteins can pro- vide much more information storage capacity than life- less matter. The upper limit of information Iuniv forbids L(n), n > n0 to exist in our universe, where n0 = 94 a.a. such that I(n0) ∼ Iuniv. Interestingly, the most frequent protein length for the life on our planet is about n0. The actual system of life on the planet is not one of L(n), because the average protein length for different species, ranging from 250 a.a. to 500 a.a., are greater than n0. Let the actual living system Learth be the set of all possible proteins on the planet, and suppose n∗ (> n0) be the maximum protein length. Learth must be a proper subset of L(n∗) because the information of Learth is bounded by Iuniv . According to the linguis- tic theory, Learth is a language over the alphabet of 20 amino acids. So we have demonstrated the existence of protein linguistics in terms of the holographic bound that can be derived from the second law of thermodynamics. In early evolution when delivering the genetic informa- tion from the RNA world to the DNA-protein world, the holographic bound required the grammars to allow a very small part of protein sequences and to forbid all the oth- ers. There is a easy way to implement the forbiddance: the observable universe can not accommodate all the pro- teins in L(n∗). In fact, our conclusion is based on the general principle, freedom of the subtleties of the hierar- If L(n) have an additional property, e.g., chirality, Lr(n) (Ll(n)) become the set of proteins composed of right-handed (left-handed) amino acids. Let C(n) be the set of all possible chiral L(n). Only up to n = 2 a.a., the information of C(2), i.e., IC(2) = Ω(2) ≈ 10120 bits is near Iuniv . Chirality might have brought too much redundant information; broken symmetry was the solu- tion. So entropy bound is a strong law which constrains the forms of possible life in general. Let us imagine the most complex creature with the same height of us be the one whose genetic information is stored in Plank scale. The information stored in its body can be estimated as 1/l3p ∼ 10 105 bits, which violates the holographic bound. So such complex creature can not exist. So far, we are aware that the linguistics must play a significant role in generating proteins in the primordial time. In the following, a linguistic model is proposed to simulate the variance of the amino acid frequencies and the genomic GC contents as well as the protein length distributions. G A D V P S E L T R Q I N H K C F Y M W G A D V P S E L T R Q I N H K C F Y M W (for each a.a.) 1...106 (for each a.a.) 1...30 FIG. 1: Evolution of amino acid frequencies. The 20 amino acids are aligned chronologically. The variance for each amino acid in simulation fits the experimental observation. (a) Experimental observation base on the data of 106 species. For each amino acid, the 106 species are aligned from left to right by R10/10. (b) Simulation by the linguistic model. The 30 simulated proteomes are aligned by t, which increases from 0.02 to 0.40 by equal steps. (The simulations in Fig. 1, Fig. 2 and Fig. 4 are obtains together by the linguistic model.) The model consists of three parts: (i) generate protein sequence by tree adjoining grammar [12]; (ii) set amino acid for the leaves of grammars in (i) according to the tree of genetic code multiplicity (can be obtained from sym- metry analysis, see [13]) with consideration of the amino acid chronology [14]; and (iii) translate the protein se- quences to the DNA sequences according to genetic code chronology [15]. The evolution of genetic code is the core of the model. There is a variant t in the model, which represents the time in evolution. A proteome for a species is defined as many a protein generated by the model with fixed t, so t also identifies species in the model. Thus, the amino acid frequencies and the average protein length for a species can be calculated. The evolutionary trends of the amino acid frequencies can be determined when proteomes are generated at different time t. We can also simulate the evolution of genomic GC content after trans- lating the protein sequences to DNA sequences. The evolution of amino acid frequencies can be ex- plained by the model. According to the consensus chronology of amino acids to recruit into the genetic code from the earliest to the latest [14]: G, A, D, V, P, S, E, L, T, R, Q, I, N, H, K, C, F, Y, M, W, we sort the 106 species by the ratio R10/10 of average frequency for 10 later amino acids to average frequency for 10 earlier amino acids. Then we obtain the evolutionary trends of amino acid frequencies: the frequencies of G, A, D, V, P, T, R, H, W decrease, while the frequencies of S, E, I, N, K, F, Y increase and the frequencies of L, Q, C, M do not vary obviously (Fig. 1a). The variance of amino 0.2 0.4 0.6 0.8 Genomic GC content 0.2 0.4 0.6 0.8 Genomic GC content convex FIG. 2: Evolution of genomic GC content. (a) Relation- ship between genomic GC content and R10/10 for the species in database PEP (dots) and its simulation by the linguistic model (solid line, also decreasing). (b) Simulation of the cor- relation of the GC content between total genomic DNA and the 1st, 2nd, and 3rd codon positions, which agrees with the experimental observation in detail (see fig. 5 in [17], fig. 2 in [7] and fig. 9-1 in [5]). acid frequencies are amazingly monotonic by and large. Therefore, it is reasonable to assume that a mechanism underlies the evolution of amino acid frequencies. The simulation of our linguistic model [Fig. 1b] agrees with the data of 106 species [Fig. 1a] not only in the evo- lutionary trends but also in the variance magnitudes for most of the amino acids. Note that no parameter is added on purpose in the model to alter the trend for a certain amino acid. The evolution of amino acid frequency are sensitive to the amino acid multiplicities [13], any disobe- dience of which would spoil the results. Therefore, it is the evolution of genetic code that determines the evolu- tion of amino acid frequencies. An important property of the model is that the parameters of amino acid frequen- cies are constant, which indicates that the variance of amino acid frequencies developed during a short period. It agrees that the genetic code had accomplished quickly. Recently, Jordan et al observed the contemporary amino acid gain and loss, about which there were different ex- planations [16]. We believe that the evolution of genetic code drives the amino acid gain and loss. The genomic GC content decreases linearly with R10/10 for the species in database PEP. The simulation of our model agrees qualitatively with this experimental obser- vation [Fig. 2a]. In our model, the evolution of amino acid frequency and the evolution of genomic GC content are driven by a common variant t. A protein sequence generated at later time t corresponds to the DNA se- quence translated using the later codons [15], which re- sults in the relationship between genomic GC content 250 350 450 550 Average protein length Archaebateria Eubacteria Eukaryote Mycoplasma Virus FIG. 3: Rainbow distribution. Relationship between the average protein length and the highest frequency of the dis- crete fourier transformation of protein length distribution (the cutoff is 3000) for each of the 106 species. The distribution of the species from three domains likes a rainbow. Even for the group of closely related species such as mycoplasmas (be- longing to eubacteria), their distribution also form an “arch” of the rainbow. and R10/10. And the simulation of the correlation of GC content between total genomic DNA and the first, second, and third codon positions [Fig. 2b] also agrees with the re- sults based on the data of completed genomes [7][5][17], where the correlation slope of the third codon position is much greater than that of the first and the second positions. There is a characteristic convex in the mid- dle of the line of the simulated GC content for the first codon position, which agrees dramatically with the ex- perimental observations [7][5][17]. In the table of codon chronology [15], G and C (A and U) occupy all the third positions of earliest (latest) codons for 20 amino acids, while the bases appear about equally for the first and second positions. Therefore, the correlation slope for the first and second positions vary slightly while the slope for the third position varies greatly. And the lower limit l ∼ 0.3 and upper limit u ∼ 0.7 of the GC content among species can be explained similarly; l + u = 1 is required in theory. The linguistic mechanism can also be supported by the distribution of protein length. When observing the dis- tribution of species in the space of average protein length and the highest frequency of discrete fourier transfor- mation of protein length distribution, we unexpectedly found that the species for the three domains gathered in three parallel arches respectively, which likes a rainbow [Fig 3]. This indicates that the fluctuations in protein length distributions can not be products of a stochastic process. The characters of the protein length distribu- tion (bell-shape profile, periodic-like fluctuations) have 250 350 450 550 Average Protein Length 10 20 Average mini−protein length Archaebateria Eubacteria Eukaryote Virus increasing t FIG. 4: The evolutionary flow. Relationship between av- erage protein lengths and RHQW/GV for the 106 species. The species of three domains (Archaebacteria: blue square, Eu- bacteria: dot, Eukaryotes: red circle) gather together in re- spective regions and all the species form an evolutionary flow. The proteome size is represented proportionally by the tail under each species (big: red, middle: green and small: blue); species with big genome sizes locate in the midstream of the evolutionary flow. (Embedded) Simulation of the evolution- ary flow, whose (upward) bending direction agrees with the direction of the experimental observation. been simulated by the linguistic model, which should be intrinsic properties related to underlying grammars [18]. We also find the relationship between the average pro- tein length and the ratio of amino acid frequencies. The species of three domains gather in different regions in the space of the average protein length and the ratio RHQW/GV of average frequency for several later amino acids (H, Q, W) to average frequency for several earlier ones (G, V) [Fig. 4]. The points of all species form a bending line [Fig. 4], which can be explained as an evo- lutionary flow in that (i) the species with large (small) genome locate in the midstream (margin) of the flow [Fig. 4] and (ii) the (rightward) evolutionary direction paral- lels the directions of decreasing correlations of protein length distributions among groups of the closely related species. The evolutionary flow can be simulated by our model [Fig. 4, Embedded]. The evolutionary direction and the bending direction in the simulation agree with the evolutionary flow of the 106 species. In conclusion, the holographic bound improves our un- derstanding of life, which supervises the maximum com- plexity of life. Linguistics is necessary in storage of in- formation in the protein/DNA sequences. We show that the particular variance of amino acid frequencies and GC content for the contemporary species are the products of certain genetic code multiplicity and amino acid chronol- ogy evolved in primordial time. The linguistic model succeeds not only in the simulations of respective aspects (amino acid frequencies [Fig. 1], GC content [Fig. 2b], protein length distribution [Fig. 3]) but also in their re- lationships (amino acid frequencies and GC content [Fig. 2a], amino acid frequencies and protein length distribu- tion [Fig. 4]). So the thorough and detailed fit between simulations and experimental observations confirms the validity of the linguistic framework, which is grounded in general principles in physics. We thank Hefeng Wang, Liu Zhao, Donald R. Fors- dyke, Zhenwei Yao for valuable discussions. Supported by NSF of China Grant No. of 10374075. ∗ [email protected] [1] M. Gimona, Nature Rev. Mol. Cell Boil. 7, 68-73 (2006). [2] J. D. Benkenstein, Lett. Nuovo. Cim. 4, 737 (1972); S. Hawking, Commun. Math. Phys. 43, 199 (1975); G. ’t Hooft, in Salam-festschrifft, eds A. Aly et al., (World Scientific, Singapore, 1993); L. Susskind, J. Math. Phys. 36, 6377 (1995). [3] L. Susskind and J. Lindesay, An Introduction to Black Holes, Information, and the String Theory Revolution: the Holographic Universe (World Scientific, Singapore, 2005). [4] W. Gilber, Nature 349, 99 (1991); A. Wada, Bioinfor- matics 16, 663-664 (2000). [5] D. R. Forsdyke, Evolutionary Bioinformatics (Springer, New York, 2006). [6] B.-O. Küppers, Information and the origin of life (MIT Press, Cambridge Mass., 1990). [7] A. Muto and S. Osawa, Proc. Natl. Acad. Sci. USA 84, 166-169 (1987). [8] N. Sueoka, Proc. Natl. Acad. Sci. USA 47, 1141-1149 (1961); X. Gu et al., Genetica 103, 383-391 (1998). [9] R. D. Knight and L. F. Landweber, Cell 101, 569-572 (2000); E. Szathmáry, Nature Rev. Genetics 4, 995-1001 (2003); S. Osawa. et al., Microbiol. Rev. 56, 229-264 (1992). [10] The database with Predictions for Entire Proteomes on URL: http://cubic.bioc.columbia.edu/pep. [11] D. H. Haft et al., Bioinformatics 21, 293-306 (2005). [12] A. K. Joshi and Y. Schabes, in Handbook of Formal Lan- guages, eds G. Rozenberg and A. Salomma (Springer, Heidelberg, 1997), pp.69-214. [13] Jose Eduardo M. Hornos and Y. M. M. Hornos, Phy. Rev. Lett. 71, 4401-4404 (1993). [14] E. N. Trifonov, J. Biomol. Struct. Dyn. 22, 1-11 (2004). [15] E. N. Trifonov et al., J. Mol. Evol. 53, 394-401 (2001). [16] I. K. Jordan et al., Nature 433, 633-638 (2005); L. D. Hurst et al., Nature 442, E11-E12 (2006). [17] A. Gorban et al., Physica A 353, 365-387 (2005). [18] In preparation. mailto:[email protected] http://cubic.bioc.columbia.edu/pep

0704.1170

Externally-polluted white dwarfs with dust disks

Externally-Polluted White Dwarfs With Dust Disks M. Jura1, J. Farihi,2 & B. Zuckerman1 ABSTRACT We report Spitzer Space Telescope photometry of eleven externally-polluted white dwarfs. Of the nine stars for which we have IRAC photometry, we find that GD 40, GD 133 and PG 1015+161 each has an infrared excess that can be understood as arising from a flat, opaque, dusty disk. GD 56 also has an infrared excess characteristic of circumstellar dust, but a flat-disk model cannot reproduce the data unless there are grains as warm as 1700 K and perhaps not even then. Our data support the previous suggestion that the metals in the atmosphere of GD 40 are the result of accretion of a tidally-disrupted asteroid with a chondritic composition. Subject headings: circumstellar matter – asteroids – stars, white dwarfs 1. INTRODUCTION Gravitational settling of heavy elements is so effective in white dwarfs cooler than 20,000 K that their atmospheres are expected to be pure hydrogen or pure helium (Paquette et al. 1986), and the ∼20% of these stars which exhibit photospheric metals are thought to be externally polluted (Wolff et al. 2002, Zuckerman et al. 2003, Koester & Wilkin 2006). A promising model to explain these data is that a minor-body is perturbed (Debes & Sigurdsson 2002) to orbit within the tidal radius of the star where it is destroyed and a dusty disk is produced (Jura 2003). Accretion from this disk can explain atmospheric pollutions (Zuckerman et al. 2003) as seems likely to have occurred for GD 362 (Becklin et al. 2005, Kilic et al. 2005, Jura et al. 2007) and G29-38 (Jura 2003, Reach et al. 2005b). While the disrupted minor-body model is promising, despite extensive surveys, only a handful of white dwarfs are known to display an infrared excess (Kilic et al. 2006b, Kilic & Redfield 2007, Mullally et al. 2007, Farihi et al. 2007). We have therefore obtained Spitzer Space Telescope data to investigate the frequency of dusty disks orbiting white dwarfs and to assess models for the sources of the external pollution. 1Department of Physics and Astronomy and Center for Astrobiology, University of California, Los Angeles CA 90095-1562; jura, ben @astro.ucla.edu 2Gemini Observatory, 670 North A’ohoku Place, Hilo HI 96720; [email protected] http://arxiv.org/abs/0704.1170v1 – 2 – 2. OBSERVATIONS We observed eleven stars with atmospheric metals found in the SPY project (Koester et al. 2005) that also show a hint of K-band excess in 2MASS photometry. We used the IRAC (Fazio et al. 2004) and MIPS (Rieke et al. 2004) cameras on the Spitzer Space Telescope (Werner et al. 2004) to obtain photometry at wavelengths between 3.6 µm and 24 µm. IRAC observations were executed using a 20 point cycling dither pattern of medium step size, with 30 s individual exposures, yielding a total integration time of 600 s at all wavelengths. MIPS observations were performed using 10 cycles of the default 14 point dither pattern, with 10 s individual exposures, yielding a total integration time of 1400 s. The data were processed with the IRAC and MIPS calibration pipelines (both versions 14) to create single, reduced images upon which to make photometric measurements. We removed the one dimensional artificial gradient in the sky which is a common feature of MIPS pipelined data by subtracting a median collapsed image across the direction of the gradient. Aperture photometry was carried out with standard IRAF tasks, and measured fluxes were corrected for aperture size, but not for color. For IRAC detections, fluxes were typically measured in a 3-pixel radius aperture, but a 2-pixel radius aperture was used for faint sources or objects in a crowded field, and a 5-pixel radius aperture was used for GD 56, the brightest source. MIPS 24 µm flux was measured in a 2.45 pixel radius aperture for GD 56, but a smaller aperture with a radius of 1.22 pixels was necessary for GD 133 because of a nearby galaxy. The measured fluxes were corrected to the standard IRAC and MIPS photometric apertures using corrections found in the most recent versions of the IRAC and MIPS data handbooks. The results are listed in Table 1. IRAC data were not obtained for two sources since they are in other Spitzer programs; these data have not yet been published. For both the IRAC and MIPS data, the photometric errors were estimated by taking the per pixel standard deviation in the extracted sky level and multiplying by the area of an aperture with a radius of 2 pixels. Color corrections have been ignored and are typically less than 1%. Our dither pattern should remove the pixel phase dependent correction as well as the IRAC array location dependent errors described by Reach et al. (2005a). Total 1σ errors (5% calibration plus photometric measurement errors) are listed in Table 1. Because there is a somewhat larger scatter in the ratios of even the best-measured photospheric fluxes in the IRAC bands for white dwarfs (Farihi et al. 2007) than expected from model atmospheres (Tremblay & Bergeron 2007 and Bergeron 2007, private communication), to be conservative, following Hines et al. (2006) and Silverstone et al. (2006), we adopt a 10% calibration uncertainty in the reported fluxes, as reflected in the error bars displayed in Figures 1-4. Upper limits for non-detections in both IRAC and MIPS images were derived identically to the photometric errors for detections. – 3 – To assess the reliability of our IRAC photometry obtained with small apertures, we also reduced our data with CCDCAP3. Although there is no significant difference with apertures of 3 pixel-radii, it was found that IRAF reported fluxes 3% smaller than CCDCAP with 2-pixel radius apertures at the two shorter wavelength IRAC channels where undersampling of the array is most significant. We have found that this small offset is largely mitigated by the aperture corrections provided by the Spitzer Science Center, and the net uncertainty introduced by using a small aperture is well below our 1σ errors. As a further check, using 2-radius pixel aperture photometry and IRAF, we have reproduced the IRAC fluxes reported by Reach et al. (2005b) for G29-38 to within 3% at 4.5 µm and to within 1% at 7.9 µm. We used 2-pixel radius apertures to reduce the data at 3.6 µm and 4.5 µm for only one star, PG 2354+159. The data for PG 1015+161 are contaminated by a star ≈ 2′′ away (Kilic et al. 2006b). Although our IRAC images are slightly elongated by 0.′′5 or 0.4 pixels at 3.6 µm and 4.5 µm in the direction of this background star, we cannot accurately deconvolve the image, and we adopt an alternative approach to estimate its fluxes. As shown in Figure 5, a 1995.9 epoch Keck Observatory NIRC z-band image (see Farihi, Becklin & Zuckerman 2005), the angular separation of the background star was appreciably greater in the past, 2.′′9 vs. 1.′′8 in 2007.0. From the Keck data, we find values of z = 19.12 mag, J = 18.76 mag and K = 18.26 mag and infer a spectral type of approximately K0V. We extrapolate these measures to IRAC wavelengths, and we list in Table 1 our results for PG 1015+161 after the estimated contributions from the background star are subtracted from the total measured IRAC fluxes at the position of PG 1015+161. Both GD 133 and PG 1015+161 were suspected by Kilic et al. (2006b) to have an infrared excess on the basis of their 2 µm spectra; our IRAC data demonstrate that these two stars indeed have excesses. GD 56 was previously shown to have a 2 µm excess (Kilic et al. 2006b); our data significantly extend the wavelength range over which the excess is measured. Except for the slight hint in the 2MASS photometry, there is no previous measurement of an infrared excess for GD 40. 3. MODELS We first try to account for the infrared data with the passive, opaque flat disk model of Jura (2003). Here, the disk’s potential vertical stratification is ignored, so that at each distance from the central star all dust grains achieve the same temperature as determined by 3described at:http://www.noao.edu/noao/staff/mighell/ccdcap http://www.noao.edu/noao/staff/mighell/ccdcap – 4 – the balance between radiative heating and cooling. Although a more sophisticated treatment is required to account for data such as the strong silicate feature seen in both the infrared spectrum of G29-38 (Reach et al. 2005b) and GD 362 (Jura et al. 2007), this simple model accounts for the currently available data. Since all the target stars are warmer than 7000 K, the received photospheric infrared flux from the star, Fν(∗), is modeled to better than the 2σ measurement uncertainty as a single-temperature blackbody (Kilic et al. 2006a, Tremblay & Bergeron 2007), Bν , so that Fν(∗) = Bν(T∗) (1) where the star of radius, R∗, and effective temperature, T∗, lies at distance D from the Sun. For a flat, opaque disk that is passively illuminated by the host star; the expected flux, Fν(d), is (Jura 2003): Fν(d) = 12 π 1/3 R cos i 2 kB T∗ 3 h ν ∫ xmax ex − 1 dx (2) where i denotes the inclination angle of the disk and i = 0◦ corresponds to a face-on con- figuration. Here, x = (hν)/(kBTdisk) where Tdisk denotes the temperature in the disk. Since the disk is opaque, we cannot estimate its mass. Table 2 lists stellar parameters adopted from Friedrich, Jordan & Koester (2004), Koester et al. (2005), and Koester & Wilken (2006) as well as our estimated disk pa- rameters. The fits to our Spitzer data and near-infrared photometry from 2MASS or, when available, from the more accurate observations obtained at the IRTF by Kilic et al. (2006b), are shown in Figures 1-4. We find satisfactory fits for GD 40, GD 133 and PG 1015+161 with a range of possible model parameters that are listed in Table 2. The inner dust tem- perature is constrained by the excess at 3.6 µm, but the fit is not unique. To fit the 4.5 µm and 5.7 µm fluxes, there can be either a relatively broad range of temperatures with a more edge-on disk, or a narrower range of temperatures with a more face-on disk. In order not to over-predict the fluxes at 8 µm and 24 µm, the outer temperature cannot be too low. As listed in Table 2, using the thermal profile given in Jura (2003), the models lie within the tidal radius which typically is about a factor of 100 greater than the star’s radius (Davidsson 1999). We see in Figure 1 that GD 40 can be equally well fit with a disk with an inner tem- perature of 1200 K that is nearly edge-on or one with an inner temperature of 1000 K that is more nearly face-on. If, however, the inner temperature is only 800 K, then we cannot match the flux at 3.6 µm. Figure 3 displays the results for PG 1015+161. Again, there is little difference between a model which is more nearly edge-on and an inner temperature of – 5 – 1200 K and a model which is more nearly face-on with an inner temperature near 1000 K. However, a disk with an inner temperature of 800 K fails to produce enough flux at 3.6 µm. For GD 133, as shown in Figure 4, a model with an inner temperature of 1200 K that is viewed nearly edge-on is only somewhat better than a model with an inner temperature of 800 K that is viewed nearly face-on. Kilic et al. (2006b) have shown that GD 56 is unusual in having a particularly marked excess at 2 µm, and as seen in Figure 2, our model Agd56 with an inner disk temperature of 1200 K completely fails. An alternative possibility with a less than satisfactory fit is that the inner disk temperature is larger than 1200 K, and we also show in Figure 2 the results for model Bgd56 with an inner disk temperature of 1700 K and an outer disk temperature of 400 K (see Table 2). A better fit can be achieved if, rather than a flat disk, we assume that the disk is substantially warped as driven by the central star’s luminosity (Pringle 1996) or otherwise puffed up by the gravitational field of a planet. To model this scenario, we assume a single temperature blackbody at 1000 K, and, as shown in Figure 2, we can fit the data if this material is a disk with an angular radius of 1.4 × 10−10 radians as seen from the Earth. Our data are inconsistent with a simple model of interstellar accretion to explain the atmospheric metals. Assume interstellar grains are accreted at rate Ṁdust at the Bondi-Hoyle radius, Rinit, which is typically between 1 and 10 AU (Koester & Wilken 2006) and then drift inwards because of Poynting-Robertson drag to a final radius, Rfinal, where they are destroyed. In this scenario, the expected (see Jura 2006) infrared flux from interstellar (IS) accretion, Fν(IS), is: Fν(IS) ≈ Rinit Rfinal Ṁdust c This expression is valid for observations at frequencies where hν/k lies between the minimum and maximum grain temperature. To evaluate equation (3), we adopt dust accretion rates that are 0.01 of the rates of accretion of interstellar gas given by Koester & Wilken (2006) and also use their estimated distances. We adopt this value of the interstellar dust to gas ratio by mass of 0.01 from Zubko et al. (2004). For simplicity, we assume grains are destroyed by sublimation between ∼10−2 AU and ∼10−3 AU, the region where an unshielded grain that acts like a blackbody attains a temperature near 1200 K. We therefore adopt Rinit = 1000 Rfinal although the results are insensitive to the exact value of this ratio. We show in Table 3, the predicted values of Fν(24 µm) and a comparison between the predicted and observed fluxes, Fpred/Fobs, for the hydrogen-rich stars where the dwell time of metals in the atmosphere is typically less than 100 yr (Koester & Wilken 2006) and therefore accretion is almost certainly ongoing. In contrast, for the three helium rich stars in our sample (GD 40, G26-31, PG 2354+159), the atmospheric dwell times for metals are closer to 3 × 105 – 6 – yr (Paquette et al. 1986), and it is possible that accretion has stopped but there are still metals lingering in the stellar photosphere. We see that from Table 3 that the expected flux is always at least an order of magnitude larger than observed and thus this simple model of interstellar accretion is unsatisfactory. 4. DISCUSSION We find that four of nine white dwarfs with fluxes measured in the IRAC bands display evidence of circumstellar dust disks. For GD 40, the deficiency of carbon in the accreted material (Wolff et al. 2002) is naturally understood if an asteroid of at least 1023 g with a chondritic composition was tidally-destroyed (Jura 2006), in agreement with some previous qualitative suggestions (Sion et al. 1990, Aannestad et al. 1993). G29-38 was the first white dwarf found to have an infrared excess (Zuckerman & Becklin 1987) from dust while GD 362 was the second (Becklin et al. 2005, Kilic et al. 2005). With the results reported here and other recent studies (Mullally et al. 2007, Kilic et al. 2006b, Kilic & Redfield 2007, Farihi et al. 2007), there are now enough white dwarfs with infrared excesses that it is possible to begin to discern some patterns. First, all the white dwarfs with an infrared excess also display atmospheric metals, thus it is highly plausible that the stars are accreting from reservoirs of circumstellar material. Second, Kilic et al. (2006b), Farihi et al. (2007) and Kilic & Redfield (2007) have shown that the stars with relatively high calcium abundances also tend to display an infrared excess. Third, the stars with an infrared excess all have effective temperatures greater than ∼9500 K and white dwarf cooling ages less than ∼1 Gyr4. According to the simulations of Debes & Sigurdsson (2002), most of the orbital perturbations of asteroids would occur during the first several hundred million yr of the white dwarf’s cooling, consistent with the data. The scenario of a tidally-disrupted asteroid explains the data for white dwarfs with both a relatively high photospheric calcium abundance and an infrared excess. However, the source of the atmospheric metals of those white dwarfs without an infrared excess is uncertain. One possibility is that the particle density is sufficiently low that mutual collisions lead to effective dust destruction. At, for example, the typical tidal radius of a white dwarf of ∼0.01 AU, 4GD 362 previously was thought to be hydrogen-rich and have a relatively low luminosity and therefore a cooling age well in excess of 1 Gyr (Gianninas, Dufour & Bergeron 2004). However, the star is now known to be helium-rich and have a much larger luminosity and correspondingly shorter cooling age (Zuckerman et al. 2007). G167-8 has an effective temperature of 7400 K, but its infrared excess, if real, is from much cooler dust than that found in the systems described here (Farihi et al. 2007). – 7 – the orbital speed is near 300 km s−1, and even small deviations from circular orbits can lead to mutual collision speeds in excess of 10 km s−1 which result in grain destruction. For stars with an infrared excess, the disks may be so dense that the grains act more like a granular fluid and the mutual collision speeds are small. Thus, the externally-polluted white dwarfs without an infrared excess may have gas disks as has been found for SDSS J122859.93+104032.9 (Gaensicke et al. 2006). White dwarfs with a relatively low accretion rate tend not to possess an infrared excess. Scaling the accretion rates of Koester & Wilken (2006) by 0.01 as described above, Figure 6 presents a comparison of Ṁdust vs. effective temperature for DAZs with IRAC photometry, distinguishing between those with and without excess emission. The correlation between having an infrared excess and Ṁdust shown in Figure 6 is closely related to the result found by Kilic et al. (2006b) that the stars with a greater metal abundance are the ones with an infrared excess. Although the numbers are very limited, the DAZs with Ṁdust larger than about 3 × 108 g s−1 possess an excess while the stars with lower accretion rates do not. For comparison, the dust production rate in the zodiacal cloud is about 3 × 106 g s−1 (Fixsen & Dwek 2002) and considerably larger around some other main-sequence stars. Even in their advanced evolutionary state, some white dwarfs may possess a population of eroding parent bodies. 5. CONCLUSIONS We have found evidence for dusty disks orbiting four externally-polluted white dwarfs. For GD 40, the evidence lends support to the hypothesis that tidal-disruption of a carbon- deficient asteroid has occurred. This work has been partly supported by NASA and is based on observations made with the Spitzer Space Telescope which is operated by the Jet Propulsion Laboratory, California Institute of Technology, for NASA. We thank P. Bergeron for sending us models of white dwarf atmospheres, M. Kilic for useful correspondence and T. von Hippel for a helpful referee’s report. REFERENCES Aannestad, P. A., Kenyon, S. J., Hammond, G. L., & Sion, E. M. 1993, AJ, 105, 1033 Becklin, E. E., Farihi, J., Jura, M., Song, I., Weinberger, A. J., & Zuckerman, B. 2005, ApJ, 632, L119 – 8 – Carey, S. 2006, talk presented at the Spitzer data workshop Davidsson, B. J. R. 1999, Icarus, 142, 525 Debes, J. H. & Sigurdsson, S. 2002, ApJ, 572, 556 Debes, J. H. & Sigurdsson, S. 2007, in ASP Conf. Ser., EUROWD06,: 15th European White Dwarf Workshop;, ed. R. Napiwotzki & M. Barstow (San Francisco: ASP), in press Farihi, J., Becklin, E. E., & Zuckerman, B. 2005, ApJS, 161, 394 Farihi, J., Zuckerman, B., & Becklin, E. E. 2007, ApJ, submitted Fazio, G. G. et al. 2004, ApJS, 154, 10 Fixsen, & Dwek, E. 2002, ApJ, 578, 1009 Friedrich, S., Jordan, S., & Koester, D. 2004, A&A, 424, 665 Gaensicke, B. T., Marsh, T. R., Southworth, J., & Rebassa-Mansergas, A. 2006, Science, 314, 1908 Gianninas, A., Dufour, P., & Bergeron, P. 2004, ApJ, 617, L57 Hines, D. et al. 2006, ApJ, 638, 1070 Jura, M. 2003, ApJ, 548, L91 Jura, M. 2006, ApJ, 653, 613 Jura, M., Farihi, J., Zuckerman, B., & Becklin, E. E. 2007, AJ, in press Kilic, M., & Redfield, S. 2007, ApJ, in press Kilic, M., von Hippel, T., Leggett, S. K., & Winget, D. E. 2005, ApJ, 632, L115 Kilic, M., von Hippel, T., Mullally, F., Reach, W. T., Kuchner, M. J., Winget, D. E., & Burrows, A. 2006a, ApJ, 642, 1051 Kilic, M., von Hippel, T., Leggett, S. K., & Winget, D. E. 2006b, ApJ, 646, 474 Koester, D., & Wilken, D. 2006, A&A, 453, 1051 Koester, D., Rollenhagen, K., Napiwotzki, R., Voss, B., Christlieb, N., Homeier, D., & Reimers, D. 2005, A&A, 432, 1025 – 9 – Mullally, F., Kilic, M., Reach, W. T., Kuchner, M. J., von Hippel, T., Burrows, A., & Winget, D. E. 2007, ApJS, in press Pringle, J. E. 1996, MNRAS, 281, 357 Paquette, C., Pelletier, C., Fontaine, G., & Michaud, G. 1986, ApJS, 61, 197 Reach, W. T. et al. 2005a, PASP, 117, 978 Reach, W. T., Kuchner, M. J., von Hippel, T., Burrows, A., Mullally, F., Kilic, M., & Winget, D. E. 2005, ApJ, 635, L161 Rieke, G. H. et al. 2004, ApJS, 154, 25 Silverstone, M. et al. 2006, ApJ, 639, 1138 Sion, E., Hammond, G. L., Wagner, R. M., Starrfield, S. G., & Liebert, J. 1990, ApJ, 362, Tremblay, P.E., & Bergeron, P. 2007, ApJS, in press von Hippel, T., Kuchner, M. J., Kilic, M., Mullally, F., & Reach, W. T. 2007, ApJ, in press Werner, M. W. et al. 2004, ApJS, 154, 1 Wolff, B., Koester, D., & Liebert, J. 2002, A&A, 385, 995 Zubko, V., Dwek, E., & Arendt, R. G. 2004, ApJS, 152, 211 Zuckerman, B. et al. 2007, in preparation Zuckerman, B., & Becklin, E. E. 1987, Nature, 330, 138 Zuckerman, B., Koester, D., Reid, I. N. & Hunsch, M. 2003, ApJ, 596, 477 This preprint was prepared with the AAS LATEX macros v5.2. – 10 – Table 1 – Measured Infrared Fluxesa Star Other Name Fν(3.6 µm) Fν(4.5 µm) Fν(5.7 µm) Fν(7.9 µm) Fν(24 µm) (µJy) (µJy) (µJy) (µJy) (µJy) WD 0300-013b GD 40 231(12) 199(10) 159(15) 164(17) <90 WD 0408-041 GD 56 1090(55) 1212(61) 1177(61) 1112(59) 240(80) WD 1015+161 PG 1015+161 212(11) 177(9) 145(17) 126(19) <110 PG 1015+161c 197(11) 169(9) 140(17) 123(19) <110 WD 1116+026 GD 133 592(30) 527(26) 472(27) 464(29) 310(100) WD 1124-293 ESO 439-80 <50 WD 1204-136 EC 12043-1337 158(5) 99(4) 72(14) 32(17) <70 WD 1225+006 HE 1225+0038 361(18) 223(12) 144(16) 79(20) <80 WD 1315-110 HE 1315-1105 210(11) 131(7) 90(15) 30(18) <80 WD 1457-086 PG 1457-086 <90 WD 2144-079b G26-31 191(10) 122(7) 65(19) 42(22) <100 WD 2354+159b PG 2354+159 73(5) 44(4) 27(17) <21 <110 a1σ errors for detections are given in parenthesis; the procedure for deriving the upper limits is described in §2. bHe-rich star (Koester et al. 2005) cPG 1015+161 fluxes corrected for the background star, as discussed in §2 – 11 – Table 2 – Stellar and Disk Properties Star T∗ R∗ R∗/D model cos i Tmax Tmin Rmin Rmax (K) (R⊙) (10 −12) (K) (K) (R∗) (R∗) GD 40 15,200a 0.013b 3.6 Agd40 0.2 1200 600 18 44 Bgd40 0.4 1000 700 22 36 Cgd40 0.8 800 650 30 40 GD 56 14,400a 0.015c 3.6 Agd56 1.0 1200 300 16 104 Bgd56 1.0 1700 400 10 71 PG 1015+161 19,300a 0.014c 2.7 Apg10 0.3 1200 800 24 42 Bpg10 0.6 1000 800 31 42 Cpg10 1.0 800 700 42 50 GD 133 12,200a 0.014c 7.0 Agd133 0.2 1200 300 13 83 Bgd133 0.4 1000 600 17 33 Cgd133 0.8 800 600 23 33 aKoester et al. (2005) bFriedrich et al. (2004) cderived from Koester & Wilkin (2006) – 12 – Table 3 – Predicted Fν(24 µm) for Simple Interstellar Accretion followed by Poynting-Robertson drag Star D dMdust/dt Fν(24 µm) Fpred/Fobs (pc) (109 g s−1) (mJy) GD 56 74 0.26 9.9 41 PG 1015+161 95 2.0 46 >420 GD 133 43 0.37 42 140 ESO 439-80 34 0.13 23 >460 EC 12043-1337 52 0.23 18 >260 HE 1225+0038 29 0.0059 1.5 >19 HE 1315-1105 40 0.022 2.9 >36 PG 1457-086 117 1.5 23 >260 Distances are taken from Koester & Wilkin (2006) while the dust accretion rates are scaled from their values as described in §3. – 13 – Fig. 1.— Comparison of models from Table 2 and data (2σ errors) for GD 40. The plot shows the contribution from the stellar photosphere (dotted line), one disk model (blue dashed line for model Agd40) and totals (solid blue for model Agd40, solid green for model Bgd40, solid red for model Cgd40). Models Agd40 and Bgd40 are essentially indistinguishable while model Cgd40 predicts somewhat too little flux at 3.6 µm because the warmest grains are only 800 K. – 14 – Fig. 2.— Comparison of three models and data (2σ errors) for GD 56. The dotted line is the contribution from the stellar photosphere. The sums of the circumstellar and stellar fluxes are shown for models Agd56 (solid blue line) and Bgd56 (solid magenta line) listed in Table 2. We also show (solid black line) the flux for a model summing both the photosphere and excess emission for an opaque dust cloud at a temperature of 1000 K with an angular radius 1.4 × 10−10 radians as seen from Earth. – 15 – Fig. 3.— Similar to Figure 1, but for PG 1015+161 with blue, green and red for the totals of models Apg10, Bpg10 and Cpg10, respectively. The emission from only the disk is shown for model Apg10 – 16 – Fig. 4.— Similar to Figure 1, but for GD 133 with blue, green and red for the totals of models Agd133, Bpg13s and Cpg133, respecitvely. The emission from only the disk is shown for model Agd133. – 17 – Fig. 5.— KECK/NIRC z-band image, epoch 1995.9, obtained of PG 1015+161. Up lies at P. A. = 96◦; the background star lies 2.′′9 from the white dwarf at P. A. = 270◦. The proper motion proper motion of µ = 0.′′13 yr−1 at P. A. = 239◦ (Farihi et al. 2005) of PG 1015+161 is consistent with its position relative to the background source in our 2007.0 IRAC images. – 18 – Fig. 6.— Mass accretion rates (§3) vs. effective temperature for hydrogen-rich white dwarfs with an excess (solid circles) in either ground-based or Spitzer data and without an excess (open triangles) in published Spitzer data. GD 133 and PG 1015+161 (this paper), GD 56 (this paper and Kilic et al. 2006b), G29-38 (Zuckerman & Becklin 1987), WD 2115-560 (Mullally et al. 2007, von Hippel et al. 2007), WD 1150-153 (Kilic & Redfield 2007) have an excess while EC 12043-1337, HE 1225+038, and HE 1315-1105 (this paper), WD 1202-232, WD 1337+705 and WD 2149+021 (Mullally et al. 2007), and WD 0208+396, WD 0243-026, WD 0245+541, and WD 1257+278 (Debes & Sigurdsson 2007) do not. Two helium-rich white dwarfs (GD 40 and GD 362) that also display an excess are not shown here. INTRODUCTION OBSERVATIONS MODELS DISCUSSION CONCLUSIONS

0704.1171

Highly synchronized noise-driven oscillatory behavior of a FitzHugh-Nagumo ring with phase-repulsive coupling

Highly synchronized noise-driven oscillatory behavior of a FitzHugh–Nagumo ring with phase-repulsive coupling Gonzalo Izús∗,†, Roberto Deza∗ and Alejandro Sánchez∗,† ∗Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Deán Funes 3350, 7600 Mar del Plata, Argentina. †Member, CONICET Abstract. We investigate a ring of N FitzHugh–Nagumo elements coupled in phase-repulsive fashion and submitted to a (subthreshold) common oscillatory signal and independent Gaussian white noises. This system can be regarded as a reduced version of the one studied in [Phys. Rev. E 64, 041912 (2001)], although externally forced and submitted to noise. The noise-sustained synchronization of the system with the external signal is characterized. Keywords: synchronization, signal transduction, chemical waves, neuroscience PACS: 05.45.Xt, 87.16.Xa, 87.18.Pj, 87.19.La INTRODUCTION In Ref. [1]—through comparison with the synchronization patterns arising in two- dimensional arrays of FitzHugh–Nagumo (FHN) elements with phase-repulsive linear nearest-neighbor coupling—the authors were able to conclude that intracellular cal- cium oscillations in cultures of human epileptic astrocytes do interact, since the phases of nearby oscillating astrocytes maintain a nontrivial relationship. It is a fortunate fact that the (space-independent) FHN model is one of the very few multicomponent systems for which a nonequilibrium potential (NEP) has been found [2, 3], since NEPs allow in general for a deep insight on the dynamical mechanisms leading to pattern formation and other phenomena where fluctuations play a constructive role [4]. The (albeit minimal) extension of the result in Refs. [2, 3] towards extended systems carried out in this work is however enough to shed light on the dynamical cause of the conclusion in Ref. [1]: a dynamical symmetry breakdown takes place because the phase-repulsive coupling min- imizes the corresponding NEP. When the system is externally forced with a frequency less than the typical inverse deterministic time the cycle duplicates, breaking down into an “excited” phase and an “inhibited” one. These phases force neighbor elements to al- ternate with the one in between, thus creating a nontrivial phase relationship between nearby oscillating elements. The system we consider is sketched in Fig. 1: a ring of N = 256 identical FHN el- ements with phase-repulsive nearest-neighbor coupling and submitted to a (subthresh- old) common oscillatory signal and independent Gaussian white noises ξui(t), ξvi(t) with 〈ξm(t)ξn(t ′)〉 = 2ηδmnδ (t − t ′), m,n = 1, . . . ,2N. The set of equations governing ξ iI (t)+ (t) FIGURE 1. Sketch of the system and of its response Ac(t). its dynamics is u̇i = ac ui (1−u2i )− vi +Sg(t)−D(ui+1 +ui−1)+ r1 ξui(t)+ r2 ξvi(t) (1) v̇i = ε (β ui− vi +C)+ r3 ξui(t)+ r4 ξvi(t), i = 1, . . . ,N, uN+1 = u1. where ε = 0.01 is the ratio between the relaxation rates of ui and vi, β = 0.01, ac = 0.06 and C = 0.02 is a suitable constant to set the rest point in Fig. 2a. D = 0.01 is the phase-repulsive coupling constant, and the ri (which determine the transport matrix) are r1 = 0.998×102, r2 = 0.499×101, r3 = 0.998, r4 = 0.499×10−1. Moreover, taking the Milshtein integration step as dt = 5×10−3, we estimate the typical inverse deterministic time as 0.838× 10−3 and so we take the excitation frequency Ω0 as a fraction of that value (typically 0.1–0.4). Given that, Sg(t) = 0.0275sinΩ0t. THE NONEQUILIBRIUM POTENTIAL Excitable dynamics can be conceptually decomposed into two phases, a fluctuation- dominated one and a deterministic one. It would be highly desirable to find a Lyapunov function, since it greatly simplifies the dynamical analysis. However, the existence of non-variational (or conserving) components in the phase-space flow is a hint that the integrability conditions fail for the purely deterministic system. This apparently insur- mountable drawback was partially solved two decades ago by Graham and collaborators (see references in [2]) who defined the NEP for Langevin-type dynamics as the zero- noise limit of the logarithm of the stationary probability density function (pdf). The extra freedom in the choice of the transport matrix can render in some cases the prob- lem integrable. That is precisely the case for the space-independent FitzHugh–Nagumo model in its bistable and excitable regimes [2, 3]. This approach can be generalized to extended systems and the NEP associated to Eq.(1) (in the adiabatic limit, i.e. for slow FIGURE 2. a) Phase-space excursions in excitable regime (the nullclines are indicated in dashed line); b) Time evolution of u (dashed line), v (dotted line), and the NEP (full line) during a phase-space excursion. The scales of v and the NEP were adjusted for better comparison. FIGURE 3. a) NEP in excitable regime; b) Stationary pdf in excitable regime. signal) is [5] Φ(t) = Φ{t,ui(t),vi(t)}= (v2i −2β uivi−2Cvi)+ (β u2i +2Cui) u2i − u4i +Sg(t)ui uiui+1 , (2) FIGURE 4. a) NEP in bistable regime; b) Stationary pdf in bistable regime. FIGURE 5. Time evolution of u for two neighbor neurons. which must obey the integrability condition βλ1 +λ2/ε = 2λ [2]. Figure 2b depicts (in full line) the time evolution of the NEP during the phase-space excursion starting at the upper initial condition in Fig. 2a, together with that of u (dashed line) and v (dotted line). We remark that in Figs. 2a and 2b there is no noise and Φ(t) is the Lyapunov functional of the deterministic dynamics. Figures 3a and 3b (respectively 4a and 4b) are 3D and contour plots of the NEP and the corresponding stationary pdf for the excitable (respectively bistable) regime. RESULTS FOR THE COUPLED SYSTEM Synchronization between the coupled system and the external signal is observed above some noise-intensity threshold. Figure 5 is a plot of the time evolution of ui (full line), together with that of ui+1 (dashed line) for a given neuron i, showing their phase relation to the signal (dotted line)1 According to Fig. 2, we may call “active” those cells i for which ui(t) exceeds some threshold value uth. Because of the coupling, as one neuron becomes active, it inhibits the activation of its nearest neighbors. The perfect alternance seen in the figure may fail because of the noise, a necessary ingredient for the activation. A detail of the alternance can be seen in Fig. 6 for an N′= 21 subset of the ring. Figure 6a shows a situation (snapshot) of poor synchronization, in which only two neurons are active; Fig. 6b exhibits a case of a “kink” in the synchronized configuration, induced by the fact that noises are local. Note that the kinks break locally the observed coherence, and the complete history of the time evolution can be followed as a record of activity (see Fig. 7). 1 The signal has been augmented in about two orders of magnitude and shifted to aid the sight. FIGURE 6. Two snapshots of {ui} showing different degrees of synchrony. FIGURE 7. Synchronization of the ring. White corresponds to activation and black to inhibition. Horizontal dimension corresponds to time and vertical to space. A measure of “activity” for the whole ring is Ac(t) = θ [ui(t)−uth]. (3) In perfect synchrony, Ac = 0.5. Note that since the signal is subthreshold for the coupled system, Ac = 0 below threshold. Figure 8a depicts the activity as a function of time for a fixed noise intensity, showing again its phase relationship with the signal (dashed line). In Fig. 8b we show the NEP for the whole ring as a function of time, together with the (scaled) signal for reference. We remark that the observed dynamical symmetry breakdown decreases the Lyapunov function of the whole ring with respect to that of the homogeneous state, providing the route to stable synchronization. A global estimator of synchronization can be defined as ∫ t f 0 Ac(t)dt 0.5N t f . (4) Figure 9a is a plot of Ga as a function of the noise intensity. The existence of a threshold value of noise intensity and of a saturation effect can be clearly seen. The noise intensities are low enough not to degrade the excitable dynamics. FIGURE 8. a) Ac vs t for high synchronization; b) Time evolution of the NEP FIGURE 9. a) Ga vs η ; b) C vs η . Numerical simulations indicate that the coherence of firing decreases with the noise intensity although the global activity (representative of global estimators) keeps the order of magnitude. To quantify this phenomena we have calculated the normalized self- correlation C = 〈uiui+2〉 as a function of the noise intensity η . As we show in Fig. 9b the system shows a kind of “stochastic resonance in coherence” that cannot be inferred from measures of global activity. CONCLUSIONS We have investigated the noise-induced synchronization with an external signal of a ring of phase-repulsively coupled FHN elements. We have derived the exact NEP of the ex- tended system and the observed symmetry breakdown was related with the Lyapunov- functional properties of the NEP. We remark that the same conclusion holds qualita- tively for the work in Ref. [1]. Although the observed phenomenon is noise-sustained and global activity increases with noise intensity, a degradation of coherence can be appreciated. ACKNOWLEDGMENTS Financial support from CONICET, ANPCyT and the National University of Mar del Plata is acknowledged. REFERENCES 1. G. Balázsi, A. Cornell-Bell, A. B. Neiman, and F. Moss, Phys. Rev. E 64, 041912 (2001). 2. G. Izús, R. Deza, and H. S. Wio. Phys. Rev. E 58, 93–98 (1998). 3. G. Izús, R. Deza, and H. S. Wio. Comp. Phys. Comm. 121–122, 406–407 (1999). 4. H. S. Wio, in Fourth Granada lectures in computational physics, P. L. Garrido and J. Marro, Eds.; LNP 493 (Springer-Verlag, Berlin, 1997), p. 135. 5. A. Sánchez, G. Izús, and R. Deza, in preparation.

0704.1172

Disentanglement in a quantum critical environment

Disentanglement in a quantum critical environment Zhe Sun and Xiaoguang Wang∗ Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China C. P. Sun† Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, China (Dated: October 25, 2018) We study the dynamical process of disentanglement of two qubits and two qutrits coupled to an Ising spin chain in a transverse field, which exhibits a quantum phase transition. We use the concurrence and negativity to quantify entanglement of two qubits and two qutrits, respectively. Explicit connections between the concurrence (negativity) and the decoherence factors are given for two initial states, the pure maximally entangled state and the mixed Werner state. We find that the concurrence and negativity decay exponentially with fourth power of time in the vicinity of critical point of the environmental system. PACS numbers: 05.40.-a, 03.65.Yz, 32.80.-t,03.67.Mn I. INTRODUCTION Entanglement is one of the most essential features in quantum mechanics [1] and in recent decades has been focused by people in many fields of physics. Motivated by the progress of quantum information, entanglement has become a basic resource in the quantum technologies such as quantum teleportation and quantum cryptography [2]- [4]. On the other hand, generally a realistic system is surrounded by an environment. The coupling between a quantum system and its environment leads to decoher- ence of the system. Thus, it is natural for us to consider the process of degradation of entanglement due to the decoherence. More recently, Yu and Eberly [5] showed that two entangled qubits become completely disentan- gled in a finite time under the influence of pure vacuum noise. Surprisingly, they found that the behaviors of lo- cal decoherence is different from spontaneous disentan- glement. The decoherence effects take an infinite time evolution under the influence of vacuum while the entan- glement vanishes suddenly in a finite time. Some other researchers also investigated the process of disentangle- ment in the open quantum systems [6]-[9]. The problem of decoherence from spin environments was studied by Cucchietti et al [10], while they considered the spin en- vironments consisting of N independent other than cor- related spins . In most of the previous studies, uncorrelated environ- ments are usually considered, and modelled by a reser- voir consists of harmonic oscillators. Although a collec- tion of harmonic oscillators is a well approximated mod- elling to represent the environment weakly coupled to system, however, in the practical situation, particles in the environment may have interactions with each other. Consequently, a problem comes out: How does the en- ∗Electronic address: [email protected] †Electronic address: [email protected] tanglement evolves in a correlated environment? In this paper, we consider this problem and choose a correlated spin chain, the Ising model in a transverse field, as the surrounding system. Moreover, this surrounding system displays quantum phase transition (QPT) at some criti- cal point and thus it possesses the dynamic hypersensi- tivity with respect to the perturbation even induced by a single qubit [11]. As a quantum critical phenomenon, QPT happens at zero temperature, at which the thermal fluctuations van- ish. Thus, QPT is driven only by quantum fluctuation. Usually, at the critical point there exists degeneracy be- tween the energy levels of the systems when QPT hap- pens. Therefore, it can be excepted that, when we study the dynamic evolution of the system coupled to a envi- ronment with QPT, some special dynamic features will appear at the critical point. Quan et al [11] have studied the decoherence induced by the correlated environment. It was shown that at the critical point of a QPT the de- coherence is enhanced. Following this work, Cucchietti et al [12] discovered that the decoherence induced by the critical environment possesses some universality with the Boson-Hubbard model as an illustration. Now, we consider two spins coupled to the Ising spin chain in a transverse field, and the purpose is to reveal the effect of the correlated environment on the dynamic evolution of the two-spin entanglement. We will study different cases including two qubits and qutrits. More- over, we will consider cases that the two spins initially start from a pure maximally entangled state and a mixed Werner state [13]. The ‘sudden death’ of entanglement is found to be a quite common phenomenon. This paper is organized as follows. In Sec. II, we intro- duce the model of two-spin system coupled to Ising spin chain with a transverse field. By exactly diagonalizing the Hamiltonian, we give expression of the time evolu- tion operator. In Sec. III, the analytical results of the concurrence [14] of the two qubits are calculated to show the dynamics of entanglement. Numerical results are also given to illustrate the details of the dynamical behaviors http://arxiv.org/abs/0704.1172v1 mailto:[email protected] mailto:[email protected] of entanglement. In Sec. IV, two qutrits are coupled to the Ising spin chain. The analytical and numerical re- sults of the negativity [15, 16] are given. At last we give the conclusion in Sec. V. II. MODEL HAMILTONIAN AND EVOLUTION OPERATOR We choose the engineered environment system to be an Ising spin chain in a transverse field which displays a QPT. Two spins are transversely coupled to the chain. The corresponding Hamiltonian reads σxl σ l+1 + (s1z + s2z) , (1) where λ characterizes the strength of the transverse field, g denotes the coupling strength between the Ising chain and the two spins, s1 and s2, σ (α = x, y, z) are the Pauli operators defined on the l-th site, and the total number of spins in the Ising chain is L = 2M + 1. The Ising model is the simplest model which exhibits a QPT, and can be exactly calculated. In order to diagonalize the Hamiltonian, firstly we no- tice that [s1z + s2z, σ ] = 0, thus it is convenient to define an operator-valued parameter Λ̂ = λ+ (s1z + s2z), (2) which is a conserved quantity. When we diagonalize the Ising spin chain, the parameter Λ̂ can be treated as a c- number with different values corresponding to the eigen- values of s1z + s2z in the two-spin subspace. By combining Jordan-Wigner transformation and Fourier transformation to the momentum space [17], the Hamiltonian can be written as [18] σkx (Ωkσkz) e σkx + σ0z (3) where we have used the following pseudospin operators σkα (α = x, y, z) [18] σkx = d + d−kdk, (k = 1, 2, ...M) σky = −id†kd + id−kdk, σkz = d dk + d d−k − 1, σ0z = 2d 0d0 − 1, (4) and d , dk{k = 0, 1, 2, ...} denote the fermionic creation and annihilation operators in the momentum space, re- spectively. Here, −Λ̂+2 cos (2πk/L) + 4 sin2 (2πk/L), (5) θk = arcsin −2 sin . (6) From Eq. (3) and the units where h̄ = 1, the time evolu- tion operator is obtained as: U(t) = e−i(− +1)σ0zt σkxe−itΩkσkze−i σkx . (7) Having explicitly known the evolution operator, we now consider the entanglement dynamics of the two qubits and two qutrits. III. DYNAMICAL DISENTANGLEMENT OF TWO QUBITS A. The case with initial pure entangling state We investigate the dynamic evolution of two-qubit en- tanglement and assume that the two qubits initially start from a maximally entangled state. |Φ〉 = 1√ (|00〉+ |11〉) . (8) Here, |0〉 and |1〉 denote the spin up and down, respec- tively. The initial state of environment is assumed to be the vacuum state in the momentum space, namely, |ψE〉 = |0〉k=0 ⊗k>0 |0〉k|0〉−k, and the vacuum state |0〉k satisfies dk|0〉k = 0. We may write a more general initial state of this composite system as |Ψ(0)〉 = (a |00〉+ b |11〉)⊗ |ψE〉 . (9) From the evolution operator (7), the state vector at time t is given by |Ψ(t)〉 = a |00〉 ⊗ U0 |ψE〉+ b |11〉 ⊗ U1 |ψE〉 , (10) where the unitary operator U0 and U1 can be obtained from the unitary operator U(t) by replacing operator Λ̂ with number λ+g/2 and λ−g/2, respectively. Tracing out the environment, in the basis spanned by {|00〉 , |11〉 , |01〉 , |10〉}, the reduced density matrix of the two-spin system is obtained as ρ1,2 = |a|2 ab∗F (t) a∗bF ∗(t) |b|2 ⊕ Z2×2, (11) where F (t) = 〈ψE |U †1U0 |ψE〉 is the decoherence factor, and Z2×2 denotes the 2×2 zero matrix. Now, the concur- rence [14] of the reduced density matrix can be readily given by C = 2|ab∗F (t)| = C0|F (t)|, (12) where C0 is the concurrence of the initial state. We see that the concurrence is proportional to the norm of the decoherence factor, and when the initial state is in a max- imally entangled state (8), C = |F (t)|, namely, the con- currence is equal to the norm of the decoherence factor. Let us consider the decoherence factor F (t) = 〈ψE |U †1U0 |ψE〉 = Fk, (13) where Un(n = 0, 1) is generated from Hamiltonian Hn with Λ̂ = Λn(a number). From the unitary operator (7) and the initial vacuum state, we obtain |F (t)| = sin(Ω t) cos(Ω t) sin θ − cos(Ω(0) t) sin(Ω t) sin θ − sin2(Ω(0) t) sin2(Ω t) sin2(θ − θ(1) where Ω and θ are obtained by replacing Λ̂ with Λn in Eqs. (5) and (6), respectively. Here, Λ0 = λ+g/2 and Λ1 = λ−g/2. This is one of our main results. We see that the zero mode (k = 0) has no contribution to the decoherence factor. Clearly, every factor Fk is less than unit. So it can be well expected that in the large L limit, |F (t)| will go to zero under some reasonable conditions. By carrying out similar analysis of Ref. [11], we intro- duce a cutoff number Kc and define the partial product for the decoherence factor |F (t)| Fk ≥ |F (t)| , (15) from which the corresponding partial sum S (t) = ln |F (t)| |lnFk| . (16) For the case of small k and large L, we have Ω |2− Λn|, consequently − θ(1) ≈ 16k 2π2 (Λ0 − Λ1)2 L2 (2− Λ0)2 (2− Λ1)2 . (17) As a result, if L is large enough and Λ0−Λ1 is very small perturbation the approximation of S can be obtained as S (t) ≈ −2E (Kc) (2− Λ0)−2 (2− Λ1)−2 ×{(Λ0 − Λ1)2 sin2 (|2− Λ0| t) sin2 (|2− Λ1| t) +[sin (|2− Λ0| t) cos (|2− Λ1| t) |2− Λ1| − sin (|2− Λ1| t) cos (|2− Λ0| t) |2− Λ0|]2}, where E (Kc) = 4π 2Kc (Kc + 1) (2Kc + 1) / . (19) In the derivation of the above equation, we have used ln(1− x) ≈ −x for small x and k2 = n(n+ 1)(2n+ 1)/6. 0 50 100 0 2 4 6 8 10 λ=0.1 (a) (b) FIG. 1: (a) Concurrence versus time t with different λ in the case of weak coupling strength g = 0.1. The size of the environment is L = 300. (b) shows the cases of larger λ. 0 20 40 60 80 L=200 L=600 L=1000 FIG. 2: Concurrence versus time with different environment size L = 200, 600 and 1000. The transverse field λ = 4, and the coupling strength g = 0.1. For our two-qubit case, Λ0 = λ+g/2, Λ1 = λ−g/2. When λ→ 2, and with a proper small g we have |F (t)| ≈ e−γt with γ = 2E (Kc) g 2. Notice that |F (t)| is larger than |F (t)| = C. Therefore, from the above heuristic analysis we may expect that when the parameter λ is adjusted to the vicinity of the critical point λc = 2, the concurrence (or the decoherence factor) will exponentially decay with the fourth power of time. Moreover, for short times, from Eq. (14), the concurrence becomes C ≈ e−Γt with Γ = 1/2 sin2(θ − θ(1) Now we resort to numerical analysis of the dynamical sensitivity and the concurrence decay. In the Fig. 1 (a) and (b), we plot the concurrence versus time for different λ. We find that in the vicinity of the critical point about λ ∈ [2 − 0.3, 2 + 0.3], concurrence decays monotonously with time. And extending the time range, however there are not the revivals of concurrence. Figure 1 (a) shows the cases of λ ≤ 2. We can see that concurrence for 0 2 4 6 8 10 g=0.1 g=100 FIG. 3: Concurrence versus time at the critical point λ = 2 with different coupling strength g. the case λ = 2 decays more rapidly than other cases. It should be noted that, the dynamics of the two-qubit en- tanglement in Eq. (12) is absolutely determined by the decoherence factor in Eq. (14), thus from a theoretical point of view, the complete disentanglement cannot be realized in a finite time. When parameter λ becomes larger than λc,(g = 3, 4 and 5), the numerical results of the concurrence are shown in Fig. 1 (b). The concur- rence oscillates with time, and collapses and revivals are observed. This is in contrast with the case of small λ, where no revivals are found. The surrounding system displays a QPT near the crit- ical point, and there exists a competition between differ- ent order tendencies [17]. From another point of view, near the critical point quantum chaotic behaviors may emerge [19]. For a system with quantum chaos, though it is prepared in identical initial state, two slightly differ- ent interactions can lead to two quite different quantum evolutions. In our system the decoherence factor can act as a fidelity and quantify the difference between the two states which are produced through two different evolu- tions. Decay of the fidelity can indicate the presence of the quantum chaos [20], and here the monotonous de- cay of the decoherence factor (concurrence) at the crit- ical point may be considered as a signature of quantum chaos. In Fig. 2, for weak coupling g = 0.1 and λ = 4, the oscillation of concurrence is suppressed by enlarging the size of environment. The larger environment prevents the revival of entanglement. In the short-time region, we can see the larger size of environment will accelerate the monotonous decay of concurrence. From Eq. (14), each factor Fk is smaller than 1, thus it is reasonable that large size of environment will be more effective to suppress the factor F (t), and consequently suppress the concurrence. In Fig. 3, we consider the effects of coupling g on the dynamics of entanglement. At the critical point λ = 2, we adjust g from a small one g = 0.1 to a strong one g = 100. It can be found that when we properly en- large the coupling, e.g. g = 1, the concurrence decays more sharply than the case g = 0.1. However, when we continue enlarging the coupling to about g > 10, e.g. g = 25, concurrence will oscillate quickly and does not decay monotonously to zero any more. For the case of very large coupling g = 100, concurrence behaves as a weak oscillation near the initial value of C = 1. It can be expect that to the strong coupling limit of g, the concur- rence will stay at C = 1 without changing with time. The above behaviors remind us of the quantum Zeno effects in process of quantum measurement [21]. The phenom- ena shown in Fig. 3 is similar to the decay probability which can be suppressed by the increasing coupling be- tween system and measuring apparatus in quantum Zeno effects. B. The case of mixed state Now, we study the dynamics of disentanglement of mixed entangled state and assume the two qubits being initially in a Werner state [13], which is given by ρs = P |Φ〉 〈Φ|+ I4×4, (22) where |Φ〉 is the maximally entangled state given by Eq. (8), the parameter P ∈ [0, 1], and I4×4 denotes a 4 × 4 identity matrix. This state is a mixed state except the extreme case of P = 1. Only when P > 1/3, the Werner state ρs is entangled. We assume the initial state of the whole system ρtot is in a direct product form as ρtot = ρs ⊗ |ψE〉 〈ψE | , (23) where |ψE〉 is the initial state of the environment. Af- ter the time evolution, we can obtain the reduce density matrix of the two-qubit system in the basis spanned by {|00〉 , |11〉 , |01〉 , |10〉} as follows ρ1,2 = PF (t) PF ∗(t) 1+P I2×2, (24) where the decoherence factor F (t) is the same as Eq. (14). From Eq.(24), the concurrence is derived as C = max |F |+ 1 . (25) When P = 1, it reduces to Eq. (12) for the pure maxi- mally entangled state. While in the region 1/3 < P < 1, the concurrence vanishes when the decoherence factor |F | ≤ (P−1 − 1)/2. (26) Thus there exists a finite disentanglement time td, after which the entanglement is zero. According to the results of heuristic analysis in Eq. (20), |F (t)| ≈ e−γt4 , in the condition of weak coupling and λ → 2, we can approxi- mately give the disentanglement time . (27) 0 2 4 6 8 10 P=0.5 P=0.7 FIG. 4: Concurrence versus time at the critical point λ = 2 and coupling strength g = 0.1 for parameters P = 0.5, 0.7 and 1. Then, the disentanglement time increases as the proba- bility P increases from 1/3 to 1. In Fig. 4, we also numerically calculate the concurrence versus time for different probabilities. For the mixed states corresponding to P = 0.5 and 0.7, disentangle- ment process takes only a finite time, while for the pure state case (P = 1), disentanglement is only completed asymptotically, and it will take an infinite time. Numeri- cal results are consistent with the above analytical results that the disentanglement time increases with the increase of P . IV. DYNAMICAL ENTANGLEMENT EVOLUTION OF TWO QUTRITS Now, we consider the case of two qutrits and use the negativity [15] to quantify entanglement. For the sys- tems with spin larger than 1/2, a non-entangled state has necessarily a positive partial transpose (PPT) ac- cording to the Peres-Horodecki criterion [15]. In the case of two spin halves, and the case of (1/2,1) mixed spins, a PPT is also sufficient. Vidal and Werner [16] developed the Peres-Horodecki criterion and presented a measure of entanglement called negativity that can be computed ef- ficiently, and the negativity does not increase under local manipulations of the system. The negativity of a state ρ is defined as N (ρ) = |µi|, (28) where µi is the negative eigenvalue of ρ T2 , and T2 denotes the partial transpose with respect to the second subsys- tem. If N > 0, then the two-spin state is entangled. The negativity has been used to characterize the entan- glement in large spin system very well [22]-[24]. And by means of negativity, Derkacz et al. have studied the process of disentanglement in a pair of three-level atoms interacting with the vacuum [8]. A. The case with initial pure state In a similar vein as the study of two-qubit case, we write a general initial state of the many-body system as |Ψ(0)〉 = (a |00〉+ b |11〉+ c|22〉)⊗ |ψE〉 . (29) where |0〉, |1〉 , |2〉 denote the spin-one state with mag- netic quantum number 1, 0, -1 respectively. From the evolution operator (7), the state vector at time t is given |Ψ(t)〉 = a |00〉 ⊗ U0 |ψE〉+ b |11〉 ⊗ U1 |ψE〉 +c|22〉 ⊗ U2 |ψE〉 , (30) where the unitary operator U0, U1,and U2 are obtained from the unitary operator U(t) by replacing operator Λ̂ with number λ+g, λ and λ− g, respectively. In the basis spanned by {|00〉, |11〉, |22〉, |01〉, |10〉, |02〉, |20〉, |12〉, |21〉}, the reduced density matrix of the two-qutrit system is ρ1,2 = |a|2 ab∗F1(t) ac∗F2(t) a∗bF ∗1 (t) |b|2 bc∗F3(t) a∗cF ∗2 (t) b ∗cF ∗3 (t) |c|2 ⊕Z2×2 ⊕ Z2×2 ⊕ Z2×2, (31) where F1(t) = 〈ψE |U †1U0 |ψE〉 , F2(t) = 〈ψE |U †2U0 |ψE〉 , F3(t) = 〈ψE |U †2U1 |ψE〉 (32) are the decoherence factors. The partial transpose with respect to the second sys- tem gives ρT21,2 = diag(|a|2, |b|2, |c|2)⊕B1 ⊕B2 ⊕B3, (33) where the three 2× 2 matrices 0 ab∗F1(t) a∗bF ∗1 (t) 0 0 ac∗F2(t) a∗cF ∗2 (t) 0 0 bc∗F3(t) b∗cF ∗3 (t) 0 . (34) Then, from the above matrix ρT21,2, one can obtain the negativity as N = |ab∗F1(t)|+ |ac∗F2(t)|+ |bc∗F3(t)|. (35) For the maximally entangled state, a = b = c = 1/ and the negativity simplifies to N = 1 (|F1(t)|+ |F2(t)|+ |F3(t)|) . (36) 0 2 4 6 8 10 λ=0.1 0 20 40 60 80 100 (a) (b) FIG. 5: (a) Negativity versus time with different cases of λ = 0.1, 1 and 2. The coupling g = 0.1 and the size of environment L = 300. (b) shows the cases of λ = 3, 4 and 5. The highest one (solid line with up triangles) corresponds to the case λ = 5, and the lowest one (dashed line with points) corresponds to λ = 3. 0 2 4 6 8 10 g=0.1 g=100 FIG. 6: Negativity versus time with different coupling strengths g = 0.1, 1, 15 and 100 at the critical point λc = 2. From the above equation, we can find the negativity is a linear combination of three decoherence factors. Also with the vacuum state of environment, the decoherence factors |Fν(t)| = 〈ψE |U †jUi |ψE〉 are given by Eq.(14) by the replacements Ω → Ω(i) → Ω(j) → θ(j) . Here, Fν(t) denotes the three factors F1(t), F2(t) and F3(t). U jUi correspond to U 1U0, U and U 2U1 in the three factors Eq. (32). The parameters and θ (n = 0, 1, 2) can be obtained by substitut- ing Λ0 = λ+g, Λ1 = λ and Λ2 = λ−g into Eq. (5) and During the similar analysis in the case of two qubits, we can also introduce the cutoff number Kc and de- fine the partial product for the three decoherence fac- tors. Through the small k approximation, we can obtain the three partial sums corresponding to the three fac- tors. Therefore, under the condition of weak coupling g and λ→ 2, in a finite time the three factors F1(t), F2(t) and F3(t) will decay exponentially with time in a similar form as Eq. (20). We numerically calculate the dynamics of negativity. In Fig. 5 (a), it shows the similar phenomena in Fig. 1 (a). When the coupling g is weak and λ→ 2, the dynam- ical behaviors of the three decoherence factors in nega- tivity (36) are nearly identical. Each of the factors decay with time just as in Eq. (20), thus it can be understood that negativity also decays monotonously with time in the vicinity of λ = 2. In Fig. 5 (b), we consider the cases of larger couplings. Comparing it with Fig. 1 (b), the behaviors of negativity have some differences with con- currence. More revivals are found in the behavior of the negativity, and they result from the linear superposition of the three decoherece factors. In Fig. 6, we numerically study the effects of different couplings g on the dynamics of negativity. Similar to the dynamic behaviors of the concurrence. With a properly large coupling such as g = 1, the decay of negativity will be much sharper. But very strong coupling (g = 15) will make negativity oscillate rapidly. To the strong coupling limit case of g = 100, negativity decays from the initial value N = 1 to a steady value 1/3, which is different from the concurrence of the two qubits. Let us carry out the approximate analysis just like in the case of two qubits. We can obtain three partial sum S1, S2 and S3, corresponding to the three decoherence factors in Eq. (32), which are similar to Eq. (18). When g → ∞ and λ → 2, we have S2 → 0 and S1 = S3 ≈ −2E (Kc) t2 where E (Kc) is in Eq. (19), thus negativity will decay sharply to a steady value of 1/3. We can see that different dynamic properties of the factors cause the behaviors of negativity shown in Fig. 6 is different from concurrence in Fig. 3. B. The case of mixed state We then consider the mixed state, namely, the two- qutrit Werner state ρs = P |Φ〉〈Φ|+ I9×9, (37) where |Φ〉 is the maximally entangled state of two qutrits and |Φ〉 = (|00〉+ |11〉+ |22〉) / 3. Assume that the whole system is initially in ρ tot = ρs ⊗ |ψE〉 〈ψE |. After time evolution operator in Eq. (7), we can obtain the re- duce density matrix of the two qutrits at arbitrary time t. Then, we make the partial transpose with respect to the second system on the reduce density matrix, and obtain ρT21,2 = diag(1 + 2P, 1 + 2P, 1 + 2P ) ⊕B1 ⊕B2 ⊕B3, (38) where the three 2× 2 matrices PFk(t) (t) 1−P k = {1, 2, 3} (39) From partially transposed reduced density matrix, the negativity is given by N = 1 |Fk(t)|+ . (40) Since |Fk(t)| ≤ 1, the existence of nonzero negativity needs the parameter P satisfying the condition 1/4 < P ≤ 1. From the above equation, we can also reads that the disentanglement occurs only when all the three factors satisfy |Fk(t)| ≤ (P−1 − 1)/3. Furthermore, we study the case of a d-dimension Werner state being the initial state. Thus we give the initial state of the system as i,j=0 |ii〉 〈jj|+ 1− P Id2×d2 , (41) where the basis vector |ii〉 is the eigenvector of sz = s1z + s2z with the eigenvalue 2i+1− d. Then the initial state of the whole system is also performed by a direct product form as ρtot = ρs ⊗ |ψE〉 〈ψE | . After the simi- lar process mentioned in the former parts, we have the matrix ρT21,2 denoting the reduce density matrix after the partial transpose over the second subsystem at time t, which is shown as: ρT21,2 = i,j=0 |ij〉 〈ji|Fi,j(t) + Id2×d2 diag [1 + (d− 1)P, ..., 1 + (d− 1)P ] PFi,j(t) PF ∗i,j(t) , (42) where the decoherence factors Fi,j(t) = 〈ψE |U †jUi |ψE〉 , and the corresponding time evolution operator Ui can be obtained from Eq. (7) by replacing operator Λ̂ with value λ+g/2(2i + 1 − d), respectively. It is apparent that we should only focus on the 2 × 2 matrices and obtain the negativity as N = 1 |Fi,j(t)|+ , (43) from which we can see that negativity will be complete vanishes when all the norms satisfy |Fi,j(t)| ≤ (P−1 − 1)/d simultaneously. V. CONCLUSION In summary, we have studied the dynamics of entan- glement in a pure dephasing system. By making use of the concept of concurrence, we studied two qubits cou- pled to an Ising spin chain in a transverse field. When the two qubits initially started from a pure entangled state, we obtained the analytical results of concurrence which is just a simple product of the initial concurrence C(0) and the decoherence factor F (t). Thus the dynamic properties of concurrence is absolutely determined by the decoherence factor. Specially, in the case of weak cou- pling, the concurrence decays exponentially with time when λ → λc. Moreover, we found the decay of deco- herence factor is of the form exp(−Γt4), which is not a Gaussian form like in Ref. [11] and [12]. Certainly this is due to the initial state of the environment we have chosen. Furthermore, when the two qubits are initially in the Werner state, we have found that the complete disentan- glement takes place in a finite time just as the ‘sudden death’ of entanglement discovered in Ref. [5]. In [5], due to the process of spontaneous emission, the sudden death of entanglement can occur in an arbitrary entangled state (pure or mixed). However, in our system with dephas- ing effects, when the two entangled qubits are in a pure state, there does not exist such a phenomena. We also considered two qutrits coupled to the Ising spin chain. When the qutrits initially start from a pure state, we have obtained the expression of negativity which is a linear combination of three decoherence factors. With weak coupling, negativity also decays monotonously in the condition λ → 2. When the qutrits are initially in a Werner state, the complete disentanglement could oc- cur in a finite time, and then the properties of negativity are the three decoherence factors. Indeed, the correlated environment, especially when QPT happens, greatly af- fects the decoherence and the disentanglement process. The entanglement decay in other environment which dis- plays a QPT [25], or quantum chaos [26] deserves further investigations. Acknowledgments This work is supported by NSFC with grant Nos.10405019 and 90503003; NFRPC with grant No. 2006CB921206; Specialized Research Fund for the Doc- toral Program of Higher Education (SRFDP) with grant No.20050335087. [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). [2] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information Cambridge University Press, Cambridge, England, 2000). [3] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). [4] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991) [5] T. Yu and J.H.Eberly, Phys. Rev. Lett. 93, 140404 (2004). [6] M. S. Zubairy, G. S. Agarwal, and M. O. Scully, Phys. Rev. A 70, 012316 (2004). [7] P. J. Dodd, Phys. Rev. A 69, 052106. [8] L. Derkacz and L. Jakóbczyk, Phys. Rev. A 74, 032313(2006). [9] K. Roszak and P. Machnikowski, Phys. Rev. A 73, 022313 (2006). [10] F. M. Cucchietti, J. P. Paz, and W. H. Zurek, Phys. Rev. A 72, 052113 (2005). [11] H. T. Quan, Z.Song, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. 96, 140604 (2006). [12] Fernando Martin Cucchietti, Sonia Fernandez Vidal and Juan Pablo Paz, Phys. Rev. A 75, 032337 (2007). [13] R. F. Werner, Phys. Rev. A 40, 4277 (1989). [14] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [15] A. Peres, Phys. Rev. Lett. 77, 1413 (1996); M. Horodecki, P. Horodecki and R. Horodecki, Phys. Lett. A 223 1 (1996). [16] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002). [17] S. Sachdev, Quantum Phase Transition (Cambridge Uni- versity Press, Cambridge England, 1999) [18] Y. D. Wang, Fei Xue and C. P. Sun, quant-ph/0603014. [19] Clive Emary and Tobias Brandes, Phys. Rev. Lett. 90, 044101 (2003). [20] Joseph Emerson, Yaakov S.Weinstein, Seth Lloyd and D.G. Cory, Phys. Rev. Lett. 89, 284102 (2002). [21] Kazuki Koshino and AkiraShimizu, Physics Reports 412 (2005) 191C275. [22] J. Schliemann, Phys. Rev. A 68, 012309 (2003). [23] X. Wang, H. B. Li, Z. Sun and Y. Q. Li, J. Phys. A: Math. Gen. 38 8703 (2005). [24] Z. Sun, X. Wang and Y. Q. Li, New J. Phys. 7, 83 (2005). [25] Neill Lambert, Clive Emary and Tobias Brandes, Phys. Rev. Lett. 92, 073602 (2004). [26] H. Fujisaki, T. Miyadera and A. Tanaka, Phys. Rev. E 67, 066201 (2003). http://arxiv.org/abs/quant-ph/0603014

0704.1173

The Deviation of the Vacuum Refractive Index Induced by a Static Gravitational Field

The Deviation of the Vacuum Refractive Index Induced by a Static Gravitational Field Xing-Hao Ye, Qiang Lin∗ Department of Physics, Zhejiang University, Hangzhou 310027, China (Dated: October 29, 2018) We analyzed the influence of static gravitational field on the vacuum and proposed the concept of inhomoge- neous vacuum. According to the observational result of the light deflection in solar gravitational field as well as the corresponding Fermat’s principle in the general relativity, we derived an analytical expression of the refrac- tive index of vacuum in a static gravitational field. We found that the deviation of the vacuum refractive index is composed of two parts: one is caused by the time dilation effect, the other is caused by the length contraction effect. As an application, we simulated the effect of the gravitational lensing through computer programming and found that the missing central imaging could be interpreted in a reasonable way. PACS numbers: 42.25.Bs, 42.50.Lc, 04. I. INTRODUCTION Vacuum is usually considered as “homogeneous” and “isotropic”, i.e., vacuum does not differ from place to place, and the refractive index of vacuum is always equal to 1. However, the recent theoretical and experimental progresses demonstrate that such concept of vacuum turns out to be in- appropriate when there are matters or fields within finite dis- tance. For example, the vacuum inside a microcavity is modi- fied due to the existence of the cavity mirrors, which will alter the zero-point energy inside the cavity and cause an attrac- tive force between the two mirrors known as Casimir effect [1, 2], which has been verified experimentally [3, 4]. A sec- ond example is that, under the influence of electromagnetic field, vacuum can be polarized, which has led to astonishingly precise agreement between predicted and observed values of the electron magnetic moment and Lamb shift, and may in- fluence the motion of photons [5]. Dupays et al. [6] studied the propagation of light in the neighborhood of magnetized neutron stars. They pointed out that the light emitted by back- ground astronomical objects will be deviated due to the optical properties of quantum vacuum in the presence of a magnetic field. Also in [7], Rikken and Rizzo considered the anisotropy of the optical properties of the vacuum when a static mag- netic field B0 and a static electric field E0 are simultaneously applied perpendicular to the direction of light propagation. They predicted that magnetoelectric birefringence will occur in vacuum under such conditions. They also demonstrated that the propagation of light in vacuum becomes anisotropic with the anisotropy in the refractive index being proportional to B0 × E0. The facts that the propagation of light in vacuum can be modified by applying electromagnetic fields to the vacuum implies that the vacuum is actually a special kind of opti- cal medium [5, 6]. This is similar to the Kerr electro-optic effect and the Faraday magneto-optic effect in nonlinear di- electric medium. This similarity between the vacuum and the dielectric medium implies that vacuum must also have its in- ner structure, which could be influenced by matter or fields as ∗Electronic address:[email protected] well. Actually, the structure of quantum vacuum has already been investigated in quite a number of papers [8, 9, 10]. In this paper, with the analysis of the influence of static gravitational field on the vacuum, we put forward a new con- cept that the curved spacetime around a certain matter can be treated as an optical medium with a graded refractive index. We suggest that the so-called curved spacetime is a reflec- tion of the vacuum inhomogeneity caused by the influence of gravitational matter. Based on this idea, the refractive index of vacuum is derived. We will also apply this concept to un- puzzle the problem of the central image missing in almost all the observed cases of gravitational lensing [11]. II. THE DEVIATION OF THE VACUUM REFRACTIVE INDEX According to the astronomical observation, the light prop- agating through a space with a celestial body nearby will be deflected. It can be interpreted with the curved spacetime in general relativity. As a matter of fact, it can also be inter- preted with the assumption that the vacuum around matter is inhomogeneous with refractive index deviated from 1. Here we put forward a theoretical model to describe the refractive index profile based on the Fermat’s principle for the propaga- tion of light in a static gravitational field, which was given by Landau and Lifshitz [12]: −1/2dl = 0, (1) where dl is the local length element passed by light and mea- sured by the observer at position r in the gravitational field, r is the distance from this element of light to the center of grav- itational matter M, g00 is a component of the metric tensor gµν, g 00 dl corresponds to an element of optical path length. = dt/dτ, where dτ represents the time interval mea- sured by the local observer for a light ray passing through the length dl, while dt is the corresponding time measured by the http://arxiv.org/abs/0704.1173v1 mailto:[email protected] observer at infinity. Eq.(1) could then be rewritten as −1/2dl = δ nds = 0, (2) where ds is the length element measured by the observer at infinity, corresponding to the local length dl. Eq.(2) shows that if we set the scale of length and time at infinity as a standard scale for the whole gravitational space and time, the propagation of light then satisfies the standard representation of Fermat’s principle, with the space — actu- ally the vacuum — possessing a refractive index given by = n1n2. (3) The factor n1 of the refractive index relating to the time transformation effect dt/dτ can be derived from the Newto- nian attraction, which contributes partially to the deflection of light. Considering a photon of relativistic mass m∞ at the infinity moving down to position r, the work done on to the photon by the Newtonian gravity is dr = d(mc2), (4) where G is the gravitational constant, c is the velocity of light, M is the mass of a star (say the Sun), r is the distance to the center of the star. Integrating Eq.(4) gives mr = m∞e rc2 , (5) where mr is the relativistic mass of the photon at position r. Since the photon energy is E = hν = mc2, where h is the Planck constant, ν is the photon frequency, then we have m = hν/c2. Substituting it into Eq.(5) gives νr = ν∞e rc2 . (6) It is just the frequency shift caused by the gravitational force, which reflects that a clock in a gravitational field runs slower than that far away from the gravitational center. That is dτ = e− rc2 dt, (7) where dτ denotes the time measured by a clock at position r, dt is the converted time of dτ, i.e., the time measured by the clock at infinity. This relation indicates that, if the length scale is the same, i.e., dl = ds, when an observer at position r reports a light velocity c1 = dl/dτ, it should be converted by the observer at infinity into c′1 = rc2 dτ = c1e rc2 . (8) FIG. 1: Light deflection caused by a graded refractive index. This change of light velocity will certainly bring a deflection to the light propagation. The corresponding refractive index rc2 . (9) Let us now consider the deflection angle caused by this graded refractive index. In Fig.1, the curve AP represents the light ray, β is the angle between the position vector r and the tangent at the point P on the ray, ϕ is the deflection angle of light. Since the refractive index shown in Eq.(9) has spheri- cal symmetry, i.e., depends only on the distance r for a given mass M, according to the Fermat’s principle nds = 0, (10) where ds = dr 1 + (rα̇)2, α̇ = dα/dr, n = n(r), we have the corresponding Lagrangian function L(α, α̇; r) = n(r) 1 + (rα̇)2. (11) Using the Lagrangian equation = 0, (12) we get [13] nr sin β = constant, (13) nr sin β = n0r0, (14) where r0 and n0 represent the radius and refractive index at the nearest point A respectively. Since tan β = , (15) associating with Eq.(14) reaches ( nrn0r0 ) 2 − 1 . (16) FIG. 2: Light deflection in solar gravitational field. For a light ray passing by the Sun as shown in Fig.2, the total angular displacement of the radius vector r reads ∆α = 2 ( nrn0r0 ) 2 − 1 , (17) where r0 represents the nearest distance to the center of the Sun. Because the gravitational field of the Sun is a weak field, the value of GM/rc2 is quite small, so substituting Eq.(9) into Eq.(17) gives a solution of first order approximation ∆α = π + . (18) Then the total deflection angle of light caused by the refrac- tive index n1 in solar gravitational field is ∆ϕ1 = ∆α − π = . (19) In fact, this result was obtained early in 1911 by Einstein [14], who also investigated the effect of red shift and the cor- responding slowing down of the light velocity in gravitational field and then figured out the light deflection as shown in Eq.(19) with the use of Huygens’ principle. Since the ac- tual total deflection angle of light propagation calculated by the general relativity [15, 16] and measured by the astronomi- cal observation [17] is twice that value, we then come to know that the length transformation effect dl/ds in Eq.(2) must have the same relation as that of the time transformation effect dt/dτ expressed in Eq.(9), namely rc2 . (20) This relation indicates that a ruler in a gravitational field is shorter than that far away from the gravitational center. So when an observer at position r reports a length dl, it should be converted by the observer at infinity into ds = e− rc2 dl. (21) For a light passing through a length dl, if the time scale is the same, i.e., dτ = dt , the light velocity c2 = dl/dτ reported by the observer at position r should then be converted by the observer at infinity into c′2 = rc2 dl = c2e rc2 . (22) FIG. 3: The dependence of the vacuum refractive index n on the distance r. This change of light velocity will also bring a deflection to the light propagation. The refractive index corresponding to this kind of deflection is rc2 , (23) which also causes a deflection angle of light ∆ϕ2 = . (24) Therefore, the total deflection angle of light in solar gravita- tional field is ∆ϕ = ∆ϕ1 + ∆ϕ2 = . (25) The above result shows that, if the two refraction effects are considered simultaneously, then the gravitational space — ac- tually the vacuum in the gravitational field — can be regarded as an optical medium with a total refractive index given by n = n1n2 = e rc2 . (26) n is composed of two factors: n1 — related with the time trans- formation or “curved time”; n2 — related with the space trans- formation or “curved space”. So the curved spacetime of gen- eral relativity is reflected in the synthesized refractive index n, which is also a reflection of the inhomogeneity of the vac- uum, showing that the vacuum near the matter is influenced more than that far away from the matter. The above expression of n shows that the refractive index of the vacuum at the infinity from the gravitational matter is 1, i.e., the usual refractive index of vacuum. The closer of the position to the center of matter M, the higher the refractive index of the vacuum. The relation between n and r is depicted in Fig.3, where 2GM/c2 is taken as the unit of r. For exam- ples, the corresponding radii for the surface of the Sun in solar gravitational field and the surface of the Earth in earth gravita- tional field are 2.36× 105 and 7.20× 108 respectively — both are far beyond the r-axis illustrated in Fig.3. The deviation of the vacuum refractive index from the usual value 1 is given by ∆n = n − 1 = e rc2 − 1. (27) FIG. 4: The two images I1, I2 of a gravitational lens. In weak field it becomes . (28) In order to provide the readers with a quantity impression, let us give two examples. For the solar gravitational field (M = 1.99 × 1030kg), the deviation of n on the surface of the Sun (r = 6.96 × 108m) is 4.24 × 10−6. For the earth grav- itational field (M = 5.98 × 1024kg), the deviation of n on the surface of the Earth (r = 6.38 × 106m) is only 1.39 × 10−9, which is so small that it can hardly be observed in usual ex- periments. Nevertheless, for a massive celestial body such as a heavy star, a galaxy or a cluster of galaxies, the deviation is not only observable, but also important and useful in gravita- tional astronomy. III. APPLICATIONS The deflection of light by massive bodies leads to the effect of gravitational lensing. Formerly, this effect should be calcu- lated complicatedly with the general relativity [18]. Once we have introduced the concept of graded vacuum refractive in- dex and obtained its relation with mass M and position r, the problem of gravitational lensing could then be treated easily with the conventional optical method. Considering a source S and a lens L of mass M, the light emitted from S is bent due to the gravitational field of the lens. The bent light could be figured out through Eq.(13) and Eq.(26). Drawing the extension line of the light from the ob- server O, the apparent (observed) position of the source image I could then be found out. The result is shown in Fig.4. This method could also be applied in studying the central imaging. In doing this, the vacuum refractive index profile inside the lensing body should be considered as well. Noticing that Eq.(26) could be virtually rewritten as n = e− c2 , (29) where Pr represents the gravitational potential at position r from the center of the lens. As a model for discussion, we suppose a lens (for example, a galaxy or a cluster of galaxies) of radius R with a density FIG. 5: A ray tracing result for the central imaging. distribution ρ = ρ0[1 − ( )k], (30) where ρ0 is the central density of the lens, 0 6 r 6 R, k > 0. The density ρ decreases with the distance r from the center of mass; the decreasing varies with the parameter k. This model gives the distribution of gravitational potential as Po = −4πρ0G 3(3 + k) Pi = −4πρ0G{ 2(2 + k) R2 − [ (2 + k)(3 + k) )k]r2},(31) for outside (r > R0) and inside (r 6 R0) the gravitational lens respectively. The vacuum refractive index profile outside and inside the gravitational lens then reads no = exp 8πρ0G 3(3 + k) ni = exp 8πρ0G 2(2 + k) R2 − [ (2 + k)(3 + k) )k]r2} .(32) Fig.5 shows a ray tracing result for the imaging of a gravita- tional lens with the above described vacuum refractive index profile. In the figure, only three paths of ray (the three thick lines) could pass through the observer O, forming the upper, lower and central images respectively. From the figure, we find that, under the same conditions, the larger the distance OL from the observer to the lens, the closer the central imag- ing light to the center of the lens. If the source S and the observer O are counterchanged, it could also be known from the figure that, the larger the distance S L from the source to the lens, the closer the central imaging light to the center of the lens. In addition, through the change of the lens mass M = 43πR 3ρ0k/(3 + k), we also find that, when the mass M increases, the distance from the central imaging light to the center of the lens decreases (Fig.6, where the mass ratio of the lenses corresponding to the four central imaging rays from bottom to top is 2 : 3 : 4 : 5 ). For the actual condition of gravitational imaging, the dis- tances OL, S L and the mass M are all astronomical figures; FIG. 6: Tracing the central imaging rays for lenses of different mass. therefore, the light of central imaging is extremely close to the center of the lens. However, for a lensing body with a density increasing towards the center, it is possible that there are barrier matters near the center which will destroy the for- mation of the central image. Besides, the relatively longer inner path of the central imaging light adds the possibility of light being held back by the lens matters on the way. These and some other factors such as the relative faintness of the central imaging light and the possibly higher brightness of the lens core itself, all decrease the possibility of central imaging being actually observed. This analysis is firmly supported by the fact that the number of observed images is not “odd” as expected by the existed theories but “even” in almost all cases of gravitational lensing [11]. IV. CONCLUSIONS We have proposed the concept of inhomogeneous vacuum with graded refractive index based on the analysis of the influ- ence of static gravitational field on the vacuum. we derived the expression of this refractive index analytically. By using this expression, we investigated the effect of gravitational lensing in a conventional optical way and provided a reasonable inter- pretation for the problem of central image missing. The result indicates that, the concept of inhomogeneous vacuum is mathematically equivalent to the curved spacetime in the general relativity; therefore, an effective and convenient alternative method (i.e., optical method) could be established to solve the so complicated problems in gravitational astron- omy. Physically, under such point of view, the motion of light in gravitational space is a motion of light wave in a quantum vacuum with graded refractive index. And as we know that, in conventional optics, the Fermat’s principle says that the op- tical path between two given points is an extremum. This is also equivalent to the theorem in the general relativity that a particle always moves along a geodesic line in a curved space- time. Acknowledgments We wish to acknowledge the supports from the Na- tional Key Project for Fundamental Research (grant no. 2006CB921403), the National Hi-tech project (grant no. 2006 AA06A204) and the Zhejiang Provincial Qian-Jiang-Ren-Cai Project of China (grant no. 2006R10025). [1] H. Gies and K. Klingmüller, Phys. Rev. Lett. 96, 220401 (2006). [2] T. Emig et al., Phys. Rev. Lett. 96, 080403 (2006). [3] S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997). [4] H. B. Chan et al., Science 291, 1941 (2001). [5] N. Ahmadi and M. Nouri-Zonoz, Phys. Rev. D. 74, 044034 (2006). [6] A. Dupays et al., Phys. Rev. Lett. 94, 161101 (2005). [7] G. L. J. A. Rikken and C. Rizzo, Phys. Rev. A. 63, 012107 (2000) ; 67, 015801 (2003). [8] A. Armoni, A. Gorsky, and M. Shifman, Phys. Rev. D. 72, 105001 (2005). [9] A. Barroso et al., Phys. Rev. D. 74, 085016 (2006) . [10] K. R. Dienes, E. Dudas, and T. Gherghetta, Phys. Rev. D. 72, 026005 (2005). [11] J. N. Winn, D. Rusin, and C. S. Kochanek, Nature 427, 613 (2004). [12] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon Press, New York, 1975). [13] M. Born and E. Wolf, Principles of Optics (7th edition) (Cam- bridge University Press, Cambridge, 1999). [14] H. A. Lorentz, A. Einstein, and H. Minkowski, The Principle of Relativity (Methuen, London, 1923). [15] H. C. Ohanian, Gravitation and Spacetime (W. W. Norton and Company Inc., New York, 1976). [16] S. Weinberg, Gravitation and Cosmology (John Wiley and Sons, New York, 1972). [17] E. B. Fomaleont and R. A. Sramek, Phys. Rev. Lett. 36, 1475 (1976). [18] S. Mollerach and E. Roulet, Gravitational Lensing and Mi- crolensing (World Scientific, New Jersey, 2002).

0704.1174

Deaconstructing Functions on Quadratic Surfaces into Multipoles

Deconstructing Functions on Quadratic Surfaces into Multipoles Gabriel Katz August 4, 2021 Abstract Any homogeneous polynomial P (x, y, z) of degree d, being restricted to a unit sphere S2, admits essentially a unique representation of the form λ + Lkj ], where Lkj ’s are linear forms in x, y and z and λ is a real number. The coefficients of these linear forms, viewed as 3D vectors, are called multipole vectors of P . In this paper we consider similar multipole representations of polynomial and analytic functions on other quadratic surfaces Q(x, y, z) = c, real and complex. Over the complex numbers, the above representation is not unique, although the ambiguity is essentially finite. We investigate the combinatorics that depicts this ambiguity. We link these results with some classical theorems of harmonic analysis, theorems that describe decompositions of functions into sums of spherical harmonics. We extend these classical theo- rems (which rely on our understanding of the Laplace operator ∆S2) to more general differential operators ∆Q that are constructed with the help of the quadratic form Q(x, y, z). Then we in- troduce modular spaces of multipoles. We study their intricate geometry and topology using methods of algebraic geometry and singularity theory. The multipole spaces are ramified over vector or projective spaces, and the compliments to the ramification sets give rise to a rich family of K(π, 1)-spaces, where π runs over a variety of modified braid groups. 1 Introduction This paper aims to generalize and extend some results in [KW] and [W] about deconstruction of cosmic microwave background radiation (CMBR) into multipole vectors and explain these results to a mathematically inclined audience. Our exposition, to a degree, is self-sufficient. It resembles in spirit to the paper of V. Arnold [A] which contains a nice treatment of similar structures (in a setting that is a special case of ours). In [S], [S1] Sylvester discovered the following basic fact1: Theorem 1 Every real homogeneous polynomial P of degree d in x, y, and z may be written as P (x, y, z) = (a1x+ b1y + c1z)(a2x+ b2y + c2z) ... (adx+ bdy + cdz) + + (x2 + y2 + z2) ·R(x, y, z), (1) 1rediscovered in [WK] by the authors http://arxiv.org/abs/0704.1174v1 where the remainder term R is a homogeneous polynomial of degree d − 2. The decomposition is unique up to reordering and rescaling of the linear factors in the product; in other words, R(x, y, z) is unique. For harmonic homogeneous polynomials P , a somewhat similar representation (see formula (7)) was discovered by James Clerk Maxwell in his 1873 Treatise on Electricity and Magnetism [M]. Its relation to algebraic geometry in general, and to the Sylvester theorem, in particular, is well- explained by Courant and Hilbert in [CH] VII, Sec. 5. Sylvester’s Theorem 1—an application of Bézout’s theorem—has a pleasing corollary: Theorem 2 [KW ] When restricted to a unit sphere S2, every real polynomial P in x, y, and z of degree d can be written in the form P (x, y, z) = λ+ (a11x+ b11y + c11z) + (a21x+ b21y + c21z)(a22x+ b22y + c22z) + . . . . . . + (ad1x+ bd1y + cd1z)(ad2x+ bd2y + cd2z) . . . (addx+ bddy + cddz), (2) where λ and the ajk, bjk, cjk are real numbers. The decomposition is unique up to reordering and rescaling of the linear factors in each of the products. We denote by ∆ the Laplace operator on the euclidean 3-space and by ∆S2 the Laplace operator on the unit sphere. An eigenfunction of ∆S2 which has an eigenvalue −d(d + 1) is called the d-th spherical harmonic. In fact, such an eigenfunction is a restriction to S2 of a homogeneous harmonic polynomial in x, y, and z of degree d. It is well-known that spherical harmonics form a basis in the space L2(S 2) of L2-integrable functions on a unit sphere (see [CH]). As a result, an infinite decomposition is available for any function f ∈ L2(S 2) (cf. [L], formula (26)). Theorem 3 Any real function f ∈ L2(S 2) has a representation in the form f(x, y, z) = (akjx+ bkjy + ckjz) where the series converges in the L2-norm, and the mutually orthogonal polynomials in the figure brackets are among the d-th spherical harmonics. This representation is unique, up to reordering and rescaling of the linear factors in each of the products. In the space of real continuous functions f on S2 there is a dense (in the sup-norm) and O(3)- invariant subset F(S2) comprised of functions that admit a representation in the form of a series (adjx+ bdjy + cdjz) that converges uniformly on the sphere. Each function from F(S2) admits a canonical extension into the interior of the unit ball D3 where it produces a real analytic function. Theorem 3, claiming a form of stability of decomposition (2) under the polynomial approximations, is a useful tool for analyzing patterns in the spherical sky [W]. At the first glance, the sphere seems to occupy a special place in these results. However, what is really important, is the quadratic nature of the surface. This paper is concerned with similar decompositions of functions on other quadratic affine surfaces, real and complex. For instance, it is natural to wonder whether one can deconstruct in a similar fashion polynomial or real analytic functions on a hyperboloid. The answer is affirmative, but the uniqueness of the representation is lost. Evidently, polynomial deconstructions of type (2) can be described by listing the 3-vectors (ajk, bjk, cjk), where 1 ≤ j ≤ d and 1 ≤ k ≤ j. Following Maxwell, we call them multipole vectors. There is a sizable ambiguity in choosing the multipole vectors that represent a given polynomial function. Factoring out this ambiguity produces an object that we call a multipole or, more specifically, a d-pole. A major advantage of representing functions P (x, y, z) in terms of multipole vectors is the naturality with which the multipoles behave under linear transformations, in particular, under rotations. The focal point of this paper is to describe rich algebro-geometrical structures and topology of modular spaces of d-poles M(d) and its “semi-compactification” M(d) (see formulas-definitions (10), (11) and (13)). In the process, we bring under the same roof a variety of mathematical techniques and constructions that belong to the fields of algebraic geometry, singularity theory, algebraic topology, harmonic analysis, and the polynomial approximation theory. None of our techniques is very advanced, but their natural appearance within the context of studying quadratic surfaces is pleasing and somewhat surprising... The reader can follow two distinct treads in the core of the paper. The first tread winds through the results that require methods of algebraic geometry. These results are uniquely three-dimensional, that is, applicable only to quadratic surfaces. The second tread unifies results that are linear in nature and are based on the classical theory of harmonic analysis on quadratic surfaces. These results are valid for quadratic hypersurfaces in any dimension. Now, let us describe informally the main results of the paper and its structure. In the second section, we investigate a variety of special representations of complex polynomial functions on complex quadratic surfaces Sλ := {Q(x, y, z) = λ}, where Q is a given non-degenerated quadratic form and λ is a complex number. This section, the longest, contains all the core ideas and constructions of the paper. The main results of Section 2 are contained in Theorems 6, 7, 10, and 12. Theorem 7 describes the d-pole space M(d) as a principle C∗-bundle over the d-th symmetric power Symd(CP 2) of the complex projective space CP 2. Reconstructing a homogeneous polynomial function P on the cone S0 := {Q(x, y, z) = 0} from the appropriate multipole defines a regular map ΦQ from the modular space M(d) to the vector space V Q (d) — the d-graded portion of the regular ring C[x, y, z]/〈Q〉 of S0. Remarkably, this map is a surjection onto V Q (d) ◦ := V ⊥Q (d) \ {0} with a generic fiber of cardinality (2d−1)!! := 1·3·5·. . . ·(2d−1). In other words, for a generic homogeneous polynomial P of degree d, there are (2d− 1)!! Sylvester-type representations P = Q ·R+ j=1 Lj with distinct R’s (here Lj is a linear form in x, y, and z). According to Theorem 7, ΦQ is ramified over a subvariety D(ΦQ) ⊂ V Q (d) of codimension one. Points of D(ΦQ) represent polynomials P whose zero sets contain lines in S0 of multiplicity at least two. The complement V Q (d) ◦ \ D(ΦQ) is a K(π, 1)-space, where π is an extension of the modified braid group B 2d in 2d strings by the infinite cyclic group Z. The map ΦQ is C ∗-equivariant, which leads to a regular surjection ΨQ from M(d)/C∗ = Symd(CP 2) onto the projective space V ⊥Q (d) ◦/C∗ ≈ CP 2d. Theorem 6, similar to Theorem 7, describes the ramified geometry of ΨQ. The ramification set D(ΨQ) is a discriminant subvariety of CP 2d and its complement, again, is a K(π, 1)-space, where π is the braid group in 2d strings in a spherical shell. Thus, Theorems 6 and 7 describe the possibility and the ambiguities of multipole representations for homogeneous polynomial functions on the cone S0. We notice (see Corrolary 8) that the topology of the map ΦQ puts some breaks on the multipole representations of multidimensional families of functions on S0. Theorem 11 helps us to move from the multipole representations of homogeneous polynomials on the cone S0 towards the multipole representations of general polynomial functions on any affine quadratic surface Sλ: here one needs a sequence of d multipoles to capture a given function of degree d. Next, we describe the restriction of the map ΦQ to the modular space of multipoles that are confined to a given plane in C3 (see formula (27)). In fact, being restricted to the space of coplannar multipoles, ΦQ becomes a 1-to-1 map. This observation helps us to employ the quadratic form Q for constructing interesting examples of ramified maps of degree 2d from the space CP d to itself. Let us glance at these maps. The equation Q(x, y, z) = 0 defines an irreducible quadratic curve Q in the projective space CP 2∗ with projective coordinates [x : y : z]. Pick a point p ∈ CP ∗ \ Q. Then any q ∈ Q determines a line in CP 2∗ passing through p and q, and any effective divisor D of degree d on Q gives rise to a collection of lines through p and D, taken with the appropriate multiplicitiies. Such a collection can be regarded as an effective divisor of degree d on the dual space CP 1 ⊂ CP 2. The embedding CP 1 ⊂ CP 2 corresponds to the pencil of lines through p. In other words, the choice of p and Q produces a regular map ΓQ : Sym d(Q) → Symd(CP 1). As abstract varieties, Q and CP 1 are isomorphic. Hence, each quadratic form Q generates a regular map ΓQ from CP d ≈ Symd(CP 1) onto itself with a generic fiber of cardinality 2d. Theorem 10 describes and Figure 3 depicts a rather intricate geometry of the map ΓQ. This map is ramified over the variety D(ΓQ) — the union of the classical discriminat variety Dd ⊂ CP d with two hyperplanes that are tangent to it (they depend on Q). The complement CP d \ D(ΓQ) is, again, a K(π, 1)-space with π being the braid group in d strings in a cylindrical shell. This completes our algebro-geometric treatment of the modular spaces of complex multipoles. Classical harmonic analysis on spheres provides an alternative approach to function deconstructions on quadratic surfaces (see formulas (6) and (8)). We observe that these deconstructions, based on spherical harmonics, behave in a natural way under complex linear change of coordinates that preserve a given nondegenerated quadratic form Q(~v) = 〈~vB,~v〉. Such a change calls for replacing the Laplace operator ∆ with a second order differential operator ∆Q = [~∂]B −1[~∂]T , where ~∂ := (∂x, ∂y , ∂z). Under this transformation, the homogeneous polynomial solutions of the equation ∆Q(P ) = 0 take the role of spherical harmonics. In particular, homogeneous polynomial solutions of the wave equation (∂2x+∂ z )P (x, y, z) = 0 replace the spherical harmonics on the hyperboloid x2 + y2 − z2 = 1. Along these lines, our Theorem 12 is a straightforward reformulation of well- known results of classical harmonic analysis. One can view the generalized Maxwell Formula (37) as a bridge between methods of harmonic analysis (the second tread) and methods of algebraic geometry (the first tread). The third section deals with decompositions of real polynomial functions on real quadratic surfaces. In a sense, Section 3 is a “Z2-equivariant modification” of Section 2, where the cyclic group of order two acts on functions and surfaces by complex conjugation. Theorem 17 is the real version of Theorem 12 from Section 2. Theorem 14 is the real analog of theorem 7, and Theorem 13 of Theorem 11. As a byproduct of our analysis of modular spaces MR(d) of real multipoles, we prove that the manifolds Symd(RP 2) and RP 2d are diffeomorphic (see [A], Theorem 2, and our Corollary In the fourth section, we discuss a spectacular failure of the multipole representations for affine surfaces of degrees greater than two. Lemma 9 explains the nature of this failure: the set of completely factorable polynomials on non-quadratic affine surfaces is a subvariety in the space of all polynomial functions; in other words, for a form Q of degree at least three, a generic element of the ring C[x, y, z]/〈Q〉 is irreducible. The fifth section deals with infinite decompositions of functions on quadratic surfaces, both real and complex. The main issue is whether our multipole representations converge appropriately. Theorems 19 and 21 capture the main results of this section. Theorem 22 is a real analog of Theorem 21 and a generalization of Theorem 3 from the Introduction. Now a few words about the motivation for this paper that lies outside mathematics. Cosmologists who study the Cosmic Microwave Background Radiation (CMBR) routinely decompose it into a sum of spherical harmonics. This decomposition helps to analyze the correlations between the magnitude and geometry of various harmonics. There is a hope that the lower harmonics can capture some global properties of the Universe. The ultimate goal of these investigations is to understand the geometry and topology of the visible universe and the physics of the Big Bang. The data obtained byWilkinson Microwave Anisotropy Probe (WMAP) reveal puzzling correlations between the low d-portion of the harmonic decomposition of the CMBR [B], [CHS], [TOH], [EHGL]. In the case of quadrupole-octopole (d = 2 and 3) alignment, simulations based on our fast algorithm (which employes Theorem 2) show the alignment to be unusual at the 98.7 percent level (see [KW], [W]). No explanation is yet known for these strange results. 2 Multipoles and polynomials on complex quadratic surfaces We denote by C[x, y, z] the ring of complex polynomials in the variables x, y, and z. Let Q(x, y, z) be an irreducible quadratic form over the complex numbers C. From a strictly algebraic perspective, one can interpret this section as describing the ways in which the quotient ring C[x, y, z]/〈Q − λ〉, λ ∈ C, fails to be a Unique Factorization Domain. However, the flavor of our approach to this problem is more geometrical and combinatorial. The reader will be well-adviced to ignore for a while the asterisks in our notations: they are there to distinguish between vector spaces and their duals. Let S = {Q(x, y, z) = 1} be a complex algebraic surface in C3∗. At the same time, {Q(x, y, z) = 0} gives rise to a complex projective curve Q ⊂ CP 2∗ , where CP ∗ stands for the projectivization of the 3-space with coordinates x, y, and z. As in Theorem 2, we aim to decompose any polynomial P (x, y, z) restricted to S as an “economic” sum of products of homogeneous linear polynomials. First, consider the case of an homogeneous P and the corresponding complex projective quadratic curve P in CP 2∗ it generates. Denote by Z(P,Q) the intersection P ∩ Q. We assume that P and Q do not share a common component, so that Z(P,Q) is a finite set. When Z(P,Q) is a complete intersection, it consists of exactly 2d points, where d = deg(P ). Definition 1 Let Q(x, y, z) be an homogeneous irreducible quadratic polynomial, and P (x, y, z) an homogeneous polynomial of degree d. Assume that Z(P,Q) is a complete intersection. A parcelling of the set Z(P,Q) is comprized of a number of subsets {Zν ⊂ Z(P,Q)}ν so that: • Z(P,Q) = • Zν ∩ Zν′ = ∅, provided ν 6= ν • the cardinality of each subset Zν is equal to 2. When Q is irreducible and P is not divisible by Q, the intersection set Z(P,Q) is equipped with a function µ which assigns to each point p ∈ Z(P,Q) its multiplicity µ(p). We denote by Z+ the set of non-negative integers and by N the set of positive integers. Definition 2 Let Z be a finite set equipped with a multiplicity function µ : Z → N whose l1-norm ‖µ‖1 is 2d. A generalized parcelling of (Z, µ) is a collection of functions µν : Z → {0, 1, 2}, such ν µν = µ • ‖µν‖1 = 2 Of course, each parcelling of Z(P,Q) is also a generalized parcelling, where the roles of the functions µν are played by the characteristic functions of the parcels Zν . In the case of Q = x2 + y2 + z2, the following lemma is due to Sylvester [S], [S1]. Lemma 1 Let Q(x, y, z) be an irreducible homogeneous quadratic polynomial and let P (x, y, z) be a homogeneous polynomial of degree d which is not divisible by Q. Consider a generalized parcelling ν µν of the multiplicity function µ : Z(P,Q) → N. Then the homogeneous polynomial P (x, y, z) admits a representation of the form Q(x, y, z) · R(x, y, z) + Lν(x, y, z), (5) where Lν denotes a linear homogeneous polynomial that vanishes at each point p ∈ Z(P,Q) ⊂ Q with the multiplicity µν(p). Proof There exist a linear polynomial Lν and a corresponding line Lν ⊂ CP ∗ such that the multiplicity of Lν ∩ Q at each point p ∈ Z(P,Lν) equals µν(p). When µν(p) = 2, the line must be tangent to Q at p. When the support of µν—the parcel Zν ⊂ Z(P,Q))—is comprised of two points, the line Lν is chosen to contain Zν . Let us compare the restrictions of P and ν Lν to the curve Q. Both polynomials are of the same degree d. Moreover, the curve L := { ν Lν = 0} intersects with Q at the points of Z(P,Q), where it realizes the multiplicities prescribed by the function µ. It follows that the restrictions of P and ν Lν to Q are proportional, that is, for an appropriate choice of scalar λ, P |Q = λ · ν Lν |Q. Just take λ = P (q)/ ν Lν(q) where q ∈ Q \ Z(P,Q). With this choice, the curves {P − λ ν Lν = 0} and Q intersect so that the total multiplicity of the intersection is at least 2d+ 1. By the Bézout theorem, this is possible only when Z(P − λ ν Lν) ⊃ Q. Employing the irreducibility of Q, we get that P − λ · ν Lν must be divisible by Q. This completes the proof. � Corollary 4 Any effective divisor D on the complex quadratic surface S = {Q(x, y, z) = 0} that is the zero set of a homogeneous polynomial P (x, y, z) can be represented as a sum of lines Dν. � In general, the representation (5) of P depends on a generalized parcelling, and thus, is not unique. We denote by V (d) the complex vector space of homogeneous polynomials in x, y, and z of degree d. Its dimension is (d2 + 3d + 2)/2. Consider a vector subspace VQ(d) ⊂ V (d) comprised of polynomials divisible by Q. There is a canonical imbedding βQ : V (d − 2) → V (d) whose image is VQ(d). It is produced by multiplying polynomials of degree d − 2 by Q. Thus, any space V (d) is equipped with a natural filtration FQ(d) = {VQk(d)}k by subspaces of polynomials divisible by various powers {Qk} of Q. We equip each V (d) with an Hermitian inner product. For various d’s, we insist that these inner products are synchronized by the requirement: all the imbeddings βQ must be isometries. Denote by V ⊥Q (d) the subspace of V (d) orthogonal to VQ(d). It can be identified with the quotient space V (d)/VQ(d)—the d-graded part of the algebra of regular functions on the complex cone {Q(x, y, z) = 0}. The dimension of V ⊥Q (d) is equal to (2d+ 1). For a real form Q, similar spaces based on real homogeneous polynomials of a degree d make sense. We denote them V (d;R), VQ(d;R), and V Q (d;R). When Q = x2 + y2 + z2, there is a classical interpretation for the quotient V (d;R)/VQ(d;R). It can be identified with the set of harmonic homogeneous polynomials of degree d (cf. [CH] and [M]). In other words, the kernel Har(d;R) of the Laplace operator ∆ : V (d;R) → V (d − 2;R) is complementary to VQ(d;R) in V (d;R). This leads to a decomposition similar to the one in (5) (cf. [Sh, Theorem 22.2], [L], and [W]) : V (d;R) ≈ Har(d;R)⊕ VQ(d;R) (6) The direct summands in (6) are orthogonal with respect to the inner product in V (d;R) defined by the formula 〈f, g〉 = f · g dm. The measure dm on the sphere S2 is the standard one. Moreover, according to Maxwell, any polynomial P ∈ Har(d;R) admits a beautiful representation of the form P (x, y, z) = r2d+1 · ∇ , (7) where r = (x2 + y2 + z2)1/2, {vj ∈ R 3} are some vectors,2 and ∇ stands for the directional derivative operator. Physics behind Maxwell’s representation is quite transparent: 1/r is a potential of a single electrical charge, ∇ (1/r) is a potential of a “virtual” (that is, very small) dipole formed by two opposite charges, ∇ (1/r) is a potential of a “virtual” quadropole—a close pair of v1-oriented dipoles merging along the direction of v2—, ∇v3∇v2∇v1(1/r) is a potential of a “virtual” octopole, and so on... Note that, the Laplace operator ∆ = ∂2x + ∂ y + ∂ z can act on complex polynomials in complex variables x, y, and z as well. Both over real and complex numbers, its kernel is described by the same system of linear equations with real coefficients imposed on the coefficients of polynomials of a given degree. Therefore, if ∆(P ) = 0 for some complex polynomial P , then ∆(PR) = 0 and ∆(P I) = 0. Here PR is defined by the formula α,β,γ Re(aα,β,γ)x αyβzγ and P I by the formula α,β,γ Im(aα,β,γ)x αyβzγ , where P = α,β,γ aα,β,γ x αyβzγ . For any invertible complex (3×3)-matrix A = (ajk), consider a symmetric matrix B = (bjk) = A·A and the corresponding quadratic form Q(v) = 〈vB, v〉, where v = (x, y, z), 〈∼,∼〉 stands for the inner product xx′ + yy′ + zz′ in C3, and the upper script T denotes the matrix transposition. Employing Q, we can form a second order differential operator ∆Q = 1≤i,j≤3 b jk∂j∂k acting on holomorphic functions f in the complex variables {x1 = x, x2 = y, x3 = z}. Here {b ij} denote elements of the inverse matrix B−1. Formally, ∆Q = [~∂] · B −1 · [~∂]T where [~∂] := (∂1, ∂2, ∂3) := (∂x, ∂y , ∂z). Given a non-degenerated quadratic form Q, there is a change of complex coordinates (x′, y′, z′) = (x, y, z) ·A that reduces it to the canonical form Q′ = x′2+ y′2+ z′2. Consider the complex Laplace operator ∆′ = ∂2x′ +∂ y′ +∂ z′ in the new coordinates (x ′, y′, z′). Then, for any holomorphic function f(x′, y′, z′), we have [∆′f ]((x, y, z) ·A) = ∆Q[f((x, y, z) ·A)]. Thus, there is a 1-to-1 correspondence between harmonic homogeneous polynomials f(x′, y′, z′) of degree d and degree d homogeneous polynomial solutions g(x, y, z) of the equation ∆Q(g(x, y, z)) = 0. Let us examine the intersection Ker(∆′) ∩ VQ′(d). For any homogeneous polynomial T of degree d− 2, we get ∆′(Q′ ·T ) = Q′ ·∆′(T )+∆′(Q′) ·T +2〈∇T,∇Q′〉 = Q′ ·∆′(T )+ (4d− 6)T . Therefore, if Q′ · T ∈ Ker(∆′), then T must be divisible by Q′ (note that ∆′(T ) = 0 implies T = 0). Put T = Q′ · T1 and κ(d) = 4d+ 2. The equation 0 = ∆ ′(Q′ · T ) = Q′ ·∆′(Q′ · T1) + κ(d− 2) ·Q ′ · T1 is equivalent to the equation 0 = ∆′(Q′ ·T1)+κ(d−2)·T1 = Q ′ ·∆′(T1)+[κ(d−2)+κ(d−4)]·T1 . Again, it follows that T1 must be divisible by Q ′. Continuing inductively this kind of reasoning, we see that Ker(∆′) ∩ VQ′(d) = {0}. On the other hand, one can verify that dim[Ker(∆ ′)] + dim[VQ′(d)] = dim[V (d)]. Thus, V (d) = Ker(∆′)⊕ VQ′(d). 2Later, we call them ”the leading multipole vectors” of P . A polynomial P (x′, y′, z′) is divisible by x′2 + y′2 + z′2, if and only if, P ((x, y, z) ·A) is divisible by Q(x, y, z). Therefore, V (d) = Ker(∆Q)⊕ VQ(d) as well. Let OQ(3;C) denote a subgroup of the general linear group GL(3;C) that preserves the quadratic form Q. A matrix U ∈ OQ(3;C) if and only if UBU T = B. The natural OQ(3;C)-action on the space of x, y, and z-variables induces an action on the polynomial space V (d). Evidently, VQ(d) is invariant under this action. On the other hand, for any polynomial P (x, y, z), ∆Q[P ((x, y, z) · U)] = [∆Q̃P ]((x, y, z) · U)), where the operator ∆ := [~∂] · UTB−1U · [~∂]T . Since UBUT = B, we get (U−1)TB−1U−1 = B−1. By a simple algebraic trick, it follows that UTB−1U = B−1. As a result, both the quadratic form Q and the kernel Ker(∆Q) are invariant under the OQ(3;C)-action. Consider Maxwell’s representation (7) of a real homogeneous harmonic polynomial P (x′, y′, z′) of degree d. It gives rise to a map Ξ that takes the sets of real vectors v1,v2, . . . ,vd to elements of Har(d,R). The map Ξ is evidently linear in each of the vj ’s, and thus it is a real polynomial map with a vector space of real dimension 2d + 1 for its target. By [CH], [L] and [W], Ξ is onto the vector space Har(d,R). Therefore, the complexification ΞC of Ξ must be also onto the vector space Har(d,C) ⊃ Har(d,R). Indeed, the image of ΞC in Har(d,C) is a complex algebraic set containing a totally real vector subspace Har(d,R) of dimension dimC(Har(d,C)). In other words, formula (7) must be valid for any complex homogeneous harmonic polynomial P (x′, y′, z′) of degree d and appropriate complex vectors u1,u2, . . . ,ud ∈ C 3. In fact, each vector uj = vj ·A Formula (7) describes any homogeneous complex polynomial solution of the equation ∆Q(f) = 0 in the (x′, y′, z′) coordinates. Translating them back to the (x, y, z)-coordinates with the help of the identity [∇ ′f ′](x ·A) = ∇ ′·A−1 [f ′(x ·A)] leads to a formula (9) below. Thus, we have established the following proposition: Lemma 2 The decomposition V (d) = Ker(∆Q)⊕ VQ(d) (8) holds for any non-degenerated complex quadratic form Q. It is invariant under the natural OQ(3;C)- action on V (d). Moreover, for any P ∈ Ker(∆Q) of degree d, there exist vectors {uk ∈ C 3}1≤k≤d so that the generalized Maxwell formula P (x, y, z) = Q(x, y, z)d+ 2 · ∇ . . .∇ Q(x, y, z)− is valid. In particular, with Q = x2 + y2 − z2, any degree d homogeneous complex polynomial solution P of the wave equation (∂2x + ∂ y)P = ∂ zP is given by the formula (9). � Formula (9) might pose a slight challenge: after all, square roots are multivalued analytic functions. However, the ±-ambiguities in picking a single-valued branch cancel each other. We shall see that, even up to reordering and rescaling of the multipole vectors {uk}, the complex Maxwell representation (9) of a given “Q-harmonic” P is not unique. Remark. In fact, by a similar argument, the decomposition (8) is available for homogeneous polynomials in any number of variables and for any non-degenerated quadratic form Q (see [Sh], Theorem 22.2, for the proof). At the same time, the representation (9) seems to be a 3-dimensional phenomenon. It looks like that not any Q-harmonic polynomial in n > 3 variables can be expressed in terms of d directional derivatives sequentially applied to the Q-harmonic potential Q1−n/2. This seems to be related to the failure of the Sylvester-type formula (5) for n > 3. When I raised this issue with Michael Shubin, he proposed a nice conjecture in the flavor of Maxwell’s representation (although, not a direct generalization of it). In the Maxwell representation, one employs a differ- ential operator which is a monomial in d directional derivatives, while in the conjecture below one invokes a differential operator which is a polynomial in d directional derivatives. Shubin’s Conjecture (An n-dimensional variation on the theme of Maxwell’s representation). Let n > 2. For any real homogeneous and harmonic polynomial P of degree d in n variables {xj}, there exists a unique real homogeneous and harmonic polynomial P♣ also in n variables such that P (x1, . . . , xn) = r 2d+n−2 P♣(∂x1 , . . . , ∂xn) r where r = ( j=1 x 1/2, and the differential operator P♣(∂x1 , . . . , ∂xn) being applied to the poten- tial function r2−n. Moreover, the polynomials P and P♣ are proportional with the coefficient of proportionality depending only on d and n. � Given a vector space V , we denote by V ◦ the space V \ {0}. We consider the 3-space V (1) ≈ C3 of linear forms L = ax+ by + cz (which we identify with the space of their coefficients {(a, b, c)}) and the 3-space V∗(1) ≈ C ∗ with the coordinates x, y, z. This calls for a distinction in notations for their projectivizations: CP 2 and CP 2∗ . Hence, each point (a, b, c) ∈ V (1) ◦ determines a point l = [a : b : c] in CP 2 and a complex line L in CP 2∗— the zero set of the form L. Now, let us return to the decomposition (5). Each polynomial Lν from (5) can be viewed as a vector in V (1) = V ⊥Q (1). Therefore, a given homogeneous polynomial P of degree d which is not divisible by Q, together with the appropriate generalized parcelling, produces a collection of non- zero vectors {wν ∈ V (1)}ν 3. We shall call this unordered set of vectors {wν} the leading multipole vectors of P with respect to the corresponding generalized parcelling of µ : Z(P,Q) → Z+. Lemma 3 For a given generalized parcelling µ = ν µν of µ : Z(P,Q) → Z+, the subordinate leading multipole vectors {wν ∈ V (1)}ν are unique up to reordering and rescaling of the Lν’s (equivalently, the polynomial R in (5) is unique). Proof We use the same notations as in the proof of Lemma 1. By an argument as in that proof, any linear polynomial L′ν which defines a line L ν with the property L ν ∩ Q = Lν ∩ Q (the points of intersection have multiplicities prescribed by µν) is of the form λLν . Here λ ∈ C ∗ and Lν is a preferred polynomial corresponding to a vector wν ∈ V (1). Indeed, as in the proof of Lemma 1, for an appropriate λ, by the Bézout Theorem, λLν − L ν must be divisible by Q. � As a result, the multipole is well defined by the parcelling of µ modulo some rescaling. Clearly, one can always replace each Lν in (5) with λνLν as long as ν λν = 1. Next, we analyze the exact meaning of the “reordering and rescaling” ambiguity. 3These vectors are not necasarilly distinct. Let Hq be an abelian subgroup of (C ∗)q formed by vectors {λν}1≤ν≤q subject to the restriction ν λν = 1. Its rank is q − 1. Let Sq stand for the symmetric group in q symbols {1, 2, . . . , q}. We denote by Σq an extension 1 → Hq → Σq → Sq → 1 of Sq by Hq. This group is generated by the obvious actions of Sq and Hq on (C) We introduce an orbit-space M(k) := [(V (1)◦]k/Σk (10) whose points encode the products of linear forms as in the decomposition (5). Here the group Σk acts on the products of spaces by permuting them and by rescaling their vectors. Its subgroup Hk acts freely. Because of the uniqueness of the prime factorization in the polynomial ring C[x, y, z], the space M(k) provides us with a 1-to-1 parametrization of the space of homogeneous degree k polynomials that are products of linear forms. As an abstract space, (10) can be expressed as M(k) = [C3 ◦]k Σk (11) Definition 3 The elements of orbit-space M(k) will be called k-poles, or simply, multipoles. We introduce groups Γk in a manner similar to the introduction of the groups Σk. The group Γk is an extension of the permutation group Sk by the group (C ∗)k ⊃ Hk. Thus, Γk/Σk ≈ C Therefore, the space M(k) in (10), (11) fibers over the projective variety B(k) := ν=1 V (1) = SymkCP [V (1)] ≈ Symk(CP 2) (12) with the fiber C∗. Here Symt(X) := Xt/Sn denotes the t-th symmetric power of a space X. The natural map M(k) → B(k) is a principle C∗-fibration which gives rise to a line bundle η(k) := {M(k) ×C∗ C → B(k)}. By shrinking its zero section to a point, we form a quotient space M(k) := [M(k)×C∗ C] /B(k). (13) It differs from M(k) by a single new point 0—a point which will represent the zero multipole. Evidently, M(k) is a contractible space. Given a collection of vector spaces {Vα}, their wedge product ∧αVα (not to be mixed with the exterior product!) is the quotient of the Cartesian product ×αVα by the subspace comprised of sequences {vα ∈ Vα} such that at least one vector from the sequence is zero. Thus, topologically, M(k) := [∧k(V (1)]/Σk. (14) Lemma 3 leads to the following proposition. Corollary 5 Consider an homogeneous polynomial P (x, y, z) of degree d which is not divisible by an irreducible homogeneous quadratic polynomial Q(x, y, z). Then any generalized parcelling ν µν of the multiplicity function µ : Z(P,Q) → Z+ uniquely determines a multipole in the space M(d) introduced in (10) or (11). � The space of multipoles has singularities. Its singular set sing(M(k)) arises from the sets points in [C3◦]k fixed by various non-trivial subgroups of Σk. These subgroups all are the conjugates (in Σk) of certain non-trivial subgroups of Sk (recall that Hk ⊂ Σk acts freely). The partially ordered set of the orbit-types give rise to a natural stratification of the multipole space M(k). The space sing(M(k)) is of complex codimension two in M(k). Its top strata corresponds to transpositions from Sk. Thus, a generic point from sing(M(k)) has a normal slice in M(k) which is diffeomorphic to a cone over the real projective space S3/Z2 = RP 3. The larger stabilizers of vectors from the space Ck of the obvious Sk-representation correspond to smaller strata of more complex geometry. Evidently, sing(M(k)) is invariant under the diagonal action of C∗ ≈ Γk/Σk. These observations are summarized in the lemma below. Lemma 4 The singular set sing(M(k)) ⊂ M(k) is of codimension two. It is invariant under the C∗-action. Therefore, it is a principle C∗-fibration over the singular set sing(Symk(CP 2)) ⊂ Symk(CP 2). A generic point of sing(M(k)) has RP 3 for its normal link. Points of sing(M(k)) correspond to completely factorable polynomials L = ν Lν with at least two proportional linear factors (in other words, to weighted collections of lines in CP 2∗ that contain at least one line of multiplicity greater than one). � Let Fact(d) ⊂ V (d) denote the variety of homogeneous polynomials of degree d in x, y, and z that are products of linear forms. Given a multipole w ∈ M(d) one can construct the corresponding completely factorable polyno- mial L(w) = ν Lν ∈ V (d). Note that, due to the uniqueness of the prime factorization in the polynomial ring and in view of our definition of multipoles, the correspondence w ⇒ L(w) gives rise to a 1-to-1 map Θ : M(d) → Fact(d). Consider a “Viète-type” algebraic map ΦQ : M(d) → Fact(d) → FactQ(d), (15) where ΠQ is induced by restricting polynomials in x, y, and z to the surface {Q(x, y, z) = 0}. The symbol FactQ(d) ⊂ V (d)/VQ(d) ≈ V Q (d) denotes the variety of homogeneous polynomial functions on the surface {Q = 0} that also decompose into products of linear forms. Due to formula (5) in Lemma 1, any non-zero homogeneous polynomial on the surface {Q = 0} admits a linear factorization. Hence, ΦQ is onto and FactQ(d) can be identified with the space [V (d)/VQ(d)] V ⊥Q (d) The map ΦQ, extends to an algebraic map Φ̃Q : Eη(d) −→ V Q (d) (16) defined on the space Eη(d) of the line bundle η(d). It sends the zero section B(d) of η(d) to the zero vector 0 ∈ V ⊥Q (d) and each fiber of η(d) isomorphically to a line in V Q (d) passing through the origin. In fact, Φ̂Q|Eη(d)\B(d) = ΦQ. Evidently, Φ̂Q gives rise to a continuous map ΦQ : M(d) −→ V Q (d) (17) It turns out that ΦQ has finite fibers. We need some combinatorial constructions which will help us to prove this claim and to describe the cardinality of the ΦQ-fibers. With every natural d we associate an integer κ(d) that counts the number of distinct parcellings in a finite set of cardinality 2d. Any parcelling is obtained by breaking a set Z of cardinality 2d into disjoint subsets of cardinality 2. Thus, κ(d) = (2d− 1)!! := (2d− 1)(2d− 3)(2d− 5) ... 3 · 1 is the number of possible handshakes among a company of 2d friends. There is an alternative way to compute this number: consider the standard action of the permutation group S2d on the set of 2d elements. The action induces a transitive action on the set of all parcellings. Under this action, the subgroup S 2d that preserves the parcelling {{1, 2}, {3, 4}, ... , {2d − 1, 2d}} is an extension 1 → (S2) d → S 2d → Sd → 1 (18) of the permutation group Sd that acts naturally on the pairs by the group (S2) d ≈ (Z2) d that exchanges the elements in each pair. As a result, we get an identity κ(d) = (2d − 1)!! = (2d)!/(2d · d!) As we deform a polynomial P into a polynomial P1, two or more points in Z(P,Q) can merge into a single point of Z(P1, Q). Its multiplicity is equal to the sum of multiplicities of the points forming the merging group. Through this process, any generalized parcelling p of the original µ : Z(P,Q) → N gives rise to a new and unique generalized parcelling p1 of µ1 : Z(P1, Q) → N. Thus, we can define a partial order in the set of all generalized parcellings of effective divisors on Q of degree 2d by setting µ ≻ µ1 and p ≻ p1 (see Figures 1 and 3). In the same spirit, let κ(µ) stand for the number of distinct generalized parcellings of a function µ : Z → N (recall that ‖µ‖1 = 2d) on a finite set Z. Unless µ is identically 1 and |Z| = 2d, κ(µ) < κ(d). When two intersection points merge, the number of generalized parcellings drops: there are distinct original parcellings that become indistinguishable after the merge (see the left diagram in Figure 1). For example, when two simple points in a complete intersection merge, the number of generalized parcellings changes from κ(d) to κ(d − 2) + [κ(d) − κ(d − 2)]/2 = [κ(d) + κ(d − 2)]/2, that is, it drops by [κ(d) − κ(d− 2)]/2. In general, µ ≻ µ1 implies κ(µ) > κ(µ1). For a generic L ∈ Fact(d) the set Z(L,Q), as well as all the parcels {Zν} defined by the linear factors Lν , are complete intersections. For such an L, the number of multipoles in Φ Q (ΠQ(L)) is the number κ(d) of distinct parcellings in the set Z(L,Q). For any L ∈ Fact(d), the cardinality of Φ−1Q (ΠQ(L)) (equivalently, of Π Q (ΠQ(L))) equals to the the number of generalized parcellings of the multiplicity function µ : Z(L,Q) → N. Indeed, assume that two completely factorable polynomials L,L′ coincide when restricted to the surface {Q = 0}. Then L−L′ must be divisible by Q. Therefore, Z(L,Q) = Z(L′, Q), moreover, the two multiplicities of each point in the intersection (defined by the curves L and L′) must be equal as well. Hence, L and L′ define two parcellings of the same multiplicity function µ on the intersection set. Let X,Y be topological spaces and f : X → Y a continuous map with finite fibers. For a while, the ramification set D(f) of f is understood as the set {y0 ∈ Y } such that, for any open neighborhood U of y0, the cardinality of the fibers {f −1(y)}y∈U is not constant. Figure 1: Two distinct ways in which parcellings degenerate. Each time L ∈ Fact(d) produces in CP 2∗ a union of lines with at least one pair of lines sharing its intersection point with the curve Q, the point Π(L) ∈ D(ΠQ) = D(ΦQ). Moreover, as Figure 1 demonstrates, this change in cardinality of fibers occurs due to their bifurcations in the vicinity of L. As a result, points in D(ΦQ) give rise to effective divisors of degree 2d on the curve Q with at least one point in their support being of multiplicity greater than one. On the other hand, any such divisor can be generated by intersecting a weighted set of lines with Q (just use any generalized parcelling). Note that any line tangent to Q also contributes a point of multiplicity two. However, the lines tangent to Q are not contributing to the bifurcation of the ΠQ-fibers (see the right diagram in Figure 1). Therefore, in order to conclude that any divisor on Q with multiple points corresponds to a point of D(ΦQ), we need to use generalized parcellings which favor pairs of lines that share their intersection with Q to a single tangent line (both patterns produce an intersection point of multiplicity two as shown in Figure 1). Evidently, this can be done, provided d > 1. The requirement that L ∈ Fact(d) has a pair of linear forms vanishing at a point of {Q = 0}, locally, is a single algebraic condition imposed on the coefficients of of the two forms. Therefore, taking closures, it picks a codimension one subvariety Π−1Q (D(ΠQ)) ⊂ Fact(d). Thus, D(ΠQ) ⊂ V Q (d) a subvariety of codimension one as well. Fortunately, since Q admits a rational parameterization by a map α : CP 1 → Q, the set Π−1Q (D(ΠQ)) can be described in terms of solvability of a system of two rather simple equations. In homogeneous coordinates [u0 : u1], such a parameterization α can be given by the formula α([u0 : u1]) = [α0([u0 : u1]) : α1([u0 : u1]) : α2([u0 : u1])], where α0, α1, α2 are some quadratic forms. For example, for Q = x 2 + y2 + z2, α0 = i(u 0 − u 1), α1 = 2iu0u1, α2 = u 0 + u The inverse of α is produced by the central projection of Q onto a line in CP 2∗ from a center located at Q. Therefore, as abstract algebraic curves, Q and CP 1 are isomorphic. In fact, L(x, y, z) ∈ Fact(d) belongs to Π−1Q (D(ΠQ)), if and only if, the system L(α0, α1, α2) (u0, u1) = 0 ∂xL(α0, α1, α2)∂u0α0 + ∂yL(α0, α1, α2)∂u0α1 + ∂zL(α0, α1, α2)∂u0α2 (u0, u1) = 0, (19) that guarantees an existence of a multiple zero for the polynomial L(α(u0, u1)) in CP 1, has a non- trivial solution (u0, u1). Writing down explicitly the resultant of the two polynomials in the LHS of (19), and thus the equation of Π−1Q (D(ΠQ)), seems to be cumbersome. Note that the Euler identity 2d · L = (u0∂u0α0 + u1∂u1α0)∂xL+ (u0∂u0α1 + u1∂u1α1)∂yL+ (u0∂u0α2 + u1∂u1α2)∂zL explains the asymmetry of (19) with respect to the variable u0: exchanging the roles of u0 and u1 leads to an equivalent system of equations. Recall that for a projective variety X, the set of zero-dimensional, degree d effective divisors is the projective variety Symd(X) (see [Ch]). The map ΦQ in formula (15) induces a well-defined regular map of the varieties: ΨQ : Sym d(CP 2) → Sym2d(Q). (20) This map is produced by realizing a given multipole w, or rather its C∗-orbit w̃, by a completely factorable polynomial L(w) and then forming the intersection set Z(L(w), Q) equipped with the appropriate multiplicities—the divisor ΨQ(w̃). The map ΨQ is onto: any effective divisor of degree 2d on Q admits a generalized parcelling, and thus is generated by intersecting Q with a weighted collection of lines. Since the number of generalized parcellings of Z(L(w), Q) is finite, the map ΨQ has finite fibers. All this can be seen from a different angle. The divisor ΨQ(w̃) uniquely determines the proportionality class of the function L(w)|{Q=0}—a point in CP (V Q ). We have seen that the ∗-equivariant map ΦQ in (15) is onto. Thus, ΦQ/C ∗ : Symd(CP 2) → CP (V ⊥Q ) is onto as well. Since Q is a rational curve (topologically, a 2-sphere), there is a 1-to-1 correspondence between points of CP (V ⊥Q ) ≈ Sym 2d(CP 1∗ ) and of Sym 2d(Q). Moreover, as abstract varieties, CP (V ⊥Q ) and of Sym2d(Q) are isomorphic. This isomorphism can be used to identify the maps ΨQ and ΦQ/C To simplify our notations, we will use the same symbol ΨQ for both maps; when there is a need to distinguish them, we will just indicate the relevant target space. Our combinatorial considerations imply that the locus D(ΨQ) consists of the effective divisors of degree 2d onQ with at least one point in their support being of multiplicity ≥ 2. While Ψ−1Q (D(ΨQ)) is also described by (19), the locus sing(Fact(d)) is the preimage of effective divisors of degree 2d on Q with at least one pair of points in their support being of multiplicity ≥ 2 (they can generate a double line), or at least one point being of multiplicity ≥ 4 (it corresponds to a double line tangent to Q). Employing Lemma 4, we have established Lemma 5 The ramification set D(ΦQ) for the map ΦQ : M(d) → V Q (d) is the set {ΠQ(L)} of complex codimension one, where L ∈ Fact(d) defines on Q an effective divisor with at least one point in its support being of multiplicity two at least. In other words, Π−1Q (D(ΦQ)) is defined by the resultant of the two polynomials in the LHS of (19). The ramification set D(ΦQ) contains the ΦQ-image of the singular set sing(M(d)). This image is of codimension two in V Q (d). It can be identified with completely factorable homogeneous polynomials of degree d on the the surface {Q = 0} that have at least one pair of proportional linear factors. � Note that the space Z of simple effective degree k divisors on the curve Q is homeomorphic to the space of simple effective degree k divisors on the sphere S2 = CP 1∗ . Therefore, π1(Z) ≈ Bk, where Bk stands for the braid group in k strings residing in the spherical shell S 2 × [0, 1]. In particular, Sym2d(Q) \ D(ΨQ) is a K(B2d, 1)-space. Let B be the preimage of the subgroup S ⊂ S2d in (18) (of order 2 d · d!) under the canonical epimorphism B2d → S2d. We call it the braid group in 2d strings with coupling 4. It is a subgroup of index (2d − 1)!! in the braid group B2d. Recall that a covering of a K(π, 1)-space is again a K(π′, 1)-space, where π′ is an appropriate subgroup of π. Since all our constructions are C∗-equivariant, Lemmas 4 and 5 together with the arguments above lead to Theorem 6 • The map ΨQ : Sym d(CP 2) → CP 2d ≈ V ⊥Q (d) ◦/C∗ is a finite ramified covering with a generic fiber of cardinality (2d − 1)!!. It is ramified over the subvariety D(ΨQ) whose points are the proportionality classes of homogeneous polynomials of degree d on the quadratic surface {Q(x, y, z) = 0} that define there effective divisors of degree d with at least one multiple line5. • The space CP 2d \ D(ΨQ) is a K(π, 1)-space with the group π being isomorphic to the braid group B2d in 2d strings in the spherical shell S 2 × [0, 1]. • As a result, Symd(CP 2) \Ψ−1Q (D(ΨQ)) is a K(B 2d, 1)-space, where B 2d is the braid group in 2d strings with coupling. • The space M(d) \Φ−1Q (D(ΦQ)) also is K(π, 1)-space with the group π being isomorphic to an extension of the group B by the infinite cyclic group Z. � Now, we will investigate the ramification locus of ΦQ from a more refined point of view characteristic to the singularity theory. First, we would like to understand when a non-zero vector w from the tangent cone TL of Fact(d) at a point L = j=1 Lj is parallel to the subspace VQ(d), in other words, when TL contains a vector w that is mapped to zero under the projection π : V (d) → V (d)/VQ(d) ≈ V ⊥Q (d). Away from the singularity set sing(Fact(d)) ⊂ Fact(d), the ΠQ-image of such an L belongs to a locus E ⊂ V ⊥Q (d) over which rank(dπ|Fact(d)) < dim(V Q (d)) at some point in Π Q (E). By the implicit function theorem, the ramification locus D(ΠQ) ⊂ V Q (d) for the map ΠQ : Fact(d) → V ⊥Q (d) (equivalently, for the map ΦQ) is contained in the union E ∪ ΠQ(sing(Fact(d))). It can happen that at a singularity L ∈ sing(Fact(d)) the tangent cone does not have vectors w 6= 0 with the property π(w) = 0, and still π is ramified in the vicinity of π(L). For instance, consider the obvious projection π of the real cone x2+ y2− z2 = 0 onto the xy-plane: π is ramified at the origin (0, 0), but for any w 6= 0 from the tangent cone at (0, 0, 0), π(w) 6= 0. Any w ∈ TL is of the form limt→0 j=1(Lj+tMj)− j=1 Lj , where each Mj is an appropriate linear form in xj, yj , and zj or Mj = 0 identically. This limit is equal to the polynomial P = i 6=j Li)Mj . The vector P at L is parallel to the subspace VQ if and only if the polynomial P is divisible by Q. In other words, the vector P at L is parallel to VQ if and only if the polynomial P , being restricted to the curve Q, is identically zero. Thus, we are looking for the Mj ’s subject to the constraint: the polynomial i 6=j Li)Mj = ( i Li)( ), being restricted to Q, is identically zero. For 4Note that B 2d is not the braid group in 2d strings colored with d colors, each color marking a pair of strings! 5equivalently, that define on the curve Q ⊂ CP 2∗ effective divisors with a point of multiplicity at least two. L 6= 0, this is equivalent to the constraint (21) imposed on the rational functions {Mj/Lj}: = 0. (21) Equation (21) always have obvious solutions: {Mj = αjLj}, where {αj ∈ C} and j αj = 0. These are exactly solutions that represent the zero tangent vector (the tip L of the tangent cone). Indeed, j=1(Lj + tMj) = j=1(Lj + tαjLj) to conclude that w = 0 if and only if j αj = 0. So, the proper question is how to describe all the L ∈ Fact(d) for which (21) has a solution distinct from the set of obvious solutions {Mj = αjLj} with j αj = 0. The images of such L’s under the projection π : V (d) → V ⊥Q (d) will generate the locus E ⊂ V Q (d) over which the differential dπ|Fact(d) is not of the maximal rank dim(V Q (d)) = 2d + 1. Evaluating the LHS of equation (21) at 2d+1 generic points residing in Q imposes linear constraints on 3d variables—the coefficients of the Mj ’s. If these constraints are independent, the solution space of the linear system must be of dimension 3d− (2d+1) = d− 1, which is exactly the dimension of the space formed by the obvious solutions. This indicates that, for a generic L, we should not expect any non-obvious solutions. Lemma 6 below validates this guess. Let us denote by ML the quotient of the vector space of all solutions {Mj} of (21) by the subspace of obvious solutions, as defined above. The correspondence L ⇒ dim(ML) gives rise to a new natural stratification of the space Fact(d), and thus, of the space MQ(d). In the new notations, π(L) ∈ E if and only if ML 6= 0. We suspect that this stratification is consistent with, but cruder than the stratification induced by the orbit-types of the Σd-action. Lemma 6 The loci E and D(ΠQ) coincide. As a result, the existence of a non-trivial solution for the system (19) is equivalent to the existence of a non-obvious solution for the equation (21). Also, the locus E ⊃ ΦQ(sing(M(d))). Proof. We notice that if two distinct lines, say L1 and L2, share a point p ∈ Q, then (21) has a non-obvious solution (M1,M2, 0, ... , 0). Indeed, inscribe in the quadratic curve Q any ”quadrilateral” formed by the pair of lines L1,L2 together with a new pair of lines M1,M2. The lines L1, M1 share a point q ∈ Q, the lines L2, M2 share a point r ∈ Q, and the lines M1, M2 share a point s ∈ Q, all four points p, q, r, s being distinct. Next, pick some linear forms M1 and M2 representing M1 and M2. Then one can find constants λ1, λ2 so that the polynomial Q = λ1M1L2 + λ2M2L1. The argument is very similar to the one used in the proof of Lemma 1. Thus, λ1M1L2 + λ2M2L1|Q = 0, which can be written in the form λ1M1/L1 + λ2M2/L2|Q = 0 required by (21). Here, evidently, M1 is not proportional to L1 and M2 is not proportional to L2. Therefore, for any L ∈ Fact(d) that defines on Q an effective divisor containing a point of multiplicity two, ΠQ(L) must belong to the locus E . It follows ”by continuity” that all the L’s that produce multiplicity functions µ : Z(L,Q) → N, distinct from the identity function 1, project to E via Π. According to Lemma 5, these are exactly the factorable polynomials that project to the ”combinatorial’” ramification locus D. � It follows from Lemma 6 that, for a generic P ∈ V (d), the affine subspace P +VQ(d) hits transver- sally the subvariety Fact(d) at a finite set of points whose cardinality is exactly κ(d). Therefore, κ(d) must be the degree of that variety. Let X,Y be quasi-affine varieties and let F : X → Y be a proper regular map. In Theorem 7 below, the ramification set D(F ) of a mapping F : X → Y with finite fibers is understood to be the closure of a set D◦(F ). By definition, y ∈ D◦(F ) when F−1(y) contains a point x such that there is a non-zero vector v ∈ TxX that is mapped to zero in TyY by the differential DF . The arguments above lead to our main result: Theorem 7 • The map ΦQ : M(d) → FactQ(d) = V Q (d) ◦ is onto, and its generic fiber is a finite set of cardinality (2d−1)!!. This map takes the multipole space M(d)—the space of a C∗- bundle η(k) over the variety Symd(CP 2)—onto the complex vector space V ⊥Q (d) of dimension 2d + 1 with its origin being deleted. The map ΦQ extends to a map ΦQ : M(d) −→ V Q (d) with finite fibers. • ΦQ is ramified over the discriminant variety D(ΦQ)—the set of polynomials P ∈ FactQ(d) whose zero sets are effective divisors on the surface {Q(x, y, z) = 0} with at least one of their line components being of multiplicity at least two.6 The subvariety D(ΦQ) ⊂ V Q (d) ◦ is of complex codimension one, and is described in terms of solvability of (19) or (21). It contains the ΦQ-image of the singular set sing(M(d)) as a codimension one subvariety. • In fact, ΦQ, ΦQ are C ∗-equivariant maps. Therefore, ΦQ gives rise to a surjective map ΨQ : Sym d(CP 2) → CP 2d = CP (V ⊥Q (d) of degree (2d − 1)!! which is also ramified over a subvariety D(ΨQ) ⊂ CP 2d of codimension one. The compliment CP 2d \ D(ΨQ) is a K(B2d, 1)-space, Sym d(CP 2) \ Ψ−1Q (D(ΨQ)) is a 2d, 1)-space, where B 2d is the braid group with coupling, while M(d) \ Φ Q (D(ΦQ)) is a K(π, 1)-space, where π an extension of B by Z. • The degree of the variety of completely factorizable homogeneous polynomials Fact(d) ⊂ V (d) is also (2d− 1)!!. This variety is invariant under the obvious C∗-action on V (d). � Theorem 7 has a number of topological implications, some of them dealing with interesting rami- fications over complex projective spaces. Our next goal is to describe these implications. Let the space X be of a homotopy type of a connected, finite-dimensional CW-complex. Recall that the Dold-Thom Theorem [DT] links the homotopy groups of Symd(X), d being large, with the integral homology of a space X. By picking a base point a in X, one gets a stabilization map Symd(X) → Symd+1(X), well-defined by the formula (x1, x2, ..., xd) → (a, x1, x2, ..., xd). Here {xi ∈ X}. This provides us with canonical homomorphisms πk(Sym d(X)) → πk(Sym d+1(X)) of the k-th homotopy groups. In our case, the Dold-Thom Theorem claims that limd→∞ πk(Sym d(CP 2)) = Hk(CP 2;Z). In par- ticular, limd→∞ π2(Sym d(CP 2)) = Z = limd→∞ π4(Sym d(CP 2)). Also, limd→∞πk(Sym d(CP 2)) = 0, provided k = 1, 3 or k > 4. On the other hand, π2(CP 2d) = Z, but π4(CP 2d) = 0. Therefore, at least for large d’s, there is an infinite order element α ∈ π4(Sym d(CP 2)) that is mapped by (ΨQ)∗ 6Equivalently, the set of polynomials P for which P ∩Q has a point of multiplicity at least two. to zero and an element β ∈ π2(Sym d(CP 2)), so that (ΨQ)∗(β) ∈ π2(CP 2d) is a generator. It is pos- sible to realize α and β geometrically. What is clear ratherway, that the spheroids α and β can not be pushed into the aspherical portion Symd(CP 2)\Ψ−1Q (D(ΨQ)) of Sym d(CP 2). The realization of α is based on an interesting fact that I originally learned from Blaine Lawson: the quotient of CP 2 by the complex conjugation τ : [x : y : z] → [x̄ : ȳ : z̄] is homeomorphic to the sphere S4. Later on, I have found its nice generalization in [A], Theorem 8. Therefore, the map φ : CP 2 → CP 2 ×CP 2 given by the formula φ(p) = (p, τ(p)), where p ∈ CP 2, is evidently Z2-equivariant with respect to the τ -action in the domain and the symmetrizing action in the range. This gives rise to the desired quotient map α : S4 ≈ CP 2/{τ} → Sym2(CP 2) that survives into the higher symmetric powers of CP 2. Constructing class β is straightforward: it is given by the obvious inclusion S2 ≈ CP 1 ⊂ CP 2 followed by the diagonal embedding ∆ : CP 2 → Symd(CP 2). The existence of a non-trivial α : S4 → Symd(CP 2) whose ΨQ-image is null-homotopic in CP has a curious implication: Corollary 8 For any d ≥ 2, there is a family of polynomial functions of degree d on the quadratic surface {Q = 0} which is parameterized by a 5-dimensional disk and which does not admit a continuous lifting to the multipole space M(d). At the same time, the functions parameterized by the 4-sphere forming the boundary of the disk can be continuously represented by multipoles. � Any map f : X → Y induces a natural map fk∗ : Sym k(X) → Symk(Y ) that is defined by the formula (fk∗ )[ ν µνxν ] = ν µνf(xν), where {xν ∈ X} and {µν ∈ N}. When f is 1-to-1 or onto, so is fk∗ . The construction of fk∗ provides us with a rich source of interesting ramified coverings. Consider for example, a semi-free cyclic action on a sphere S2 ⊂ R3. The group Zl acts on S 2 by rotations around a fixed axis on the angles that are multiples of 2π/l. Topologically, the orbit-space S̃2 := S2/Zl is again a 2-sphere. Let f : S2 → S̃2 be the orbit-map. Then fk∗ : Sym k(S2) → Symk(S̃2) gives an example of a ramified degree lk covering map of CP k over itself! For l = 2, we shall see later how this degree 2k ramification fk∗ : CP k → CP k is linked to the multipole spaces on quadratic surfaces. On the other hand, by a simple cohomological argument, any map F : CP k → CP k has a degree that is the k-th power of a non-negative integer l. For instance, there is no ramified map from CP 2 to itself of degrees that are not of the form l2. Question Given a closed oriented manifold M , what are possible degrees of maps from M to itself? Evidently, the answer depends on the homotopy type of M . When Z2 acts freely on S 2 by the central symmetry, the orbit-map f : S2 → RP 2 gives rise to another interesting ramified map fk∗ : CP k → RP 2k of degree 2k (its existence follows from our results in Section 3 dealing with real multipole spaces). In different terms, the map fk∗ has been described in [A]. In order to derive next few corollaries of Theorem 7, we need to take a detour aimed at constructing (with the help of f) transfer maps that take effective 0-divisors on Y to effective 0-divisors on X. These constructions are variations on the theme of the classical Hurwitz’ Theorem (see [H], pp. 299-304). Our next goal is to present constructions and arguments that lead to Theorem 10. Let X and Y be smooth complex projective varieties and f : X → Y a regular surjective map with finite fibers. With any x ∈ X we associate a multiplicity number µf (x). It is the multiplicity attached to the intersection of the f -graph Γf with the subspace X × f(x) ⊂ X × Y at the point (x, f(x)). Since all the f -fibers are finite, the intersection Γf ∩ (X × f(x)) is finite as well. Next, with any y ∈ Y we associate an effective 0-divisor Df−1(y) := x∈f−1(y) µf (x)x whose degree is y-independent and coincides with the degree df of the map f . The correspondence y ⇒ Df−1(y) produces a regular embedding f# : Y → Symdf (X) (22) For any k, the map f# gives rise to an embedding k : Sym k(Y ) → Symk·df (X) (23) defined by the formula f ν µ(yν)y) = ν µ(yν)Df−1(y). Therefore, applying this construction to the setting of Theorem 7 withX = Symd(CP 2), Y = CP 2d, and f = ΨQ, we get the following proposition: Corollary 9 Any irreducible quadratic form Q(x, y, z) determines canonical embeddings : Symk(CP 2d) → Symk·(2d−1)!!(Symd(CP 2)), where d and k are arbitrary whole numbers. In particular, Ψ Q := Ψ Q,1 embeds CP 2d into Sym(2d−1)!!(Symd(CP 2)). � For a map f : X → Y as above, one can define a map f# that takes effective 0-divisors on X into effective 0-divisors on X. By definition, each point x ∈ X is mapped to the divisor Df−1(f(x)). By linearity, we have µ(xν)xν µ(xν)Df−1(f(xν)). (24) This map transforms 0-divisors of degree k into 0-divisors of degree k · df : fk# : Sym k(X) → Symk·df (X) (25) Both maps, fk# from (25) and f k from (23), have the same targets. Moreover, their images coincide. Indeed, for each point y ∈ Y , pick any point x ∈ f−1(y). Then, f#(y) = Df−1(y) = f#(x). Recall, that f k is a 1-to-1 map and thus is invertible over its image. Therefore, the map −1 ◦ fk# : Sym k(X) → Symk(Y ) (26) is well-defined. Lemma 7 Let X and Y be smooth complex projective varieties and f : X → Y be a regular onto map with finite fibers. The map (f −1◦fk# in (26) coincides with the natural map f ∗ : Sym k(X) → Symk(Y ). It defines a ramified covering with a generic fiber of cardinality (df ) Proof The map f# takes each point x ∈ X to the divisor Df−1(f(x)). 7. At the same time, the transfer f# takes f(x) to the same divisor Df−1(f(x)). Thus, (f #)−1◦f# maps x to f(x). Extending this argument by linearity proves the claim. � Now we examine how these constructions apply to projective curves in CP 2∗ and eventually to the multipole spaces. The role of X will be played by a curve C ⊂ CP 2∗ (most importantly, by Q), the role of Y by a pencil of lines in CP 2∗ through a point p. The map f takes any point q ∈ C to a line L passing through q and p. Any linear embedding ρ : CP 1 ⊂ CP 2 induces an embedding ρd : Symd(CP 1) → Symd(CP 2). The geometry of ρd is tricky. Since the quotient space CP 2/CP 1 is homeomorphic to a 4-sphere S4, we get a surjection Symd(CP 2)/Symd(CP 1) → Symd(S4), but even the spaces {Symd(S4)} have subtle topology. For example, Sym2(S4) is homeomorphic to a mapping cylinder of a map Σ4(RP 3) → S4, where Σ4(∼) denotes the fourth suspension (see [Ha], Example 4K.5). The regular map Ψ̂Q : Sym d(CP 1) → Symd(CP 2) → Sym2d(Q) ≈ CP 2d (27) describes the role and place of C-planar multipoles (they are linked to the planarity of the quadra- poles and octapoles in the deconstruction of CMBR that was briefly mentioned in the introduction). In contrast with ΨQ, we will see that the map Ψ̂Q is 1-to-1. Since Sym d(CP 1) is homeomorphic to CP d, its image in Sym2d(Q) is homeomorphic to CP d as well (compare this with (21)). Moreover, we claim that the ΨQ-image of Sym d(CP 1) in CP (VQ(d)) is a d-dimensional projective subspace. It is sufficient to examine the case of ρ whose image is given by the equation z = 0 in the homogeneous coordinates [x : y : z]. In such a case, the ΨQ-image if formed by the proportionality classes of homogeneous polynomials that are products of d linear forms in x and y alone, the products being restricted to the cone {Q = 0}. However, any homogeneous polynomial in two variables factors over C into a product of linear forms. Therefore, the ΨQ-image of Sym d(CP 1) in CP (VQ(d)) is generated by all homogeneous polynomials in x and y, which clearly form a vector space. Alternatively, Ψ̂Q-image of Sym d(CP 1) in Sym2d(Q) could be described in terms of effective divisors on Q as follows. As a byproduct of this description, we will construct interesting examples of ramified regular maps from CP d to itself. Recall that points L ∈ CP 1 ⊂ CP 2 correspond to a pencil of lines L ⊂ CP 2∗ that pass through a particular point p ∈ CP 2∗ . That p determines the embedding ρ. Each point q ∈ Q determines a unique line Lq that passes through q and p, and therefore, a unique point Lq in the dual subspace CP 1 ⊂ CP 2. Let f : Q → CP 1 be a 2-to-1 ramified map defined by the correspondence q ⇒ Lq. As described in (24) and (26), f gives rise to maps f d : Sym d(CP 1) → Sym2d(Q) and fd# : Symd(Q) → Sym2d(Q) with f d being a 1-to-1 map. In fact, examining the construction of ΨQ, we see that Ψ̂Q = f d . Moreover, using Lemma 7, the map f ∗ : Sym d(Q) → Symd(CP 1) factors as (f −1 ◦ fd#. Therefore, the ramification locus D(f ∗ ) ⊂ Sym d(CP 1) for fd∗ is the Ψ̂ Q -image of the ramification locus D(fd#) ⊂ f #(Sym d(Q)) ⊂ Sym2d(Q) for fd#. 7Recall that the multiplicity of x in Df−1(f(x)) is µf (x). Figure 2 illustrates the case d = 3. Passing from the first to the second column depicts the map f3# : Sym 3(Q) → Sym6(Q) generated by the linear projection from a center p located at infinity. Passing from the first to the third column depicts the map f3∗ : Sym 3(Q) → Sym3(CP 1) ≈ CP 3. Here CP 1 is viewed as the pencil of lines in CP 2∗ through p. Topologically, the map f ∗ is a 8-to-1 ramification of CP 3 over itself. It is described in some detail in Example 1 and is depicted in Figure 3. The passage from the second to the third column is a 1-to-1 correspondence. Figure 2: When Lq is not tangent to Q at q, then let q ⋆ 6= q be ”the other point” in Q that belongs to the line Lq. When Lq is tangent to Q at q, then by definition, put q ⋆ = q. The correspondence τp : q ⇒ q is an involution on Q with two fixed points a and b. Its orbit-space Q/{τp} topologically is a 2- sphere. The involution τp induces an involution τ p that acts on the space Sym k(Q). By definition, any effective divisor ν µ(qν)qν ∈ Sym k(Q) is transformed by τkp into the divisor ν µ(qν)q Evidently, the image of fd# : Sym d(Q) → Sym2d(Q) is contained in the τ2dp -invariant part of the space Sym2d(Q); however, not any invariant divisor belongs to that image. Each divisor from Im(fd#) not only must be τ p -invariant, but in addition, its multiplicity at the fixed points a and b must be even. In other words, such a divisor D# must be of the form 2µaa+ 2µbb+ µ(qν)[qν + q where qν 6= a, b and µa, µb ∈ Z+. The cardinality of the f #-fiber over D # is given by the formula |(fd#) −1(D#)| = [µ(qν) + 1]. (28) Indeed, for any q ∈ Q\{a, b} there are µ(qν)+ 1 effective divisors of degree µ(qν) with the support in qν q⋆ν and whose f µ(qν) # -image is µ(qν)[qν + q ν ]; at the same time, there is a unique divisor µaa whose f µ(µa) -image is 2µaa. When all µν = 1 and µa = 0 = µb, the fiber (f −1(D#) is of cardinality 2d. Examining (28), we see that the ramification setD(fd#) for f # : Sym d(Q) → Sym2d(Q) is comprised of divisors of two kinds: 1) the ones that contain at least one summand of the form 2(q+q⋆), where q 6= a, b, and 2) the ones that contain 2a or 2b as a summand. For example, if D ∈ Symd(Q) contains a pair of distinct points q1, q2 = q 1 and the rest of points in the support of D are generic (i.e., for i, j > 2, qi 6= q j ), then (f −1(fd#(D)) consists of 2 d − 2d−2 = 3 · 2d−2 elements. At the same time, as we perturb D in order to avoid the coincidence q2 = q 1, (f −1(fd#(D)) consists of 2d elements. Therefore, in view of Lemma 7, the ramification set D(fd∗ ) for f ∗ : Sym d(Q) → Symd(CP 1) is comprised of divisors of two kinds: 1) the ones that contain at least one summand of multiplicity ≥ 2, and 2) the ones that contain points La or Lb giving rise to lines La and Lb passing through the point p and tangent to the curve Q. Given a space X, let us denote by ∆d(X) ⊂ Sym d(X) the discriminat set formed by the divisors containing points of multiplicity at least two. Also, for any point a ∈ X, we denote by Symd−k the subset of Symd(X) formed by the divisors containing the summand k · a. Thus, D(fd∗ ) = ∆d(CP 1) ∪ Symd−1a (CP 1) ∪ Symd−1 (CP 1) (29) D(fd#) = ∆2d(Q) ∪ Sym 2a (Q) ∪ Sym 2b (Q) . (30) Employing the Viéte Map V : Symd(CP 1) → CP d, we transplant the algebraic set D(fd∗ ) into the space CP d. The image of ∆d(CP 1) under the Viéte Map is the classical discriminant variety in Dd ⊂ CP d, while the images of Symd−1a (CP 1) and of Symd−1 (CP 1) form linear subspaces CP d−1a and CP d−1 in CP d. Thus, V (D(fd∗ )) = Dd ∪ CP a ∪ CP . It follows from [K], Theorem 6.1, that each of the two spaces CP d−1a and CP is tangent to the discriminant variety Dd along, respectively, the linear subspaces CP d−2a and CP Note that the complement Symd(CP 1) \ D(fd∗ ) is the configuration space of d-tuples of distinct points in the domain CP 1 \ (a∗ ∪ b∗). Here a∗, b∗ stand for the two points in CP 1 that are dual to the lines passing through p and tangent to the curve Q. Therefore, it is a K(Bannd , 1)-space, where Bannd stands for the braid group in d strings residing in a cylinder with an annulus base [S2 \ (a∗ ∪ b∗)]× [0, 1]. Using the birational identifications Symd(Q) ≈ CP d ≈ Symd(CP 1), we have constructed a ramified covering ΓQ : CP d ≈ Symd(Q) → Symd(CP 1) ≈ CP d. The following proposition summarizes the conclusions of our arguments above (centered on (27) and (29), (30)). It describes an intricate stratified geometry of this ramified covering, a geometry that is “reductive” in its nature with respect to the shift d ⇒ d− 1. Theorem 10 • Any irreducible quadratic form Q gives rise to a ramified covering ΓQ : CP d → CP d of degree 2d. The map ΓQ is ramified over the algebraic set D(ΓQ) := Dd ∪ CP a ∪ CP — the Viéte image of the set ∆d(CP 1) ∪ Symd−1a (CP 1) ∪ Symd−1b (CP • The discriminant variety Dd ⊂ CP d is Q-independent. The two linear subspaces CP d−1a and CP d−1 are tangent to the variety Dd along, respectively, subspaces CP a and CP generic point of Dd has a ΓQ-fiber of cardinality 3 · 2 d−2,8 while a generic point of CP d−1a ∪ CP d−1 has a fiber of cardinality 2d−1. • The complement CP d\D(ΓQ) to the ramification set is a K(B d , 1)-space, where B d denotes the annulus braid group in d strings. • Moreover, over to each of the two subspaces CP d−1a and CP , the map ΓQ inherits a similar stratified structure with respect to the dimensional shift d ⇒ d− 1. � Example 1. Let d = 2. Then ΓQ : CP 2 → CP 2 is of degree 4. The discriminat parabola D2 is given by {x 2 − 4yz = 0}. The ramification locus D(ΓQ) consists of that parabola together with two tangent lines CP 1a = {Ax+ y +A 2z = 0} and CP 1b = {Bx+ y +B 2z = 0} which share a point E = [−(A+B) : AB : 1]. They are tangent to the parabola at the points F = [−2A : A2 : 1] and G = [−2B : B2 : 1]. Here the parameters A and B depend on the quadratic form Q(x, y, z). The ΓQ-fibers over E,F,G are singletons; the cardinality of the fiber over D2 \ (F ∪ G) is 3; the cardinality of the fiber over CP 1a \(F ∪E) and over CP b \(G∪E) is 2. It is still a bit mysterious how all this data manage to produce CP 2 as a covering space! Anyway, the compliment CP 2 \ D(ΓQ) is a K(Bann2 , 1)-space, where B 2 is the braid group with two strings in a ”fat” annulus. Now consider the case d = 3 depicted in Figure 3. That figure exhibits a two-fold symmetry that exchanges points a and b where the pencil of parallel lines is tangent to the curve Q. Pattern 1 in Figure 3 corresponds to the generic stratum CP 3, while pattern 2 defines the discriminant surface D3 ⊂ CP 3 of degree 4. It is homeomorphic to CP 1×CP 1. Patterns 3a and 3b each corresponds to two planes CP 2a and CP b . Each of the two planes is tangent to the discriminant surface D3 along lines la and lb labeled by patterns 6a and 6b, respectively. These planes intersect along another line l labeled by pattern 5. Pattern 4 encodes the singular locus C of the discriminant surface D3. It is a rational curve of degree 3 that is homeomorphic to S2. In fact, D3 is the ruled surface spanned 8that is, 2d−2 less than a generic ΓQ-fiber. 9a 9b Figure 3: Patterns labeling the stratification of the map ΓQ : CP 3 → CP 3 and the geometry of its ramification locus. by lines tangent to C ([K]). The surface D3 intersects with the plane CP a along the union of the line la with a quadratic curve D2,a labeled by pattern 7a. The lines la and l are both tangent to the parabola D2,a. This configuration is already familiar from our description of the case d = 2. Similarly, pattern 7b labels parabola D2,b. Curves C, D2,a, and la are all tangent at a point Aa labeled by pattern 8a. Curves l and D2,a are tangent at a point Ba labeled by pattern 9b. That point also lies on the line lb. The labeling of points Ab and Ba is done by patterns 8b and 9a, respectively. In accordance with Theorem 10, the ramification locus D(ΓQ) for the 8-to-1 map ΓQ coinsides with D3 ∪ CP a ∪ CP b and its complement is a K(B 3 , 1)-space. � Now, we turn to deconstructions of both homogeneous non-homogeneous polynomials on complex quadratic surfaces {Q(x, y, z) = const}. First, consider a general homogeneous polynomial P of degree d. We can apply decomposition (5) to the homogeneous term R of degree d− 2 in (5). This will produce a new collection of vectors—a new multipole—{w1,ν} associated with an appropriate generalized parcelling of µ : Z(R,Q) → Z+. This process of producing lower order multipoles {ws,ν ∈ V (1)}s,ν , where 0 ≤ s ≤ ⌈d/2⌉ and 0 < ν ≤ d − 2s, can be repeated again and again until all the degrees are “used up”. We notice that the leading multipole (of highest degree) is determined by this algorithm in a more “direct way” than the lower degree multipoles. Also note that the choice of a generalized parcelling τ for µ : Z(P,Q) → Z+ affects the choice of a generalized parcelling for µ : Z(R,Q) → Z+, where the polynomial R = (P − ν Lν)/Q s. Here the product ν Lν is determined by the τ , and s is the maximal power of Q for which the division in the ring of polynomials is possible. Example 2. Let d = 5. Then any homogeneous polynomial P of degree 5 has a representation of the form P = L01L02L03L04L05 +Q · L11L12L13 +Q 2 · L21 (31) where all the Lij’s are linear forms (some of which might be zeros). The number of such represen- tations does not exceed (9!!)× (5!!). � Any non-homogeneous polynomial P (x, y, z) of degree d can be written in the form P (0) + P (1) where the degrees of monomials comprising P (n) are congruent to n modulo 2. Note that the decomposition P = P (0) + P (1) intrinsically makes sense on every quadratic surface Q(x, y, z) = λ, (λ ∈ C): if P belongs to the principle ideal 〈Q − λ〉, so does each term P (n), n = 0, 1. Thus, if P ≡ P̃ mod〈Q− λ〉, then we have P (n) ≡ P̃ (n) mod〈Q− λ〉. On the surface {Q(x, y, z) = 1}, any component P (n) of the polynomial P (0) +P (1) can be homog- enized by multiplying its terms of the same degree by an appropriate power of Q. We denote by Q the appropriate homogeneous polynomial. Generically, deg(P Q ) = d− n. Example 3. Let d = 5. Any polynomial P of degree 5 has a representation of the form P = L01L02L03L04L05 +Q · L11L12L13 +Q 2 · L21 + M01L02M03M04 +Q ·M11M12 +Q 2 · λ (32) where all the Lij ’s and Mij ’s are linear forms (some of which might be zeros), and λ is a number. The number of such representations does not exceed (9!!)× (5!!)× (7!!)× (3!!) = 9× 72× 53× 34. � Now one can apply recursively Lemmas 1, 3 and Corollary 5 to each homogeneous polynomial P n = 0, 1. Letting Q = 1 proves formula (2) from Theorem 2. Let us restate and generalize this theorem in terms of the multipoles: Theorem 11 • Let Q(x, y, z) be an irreducible quadratic form and let P (x, y, z) be any complex polynomial of degree d. Its restriction P |S to the complex surface S = {Q(x, y, z) = 1} admits a representation of the form P (x, y, z) = λ+ Lk,l(x, y, z), (33) where the linear forms {Lk,l} are chosen so that each non-zero product l=1 Lk,l(x, y, z) is determined, via the map ΦQ (see (15)), by an appropriate multipole wk from the variety M(k). • The representation (33) is unique, up to a finite ambiguity and up to reordering and rescaling of multipliers in each product l=1 Lk,l(x, y, z). In other words, the set of P -representing multipoles {wk ∈ M(k)}1≤k≤d is finite. Its cardinality does not exceed k=1[(2k − 1)!!]. We would like to end this section by establishing a few facts about alternative decompositions of polynomials that are based on formula (9). In achieving this goal we are guided by Theorem 22.2 from [Sh]. We claimed that the direct sum in (6) is orthogonal with respect to the inner product 〈f, g〉 = f · g dm, where dm is the standard rotationally symmetric measure on the unit sphere S2. Let us clarify this claim. Applying (6) recursively, we get that for any real homogeneous polynomial P of degree d can be written as d−2k≥0 Q k ·PHk , where P k is a real homogeneous harmonic polynomial of degree d−2k. Letting Q = 1, we see that for any homogeneous P , there is a harmonic polynomial d−2k≥0 P k such that P |S2 = P H |S2 . Now, any two harmonic homogeneous polynomials PHk and P l of different degrees are eigenfunctions with different eigenvalues −(d− 2k)(d− 2k+1) and −(d−2l)(d−2l+1) for the Laplace operator ∆S2 on the sphere. Therefore, they are orthogonal, PHk · P l dm = 0. Since any homogeneous polynomial R of degree d− 2 can be represented d−2k≥0; k>0 Q k · PHk , we get the claimed orthogonality (PH0 )(Q · R) dm = 0 of the direct sum in (6). This argument extends to complex harmonic polynomials as follows. If a polynomial P is homo- geneous, so are the polynomials PR and PI (introduced shortly after formula (7)). Also, if P is harmonic, so are the polynomials PR and PI. Note that P (x, y, z)|S2 = [PR(x, y, z)− iPI(x, y, z)]|S2 . Therefore, with Q = x2 + y2 + z2, for any complex harmonic and homogeneous polynomial P of degree d and any homogeneous polynomial T of degree d− 2, we get P · [Q · T ] dm = (PR + i PI)(TR − i TI) dm (PR · TR) dm+ (PI · TI) dm− i (PR · TI) dm+ i (PI · TR) dm. Each of the four integrals must vanish because each integrant, being restricted to S2, is a product of a real homogeneous and harmonic polynomial of degree d by a real polynomial of a lower degree. In the spherical coordinates θ, φ, the inner product is given by an integral f · ḡ dm = 0≤φ≤π; 0≤θ≤2π f(Λ(θ, φ)) · ḡ(Λ(θ, φ)) | sin φ| dθ dφ where Λ(θ, φ) = (cos θ sinφ, sin θ sinφ, cosφ). Now we are going to transfer this hermitian inner product from the sphere to its image ΥQ in a given complex quadratic surface SQ := {Q(x, y, z) = 1}. As before, let A be a complex invertible matrix that reduces Q to the sum of squares: x′2+y′2+z′2 = Q((x, y, z) ·A), where (x′, y′, z′) = (x, y, z) ·A. Let ΥQ = {Q((x, y, z) ·A) = 1}, with (x, y, z) ·A being a real vector. The ellipsoid ΥQ is the image of the unit sphere under the complex linear transformation A−1. It is a totally real 2-dimensional algebraic surface in the complex surface SQ. The real-valued measure dmQ on ΥQ is the pull-back under the map A of the standard measure on the unit sphere. It is invariant under the action of the compact subgroupA·O(3;R)·A−1 ⊂ GL(3;C), where O(3;R) denotes the orthogonal group. For any pair of (homogeneous) complex functions f, g on C3, we get a dot product f((x, y, z)A) · ḡ((x, y, z)A) dmQ = f(x′, y′, z′) · ḡ(x′, y′, z′) dm 0≤φ≤π; 0≤θ≤2π f(Λ(θ, φ)) · ḡ(Λ(θ, φ)) | sin φ| dθ dφ (34) Applying (9) recursively, gives the following proposition: Theorem 12 • The space of complex homogeneous polynomials admits an OQ(3;C)-invariant decomposition V (d) = ⊕d−2k≥ 0 Q k ·HarQ(d− 2k) (35) The summands in (35) are orthogonal with respect to the hermitian inner product defined by the formula (34). In particular, the homogeneous polynomials P ∈ HarQ(d)—the solutions of the equation ∆Q(P ) = 0—are characterized by the property: for each T ∈ VQ(d), P · T̄ dmQ = 0. • Any polynomial function F on the surface S = {Q(x, y, z) = const} can be obtained by restricting to S a polynomial P ∈ Ker(∆Q). • For any two polynomials M and N , “the complex Dirichlet problem” {∆Q(P ) = M, P |S = N |S} (36) has a unique polynomial solution P . • Any polynomial solution P of the equation ∆Q(P ) = 0, deg(P ) ≤ d, (equivalently, any polynomial P of degree d at most, being restricted to the surface S) admits a Maxwell-type representation P (x, y, z) = d−2k≥ 0 Q(x, y, z)d−k+ 2 · ∇ . . .∇ ud−2k,k Q(x, y, z)− , (37) where uj,k are appropriate vectors in C Remark. All the statements of Theorem 12, but the last one (dealing with the generalized Maxwell representation (37)), can be easily generalized for polynomials in any number of variables. Proof. Decomposition (35) follows from (8), and (37) from (35) together with (9). The second bullet is obviously implied by (33) and the linearity of ∆Q. In order to prove the third bullet, note that (8) implies that ∆Q : V (k) → V (k − 2) is onto. Therefore, for any M , there exists a polynomial T , so that ∆Q(T ) = M . On the other hand by bullet two, there exists a harmonic polynomial R ∈ Ker(∆Q) such that R|S = (N − T )|S . As a result, P = T +R solves the Dirichlet problem. In order to show that P is unique, it is sufficient to prove that no polynomial of the form (Q− 1)S belongs to Ker(∆Q). Since ∆Q is homogeneous of degree −2, ∆Q(Q · S) 6= ∆Q(S), unless S = 0. Indeed, let F be the leading homogeneous portion of S of the degree deg(S). Then ∆Q(Q · S) = ∆Q(S) implies ∆Q(Q · F ) = 0 which is, as we have shown before, impossible, unless F = 0. � 9not necessarily homogeneous, even when P is homogeneous 3 Deconstructing reality We have seen that different parcellings of the intersection Z(P,Q) led to different deconstructions of polynomial functions on a quadratic surface {Q = 1}. The next idea is to invoke symmetry in order to get a unique equivariant parcelling and thus, a unique deconstruction of symmetric functions on a quadratic surface supporting an appropriate group action. We deal with a special, but important case of Z2-action. The model example is provided by the complex conjugation τ : (x, y, z) → (x, y, z)10. If Q(x, y, z) has real coefficients, both the curve Q ⊂ CP 2∗ and the surface S = {Q = 1} ⊂ C ∗ are invariant under the τ . When Q is a definite real form, the Z2-action by conjugation on Q is free since the fixed point set Q Z2 = Q∩ RP 2∗ = ∅. For a homogeneous polynomial P with real coefficients the intersection Z(P,Q) ⊂ CP 2∗ and the multiplicity function µ : Z(P,Q) → Z+ are τ -invariant as well. We notice that a τ -invariant generalized parcelling of such a µ will produce a set of τ -invariant lines Lν . Thus, we can assume that all the linear forms Lν in (6) are with real coefficients. Indeed, given a representation P = ν Lν +Q · R, where P,Q and Lν ’s are real and λ ∈ C, we get P = λ · ν Lν +Q · R. Hence, 2P = (λ+ λ) · ν Lν +Q · (R+R) which implies the validity of (6) over the reals. With this observation in mind, we introduce real versions of the multipole spaces (10), (11). Let MR(k) = {[R3 ◦]k/ΣRk (38) where the group ΣRk is defined similar to its complex version. The only difference lies in the definition of HRk : it is now a subgroup of (R ∗)k, not of (C∗)k. The space of real multipoles MR(k) is a space of a principle R∗-fibration over the real variety BR(k) := Symk(RP 2). According to [A], Theorem 2, it is diffeomorphic to RP 2k (cf. Corollary 15). We can form an associated line bundle ηR(k) = {MR(k)×R∗ R → B R(k)}. As in the complex case, put EηR(k) = MR(k)×R∗ R and form the quotient space (k) = EηR(k)/BR(k) (39) by collapsing the zero section to a point 0. As before, M (k) is a contractible space. Real homogeneous polynomials of degree d form a totally real subspace V (d; R) in the complex space V (d). The hermitian metric on V (d) defined by (34) generates an eucledian metric on V (d; R) induced by the inner product 〈f, g〉R := Re f(x, y, z) · ḡ(x, y, z) dmQ f((x, y, z)A−1) · ḡ((x, y, z)A−1) dm Moreover, as in the complex case, the imbedding βQ : V (d− 2; R) → V (d; R) is an isometry (recall that Q is real). 10Another interesting involution to consider is η : (x, y, z) → (y,−x, z). It acts freely on any curve Q, where Q = x2 + y2 + qz2, and q 6= 0 is a real number. Many results of this section have analogues for the polynomials P , such that P (η(x, y, z)) = P (x, y, z), and their multipole spaces based on pairs of vectors ((a, b, c), (−b̄, ā, c̄)). Similarly to the complex case, we introduce a variety Fact(d; R) ⊂ V (d; R) of completely factorable real polynomials. The orthogonal projection V (d; R) → V ⊥Q (d; R) maps Fact(d; R) into the vector space V ⊥Q (d; R). Due to Theorem 12, V Q (d; R) can be identified with Ker(∆ Q). When the quadratic form Q is not definite, we can also identify the space V ⊥Q (d; R) with the d-graded portion of the polynomial function ring on the real cone {Q = 0}. In any case, we get an algebraic map ΦRQ : M R(d) → Fact(d; R) → V ⊥Q (d; R) ◦ (41) which, by Lemma 1 and the argument above, is onto. This map extends to a map Φ̃RQ : Eη R(d) −→ V ⊥Q (d; R) (42) that takes the zero section BR(d) ≈ RP 2k of the line bundle ηR(d) to the origin in V ⊥Q (d; R), and each fiber of ηR(d) isomorphically to a line through the origin. We get an R∗-equivariant surjection Q : M (d) −→ V ⊥Q (d; R) (43) with finite fibers, where M (d) := EηR(d)/BR(d). The number of τ -equivariant parcellings of Z(P,Q) is harder to determine. It depends only on the restriction of the multiplicity function µ to the subset Z(P,Q;R) := Z(P,Q)∩RP 2∗ and is equal to the number of generalized parcellings subordinate to such a restriction. However, if Z(P,Q;R) = ∅, the conjugation acts freely on Q ⊂ CP 2∗ and the τ -equivariant parcelling is unique. In such a case, we are getting a more satisfying result: Theorem 13 Let Q(x, y, z) be an irreducible quadratic form with real coefficients and the signature distinct from −3. Then any real polynomial P (x, y, z) of degree d, being restricted to the real conic SR = {Q(x, y, z) = 1} in R3∗, has a representation P (x, y, z) = λ+ (ak,jx+ bk,jy + ck,jz) When the surface SR is an ellipsoid, the represention (44) is unique, up to reordering and rescaling of the multiplyers in the products. In such a case, P gives rise to a unique sequence of multiploles {wk ∈ M (k)}k. We observe thatMR(k) happens to be a nonsingular space: locally RP 2 and CP 1 are diffeomorphic, and Symk(CP 1) ≈ CP k is non-singular. An R-version of the argument that follows Theorem 6 and is centered on formula (21) is valid. Therefore, an R-version of Lemma 6 holds: the “combinatorial” and the “smooth” ramification loci coincide. Note that one always can find a real homogeneous polynomial P of degree k for which the curves P,Q ⊂ RP 2∗ have an empty intersection: just take a linear form L such that the line L := {L = 0} misses the real quadratic curve Q; then consider any polynomial P sufficiently close to (L)k. For such a P , the Z2-equivariant parcelling is unique and, thus, (Φ −1(P |Q=0) is a singleton. Evidently, such polynomials P form an open set. Therefore, ΨRQ : Sym k(RP 2) → RP (V ⊥Q (k; R)) is a map of degree one. These remarks lead to Theorem 14 The multipole space MR(k), which parametrizes completely factorable real homoge- neous polynomials of degree k, is a space of an R∗-bundle over the real variety Symk(RP 2). The ∗-equivariant map ΦRQ induces a surjective mapping Ψ Q : Sym k(RP 2) → RP 2k = RP (V ⊥Q (k; R)) of degree one. Unless Q is a definite form, the map ΦRQ is ramified over the discriminant variety D(ΦRQ) of codimension one. Here D(Φ Q) is comprised of homogeneous polynomials P of degree k on the real cone {Q = 0} for which the surfaces {P = 0} and {Q = 0} share a line of multiplicity at least two.11 Corollary 15 • the multipole space MR(d) = [R3 ◦]d ΣRd is diffeomorphic to the space (R2d+1)◦ ≈ S2d × R. • the multipole space M (d) in (39) is homeomorphic to the space R2d+1. • the real varieties Symd(RP 2) and RP 2d are diffeomorphic12. Proof. Let Q be a positive definite form. Then, the ramification locus D(ΦRQ) = ∅. Since under the corollay’s hypotheses the τ -equivariant parcelling is unique, every real homogeneous P , not divisible by Q, gives rise to a unique leading multipole w(P ) ∈ MRQ(d)—the map Φ Q is 1-to-1. Recalling that Φ Q is onto a vector space of dimension 2d + 1 and that the smooth ramification locus E = D(Φ Q) = ∅, completes the proof. � The diffeomorphism Symd(RP 2) → RP 2d serves as yet another transparent illustration to the Dold-Thom theorem: not only it reveals Sym∞(RP 2) as a K(Z2, 1) space, but we actually know how this stabilization of the homotopy groups occurs. In fact, π1(Sym d(RP 2)) = Z2, and for k > 1, πk(Sym d(RP 2)) = πk(S 2d). In particular, {πk(Sym d(RP 2))} vanish for 1 < k < 2d. We have seen already the main advantages of the multipole representations for the polynomials on quadratic surfaces: such representations are independent on the choice of coordinates in C3∗ or R This is in the sharp contrast with the classical decompositions in terms of the spherical harmonics. In the case of Q = x2 + y2 + z2 and over the reals, the independence of the multipoles under the rotations was observed by many. When Q is not positive-definite, similar observations hold. For a non-degenerated quadratic form Q with real coefficients, let OQ(3,R) denote the group of linear transformations from GL(3;R) that preserve the form. When the signature sign(Q) = 2, then OQ(3,R) contains the Lorenz transformation group (equivalently, the isometry group of a hyperbolic plane) as a subgroup of index two. The next proposition should be compared with Lemma 2 and Theorem 12. We noticed already that, for a real quadratic form Q, the decomposition (8) is a complexification of a similar decomposition V (d;R) = HarQ(d;R)⊕ VQ(d;R) (45) over the reals. Here HarQ(d;R) := Ker(∆Q;R) ∩ V (d;R). Therefore, (45) is OQ(3,R)-equivariant with respect to the natural action on the space V (d;R) and orthogonal with respect to the inner 11equivalently, the curves P and Q in RP 2∗ share a point of multiplicity at least two. 12cf. [A], Theorem 2. product (40). At the same time, the multipole space MR(d) is also equipped with the OQ(3,R)- action induced by the obvious diagonal action on [V (3;R)]d. Furthermore, the map Fact(d,R) → HarQ(d;R) induced by the projection V (d;R) → HarQ(d;R) (defined by (45)) is equivariant. Corollary 16 Given a real polynomial P on SR together with its representation (45) and a transfor- mation U ∈ OQ(3,R), the transformed polynomial U ∗(P )(x, y, z) := P ((x, y, z) · U) on SR acquires a representation in the form U∗(P )(x, y, z) = λ+ (a′k,jx+ b k,jy + c k,jz) , (46) where each new multipole vector (a′k,j, b k,j, c k,j) = (ak,j, bk,j, ck,j) · U T . In other words, the onto ΦRQ : MR(k) → [V (k,R)/VQ(k,R)] ≈ ⊕ k=0 HarQ(d;R) is OQ(3,R)-equivariant. Combining decomposition (45) with Theorem 12 we get its real analog: Theorem 17 Let Q be a real non-degenerated quadratic form. • The space of real homogeneous polynomials admits an OQ(3;R)-invariant decomposition V (d;R) = ⊕d−2k≥ 0 Q k ·HarQ(d− 2k;R) (47) The summands in (47) are orthogonal with respect to the inner product defined by the formula (40). • Any polynomial function F on the surface SR = {Q(x, y, z) = const} can be obtained by restricting to SR a polynomial P ∈ HarQ(R). • For any two polynomials M and N , the Dirichlet problem {∆Q(P ) = M, P |SR = N |SR} (48) has a unique real polynomial solution P . • Any real polynomial P of degree at most d, being restricted to the surface SR, admits a Maxwell-type representation P (x, y, z) = d−2k≥ 0 Q(x, y, z)d−k+ 2 · ∇ . . .∇ ud−2k,k Q(x, y, z)− where {uj,k} are complex 3-vectors. These vectors are real and representation (49) is unique, provided that Q is positive-definite. 4 Why one does rarely see multipoles in non-quadratic skies? Let Q(x, y, z) be an irreducible form of degree l over C. Then S := {Q(x, y, z) = 1} is the surface in C3∗ and Q := {Q(x, y, z) = 0} is an irreducible curve in CP ∗ of degree l. As before, we denote by VQ(d) the set of homogeneous degree d complex polynomials that are divisible by Q. Let V ⊥Q (d) ≈ V (d)/VQ(d) be an orthogonal complement to VQ(d) in V (d). For any sequence of non-negative integers {di} so that 1≤i≤s di = d, consider a map V ⊥Q (di) → V Q (d) (50) which is defined by taking the product of homogeneous polynomials Pi ∈ V Q (di) restricted to the surface {Q(x, y, z) = 0}. As before, the subgroup Hs ⊂ (C ∗)s of rank s− 1 acts freely on i=1[V Q (di) ◦] by scalar multipli- cation. By the definition of Hs, the map η is constant on the orbits of this action. We view the partition {d = 1≤i≤s di} as a non-increasing function ω : i → di on the index set {1, 2, . . . , s}. Denote by Sω the subgroup of the permutation group Ss that preserves ω. Let Σω be an extension of Sω by Hs that is generated by the obvious actions of Sω and Hs on (C ∗)s ≈ i=1 C Evidently, η is an Σω-equivariant map. Thus, it gives rise to a well-defined map V ⊥Q (di) Σω → V Q (d) ◦ (51) Because Q is irreducible, the map Φω has V Q (d) ◦, and not just V ⊥Q (d), as its target. As before, we introduce the mulipole space MQ(ω) := V ⊥Q (di) Σω (52) which is a space of a principle C∗-fibration over the orbifold BQ(ω) := CP (V ⊥Q (di)) Sω (53) As in the case of quadratic forms Q, one has a map Θ : MQ(ω) → Fact(ω), where Fact(ω) ⊂ V (d) is the variety of homogeneous degree d polynomials in x, y, and z that admit a factorization as a product of polynomials of the degrees {di}1≤i≤s that are prescribed by the partition ω. The map Θ takes each multipole to the product of the corresponding polynomial factors. Unlike the case of a quadratic Q, Θ may not be a 1-to-1 map, although, its generic fiber is a singleton. 13To simplify our notations, we do not indicate (as before) the dependency of the map on Q. This conclusion follows from the unique factorization property for the ring C[x, y, z]: just consider elements of Fact(ω) that are products of irreducible factors of the degrees prescribed by ω. The same uniqueness of factorization implies that each fiber of Θ is finite: there are only finitely many ways of organizing irreducible factors, in which a polynomial P ∈ Fact(ω) decomposes, into blocks of degrees {di}. The map Θ is not surjective either. However, Φω takes the multipole space onto the space FactQ(ω) of degree d homogeneous polynomials on the surface {Q = 0} that admit factorizations subordinate to ω: Φω : MQ(ω) → Fact(ω) → FactQ(ω) ⊂ V Q (d) ◦ (54) Definition 4 Let d be a natural number and ω = {d = i=1 di} its partition. Let Z be a finite set equipped with a multiplicity function µ : Z → N whose l1-norm ‖µ‖1 is ld. A generalized ω-parcelling of (Z, µ) is a collection of functions µi : Z → N, such that i µi = µ • ‖µi‖1 = ldi When Z is comprised of ld points and each µi takes only two values 0, 1, the generalized parcelling is called just an ω-parcelling. Any polynomial P (x, y, z) of degree d that is not divisible by Q determines a multiplicity function µ : Z(P,Q) → N whose l1-norm is ld. Here Z(P,Q) := P ∩ Q ⊂ CP ∗ is a finite set. If such a polynomial P is a product i Li, where deg(Li) = di, then the Li’s give rise to a unique generalized ω-parcelling i µi. Lemma 8 Any generalized ω-parceling i µi of a given multiplicity function µ on a finite set Z ⊂ Q corresponds to at most one multipole in the space MQ(ω). Proof. Assume that, for each index i, there exist a polynomial Li that realizes µi on the finite intersection set Z ⊂ Q. Such polynomial is not divisible by Q. Put Zi = Z(Li, Q) := Li∩Q. Then, employing the Bezout Theorem, any other polynomial Mi that realizes the same multiplicity on the same intersection set Zi must be of the form Mi = λiLi+Q ·Ri. We notice that if Li ∈ V Q (di), then Mi /∈ V Q (di), provided Ri 6= 0. � Corollary 18 The map Φω has finite fibers over FactQ(ω). Proof. An element of P ∈ V ⊥Q (d) ◦ is determined, up to proportionality, by its multiplicity function µ : Z(P,Q) → N. Now the claim of the corollary follows from Lemma 8 and the observation that a given multiplicity function admits only finitely many generalized ω-parcellings. � Of course, not any generalized ω-parceling on a given pair (Z ⊂ Q, µ) is realizable by a product i Li with the properties as above. Crudly, this happens because not any ldi points on Q can be placed on a curve Ci of degree di that does not contain Q as its component. A curve of degree di can always accommodate (d2i + 3di)/2 points in CP ∗ . Thus, if an inequality (d i + 3di)/2 ≥ dil is valid, that is, if di ≥ 2l− 3, the right curve might be found; but it is still unclear how to avoid the very real possibility that Ci ⊃ Q when di ≥ l. In fact, such a possibility is a reality! Unfortunately, unless l = deg(Q) = 2 or s = 1, the image of Φω is of a smaller dimension than the one of the target space V ⊥Q (d) ◦. As a result, there is no analog of the Sylvester Theorem on non-quadratic surfaces; when deg(Q) > 2, a generic element of V ⊥Q (d) ◦ is irreducible. Let us explain these claims. Recall that for any d ≥ l, dim(V ⊥Q (d)) = {(d2 + 3d)− [(d− l)2 + 3(d− l)]} = l (2d− l+ 3). Since the dimension of the group Hs is s− 1, we get dim(V ⊥Q (d))− dim(MQ(ω)) = (2d − l + 3)− (2di − l + 3) + (s− 1) = [l2 − 3l + 2], provided all di ≥ l. Under these hypotheses, the difference of the two dimensions vanishes only when l = 1, 2 or s = 1. Hence, Lemma 9 For l = deg(Q) > 2 and {di ≥ l}1≤i≤s, the map Φω is not onto, i.e. a generic polynomial from V ⊥Q (d) ◦ is not a product of polynomials of degrees ≥ l. The codimension of Φω(MQ(ω)) in V ⊥Q (d) is [l2 − 3l + 2]. For example, on a cubic surface , dim(V ⊥Q (d))− dim(MQ(ω)) = (s− 1), provided {di ≥ 3}1≤i≤s. If we drop the hypotheses {di ≥ l}, the computation is a bit more involved: dim(V ⊥Q (d)) − dim(MQ(ω)) = (2d− l + 3)− i: di≥l (2di − l + 3)− j: dj<l (dj + 3) + (s− 1) We conjecture that the RHS of the formula above is always positive, unless l = 1, 2 or s = 1. 5 Multipoles and function approximations on quadratic surfaces In this section we will be concerned with polynomial approximations of holomorphic functions f : S → C on an irreducible complex quadratic surface SQ = {Q(x, y, z) = 1}, as well as with polynomial approximations of continuous functions f : SRQ → R on its real version. As before, we would like to represent the approximating polynomials in terms of their multipoles. As we use polynomials of higher and higher degrees to improve the approximations, the issue is stability of the multipole representations. In general, such stability is absent for several reasons: 1) the intrinsic ambiguities of the multipole representations for complex plynomials; 2) the difficulty of converting an analytic function on a surface into a “homogeneous” analytic function in the ambient space (homogenizing polynomials worked well). Even abandoning multipole representations in favor of linear methods of harmonic analysis, does not eliminate the stability issue instantly: in general, the coefficients of approximating polynomials fail to stabilize. However, if the approximating polyno- mials are linear combinations of mutually orthogonal and normalized polynomials (analogous to the Legendre polynomials), the coefficients of the combinations will stabilize. By introducing appro- priate notions of orthogonality for polynomials on a quadratic surface, we aim to establish similar facts for polynomial approximations there14. Then we combine harmonic analysis with non-linear methods of multipole representation for polynomials (as it is done in formula (3) in Theorem 3). Let C(K) denote the algebra of all continuous C-functions on a Hausdorff compact space K. Recall that a uniform algebra is a closed (in the sup-norm) subalgebra A ⊂ C(K) that separates points of K. Such an algebra is called antisymmetric, if any real-valued function from A is constant. A subset Y ⊂ K is called an antisymmetry set for A if any function from A, which is real-valued on Y , is a constant. The Bishop Theorem about antisymmetric subdivisions (cf. [G], Theorem 13.1) claims that the maximal sets of antisymmetry {Eα} are closed and disjoint, and their union is K. Moreover, if f ∈ C(K) and, for each α, f |Eα ∈ A|Eα , then f ∈ A. In particular, if each Eα is a singleton, then A = C(X). For a space X ⊂ Cn, let us denote by P̄(X) the closure in the sup-norm on compacts in X of the algebra P(X) generated by all complex polynomial functions. Note that if any two points in X can be separated by a real-valued polynomial, then Bishop’s Theorem implies that P̄(X) = C(X). For instance, consider a section H of the complex surface SQ by a totally real subspace V 3 ⊂ C3 — an image of R3 ⊂ C3 under a complex transformation A ∈ GL(3;C). Since any two points in R3, can be separated by a real-valued polynomial, the same property holds for any two points in V 3, and thus, in H. By the Bishop Theorem, any continuous function f on H admits an approximation in the sup-norm on compacts by complex polynomials. In particular, this conclusion is valid when H is one of the real surfaces ΥQ ⊂ (R 3)A−1 or SRQ ⊂ R 3 which have been employed on many occasions. For a compact set K ⊂ SQ, denote by K̂ the polynomial hull (closure) of K. It consists of all points v in C3 with the property: |P (v)| ≤ supw∈K|P (w)| for any complex polynomial P . Because for any point v /∈ S and w ∈ S, we have |(Q − 1)(v)| > |(Q − 1)(w)| = 0, the polynomial closure K̂ must be contained in S. In fact, the polynomial closure K̂ of a compact K ⊂ C3 must be a polynomially convex compact set. A theorem by Oka and Weyl (cf. [G], Theorem 5.1) claims that any complex analytic function, defined in a neighborhood of a compact polynomialy convex set, admits an approximation in the sup-norm on K by complex polynomials. Thus, any analytic function f(x, y, z) defined in a neighborhood of K̂ ⊂ SQ, K being a compact in SQ, can be uniformly approximated on K̂ by complex polynomials. When dealing with families of functions on non-compact sets, we pick the uniform convergency on compact subsets as a default topology. In this topology, the subalgebra O(SQ) of holomorphic functions is closed in the algebra of all continuous functions C(SQ) (cf. [GuR], Lemma 11). In particular, if a sequence of polynomials (in x, y, and z) is converging in the sup-norm on every compact in SQ, then its limit is a holomorphic function on SQ. One can employ any expanding 14The spherical harmonics reflect a particular case of such orthogonality. family {Kr}1≤r≤∞ (i.e., Kr ⊂ Kr+1 and ∪r Kr = SQ) of polynomially convex compacts Kr ⊂ SQ to build a sequence {Pr} of polynomials that approximate a given holomorphic function f ∈ O(SQ). Let F(SQ) ⊂ O(SQ) be a subset formed by the functions that admit a representation as a series , (55) where each Lkj is a linear form in x, y, and z. The series is required to converge uniformly on each compact K ⊂ SQ (as we remarked before, any such uniformly converging series produces a holomorphic function on SQ). In fact, one can define a similar set F(K) ⊂ C(K) for any closed K ⊂ C3. Due to Theorem 11, any polynomial on SQ is of the form (33) (which is a special case of (55)). Therefore, when P(K) = C(K) for a compact K ⊂ SQ, then F(K) is dense in C(K) as well. Examining (55), we observe that if this series converges at a point v = (x, y, z) it must converge absolutely at any other point λ · v, were the complex number λ has modulus less than 1. For any set Y ⊂ C3, we denote by Y • the set {λv| v ∈ Y, λ ∈ C, |λ| < 1} and call it the round hull of Y . Consider the set S•Q. Because any complex line through the origin that does not belong to the complex cone {Q = 0} hits SQ at a pair of antipodal points, S Q is an open domain in C 3, com- plementary to the cone whose boundary contains SQ (the origin belongs to S Q). Therefore, any function f ∈ F(SQ) must be a restriction of a function which is analytic in S Q and continuous in S•Q ∪ SQ. I doubt that the converse statement is true. Note that a given function f on SQ may have many analytic extensions in S•Q: for example, 1 extends to 1 and to 1/Q (the latter has poles along the complex cone). If a section H of SQ by a totally real subspace V 3 ⊂ C3 is an ellipsoid, then its interior in V 3 coincides with S•Q ∩ V 3. Thus, any real function f on H that admits a representation as in (55) must be real analytic in the interior of the ellipsoid. So, it is represented by its Taylor series at the origin; on the other hand, series (55), uniformly converging in the vicinity of the origin, gives rise to a very specialized Taylor series (just count the dimensions of the coefficient spaces of each degree to see how special it is). Since any function from O(SQ) admits a polynomial approximation on compacts, the subset F(SQ) is dense in in the space of all holomorphic functions. So, F(SQ) is squeezed between the vector space O(SQ) and its dense subspace P(SQ). It is not even clear whether F(SQ) is a vector space. To understand the structure of the set F(SQ) is a challenge; unfortunately, our progress towards this goal is minimal. Note that, if a holomorphic function vanishes on a totally real analytic surface H ⊂ SQ, it must vanish in a neighborhood of H in SQ. We summarize the observations above in Theorem 19 Any holomorphic function f on complex surface SQ is a limit in the topology of uniform convergence on compacts of polynomial functions in C3. The approximating polynomials each admit a representation as in (33). As a result, such an f can be described by a double-indexed set of multipoles {wjk ∈ M(k)}0≤j≤k<∞. A subset F(SQ) of functions that admit a representation as in (55) is dense in the space of all holo- morphic functions O(SQ) and invariant under multiplications by scalars and the natural OQ(3;C)- action15. Each function f ∈ F(SQ) admits a canonic holomorphic extension f(λx, λy, λz) = Lkj(x, y, z) into the round hull S•Q. Here (x, y, z) ∈ SQ and |λ| < 1. Any continuous function f : H → C on the totally real quadratic surfaces H = ΥQ or H = S a limit in the topology of uniform convergence on compacts in H of polynomial functions in C3. As a result, the set F(H) is dense in C(H). Each function from F(H) admits a canonic analytic extension into the open portion of real cone over H, a portion that is bounded by H and contains the origin. Again, f can be described by a double-indexed set of multipoles {wjk ∈ M(k)}0≤j≤k<∞. Under the hypotheses of Theorem 19, unless a given function is in the sets F(SQ) or F(H), the set of multipoles that represent it is far from being unique. In order to achieve some kind of uniqueness, we need to consider multipole representations for the L2-integrable functions and to employ the orthogonality. Hence, consider the vector space P(SQ) of all polynomial functions restricted to SQ and equipped with the inner product 〈f, g〉 defined by (34). If P is a homogeneous nonzero polynomial, then 〈P,P 〉 > 0. An homogeneous polynomial is determined by its restriction to SQ. Evidently, 〈P,P 〉 = 0 implies that the restriction of P to ΥQ is zero. Since ΥQ ⊂ SQ is a totally real analytic submanifold, P |ΥQ = 0 implies that P vanishes in the vicinity of ΥQ in SQ. By analyticity, P must vanish everywhere in SQ, and hence, P is a zero polynomial. As a result, 〈P,P 〉 gives rise to a norm on the space of homogeneous polynomials V (d). Because any function from P(SQ) is a restriction of an homogeneous polynomial, we get a non-degenerated Hermitian inner product on the vector space P(SQ). In view of Theorem 12 and by a similar line of arguments, 〈P,P 〉 = 0 implies that P = 0 for any complex polynomial P ∈ Ker(∆Q). Therefore, being restricted to a space of Q-harmonic polynomials, the inner product 〈f, g〉 in (34) gives rise to an Hermitian structure and an L2-norm ‖P‖ΥQ . In particular, each space HarQ(k) ≈ V Q (k) inherits this norm, and HarQ(k) is orthogonal to HarQ(l), provided l 6= k. Now, consider the vector space k=0 HarQ(k) formed by sequences of vector-polynomials {Pk ∈ HarQ(k)} and the subspace L 2 := ⊕ k=0 HarQ(k) formed by infinite sequences {Pk ∈ HarQ(k)} subject to the condition k=0 ‖Pk‖ < ∞. Let L2(ΥQ) denote the complex Hilbert space of L2-integrable functions on the ellipsoid ΥQ. Every function f ∈ L2(ΥQ) defines a unique system of its “Fourier components” {fk ∈ HarQ(k)}. Each fk is the unique polynomial from HarQ(k) that delivers the minimum minP∈HarQ(k) ‖f − P‖ΥQ . By Theorem 19, any f ∈ C(ΥQ) is a limit in the sup-norm on ΥQ of harmonic polynomials, it must be also the limit in L2(ΥQ) of the same sequence of polynomials. Therefore, if f ∈ C(ΥQ) is orthogonal to all the subspaces HarQ(k), it must be the zero function. Hence, as an element of L2(ΥQ), f = k=0 fk, and ‖f‖ k=0 ‖fk‖ 15By its definition, every function from F(SQ) can be described by a sequence of multipoles {wk ∈ M(k)}0≤k<∞. particular, any f ∈ O(SQ) determines a unique system of its harmonics {fk ∈ HarQ(k)}. Moreover, it is determined by f |ΥQ , and thus, by its harmonics {fk}. Evidently, the sequence of partial sums {f[d] := k=0 fk ∈ Ker(∆Q)}d converges in L2(ΥQ) to f . In terms of homogeneous polynomials, we get an analogous sequence {f{d} := ⌈(d−k)/2⌉fk}d converging in L2(ΥQ) to f . Similar arguments can be applied, instead of the ellipsoid ΥQ, to any surface H ⊂ SQ that is a section of SQ by a totally real subspace V 3 ⊂ C3. In particular, they are valid for SRQ. First, we pick a measure dm on H such that any polynomial P |H ∈ L2(H) (when H is compact, one can choose any measure of finite volume). For example, consider the area 2-form ω on H induced by the imbeddingH ⊂ C3 and multiply it by the factor exp[−(xx̄+yȳ+zz̄)] to get the “right measure” dm. Then we define a new inner product by 〈f, g〉H = f · ḡ dm. Since H is totally real, this will give rise to an Hermitian structure in each space V (k). Notice that the multiplication-by-Q imbedding V (k) → VQ(k+2) ⊂ V (k+2) is an isometry. As before, we form the orthogonal compliments V Q (k) to the subspaces VQ(k) (the only difference is that now the space V Q (d) could be different from the space HarQ(d) of Q-harmonic polynomials). Then we argue that the Hilbert space L2(H, dm) is a closure of ⊕∞k=0 V Q (k) in the L2-norm. As before, any continuous function f on H acquires a unique representation f = k=0 fk, where {fk ∈ V Q (k)}, and ‖f‖ k=0 ‖fk‖ Thanks to Theorems 7 and 11, each complex homogeneous polynomial fk ∈ V Q (k) can be repre- sented by some multipole w k ∈ M(k). Similarly, any real homogeneous polynomial fk ∈ V Q (k;R) can be represented by some multipole w k ∈ M (k). When Q is positive definite, these real multi- poles are unique. However, in general, the ambiguity of the multipole representation could cause trouble. So, we need to choose the representing multipoles with some care. Due to the embedding Θ : M(k) → V (k) (see (15)) with the image Fact(k), the multipole space M(k) acquires a metric ρ induced by the LH2 -norm ‖P‖H in V (k). The lemma below helps to estimate the size of the fiber Φ Q (u) over u ∈ V Q (k) in terms of an universal angle θk = θ(k,H, dm) and the L 2 -norm of u: Lemma 10 Consider the distance function ρ on M(k) that is induced by the LH2 -norm ‖ ∼ ‖H in V (k) via the embedding Θ : M(k) → Fact(k) ⊂ V (k). Then there exist an angle 0 < θk ≤ π/2 so that, for each u ∈ V ⊥Q (k) ⊂ V (k), the distance from any w ∈ Φ Q (u) to the zero multipole is at most ‖u‖/sin(θk), and thus the diameter of the fiber Φ Q (u) is at most 2‖u‖/sin(θk). Proof. Let S(k) denote a unit sphere (with respect to the ‖ ∼ ‖H -norm) in V (k) and centered at the origin. Because Q is irreducible, Fact(k) ∩ VQ(k) = ∅. Thus, the compact sets S(k) ∩ Fact(k) and SQ(k) = S(k)∩VQ(k) are disjoint. Therefore, there is a number 0 < θ ≤ π/2 so that the angle between any two vectors u ∈ S(k)∩Fact(k) and v ∈ SQ(k) is greater than or equal to θ. Note that S(k) ∩ Fact(k) and SQ(k) are invariant under the circle action S 1 ⊂ C∗. So, Fact(k) and VQ(k) also are real cones with their tips at the origin and bases S(k) ∩ Fact(k) and SQ(k). We conclude that the angle between any two vectors u ∈ Fact(k) and v ∈ VQ(k) has the same lower bound θ > 0. Now consider an open real cone CQ(k) ⊂ V (d) comprised of vectors that form an angle φ < θ with the subspace VQ(k) and a complementary cone C Q(k) := V (d) \ CQ(k) ⊃ V Q (k). The argument above shows that Fact(k) ⊂ C⊥Q(k). Hence, the distance from any w ∈ Φ Q (u) to the zero multipole is at most ‖u‖/sin(θk), and the diameter of the fiber Φ Q (u) is at most 2‖u‖/sin(θk). Corollary 20 Consider a continuous function f ∈ L2(H) and its orthogonal decomposition k=0 fk, where fk ∈ V Q (k). Then, for any choice of the multipoles wk ∈ Φ Q (fk), sin2(θk) · ρ(wk,0) 2 < ∞, (56) where ρ(wk,0) = ‖Θ(wk)‖H . � It seems to be far from trivial to understand the asymptotic behavior of {sin(θk)} as k → ∞. Perhaps, the lack of understanding of this asymptotics it is the most significant gap in our analysis. To state the last claim in the next theorem, we need one technical definition that likely has very little to do with the essence of the statement. The set D(ΦQ) ⊂ V Q (k) is a complex algebraic variety. Itis stratified by algebraic sets {Dk,π} which are labeled by various partitions π of 2k. This labeling is done by attaching the divisor P ∩Q ∈ Sym2k(Q), or rather the partition π of 2k defined by the multiplicity function of P ∩Q, to each homogeneous polynomial P (x, y, z) restricted to the cone {Q = 0} and viewed as an element of V ⊥Q (k). In particular, when π = {2d = 1+1+1+ . . .+1} or {2d = 2 + 1 + 1 + . . . + 1}, then Dk,π = V Q (k) or Dk,π = D(ΦQ), respectively. The natural partial order among partitions reflects the inclusions of the corresponding strata. If we delete all the substrata from a given stratum Dk,π, we get a “pure” stratum that we denote D k,π. The variety D(ΦQ) is a Whitney stratified space; as a result, the vicinity of every stratum D k,π has a structure of a bundle whose fiber is a real cone over another stratified space Lkk,π —the link of D k,π. We will make use of this fact together with another important feature of the stratification Dk,π: namely, all the strata of Lkk,π have even real codimensions. We say that a parametric curve γ : [0, 1] → V ⊥Q (k) ≈ HarQ(k) is tame if it consists of a finite number of arcs, each of which has the following property: the interior of each arc is contained in some stratum D◦k,π. We say that a continuous function family {ft ∈ L2(ΥQ)}0≤t≤1 is tame, if for each k, the path {(ft)k ∈ HarQ(k)}0≤t≤1 is tame. First, consider the functions f ∈ O(SQ) such that, for each k, the polynomial fk ∈ HarQ(k) ≈ V ⊥Q (k) does not belong to the ramification locus D(ΦQ) ⊂ V Q (k) of the map ΦQ (the rest of the functions form a complex codimension one subset D ⊂ O(SQ)). Over the compliment to the variety D(ΦQ), the map ΦQ is a covering map. Thus, for each initial lifting, the deformation (ft)k admits a unique lifting to the multipole space, as long as (ft)k ∈ V Q (k) \ D(ΦQ). Next, for any tame t-family {ft ∈ O(SQ)}0≤t≤1, consider the tame curve {(ft)k ∈ V Q (k)}0≤t≤1 and the first arc γ in a finite sequence of arcs that form this curve. There are two possibilities: 1) the arc starts at a stratum D◦k,π and is confined to it for a while, 2) the arc starts at a stratum D but moves instantly into an ambient stratum D◦k,π′ . In the first case, over D k,π, ΦQ is a covering map and there is a unique lifting of γ extending each lifting γ̃(0) of γ(0). In the second case, we claim that, for any lifting γ̃(0) of γ(0), in the vicinity of γ̃(0), the map ΦQ is onto. Indeed, it is a proper holomorphic map, and thus, its image must be an analytic space (see [N], Theorem 2, page 129). Because ΦQ is finite, the image of a neighborhood of γ̃(0) under ΦQ must be of the maximal dimension, and hence, must contain a neighborhood of γ(0). As a result, the set Φ−1Q (γ) (it is a finite graph) must be present in any neighborhood of γ̃(0); so, we can lift γ to an arc γ̃ that starts at γ̃(0). An induction by the number of arcs in the curve (ft)k proves the existence of its lifting to the multipole space. Note that an analogous argument fails for ΦRQ: a finite image of a real analytic set is a real semi-analytic set that can miss the arc γ. Therefore, in Theorem 22 the lifting property for tame deformations is absent. The arguments above prove the following theorem: Theorem 21 Let Q(x, y, z) be an irreducible complex quadratic form, and let SQ be a complex quadratic surface defined by the equation {Q = 1}. Denote by A a complex change of coordinates that reduces the form Q to the sum of squares. Let ΥQ ⊂ SQ be a totally real ellipsoid defined by the equations {Q((x, y, z)A) = 1, Im((x, y, z)A) = 0} and equipped with the measure defined by (34). Let f be an analytic function on SQ. Then there exists sequence of multipoles {w k ∈ M(k)}0≤k<+∞ such that: • the sequence gives rise, via the maps {ΦQ}, to complex Q-harmonic polynomials P d (x, y, z) = k=0ΦQ(w k ), where the mutually orthogonal polynomials {ΦQ(w k ) ∈ HarQ(k)} are uniquely determined by f . • as d → ∞, the polynomials {P } converge in the space L2(ΥQ) to the function f |ΥQ , and therefore, uniquely determine f ∈ O(SQ). • for a given f , there are at most (2k − 1)!! choices for each multipole w • the multipoles {w } satisfy property (56) from Corollary 20. • for any tame deformation {ft ∈ O(SQ)}0≤t≤1 of the function f = f0, there exists a continuous deformation {w k ∈ M(k)} of the ft-representing multipoles, such that {w k = w k}. For functions f and their continuous deformations ft outside a subspace D ⊂ O(SQ) of complex codimension one and for each choice of the appropriate multipoles {w }, the lifting of the deformation ft to the multipole spaces is unique. � Similarly, we get Theorem 22 Let Q(x, y, z) be an irreducible real quadratic form. Let SRQ := {Q = 1} be a real quadratic surface, equipped a measure dm for which any polynomial in x, y, and z is an L2-integrable function on the surface. Let f be an L2-integrable continuous function on S Q. Then there exists sequence of multipoles {w k ∈ M(k)}0≤k<+∞ such that: • the sequence gives rise, via the maps {Φ Q}, to real polynomials P d (x, y, z) = where the mutually orthogonal polynomials {Φ )} are uniquely determined by f ; • As d → ∞, the polynomials {P d } converge to f in the space L2(S • The multipoles {w } satisfy property (56) from Corollary 20. When Q is positive-definite, • the mutually orthogonal polynomials {Φ ) ∈ HarQ(k;R)}; • each multipole w k is uniquely determined by f ; • for any continuous deformation {ft ∈ C(S Q)}0≤t≤1, of the function f = f0, there exists a unique continuous deformation {w ∈ M(k)} of the ft-representing multipoles, such that Acknowledgments. I am grateful to Jeff Weeks for introducing me to the subject. My conversations with Michael Shubin about the aspects of this investigation, motivated by harmonic analysis, were equally enlightening. I also would like to thank Blaine Lawson for explaining to me a few facts and constructions that turned out to be very helpful. 6 References [A] Arnold, V., Topological Content of the Maxwell Theorem on Multipole Representation of Spher- ical Functions, Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center, 7 (1996), 205-217. [B] Bennett, C. et al., First year Wilkinson Microwave Anisotropy Probe (WMAP 1) observations: preliminary maps and basic results, Astrophysical Journal Suppliment Series 148 (2003), 1-27. [Ch] Chow, W.-L., On the equivalence classes of cycles in an algebraic variety, Ann. of Math. 64 (1956), 450-479. [CHS] Copi C.J., Huterer D., Starkman, G.D., Multipole vectors—a new representation of the CMB sky and evidence for statistical anisotropy or non-Gaussianity at 2 ≤ l ≤ 8, to appear in Phys. Rev. D. (astro-ph/0310511). [CH] Courant, R., Hilbert, D., Methods of Mathematical Physics, v.1, Interscience Publishers, 1953, 514-521. [D] Dennis M. R., Canonical representation of spherical functions: Sylvester’s theorem, Maxwell’s multipoles and Majorana’s sphere (arXiv:math-ph/0408046 v1). http://arxiv.org/abs/astro-ph/0310511 http://arxiv.org/abs/math-ph/0408046 [DT] Dold, A., Thom, R., Quasifaserungen und unendliche symmetrische producte, Ann. of Math. (2) 67 (1956), 230-281. [EHGL] Erisen, H.K., Banday, A.J., Górski, K.M., and Lilje, P.B., Asymmetries in the cosmic mi- crowave background anisotropy field, Astrophysics. J. 605 (2004) 14-20, (arXiv:astro-ph/0307507). [G] Gamelin, T. W., Uniform Algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969. [GuR] Gunning, R.C., Rossi, H., Analytic Functions of Several Complex Variables, Prentice-Hall, 1965. [H] Hartshorne, R., Algebraic Geometry, Springer-Verlag , 1983. [Ha] Hatcher, A., Algebraic Topology, Cambridge University Press, 2002. [K] Katz., G., How Tangents Solve Algebraic Equations, or a Remarkable Geometry of Discriminant Varieties, Expositiones Mathematicae 21 (2003) 219-261. [KW] Katz, G. Weeks, J., Polynomial Interpretation of Multipole Vectors, Phys. Rev. D. 70, 063527 (2004) (arXiv:astro-ph/0405631). [L] Lachièze-Rey, M., Harmonic projection and multipole vectors, preprint (arXiv:astro-ph/0409081). [M] Maxwell, J.C., 1891 A Treatise on Electricity and Magnetism, v.1, (3rd edition) Clarendon Press, Oxford, reprinted by Dover, 1954. [N] Narasimhan, R., Introduction to the Theory of Analytic Spaces, Lecture Notes in Mathematics 25 (1966), Springer-Verlag. [Sh] Shubin, M. A., Pseudo-differential Operators and Spectral Theory, Nauka, Moscow, 1978. [S] Sylvester, J.J., Note on Spherical Harmonics, Philosophical Magazine, v. 2m, 1876, 291-307 & [S1] Sylvester, J.J., 400. Collected Mathematical Papers, v.3, 37-51, Cambridge University Press, Cambridge, 1909. [TOH] Tegmark, M., de Oliveira-Costa and Hamilton A.J.S., A high resolution foreground cleaned CMB map from WMAP, Phys. Rev. D. 68 (2003) 123523, (arXiv:astro-ph/0302496). [W] Weeks, J., Maxwell’s Multipole Vectors and the CMB, preprint (arXiv:astro-ph/ 0412231). William Paterson University, Wayne, NJ 07470-2103, U.S.A. e-mail: [email protected] & [email protected] http://arxiv.org/abs/astro-ph/0307507 http://arxiv.org/abs/astro-ph/0405631 http://arxiv.org/abs/astro-ph/0409081 http://arxiv.org/abs/astro-ph/0302496 http://arxiv.org/abs/astro-ph/0412231 Introduction Multipoles and polynomials on complex quadratic surfaces Deconstructing reality Why one does rarely see multipoles in non-quadratic skies? Multipoles and function approximations on quadratic surfaces References

0704.1177

Temperature effects on quantum cloning of states and entanglement

Temperature effects on quantum cloning of states and entanglement S. Baghbanzadeh1,2 and A. T. Rezakhani3, 4 1Department of Physics, Sharif University of Technology, P. O. Box 11155-9161, Tehran, Iran 2Department of Physics, Iran University of Science and Technology, Narmak, P. O. Box 16765-163, Tehran, Iran 3Center for Quantum Information Science and Technology, and Departments of Chemistry and Physics, University of Southern California, Los Angeles, CA 90089, USA 4Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada Performances of the symmetric universal and phase-covariant cloning transformations and entanglement clon- ers — qubit case — are investigated when the initial state of the hardware or the original state to be cloned is weakly coupled to a thermal environment. Different behaviors of each of these transformations are analyzed and contrasted with the ideal cases. PACS numbers: 03.67.-a, 03.67.Mn, 03.65.Ud Introduction.— It has been known that based on the prin- ciples of quantum mechanics, accurate cloning of arbitrary quantum state is impossible [1]. Nevertheless, on the account of the significant role of copying in quantum computation and quantum communication, a variety of approximate quantum cloning transformations have been proposed, e.g., universal cloning (UC) machines producing identical copies from ar- bitrary input qudits [2, 3, 4, 5, 6], phase-covariant cloning (PCC) machines of a class of partially known qudits [7, 8], and optimal entanglement cloning machines [9, 10]. For re- cent reviews see Refs. [11]. A question of practical relevance is how uncontrollable en- vironmentally induced decoherence or dissipation can affect performance of quantum cloning machines. In closed sys- tems, an initially pure state evolves to another pure state. In practice, however, preparation of pure states and/or keeping them pure are not generally easy tasks. In general, interac- tion with an environment degrades purity of quantum systems and makes their states mixed. A usual effect that a thermal environment can cause is thermalization (as a kind of dissipa- tion) [12, 13]. That is, because of interaction with the environ- mental degrees of freedom which are in thermal equilibrium, the quantum system will also be driven toward equilibrium. It should be noted that a generic isolated quantum many-body system does also relax to a state well described by the standard statistical-mechanical prescription [14]. In this paper, our aim is to investigate temperature effects on the performance of the cloning machines. It has been known that decoherence can put a limitation on the number of clones that a quantum cloner can generate [15]. There is also a model in which the robustness of the cloner increases with the number of qubits [16]. Through a thermalizing process, the density matrix of the system ̺ in long time will approach the Boltzmann state ̺th = e −βH/Z , where kBβ is inverse temperature (kB is the Boltzmann constant), H is the Hamiltonian of the open quantum system, and Z = Tr(e−βH) is the partition func- tion. Energy and phase relaxation processes, with the time- scales T1 and T2, respectively, are common processes present when approaching an induced equilibrium state. For a more precise and elaborate discussion of dissipation, thermaliza- tion, and decoherence see Ref. [12]. Some (phenomenolog- ical) models for the underlying dynamics of the thermaliza- tion have already been proposed [13]. We assume that the time-scale in which typical correlation functions of the en- vironment decay, tenv., is much smaller than all other time- scales, i.e., tenv. ≪ τc, Tdiss. = min{T1, T2, TO}, where τc is the time-scale of the cloning process and TO is the time-scale dictated by all other relaxation mechanisms. This assumption is important for the Markovian analysis of the dynamics of the thermalization [17, 18]. This implies that during the cloning process, a negligible amount of information flows from the environment to the system (or vice versa). Here, we also as- sume that τc . Tdiss.. This extra condition allows us to ig- nore dynamical effects of the thermalization, hence consider a simple static (toy) model — explained below — to bring temperature into play. Despite these simplifying assumptions, we will argue that the result is still reliable enough to give a hint about how temperature effects can change performance of different cloning machines such as the universal cloners, phase-covariant cloners, and entanglement cloners. Indeed, such investigation has an immediate importance in attempts to realize quantum cloning in systems where (due to thermal and perhaps other noise effects) the preparation of pure states, whether initial state of the system to be cloned or the quantum hardware, is difficult, such as in NMR systems [19, 20]. For another study using a different approach, see Refs. [21, 22]. For the purpose of illustration, we only consider the case of symmetric 1 → 2 qubit cloners. Extension to M → N qudits is straightforward as well. Optimal universal and phase-covariant cloning transfor- mations.— In the universal cloning transformation, it is usu- ally assumed that the qubit state to be cloned is a pure state, |Ψ〉a = cos θ2 |0〉 + e iφ sin θ |1〉, and the blank copy (b) and the quantum cloning machine (also called ancillary system, c) are each in a known pure state, say |0〉 [2, 3, 4, 23]. The symmetric cloning transformation, then, acts in this way: U (|Ψ〉a|0〉b|0〉c) = |Υ〉abc, where Trbc(|Υ〉abc〈Υ|) = Trac(|Υ〉abc〈Υ|). The latter condition guarantees that the fi- nal clones both have the same states, ̺outa = ̺ b . A mea- sure to quantify performance of a cloning machine is the fi- delity between the original and the output states, F (ρ, σ) = ρ1/2σρ1/2 . Optimization of the fidelity over all input states on the Bloch sphere results in the qubit optimal univer- sal cloner, in which F = 5/6 [2, 3]. For orbital states, where θ is an a priori known constant and φ ∈ [0, 2π), a class of http://arxiv.org/abs/0704.1177v3 θ0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 PSfrag replacements (a) (b) 0 00 0 FIG. 1: Fidelity (F ) of UC vs θ for some values of η: (a) ǫ = 5/11 and (b) ǫ = 2/3. phase-covariant cloning machines has been suggested [8]. Af- ter the cloning process, in the computational basis {|0〉, |1〉} (the eigenvectors of σz = diag(1,−1)) each of the clones can be identified by the density operator: ̺out00 = µ 2̺in00 + ν 2 and ̺out01 = 2µν̺ 01, where µ 2 + 2ν2 = 1, and ν2 = 1/6 for UC and ν2 = (1 − 1√ 1+2 tan4 θ )/4 for PCC. Most of this descrip- tion is also valid when the original quantum system is initially mixed. Our main assumption is that preparation of the initial pure state |Ψ〉 is diluted by a thermal bath in the following special and simple form: ̺in = (1− ǫ)|Ψ〉〈Ψ|+ ǫ̺th, 0 6 ǫ < 1. (1) The parameter ǫ, which measures how thermally perturbed the preparation is, may in general be time-dependent. Nonethe- less, based on our earlier assumptions, it would be a fairly slow-varying time-dependent function so that with a good ap- proximation we can take it a relatively small constant of the order of τc/Tdiss.. This state does not seem to arise natu- rally from a typical thermalization dynamics. Nevertheless, in Ref. [18] it has been illustrated that general behaviors ob- tained from such a simple preparation assumption (in the con- text of the geometric phases) have general features similar to those obtained from the Lindblad equation for the dynamics. It is worth mentioning that in the limit of infinite tempera- ture, the thermalized density matrix ρth is equivalent to pure noise [24]. In that case, ǫ represents the degree of pure noise existing during the process (for example, in the case of NMR systems, due to fluctuations of the external magnetic fields and similar reasons). A more general analysis of quantum cloning in the presence of a thermalization mechanism is yet lacking, but our simple analysis may also shed some light before hav- ing a more complete analysis at hand. First, we consider the effect of the thermal term only on the state of the cloner, that is, the quantum cloning hardware is thermally diluted as in Eq. (1). In this case, the initial state of the machine is mixed. Considering the fact that in the op- timal UC and PCC, the initial state of the cloning machine can be any pure state [4, 7, 23], one can conclude the opti- mal fidelity here is achieved by the existing optimal cloning transformations. By a similar analysis, it appears that for the case of diluted joint blank and ancillary systems, one can con- sider the joint state as a new blank copy and attach some new reservoir to the whole Hilbert space of the input states (i.e., the information qubit, the blank copy, and the ancilla state) as a new ancillary system and then define a new transformation θ0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 PSfrag replacements (a) (b) 0 00 0 FIG. 2: Variation in the fidelity of PCC with θ for: (a) ǫ = 5/11 and (b) ǫ = 2/3, and different values of η. for cloning [23]. This would in fact be the existing optimal cloning transformation, now acting on a larger Hilbert space, and hence one obtains the same optimal fidelity again. How- ever, from an experimental point of view, thermalization ef- fects are likely to occur during the cloning process rather than at the initial preparation level — for instance in NMR sys- tems [19, 20]. Therefore, to be more precise, thermal effects during the preparation should also be taken into account. Now, we consider the case in which the input state |Ψ〉 is thermally diluted as in Eq. (1). Our aim now is to compare the similarity between the clones and the input state of our in- terest, i.e., |Ψ〉. Indeed, here we assume that the model of the cloning machine consists of two parts: the first is the dilution of the input pure state which models the imperfect feature of the machine, and the second is some known UC or PCC trans- formation which is applied to this diluted state. The Hamilto- nian of the qubit system is taken to be H = ω0σz/2 (ω0 > 0), whence, Z = 2 cosh η, where η = ω0β/2. More general cloning transformations in spin networks with more compli- cated Hamiltonians can be found, for example, in Ref. [21]. The fidelity of the output state and the unperturbed initial state can be calculated as follows: F (θ, ǫ, η) =µ2[1− ǫ+ ǫ(e−η cos2 θ + eη sin2 + (µν − µ2/2)(1− ǫ) sin2 θ + ν2. Figure 1 illustrates how the fidelity in the UC behaves in terms of θ (orbit of the state) in thermally diluted states, for two different values of ǫ (the degree of thermalization) and η (∝ 1/T ). It can be seen that when ǫ < cosh η /(e−η sin2 + eη cos2 ), (2) the fidelity of the UC is higher than the classical value 1/2. This threshold is the fidelity of a classical-like 1 → M uni- versal cloning in which with a given probability, an unknown input state is sent to one of the M parties and a completely randomized state is transmitted to any of the other ones, of course, in the limit of large M [25]. In the literature, however, “classical cloner” has been attributed to some other cloning transformations as well — see [3, 26]. In other words, in some cases thermal noise (even in the simple form of Eq. (1)) can result in a lower performance than a classical machine. For θ > π/2, the condition (2) implies that for all 0 6 ǫ < 1, the fidelity of the output of the UC is always greater than that of PSfrag replacements FIG. 3: Fidelity vs ǫ and θ in low temperature limit (η → ∞): UC (left) and PCC (right). the classical cloner (if ω0 was negative, this would occur for θ 6 π/2). Equation (2) can also be interpreted as a condition on temperature for a given θ and ǫ in order to outperform a classical cloner. Figure 2 shows the variation of the fidelity of the outputs of the PCC machines in terms of θ, for some fixed values of ǫ and η. As is clear from this figure, in the case of equatorial qubits, similar to the case of the UC, the fidelity of the outputs does not vary with temperature — according to Eq. (2), this feature is due to the symmetry property of such states. Low temperature limits of the fidelity for both UC and PCC have been depicted in Fig. 3. In the case of the UC, for all θ in [0, π), the fidelity is a decreasing function of ǫ. The corresponding graph for the PCC also shows a decrease in the fidelity for different values of θ ∈ [0, π/2) with the pertur- bation factor ǫ. However, a closer inspection shows that here there are also some θs (& 2.52 and less than π rad) in which the fidelity of the PCC is an increasing function of ǫ. At high temperature limit, the fidelity of both UC and PCC, for all θs, is a decreasing function of ǫ. Another important point that can be concluded from the figures is that in some cases, the quality of the clones at the output of the UC can be better than that of the PCC — see for example those regions of Fig. 3 in which ǫ and θ are large and small, respectively. This is indeed con- trary to what happens when the cloning is performed perfectly without any external noise. Entanglement cloning.— Quantum cloning can be used to clone or broadcast entanglement as well [5, 9, 10, 27, 28]. Let us assume that we have an initial state in the form of |Ψ−α 〉ab = α|01〉ab − 1− α2|10〉ab, where α is real and |α| 6 1. As in the cases of the UC and the PCC, sup- pose that because of a thermal environment, the initializa- tion is diluted as in Eq. (1). Let us take our system to be two spin-1/2 particles interacting via the XX Hamiltonian: H = J(σaxσ x + σ y), where σx and σy are Pauli matri- TABLE I: Inseparability conditions of the output states in the three different scenarios of cloning. γ α ǫ γ > γc |α2 − 1/2| < αc 0 < α < 1 0 6 ǫ < 1 −1 < α < 0 0 6 ǫ < ǫ1 or ǫ2 < ǫ < 1 |α2 − 1/2| > αc ǫ2 < ǫ < 1 0 < γ 6 γc |α2 − 1/2| < αc 0 < α < 1 0 6 ǫ < ǫ2 −1 < α < 0 0 6 ǫ < ǫ1 ces. Now, we want to compare performances of the following schemes of entanglement broadcasting between two parties in the presence of thermal noise: (i) Local cloning by means of two optimal UC machines copying each qubit separately [27]. In this scenario, after the cloning process and discarding the ancillas, we will have the overall state ̺aa′bb′ whose two first (last) qubits are the copies of a (b). (ii) Non-local cloning of the two-qubit state as a whole with the UC machine of 4-level quantum states [5]. (iii) Cloning by an optimal entanglement cloner [9]. After some algebra, it can be seen that the density matrices of the clones in cases (ii) and (iii), and ̺a′b (also ̺ab′ , ̺ab, and ̺a′b′ ) — nonlocal copies — in case (i), read as follows: ̺out = (Mǫ + 1−M )(|00〉〈00|+ |11〉〈11|) +[M(1−ǫ + ǫ cosh γ ) + 1−M + L(1− ǫ)(2α2 − 1)]|01〉〈01| +[M(1−ǫ + ǫ cosh γ ) + 1−M − L(1− ǫ)(2α2 − 1)]|10〉〈10| −M [(1− ǫ)α 1− α2 + ǫ sinh γ](|01〉〈10|+ |10〉〈01|), (3) in which γ = 2βJ, Z = 2(1 + cosh γ), L = 3(1 + 2M +√ 1 + 4M − 9M2)/26, Mi = (2/3)2, Mii = 3/5, Miii = 6A2 + 4AC, A = (1/2 + 1/ 13)/3, and C = A( 3)/2. Note that, the output states of case (ii) for all values of ǫ, α, and γ, the nonlocal copies of case (i) ̺a′b, and the output states of case (iii) for ǫ = 1 and ∀γ or α = ±1/ 2 (for all ǫ and γ) all can be written in the following compact form: ̺out = M̺in + (1 − M)I/4, where I is the 4 × 4 identity matrix. To determine the regions in which the output states are separable or inseparable, we use the well-known Peres- Horodecki positive partial transposition criterion [29]. Ac- cording to this criterion, in the case of 2 × 2 and 2 × 3 sys- tems, a density matrix ̺AB is inseparable (i.e., entangled) iff (̺AB) TA (TA: partial transposition with respect to system A) is not positive. Tables I and II show the results for anti- ferromagnetic case (J > 0). The parameters in the tables are TABLE II: Inseparability conditions of the output states in the three different scenarios of cloning, at low and high temperature limits. γ ǫ, α γ → ∞ C1 and 0 6 ǫ 6 1−M and |α2 − 1/2| < α∞ C1 and 1−M < ǫ < 1 and α ∈ C1 C2 and 0 6 ǫ < 3M−1 and |α2 − 1/2| < α∞ C2 and 1−M < ǫ 6 M+1 and |α2 − 1/2| > α∞ C2 and M+1 < ǫ < 1 and α ∈ C2 γ → 0 0 6 ǫ < (1 − 1 ) and |α2 − 1/2| < α0 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 PSfrag replacements (ii) (iii) (input) FIG. 4: Entanglement phase diagrams of input and output states (achieved from three different schemes of entanglement cloning/broadcasting introduced in the text), when α = 1/ 2. The regions labeled by 1 are the regions in which entanglement exists, whilst the regions labeled by 2 indicate no-entanglement regions. This figure shows that for γ < γc (T > Tc), depending on the value of ǫ, we may or may not have entanglement. γc is a decreasing func- tion of M . In other words, the area of region 1 increases when M increases, as well. This may imply the advantage of the entanglement cloner Miii over the other entanglement broadcasting schemes. as follows: α∞1 = (3M − 1)(M + 1− 4Mǫ) 4M(1− ǫ) α∞2 = (M + 1)(3M − 1− 4Mǫ) 4M(1− ǫ) 3M2 + 2M − 1 γc = ln( M + 1 + 2 M2 +M 3M − 1 ǫ1(2) = (M − 1∓ 4Mδ)(1 + cosh γ) 2M [1± sinh γ ∓ 2δ(1 + cosh γ)] , (3M(1− ǫ)− 1) (M(1− ǫ) + 1) 4M(1− ǫ) where δ = α 1− α2, C1 ≡ 0 < α 6 1, and C2 ≡ −1 6 α 6 0. When γ → ∞ and M = Miii, since (3M − 1)/4M > (1 −M)/2M , there exists an overlap be- tween the ǫ-inequalities in the third and fourth sub-rows of Ta- ble II. In this case, one should notice that for (1−M)/2M < ǫ < (3M − 1)/4M , clones are entangled if |α2 − 1/2| < α∞2 or |α2 − 1/2| > α∞1 . This removes the ambiguity in such cases. Tables I and II imply that in most temperature regions, the inseparability inequalities are not symmetric with respect to α → −α. In other words — unlike the case of ǫ = 0 — de- pending on the sign of α, the parameter regions over which the cloned pairs are entangled may be different. Another im- portant point (see the second row of Table I) is the existence of a critical temperature Tc (∝ 1/γc) beyond which the cloned pairs for some α regions, |α2 − 1/2| > αc, for all ǫs are not entangled. Overall, by taking into account the behaviors of the upper and lower bounds of the inseparability inequalities we can find that in some temperature regions, in Table I (Table II), there exist intervals of α2 (ǫ) in which the cloned pairs are sepa- rable. The length of these intervals decreases when M in- creases (recall that Miii > Mii > Mi). Furthermore, for a given α2 (ǫ) at intermediate (two limits of) temperatures, the range of ǫ (α2) in which the clones are entangled increases when M increases as well. Indeed, for some temperature re- gions, in Table I (Table II) there exist some α2 (ǫ) in which clones for all ǫ (α in C1 or C2) and all three Ms are entan- gled — e.g., see first sub-row of Table I or second and fifth sub-rows of Table II. These facts together with the entan- glement phase diagrams in Fig. 4, whose regions show ex- istence of entanglement or its non-existence for α = 1/ indicate advantage of entanglement cloner Miii, over the other cloning schemes. That is, the optimal entanglement cloner has an advantage over other mentioned schemes of entanglement broadcasting in the sense of robustness against thermal noise. Conclusion.— We have studied the role of thermal noise in some quantum cloning schemes through a simple model of temperature effect on spin states at the input of the cloning machines. The performance of the cloning machines depends generally on the values of the thermal perturbation coefficient, the orbit of the original state on the Bloch sphere, as well as on the temperature. In addition, three scenarios of entanglement cloning of thermally diluted two-qubit states have been inves- tigated. Our analysis shows that the clones generated from non-local transformations, in particular those out of the opti- mal entanglement cloner, remain entangled for wider regions of parameters. I.e., the optimal entanglement cloner shows a relatively larger region of entanglement in the parameter space. This can be considered as an advantage of optimal en- tanglement cloner over the other scenarios in the sense of ro- bustness against thermal perturbations. This statement, how- ever, is subject to the thermalization model we have used; so for a general conclusion a more detailed study is still needed. Our results may be of importance in practical implementations of quantum cloning in systems in which thermal effects are unavoidable, e.g., nuclear spin systems [19, 20]. Indeed, the large ǫ regime of our approach — when τc is of the same or- der of magnitude as Tdiss. — has already been experimentally realized in a different context [19]. This can be considered as a non-economic cloning process [30]. Acknowledgments.— Supports by the Center of Excellence in Complex Systems and Condensed Matter (CSCM) at Sharif University of Technology, iCORE, MITACS, and PIMS are gratefully acknowledged. [1] W. K. Wootters and W. H. Zurek, Nature 299 (1982) 802. [2] V. Bužek and M. Hillery, Phys. Rev. A 54 (1996) 1844. [3] N. Gisin and S. Massar, Phys. Rev. Lett. 79 (1997) 2153. [4] D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchi- avello, and J. A. Smolin, Phys. Rev. A 57 (1998) 2368. [5] V. Bužek and M. Hillery, Phys. Rev. Lett. 81 (1998) 5003: in Quantum Computing and Quantum Communications, Lec- ture Notes in Computer Science 1509, edited by C. P. Williams (Springer-Verlag, Berlin, 1999), p. 235. [6] R. F. Werner, Phys. Rev. A 58 (1998) 1827; P. Zanardi, ibid 58 (1998) 3484; H. Fan, K. Matsumoto, and M. Wadati, ibid 64 (2001) 064301. [7] D. Bruß, M. Cinchetti, G. M. D’Ariano, and C. Macchiavello, Phys. Rev. A 62 (2000) 012302. [8] H. Fan, K. Matsumoto, X. -B. Wang, and M. Wadati, Phys. Rev. A 65 (2002) 012304; V. Karimipour and A. T. Rezakhani, ibid 66 (2002) 052111; H. Fan, K. Matsumoto, X. -B. Wang, and H. Imai, J. Phys. A: Math. Gen. 35 (2002) 7415. [9] L. P. Lamoureux, P. Navez, J. Fiurášek, and N. J. Cerf, Phys. Rev. A 69 (2004) 040301; E. Karpov, P. Navez, and N. J. Cerf, ibid 72 (2005) 042314. [10] A. Kay and M. Ericsson, Phys. Rev. A 73 (2006) 012343. [11] V. Scarani, S. Iblisdir, N. Gisin, and A. Acı́n, Rev. Mod. Phys. 77 (2005) 1225; N. J. Cerf and J. Fiurášek, Progress in Optics Vol. 49, edited by E. Wolf (Elsevier, Amsterdam, 2006), p. 455. [12] M. Merkli, I. M. Sigal, and G. P. Berman, Phys. Rev. Lett. 98 (2007) 130401. [13] R. Alicki and K. Lendi, Quantum Dynamical Semigroup and Applications (Springer, Berlin, 1987); V. Scarani, M. Ziman, P. Štelmachovič, N. Gisin, and V. Bužek, Phys. Rev. Lett. 88 (2002) 097905. [14] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452 (2008) 854. [15] K. Maruyama and P. L. Knight, Phys. Rev. A 67 (2003) 032303. [16] S. -B. Zheng and G. -C. Guo, Phys. Rev. A 72 (2005) 064303. [17] R. S. Whitney, Y. Makhlin, A. Shnirman, and Y. Gefen, Phys. Rev. Lett. 94 (2005) 070407; K. M. Fonseca Romero, A. C. Aguiar Pinto, and M. T. Thomaz, Physica A 307 (2002) 142. [18] A. T. Rezakhani and P. Zanardi, Phys. Rev. A 73 (2006) 052117. [19] H. K. Cummins, C. Jones, A. Furze, N. F. Soffe, M. Mosca, J. M. Peach, and J. A. Jones, Phys. Rev. Lett. 88 (2002) 187901. [20] T. Durt and J. -F. Du, Phys. Rev. A 69 (2004) 062316; J. -F. Du, T. Durt, P. Zou, H. Li, L. C. Kwek, C. H. Lai, C. H. Oh, and A. Ekert, Phys. Rev. Lett. 94 (2005) 040505; T. Durt, J. Fiurášek, and N. J. Cerf, Phys. Rev. A 72 (2005) 052322. [21] G. De Chiara, R. Fazio, C. Macchiavello, S. Montangero, and G. M. Palma, Phys. Rev. A 70 (2004) 062308; ibid 72 (2005) 012328. [22] A. Ferraro and M. G. A. Paris, J. Opt. B 7 (2005) 532. [23] A. Roy, A. Sen(De), and U. Sen, Phys. Lett. A 286 (2001) 1. [24] N. Linden and S. Popescu, Phys. Rev. Lett. 87 (2001) 047901. [25] R. Filip, Phys. Rev. A 69 (2004) 052301. [26] N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf, Phys. Rev. Lett. 95 (2005) 070501; S. Koike, H. Takahashi, H. Yonezawa, N. Takei, S. L. Braunstein, T. Aoki, and A. Furu- sawa ibid 96 (2006) 060504. [27] V. Bužek, V. Vedral, M. B. Plenio, P. L. Knight, and M. Hillery, Phys. Rev. A 55 (1997) 3327. [28] S. Bandyopadhyay and G. Kar, Phys. Rev. A 60 (1999) 3296. [29] A. Peres, Phys. Rev. Lett. 77 (1996) 1413; M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223 (1996) 1. [30] Other economic cloning transformations with small ǫ have also been analyzed in Ref. [20].

0704.1178

Large Gauge Hierarchy in Gauge-Higgs Unification

TU-788 Large Gauge Hierarchy in Gauge-Higgs Unification Kazunori Takenaga(a) 1 (a) Department of Physics, Tohoku University, Sendai 980-8578, Japan Abstract We study a five dimensional nonsupersymmetric SU(3) gauge theory compacti- fied on M4 × S1/Z2. The gauge hierarchy is discussed in the scenario of the gauge- Higgs unification. We present two models in which the large gauge hierarchy is realized, that is, the weak scale is naturally is obtained from an unique large scale such as a GUT and the Planck scale. We also study the Higgs mass in each model. keywords: gauge symmetry breaking, extra dimensions, boundary conditions 1E-mail: [email protected] Collaboration with M. Sakamoto (Kobe), [email protected] Talk given at the SCGT06, International Workshop on ”Origin of Mass and Strong Coupling Gauge Theories” 21-24 November 2006, Nagoya, Japan http://arxiv.org/abs/0704.1178v1 1 Introduction Higher dimensional gauge theory has been paid much attention as a new approach to over- come the hierarchy problem in the standard model without introducing supersymmetry. In particular, the gauge-Higgs unification is a very attractive idea [1, 2, 3]. The higher dimensional gauge symmetry plays a role to suppress the ultraviolet effect on the Higgs mass. The Higgs self interaction is understood as part of the original five dimensional gauge coupling, so that the mass and the interaction can be predicted in the gauge-Higgs unification. The gauge-Higgs unification has been studied extensively [4]. In the gauge-Higgs unification, the Higgs field corresponds to the Wilson line phase, which is nonlocal quantity. The Higgs potential is generated at the one-loop level after the compactification. Because of the nonlocality, the Higgs potential never suffers from the ultraviolet effect [5], which is the genuine local effect, and the Higgs mass calculated from the potential is finite as well. In other words, the Higgs potential and the mass are calculable in the gauge-Higgs unification. This is a remarkable feature rarely happens in the usual quantum field theory. It is understood that the feature entirely comes from shift symmetry manifest through the Wilson line phase, which is a remnant of the higher dimensional gauge symmetry appeared in four dimensions. The Higgs mass does not depend on the cutoff at all, so that two tremendously separated energy scales can be stable in the gauge-Higgs unification. We study a five dimensional nonsupersymmetric SU(3) gauge theory, where one of spatial coordinates compactified on an orbifold S1/Z2. We find two models (model I, II) which realize the large gauge hierarchy [6]. 2 Gauge-Higgs unification As the simplest example of the gauge-Higgs unification, we study a nonsupersymmetric SU(3) gauge theory on M4 × S1/Z2, where M4 is the four dimensional Minkowski space- time and S1/Z2 is an orbifold which has two fixed points, y = 0, πR. We impose the twisted boundary condition of the field for the S1 direction and at the fixed points by using the gauge degrees of freedom, Aµ̂(x, y + 2πR) = UAµ̂(x, y)U †, (1) (x, yi − y) = Pi (x, yi + y), P i , (i = 0, 1) (2) where U † = U−1, P i = Pi = P i and y0 = 0, y1 = πR and µ̂ stands for µ̂ = (µ, y). The minus sign for Ay is needed to preserve the invariance of the Lagrangian under these transformations. A transformation πR + y P1→ πR − y must be the same as πR + y P0→ −(πR + y) U→ πR − y, so that we obtain U = P1P0. Here we choose P0 = P1 = diag.(−1,−1, 1). The gauge symmetry at low energies consists of the zero modes for A(0)a=1,2,3,8µ . We see that the orbifolding boundary condition Pi breaks the original gauge symmetry SU(3) down to SU(2) × U(1) at the fixed points [7]. On the other hand, we observe that the zero mode for A(0)y transforms as an SU(2) doublet, so that we identify the Higgs field as A(0)4y − iA(0)5y A(0)6y − iA(0)7y . (3) The VEV of the Higgs field is parametrized, by using the SU(2)×U(1) gauge degrees of freedom, as 〈A(0)y 〉 ≡ = A(0)6y , (4) where a is a dimensionless parameter. In order to determine a, one usually valuates the effective potential for a [2]. The gauge symmetry breaking depends on the values of a0. It has been known that the matter content is crucial for the correct gauge symmetry breaking SU(2)× U(1) → U(1)em. 3 Large gauge hierarchy in the gauge-Higgs unifica- If the Higgs acquires the VEV, the W -boson becomes massive whose mass is given by MW = a0/2R. This relation defines an important ratio, = πa0, (5) whereMc ≡ (2πR)−1. Once the values of a0 is determined as the minimum of the effective potential, the compactification scale Mc is fixed through Eq.(5). In the usual scenario of the gauge-Higgs unification, the VEV is of order of O(10−2) for appropriate choice of the flavor set[8] and this yields Mc ∼ a few TeV. In order to realize the large gauge hierarchy such as Mc ∼MGUT ,MP lanck, one needs the very small values of a0. For the very small values of a, the effective potential can be expanded as V̄eff(a) = − C(2)(πa)2 + (πa)4 −ln(πa) + + C(4)(ln2) + · · · , (6) where Veff(a) ≡ CV̄eff(a) with C ≡ Γ(52)/π 2 (2πR)5, and the coefficient C(i)(i = 2, 3, 4) is defined by C(2) ≡ 24N (+)adj + 4N adj + 18 + 6dN adj + 2N fd + 18N adj + 3N , (7) C(3) ≡ 72N (+)adj + 4N 54 + 18dN adj + 2N , (8) C(4) ≡ 48 + 16dN (+)sadj + 18dN adj + 2N adj + 4N fd + 72N . (9) We emphasize that each coefficient in the effective potential is given by the discrete values, that is, the flavor number of the massless bulk matter. This is the very curious feature of the Higgs potential, which is hardly seen in the usual quantum field theory, and is a key point to discuss the large gauge hierarchy in the gauge-Higgs unification. 3.1 Model I We impose a condition C(2) = 0 in order to obtain the hierarchically small VEV. One should note that the condition is not the fine tuning of the parameter usually done in the quantum field theory. This condition is fulfilled by the choice of the flavor set. By minimizing the Higgs potential, we obtain that MW =Mc exp |C(3)| ln2 + . (10) We see that the large gauge hierarchy Mc ∼MGUT ,MP lanck is realized for the large ratio, C(4)/ ∣C(3) ∣ ≫ 1. The magnitude of the ratio for the values of p = 11, 19, where p is defined by Mc ≡ 10p GeV, is C(4)/ ∣C(3) ∣ ≃ 32.54 (p = 11), 59.12 (p = 19). The large gauge hierarchy is realized if we have C(2) = 0 and the large ratio C(4)/ ∣C(3) ∣ at the same time. Let us present a few examples of the flavor set in the model I. We choose (k,m) = (1, 0) as a demonstration. Then, we find that adj , dN adj ) = (1, 1), (2, 5), · · · , (N fd , N fd ) = (0, 3), (1, 5), · · · . For (k,m, p) = (1, 0, 19), adj , dN adj ) = (0, 29), (1, 33), · · · , (N fd , N fd ) = (42, 1), (43, 3), · · · . For (k,m, p) = (1, 0, 11), adj , dN adj ) = (0, 16), (1, 20), · · · , (N fd , N fd ) = (22, 1), (23, 3), · · · . We observe that the flavor numbers dN adj , N fd are of order O(10). One has to take care about the reliability of perturbation theory for such the large number of flavor because an expansion parameter in the present case may be given by (g24/4π 2)Nflavor, and it must be (g24/4π 2)Nflavor ≪ 1 for reliable perturbative expansion. Now, let us study the Higgs mass in the model I. The Higgs mass squared is obtained by the second derivative of the effective potential evaluated at the minimum of the potential m2H = M2W k < M . (11) The choice k = 1 is the most desirable one for the large gauge hierarchy, so that the Higgs mass is lighter than MW , which is the same result in the original Coleman-Weinberg’s paper [9]. Therefore, one concludes that the large gauge hierarchy and the sufficiently heavy Higgs mass are not compatible in the model I. 3.2 Model II We study another model called Model II in this subsection. We introduce massive bulk fermions [10, 11, 12] in addition to the massless bulk matter in the model I. We introduce a pair of the fields, ψ+ and ψ− whose parity is different to each other, ψ±(−y) = ±ψ±(y). Then, a parity even mass term is constructed like Mψ̄+ψ−. The contribution to the mass term from the massive fermions is given by ζ(3)C(2)(πa)2 → −1 ζ(3)C(2) + 8NpairB (πa)2 with B(2) = 1 + nz + e−nz, where Npair stands for the number of the pair (ψ (+), ψ(−)) and we have defined a dimen- sionless parameter z ≡ 2πRM = M/Mc. We observe that the potential is suppressed by the Boltzmann-like factor e−nz, reflecting the fact that the effective potential shares similarity with that in finite temperature field theory [13]. The essential behavior of the VEV is governed by the factor B(2), i.e. πa0 ≃ γB(2) with some numerical constant γ of order 1. If we write πa0 = e −Y , then, one finds, remembering Eq.(5), that − Y = ln(πa0) = (2− p)ln10 ≃ −34.539 for p = 17, −20.723 for p = 11. (12) The gauge hierarchy is controlled by the magnitude of Y , in other words, the bulk mass parameter z, and the large gauge hierarchy is achieved by |z| ≃ 30 ∼ 40. The large gauge hierarchy is realized by the presence of the massive bulk fermion. We notice that the flavor number of the massless bulk matter is not essential for the large gauge hierarchy in the model II. Now, let us next discuss the Higgs mass in the model II. The Higgs mass is given by m2H = −C(3)ln(πa0) + C(3) + C(4)ln2 M2W F, (13) where we have defined F ≡ −C(3)ln(πa0) + C(3) + C(4)ln2. (14) The Higgs mass depends on the logarithmic factor. We observe that the larger the gauge hierarchy is, the heavier the Higgs mass is. An important point is that the coefficient C(3) is not related with the realization of the large gauge hierarchy, so that it is not constrained by the requirement of the large gauge hierarchy at all. In order to demonstrate the size of the Higgs mass in the model II, let us choose (k, l,m) = (−4,−1,−1). Then, the flavor set is given by adj , dN adj ) = (1, 3), (2, 7), · · · , (N fd , N fd ) = (3, 1), (4, 3), · · · , fd , N fd ) = (2, 1), (3, 3), · · · , (N adj , dN adj ) = (1, 0), (2, 4), · · · . And the Higgs mass in GeV unit is calculated as 119.5 for p = 17, 92.6 for p = 11, where we have used g24 ≃ 0.42. Here, we note that in the usual scenario of the gauge- Higgs unification, one requires g4 ∼ O(1) in order to have the heavy enough Higgs mass [14, 8]. The large gauge hierarchy enhances the Higgs mass sizably even for the weak coupling. We observe that for the fixed integers (k, l), the large gauge hierarchy, that is, large ln(πa0) = −Y enhances the size of the Higgs mass. The larger the gauge hierarchy is, the heavier the Higgs mass tends to be. 4 Conclusions and discussions We have studied the five dimensional nonsupersymmetric SU(3) model compactified on M4 × S1/Z2, which is the simplest model to realize the scenario of the gauge-Higgs uni- fication. We have discussed whether the large gauge hierarchy is realized in the scenario or not. The Higgs potential is generated at the one-loop level and is obtained in a fi- nite form, reflecting the nonlocal nature that the Higgs field is the Wilson line phase in the gauge-Higgs unification. The Higgs potential is calculable and accordingly, the Higgs mass, too. We have found two models (model I, II), in which the large gauge hierarchy is realized. The condition C(2) = 0 is crucial for our discussions. In connection with the condition, it may be worth mentioning that there are examples, in which the loop correction is exhausted at the one-loop level (without supersymmetry). They are the coefficient of the axial anomaly[15] and the Chern-Simons coupling [16]. As for the latter case, a simple reason for the two (higher) loop correction not to be generated comes from the invariance of the action under the large gauge transformation. Since the shift symmetry of the Higgs potential can be regarded as the invariance under the large gauge transformation, one may be able to prove that there is no two (higher)-loop correction to the mass term of the Higgs potential. In order to confirm it, one needs more studies of the higher loop corrections to the Higgs potential (mass) in the gauge-Higgs unification [17]. Acknowledgements I would like to thank Professor K. Yamawaki and other members of the organizing com- mittee for their hospitality and for inviting me to participate in SCGT 2006. This work was supported by the 21st Century COE Program at Tohoku University. References [1] N. S. Manton, Nucl. Phys. B158 141 (1979), D. B. Fairlie, Phys. Lett. 82B (1979) 97, N. V. Krasnikov, Phys. Lett. 273B (1991) 246, [2] Y. Hosotani, Phys. Lett. 126B (1983) 309, Ann. Phys. (N.Y.) 190, 233 (1989). [3] H. Hatanaka, T. Inami and C.S. Lim, Mod. Phys. Lett. A13 (1998) 2601, G. R. Dvali, S. Randjbar-Daemi and R. Tabbash, Phys. Rev. D65 (2002) 064021, N. Arkani- Hamed, A. G. Cohen and H. Georgi, Phys. Lett. B513 (2001) 232, I. Antiniadis, K. Benakli and M. Quiros, New J. Phys. 3, (2001),20. [4] K. Takenaga, Phys. Rev. D64 (2001) 066001, Phys. Rev. D66 (2002) 085009, Y. Hosotani, S. Noda and K. Takenaga, Phys. Rev. D69 (2004) 125014, Phys. Lett. B607 (2005) 276, N. Haba, K. Takenaga and T. Yamashita, Phys. Rev. D71 (2005) 025006, N. Maru and K. Takenaga, Phys. Rev. D72 (2005) 046003, Phys. Rev. D74 (2006) 015017, K. Takenaga, Phys. Lett. 425B (1998) 114, Phys. Rev. D58 (1998) 026004. [5] A. Masiero, C. A. Scrucca, M. Serone and L. Silvestrini, Phys. Rev. Lett. 87 251601 (2001). [6] M. Sakamoto and K. Takenaga, Phys. Rev. D75 (2007) 045015. [7] M. Kubo, C. S. Lim and H. Yamashita, Mod. Phys. Lett. A17 (2002) 2249, L. J. Hall, Y. Nomura and D. R. Smith, Nucl. Phys. B639 (2002) 307, G. Burdman and Y. Nomura, Nucl. Phys. B656 (2002) 3. [8] N. Haba, K. Takenaga and T. Yamashita, Phys. Lett. B615 (2005) 247. [9] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. [10] K. Takenaga, Phys. Lett. B570 (2003) 244. [11] N. Haba, K. Takenaga and T. Yamashita, Phys. Lett. B605 (2005) 355. [12] N. Maru and K. Takenaga, Phys. Lett. B637 (2006) 287. [13] L. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 3320. [14] N. Haba, Y. Hosotani, Y. Kawamura and T. Yamashita, Phys. Rev. D70 (2004) 01510. [15] S. L. Adler and W. A. Bardeen, Phys. Rev. 182 (1969) 1517. [16] H. So, Prog. Theor. Phys. 73 (1985) 528, S. Coleman and B. Hill, Phys. Lett. 159B (1985) 184. M. Sakamoto and H. Yamashita, Phys. Lett. B476 (2000) 427. [17] Y. Hosotani, in the Proceedings of “Dynamical Symmetry Breaking”, ed. M. Harada and K. Yamawaki (Nagoya University, 2004), P17. (hep-ph/0504272), hep-ph/0607064, Y. Hosotani, N. Maru, K. Takenaga and T. Yamashita, work in progress. http://arxiv.org/abs/hep-ph/0504272 http://arxiv.org/abs/hep-ph/0607064 Introduction Gauge-Higgs unification Large gauge hierarchy in the gauge-Higgs unification Model I Model II Conclusions and discussions

0704.1179

Self-organized metal nanostructures through laser driven thermocapillary convection

Self-organized metal nanostructures through laser driven thermocapillary convection C. Favazza, J. Trice, R. Kalyanaraman Department of Physics, Washington University in St. Louis, MO 63130 and Center for Materials Innovation, Washington University in St. Louis, MO 63130 R. Sureshkumar Department of Energy, Environmental, and Chemical Engineering, Washington University in St. Louis, MO 63130 and Center for Materials Innovation, Washington University in St. Louis, MO 63130 When ultrathin metal films are subjected to multiple cycles of rapid melting and resolidification by a ns pulsed laser, spatially correlated interfacial nanostructures can result from a competition among several possible thin film self-organizing processes. Here we investigate self-organization and the ensuing length scales when Co films (1 − 8nm thick) on SiO2 surfaces are repeatedly and rapidly melted by non-uniform (interference) laser irradiation. Pattern evolution produces nanowires, which eventually break-up into nanoparticles exhibiting spatial order in the nearest neighbor spac- ing, λNN2. For films of thickness h0 > 2nm, λNN2 ∝ h while the particle radius varies as rp2 ∝ h . This scaling behavior is consistent with pattern formation by thermocapillary flow and a Rayleigh-like instability. For h0 ≤ 2nm, a hydrodynamic instability of a spinodally unstable film leads to the formation of nanoparticles. Metal nanostructures, possessing spatial order and size uniformity are attractive for a variety of ap- plications including: optoelectronics, plasmonics, chemical and biomedical sensing, and catalytic devices [1, 2, 3, 4, 5]. In recent years there has been tremendous research emphasis on finding self-organization strategies to make strongly correlated metal and semiconductor nanostructures. In the area of thin film pattern formation, epitaxial strain driven self-organization in crystallographic systems has been somewhat successful [6, 7, 8, 9]. However, this strategy of exploiting epitaxial strain cannot be applied to materials on important amorphous surfaces, such as SiO2. One possibility for self-organization on amorphous surfaces is via dewetting instabilities, such as those observed in liquid polymer films on various inert surfaces, in which preferred length scales appear when destabilizing long range attractive intermolecular forces over- come stabilizing surface tension forces [10, 11, 12]. We have previously shown that multiple pulses of single beam ns laser irradiation of Co films with thickness 1 ≤ h0 ≤ 8nm on SiO2 surfaces demonstrate self-organization by a thin film hydrodynamic (TFH) instability with length scales characterized by the metal surface tension (γ), the dispersion force characterized by the Hamaker constant A and film thickness h0 [13]. The ensuing pattern had short range spatial order (SRO) in the nearest-neighbor (NN ) particle http://arxiv.org/abs/0704.1179v1 spacing λNN1, which scaled with thickness as h , while the particle radius varied as rp1 ∼ h . Recently we have also provided preliminary evidence that 2-beam laser interference irradiation can give rise to quasi- 2D ordering of the nanoparticles with both long range order (LRO) and SRO [14]. However the evolution of the 2-beam pattern from the flat film to nanoparticle state was not investigated, and the mechanisms for spatial ordering were not identified. Here we report detailed experiments of 2-beam interference irradiation of Co on SiO2 as a function of film thickness for 1 ≤ h0 ≤ 8nm that demonstrate that the pattern evolution and resulting length scales are consequences of different mechanisms, including the TFH instability, ther- mocapillary (TC) flow and a Rayleigh-like instability, based on the timescales of these processes. One can access a particular mechanism via the choice of film thickness and/or thermophysical parameters such as γ, A and the temperature gradient. Cobalt films ranging in thickness from ∼ 1−8nm were deposited by e-beam evaporation onto optically smooth SiO2 surfaces in vacuum. Specific details of the experiments are published elsewhere [14, 15]. In brief, following deposition, the Co films were irradiated in vacuum with a 266nm pulsed laser operating at a repetition rate of 50Hz with a pulse time τ ∼ 9ns by single beam [13] or 2-beam interference irradiation. In the single beam case, films were exposed to an unfocused beam at normal incidence. For the 2-beam condition, an interference angle of ∼ 45◦ generated a periodic laser intensity profile whose contrast was maximized by adjusting the energy density of the off-normal beam with a 750mm lens positioned 300mm away from the sample at angle of ∼ 45◦. In both instances, Co films were irradiated for similar times as measured by the number of laser pulses, n, for 5 ≤ n ≤ 10, 500. Also, comparable laser energy densities were used for both types of irradiation, with the energy density, Elaser between EThreshold < 60 ≤ Elaser ≤ 200mJ [16]. The resultant morphological evolution was investigated as a function of n and was examined and characterized using a Hitachi S-4500 scanning electron microscope (SEM). It was also verified that all the films studied had minimal evaporation, as confirmed by performing energy dispersive X-ray spectrometry measurements of the Co concentration after irradiation. As detailed in prior studies for single beam irradiation [13, 16], the film shows several distinct patterns enroute to a final robust state of nanoparticles characterized by a spatially correlated nearest neighbor (NN ) spacing. Here we briefly summarize the trends for the purpose of comparison with the pattern evolution under 2-beam interference irradiation. More detailed analysis the single beam data can be found elsewhere [13, 16]. At the early stages (n = 10) regular holes with a characteristic diameter form, followed by a cellular network of polygonal structures. Eventually, the polygonal networks evolve into nanoparticles, which form predominantly at the vertices of the polygons. An important property of the evolution is that the patterns possessed a characteristic length scale at every stage. In Fig. 1(a), the scaling behavior of the spatial correlation in the final stable nanoparticle state is shown for varying initial film thicknesses. The observed trend with h0 was in agreement with classical linear TFH dewetting theory [10, 11] in which: ΛTFH = 16π3γ h2o (1) where ΛTFH represents the average NN nanoparticle spacing, λNN1, A = 1.4 × 10−18J is the experi- mentally estimated Hamaker constant for the SiO2/Co/V acuum system , γ = 1.88 is the Co liquid surface tension and h0 is the film thickness [10, 11]. Furthermore, volume conservation implied that the nanoparticles will have a radius that will vary with the initial film thickness h0 as: rp1 = 24π3γ f(θ)A o (2) where f(θ) accounts for the contact angle of the nanoparticles [13]. We experimentally observed a monomodal particle size distribution with the average radius shown in Fig. 1(b) and a trend consistent with the above theoretical prediction from linear TFH theory. The results of 2-beam irradiation were qualitatively different from single beam irradiation. In Fig. 2(a- c), the typical pattern evolution for films with h0 > 2nm is shown as a function of n. In this case, the early stages are comprised of spatially periodic film rupture at length scales comparable to the interference spacing. Longer irradiation yielded the formation of long, cylindrical-like “nanowires” and continued irra- diation resulted in the break-up of these nanowires into particles. The final particle state is characterized by a quasi-2D, comprised of the LRO due to periodic laser intensity and SRO resulting from the break-up of the nanowires. The observed differences from the two processing conditions can be explained on the basis of the various mechanisms of fluid motion operating under the two irradiation conditions. When irradiating the metal film with a 2-beam interference pattern, the resulting periodic laser intensity induces a transient and periodic thermal gradient along the plane of the film which creates a surface tension gradient not present in the single beam irradiation. Consequently, TC or Marangoni convection of the molten Co can occur. To contrast the results of single beam and 2-beam irradiation, the time scales for the various mechanisms were estimated. The timescale for TC flow can be expressed as [17]: τMa = Λ2laserη where Λlaser = 350nm is the laser fringe spacing, η = 4.45 × 10−3 Pas, is the Co metal viscosity, = −0.34× 10−3Jm−2K−1 is the rate at which the surface tension of the Co changes with temperature and ∆T is the maximum temperature difference between the peak and valley of the laser fringe. The timescale for the TFH instability can be expressed as [10, 11]: 96πγη An important aspect of the above equations is that the time scale of the TC flow decreases with increasing h0 while the TFH instability increases with increasing h0. We determined that typical liquid lifetimes range from 1 ≤ τL ≤ 10ns and thermal gradients range from 0.5−0.7 Knm [14, 16]. From Eq. 3 and 4, τs > τMa for films with ho ≤ 2nm and τs < τMa for films with h0 > 2nm. This implies that pattern formation is dominated by the TC flow for h0 > 2nm with the TFH flow dominating for h0 < 2nm, as shown in the comparative timescale plot in Fig. 3. This result was confirmed experimentally as nanowire formation was observed only above 2nm while it was absent for films < 2nm, as shown in Fig. 3(inset A). For film thicknesses near the cross-over point of h0 = 2nm, the pattern consisted of particle formation with some evidence for lateral movement of the metal (Fig. 3(inset B)) indicating that both mechanisms are operative [18]. For films with h0 > 2nm nanowire formation under the 2-beam irradiation also permitted access to a Rayleigh-like instability in which cylinders are unstable to wavelengths ≥ 2πrcyl, with the fastest growing wavelength scaling as ΛR ∝ rcyl , where rcyl is the radius of the cylinder. The characteristic time scale for this process can be expressed as [12, 19]: ρr3cyl where ρ = 7.8 g is the liquid density of Co [20]. Assuming these are partial cylinders with contact angles > 90◦and using the projected width of these regions from SEM images, we estimated rcyl, preceding the break-up into nanoparticles, to be in the range of 35 − 55nm. Based on these values of rcyl, the typical magnitude for the cylinder break-up time was estimated as τRay ∼ 1 − 2ns. Since this time scale is comparable to or smaller than the typical liquid lifetimes, the Rayleigh process is clearly accessible whenever a cylinder is formed. Also, according to the Rayleigh formulation, the particle radius scales linearly with both the spacing between particles and the radius of the original cylinder [19, 21, 22]. Through volume conservation, the cylinder radius varies as rcyl ∝ h o , resulting in a similar scaling relation for the size of the particle as rp2 ∝ h o . As shown in Fig. 1(a-b), the particle spacing and the size both scale as h o and the ratio of particle spacing to radius is independent of film thickness. While, the ratio in the classical Rayleigh break-up of a perfect cylinder is expected to be λNN2 ∼ 4.7 [19, 21, 22] our measured ratio was ≃ 5.6. We have ruled out previously existing explanations for how this ratio can be modified [23, 24, 25, 26, 27, 28]. Nichols et al. [23] have shown that if the break-up is dominated by solid- state surface diffusion, then the ratio is maintained, however, if the flow is generated by external volume diffusion, then the ratio grows to ∼ 6.1. However, the timescale over which external volume diffusion can cause break-up has been estimated as ∼ 0.3 s (τvol ∼ 13DV γΩ , where Dv is the volume self-diffusion coefficient, Ω is the atomic volume and f is the correlation factor [23]) for cylinder with rcyl = 40nm. This timescale is well outside the realm of the laser processing time (for t = 0.3 s, n ∼ O(106)), and as such can be disregarded as a possible break-up mechanism. We have also estimated the correction to the ratio for situations in which truncated cylinders break-up based on the work by McCallum et al. [24] and determined that only a small change to the ratio will be introduced. While there have also been papers examining the effect of having a non-uniform contact line or an anisotropic surface, these models also do not yield the observed 5.6 ratio [27, 28]. It is possible that the thermocapillary flow and a Rayleigh-like instability can drive self-organization of the film such that the liquid profiles are stationary solutions to the non-linear equations describing the pattern evolution [29, 30]. However, a nonlinear analysis along with further experiments are required to verify such a hypothesis. In conclusion, self-organization leading to spatially correlated nanostructures under ns laser irradiation of ultrathin Co films on SiO2 has been investigated. For films with thickness 2 − 8nm, non-uniform laser irradiation by 2-beam interference leads to pattern formation characterized by the formation of nanowires via thermocapillary flow and eventually to nanoparticles via a Rayleigh-like break-up of the nanowires. The nanowire break-up results in an average nearest-neighbor particle spacing of λNN2 ∝ h with radius rp2 ∝ h in contrast to the TFH instability in which λNN1 ∝ h20 and rp1 ∝ h . For films with h0 ≤ 2nm nanowire formation was absent because the TFH time scales were much shorter than the TC timescale. These results show that self-organization through laser-induced hydrodynamic flow can be used to make a variety of strongly correlated surface nanostructures on amorphous substrates by selecting the appropriate film thickness based upon thermophysical parameters. RK and RS acknowledge support by the NSF through a CAREER grant DMI-0449258 and grant CTS 0335348 respectively. *Electronic address: [email protected], †Electronic address: [email protected] [1] M. Quinten, A. Leitner, J. Krenn, and F. Aussenegg, Optics Lett. 23, 1331 (1998). [2] H. Garcia, J. Trice, R. Kalyanaraman, and R. Sureshkumar, Phys. Rev. B 75, 045439 (2007). [3] A. N. Shipway, K. Eugenii, and W. Itamar, Chem. Phys. 1, 18 (2000). [4] Y. Xia, P. Yang, Y. Sun, Y. Wu, B. Mayers, B. Gates, Y. Yin, F. Kim, and H. Yan, Adv. Mat. 15 (2003). [5] J.-Y. Raty, F. Gygi, and G. Galli, Phy. Rev. Lett. 95 (2005). [6] H. Brune, M. Giovannini, K. Bromann, and K. Kern, Nature 394 (1998). [7] B. S. Kwak, A. Erbil, J. Budai, M. F. Chisholm, L. A. Boatner, and B. J. Wilkens, Phys. Rev. B 49 (1994). [8] C. Teichert, M. G. Lagally, L. J. Peticolas, J. C. Bean, and J. Tersoff, Phys. Rev. B 53 (1996). [9] D. K. Goswami, B. Satpati, P. V. Satyam, and B. N. Dev, Curr. Sci. 84 (2003). [10] A. Vrij, Disc. Farad. Soc. 42 (1966). [11] A. Vrij and J. T. G. Overbeek, J. Am. Chem. Soc. pp. 3074–3078 (1968). [12] P. de Gennes, F. Brochard-Wyart, and D. Quere, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, 2004). [13] C. Favazza, R. Kalyanaraman, and R. Sureshkumar, Nanotechnology 17, 4229 (2006). [14] C. Favazza, J. Trice, H. Krishna, R. Kalyanaraman, and R. Sureshkumar, Appl. Phys. Lett. 88, 1531181 (2006). [15] C. Favazza, J. Trice, A. Gangopadhyay, H. Garcia, R. Sureshkumar, and R. Kalyanaraman, J. Elec. Mat. 35, 1618 (2006). [16] J. Trice, D. Thomas, C. Favazza, R. Sureshkumar, and R. Kalyanaraman (2006), under Review (Phys. Rev. B) Can be obtained at http://arxiv.org/ paper Id: cond-mat/0609182. [17] T. Schwarz-Selinger, D. G. Cahill, S.-C. Chen, S.-J. Moon, and C. P. Grigoropoulos, Phys. Rev. B 64 (2001). [18] P. Moriarty, M. D. R. Taylor, and M. Brust, Phys. Rev. Lett. 89, 248303 (2002). [19] Lord Rayleigh, Proc. Lon. Math. Soc. (1879). [20] W. F. Gale and T. C. Totemeier, eds., Smithells Metals Reference Book (Elsevier, 2004), 8th ed. [21] Lord Rayleigh, Proc. Roy. Soc. Lon. 29 (1879). [22] J. Eggers, Rev. Mod. Phys. (1997). [23] F. A. Nichols and W. W. Mullins, Trans. Metall. Soc. AIME (1965). [24] M. S. McCallum, P. W. Voorhees, M. J. Miksis, S. H. Davis, and H. Wong, J. Appl. Phys. (1996). [25] L. M. Hocking, J. Fluid Mech. (1990). [26] L. M. Hocking and M. J. Miksis, J. Fluid Mech. 247, 157 (1993). [27] S. H. Davis, J. Fluid Mech. (1980). [28] K. F. Gurski and G. B. McFadden, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci (2003). [29] V. S. Mitlin, J. Coll. Inter. Sci. 156 (1993). [30] M. J. Tan, S. G. Bankoff, and S. H. Davis, Phys. Fluids A 2, 313 (1990). http://arxiv.org/ http://arxiv.org/abs/cond-mat/0609182 Figure captions • Figure 1: Self-organized length scales versus initial film thickness: (a) h0 dependency for the parti- cle radius from 2-beam interference irradiation (open squares); and h dependency from single beam irradiation (closed triangles). (b) h0 dependency for the particle spacing from 2-beam irradiation (open squares) and h2 dependency from single beam irradiation (closed triangles). • Figure 2: SEM images depicting the stages of pattern formation for 2-beam interference irradiation of a ∼ 6 nm film as a function of increasing number of laser pulses n: (a) periodic rupture (b) nanowires and (c) the final nanoparticle state which exhibits both LRO and SRO. (Co rich and SiO2 rich regions correspond to bright and dark contrast respectively). • Figure 3: Comparison of the time scales for TFH dewetting and thermocapillary (TC) flow as a function of initial film thickness with the morphology for various film thickness following irradiation (insets). (A) h ≤ 1nm film showing that the entire film dewets; (B) h ∼ 2nm film showing that while TC flow causes the film to split there is still evidence for TFH dewetting and (C) h ∼ 4.5nm film showing that TC forces completely dominate morphology change. (Co rich and SiO2 rich regions correspond to bright and dark contrast respectively). (a) (b) Figure 1: (a) (b) (c) Figure 2: Figure 3: References Figure captions

0704.1182

An Optical Source Catalog of the North Ecliptic Pole Region

Accepted for Publication in ApJS An Optical Source Catalog of the North Ecliptic Pole Region1 Narae Hwang2,11, Myung Gyoon Lee2, Hyung Mok Lee2, Myungshin Im2, Taehyun Kim2, Hideo Matsuhara3, Takehiko Wada3, Shinki Oyabu3, Soojong Pak4, Moo-Young Chun5, Hidenori Watarai6, Takao Nakagawa3, Chris Pearson3,7, Toshinobu Takagi3, Hitoshi Hanami8, and Glenn J. White9,10 ABSTRACT We present a five (u∗,g′,r′,i′,z′) band optical photometry catalog of the sources in the North Ecliptic Pole (NEP) region based on deep observations made with MegaCam at CFHT. The source catalog covers about 2 square degree area cen- tered at the NEP and reaches depths of about 26 mag for u∗, g′, r′ bands, about 25 mag for i′ band, and about 24 mag for z′ band ( 4 σ detection over an 1 ′′aperture). The total number of cataloged sources brighter than r′ = 23 mag 1Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Re- search Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. 2Astronomy Program, Department of Physic and Astronomy, Seoul National University, Seoul 151-747, Korea 3Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Kana- gawa 229-8510, Japan 4Department of Astronomy and Space Science, Kyung Hee University, Yongin-si, Gyeonggi-do 446-701, Korea 5Korea Astronomy and Space Science Institute, 61-1, Hwaam-Dong, Yuseong-Gu, Daejeon 305-348, Korea 6Office of Space Applications, Japan Aerospace Exploration Agency, Tsukuba, Ibaraki 305-8505, Japan 7ISO Data Centre, ESA, Villafranca del Castillo, Madrid, Spain 8Iwate University, 3-18-8 Ueda, Morioka 020-8550, Japan 9Astrophysics Group, Department of Physics, The Open University, Milton Keynes, MK7 6AA, UK 10Space Science & Technology Department, CCLRC Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, UK 11e-mail: [email protected] http://arxiv.org/abs/0704.1182v1 – 2 – is about 56,000 including both point sources and extended sources. From the investigation of photometric properties using the color-magnitude diagrams and color-color diagrams, we have found that the colors of extended sources are mostly (u∗ − r′) < 3.0 and (g′ − z′) > 0.5. This can be used to separate the extended sources from the point sources reliably, even for the faint source domain where typical morphological classification schemes hardly work efficiently. We have de- rived an empirical color-redshift relation of the red sequence galaxies using the Sloan Digital Sky Survey data. By applying this relation to our photometry catalog and searching for any spatial overdensities, we have found two galaxy clusters and one nearby galaxy group. Subject headings: galaxies: general — galaxies: photometry — galaxies: clusters — catalogs 1. Introduction The North Ecliptic Pole (NEP) is an undistinguished region in the sky, located at α = 18h00m00s, δ = +66◦33′38′′. It is, however, a very special region since many astronomical satellites have accumulated a large number of exposures over this location since Earth- orbiting satellites must point their fixed solar panels to the Sun and be in their orbits over the ecliptic poles. Unlike the South Ecliptic Pole (SEP) region where the South Atlantic Anomaly and the LMC prevent the clear view of the extragalactic sky at certain wavelengths, the NEP region suffers very little or no obscuration by foreground Galactic sources. The galactic coordinates of the NEP are l ≈ 96.4◦, b ≈ +29.8◦ and the foreground extinction in this direction is E(B − V ) = 0.047 (Schlegel et al. 1998). Therefore, the NEP is a good target for deep, unbiased, contiguous surveys for extragalactic objects such as galaxies, galaxy clusters, and AGNs. The ROSAT All-Sky Survey (RASS) is one of the most representative surveys of the NEP region (Voges et al. 1999). The ROSAT was the first X-ray imaging satellite to survey the entire sky. A source catalog of X-ray sources that were extracted from a large area surrounding the NEP was constructed from this RASS data (Henry et al. 2001, 2006). From the followup observations and optical counterpart investigations, Gioia et al. (2003) reported that most X-ray sources in the NEP region are AGNs (∼ 49%), stars (∼ 34%), and galaxy groups/clusters (∼ 14%). Mullis et al. (2001) also found a supercluster of galaxies in the NEP region using the RASS data and suggested that some galaxy clusters in this region are part of the supercluster at z = 0.087. The NEP region was also observed in the 1.5 GHz band by Kollgaard et al. (1994). Using this radio source catalog, Brinkmann et al. (1999) – 3 – investigated the correlation between the radio sources and the RASS X-ray sources, and identified optical counterparts of the radio/X-ray sources. They found that a significant number of radio loud sources are also bright in the X-ray band, and that X-ray selection is an effective way for the search of galaxy clusters and groups. While the X-ray luminosity is an efficient measure of the hot ionized gas in the galaxy clusters, the star formation activities and the resultant stellar mass in galaxies can be esti- mated from the optical and infrared flux. Recently, a new infrared space telescope, named AKARI, was launched in February 2006. AKARI is expected to give exceedingly higher qual- ity data than the previous infrared space missions such as Infrared Astronomical Satellite (Neugebauer et al. 1984) and Infrared Space Observatory (Kessler et al. 1996). The AKARI data will also complement the successful Spitzer Space Telescope (Werner et al. 2004) data by providing better wavelength coverage over the 8−24µm band, which is not available from the Spitzer. As one of its major science programs, AKARI is currently carrying out deep near-to-mid infrared surveys over the wide area around the NEP through two major survey programs, ‘NEP-Deep’ and ‘NEP-Wide.’ The key scientific objectives of these NEP surveys are to unveil the dusty star formation history of the Universe, the mass assembly and large scale structure evolution, and the nature of the cosmic infrared background (CIRB). For further information on these NEP Programs, please refer to Matsuhara et al. (2006). Combined with these space-borne infrared surveys, the optical survey of the NEP re- gion constitutes the multifrequency dataset that is essential for the studies of cosmic star formation history. Many researchers have investigated optical counterparts of radio and X- ray sources in this region, but, for the imaging data, they mostly relied on the COSMOS scan (e.g., Brinkmann et al. 1999) or Digitized Sky Survey images of the second Palomar Observatory Sky Survey (POSS-II) plates, only complemented by some independent but patchy observations (e.g., Gioia et al. 2003). Although the POSS-II data are excellent in complete coverage of the whole sky, the usefulness of the data are still limited by the low resolution with ∼ 1.0′′ per pixel scale and the shallow depth with the limiting magnitude of rF ∼ 21.0 mag (Gal et al. 2004). In this study, therefore, we provide the first optical catalog of sources in the NEP region based on deep observations in u∗, g′, r′, i′, z′ bands obtained with MegaCam at CFHT. This optical catalog will be an important part of multifrequency datasets of galactic and extragalactic sources in the NEP region. It will also serve as a basis of the AKARI NEP survey mission by providing the information on the optical counterparts of the infrared sources. This paper is composed as follows: We describe our observations in Section 2 and data reduction procedures in Section 3. We present a bright source catalog in Section 4.1, photometric properties of the sources in the NEP region in Section 4.2, and galaxy number counts in Section 4.3. The result of galaxy cluster search using the color of red sequence galaxies is given in Sections 4.4 and 4.5. Some photometric properties of other – 4 – X-ray sources are presented in Section 4.6 and our primary results are summarized in the last section. 2. Observation The observation of the North Ecliptic Pole (NEP) region was carried out over nine photo- metric nights between August 22 and September 22, 2004 using the 3.6m CFHT telescope lo- cated at Mauna Kea, Hawaii. We used a wide field imager MegaCam at the telescope primary focus MegaPrime. The MegaCam is composed of 36 2048 × 4612 CCD’s covering about 1◦× 1◦ area with 0.185 arc seconds resolution per pixel (Boulade et al. 2003). The u∗, g′, r′, i′, z′ filter system provided with MegaCam is basically the same as that used by the Sloan Digital Sky Survey (SDSS) (York et al. 2000) except for the u∗ filter which is designed to take advan- tage of the improved UV extinction condition at the CFHT site. Therefore, the photometric data generated from this observation are presented in the CFHT unique u∗ and the SDSS g′, r′, i′, z′ system. For further information on these filter systems used with MegaCam, see the ‘Filter set’ section in http://www.cfht.hawaii.edu/Instruments/Imaging/MegaPrime/specsinformation.html page. As shown in Figure 1, the observed fields are composed of two fields separated by about 1◦ away from each other in the East-West direction, covering about two square degrees in total. The NEP-E (East) field was observed with five filters, u∗, g′, r′, i′, z′ and the other field, NEP-W (West), was observed with only four filters, g′, r′, i′, z′. Each field was covered twice with a 15′′ offset for dithering in each filter with total exposure times of 3600 sec for u∗ (1800 × 2), 2400 sec for g′ (1200 × 2), 3000 sec for r′ (1500 × 2), 3000 sec for i′ (1500 × 2), and 3600 sec for z′ (1800 × 2). The raw images were processed using Elixir system by CFHT staff. The Elixir is a collection of programs, databases, and other tools specialized in processing and evaluation of the large mosaic data (Magnier & Cuillandre 2004). The overall quality of the preprocessed images is good with a typical seeing of 0.7 ∼ 1.1′′. Detailed information on the observation is listed in Table 1. 3. Data Reduction 3.1. Pre-Photometry Processing For the reliable source detection and photometry, it is essential to prepare a master image that satisfies the following two factors: (1) to obtain the required photometric depth http://www.cfht.hawaii.edu/Instruments/Imaging/MegaPrime/specsinformation.html – 5 – and (2) to give the complete areal coverage. We used a software package SWarp1 written by E. Bertin at Terapix to transform each Multi Extension Fits (MEF) file containing 36 individual CCD frames into a single fits image data. SWarp also corrects different distortion effects between input data through the resampling process, and then combines the resampled images to create a deep output image after flux scaling and background subtraction. To make a master image, we combined the g′, r′, i′, z′ four band data using SWarp for the NEP-E and the NEP-W field respectively. During the SWarp run, BILINEAR resampling method was used since this method was found to be effective in suppressing the discontinuity effects around the chip boundaries, while other methods such as LANCZOS3 produced more noticeable discontinuities. We also used weighted images during SWarp run to enhance the quality of the output image. In the weight maps we assigned zero weights to pixels with negative values. We excluded the u∗ band data from the master image construction since we do not have the u∗ band data for the NEP-W and the S/N is relatively lower compared to other bands. Finally, we created two master detection images with high S/N’s, one for the NEP-E and the other for the NEP-W region. Two MEF images for each band were processed using SWarp with the same parame- ters to generate a final photometry image for the corresponding filter. These u∗, g′, r′, i′, z′ band images were made to have the same dimension and coordinates with the master de- tection image of the corresponding field. The depth of photometry attained by combin- ing two raw images was calculated as 4σ flux over a circular aperture with 1 ′′ diameter. The measured limiting magnitudes are u∗ ∼ 26.0, g′ ∼ 26.1, r′ ∼ 25.6, i′ ∼ 24.7, and z′ ∼ 23.7 mag, as listed in Table 1. After all these processings, two master detection im- ages and nine photometry images were prepared for the source detection and photometry. All the information on the reduced photometry images will be available from an web site http://astro.snu.ac.kr/˜ nhwang/index.files/nep.html. 3.2. Source Detection & Photometry We have used SExtractor (Bertin & Arnouts 1996) to detect sources from the master detection images of the NEP-E and NEP-W fields. A source is confirmed if it has more than five contiguous pixels above four times the background sky fluctuation. The signals from each source were measured in the u∗, g′, r′, i′, z′ band photometry images over the isophotal area previously defined during the source detection. The photometry was made using SExtractor in dual mode operation. This scheme enables the detection and photometry of any source 1See http://terapix.iap.fr/rubrique.php?id rubrique=49 for further information http://astro.snu.ac.kr/~~nhwang/index.files/nep.html http://terapix.iap.fr/rubrique.php?id_rubrique=49 – 6 – that is registered at least once in any of the g′, r′, i′, z′ band images. The total number of sources detected in our two observed NEP fields is about 130,000. The instrumental magnitudes were transformed into standard magnitudes using the transformation information provided and recorded in the header of the images by the CFHT staff. During this transformation, only the calibrated magnitudes of sources that have avail- able color information required for the calibration are calculated and kept in the catalog. Otherwise, we assigned dummy values (99.000) to the magnitudes of sources in the final source catalog. 4. Results Figure 2 displays a source count histogram in the u∗, g′, r′, i′, z′ band along with the r′ band error and stellarity distributions. This shows that the number of sources in our data increases up to u∗ ∼ 24.3, g′ ∼ 24.0, r′ ∼ 23.5, i′ ∼ 23.0, and z′ ∼ 22.2 mag before the incompleteness effect starts to take place and the number of the detected sources starts to decrease. The r′ band magnitude error is estimated to be about 0.1 mag or less at r′ ∼ 23.5 mag where the source count reaches its maximum. The stellarity distribution shows how efficiently the star/galaxy classifier works in our dataset. From this distribution, it is clearly seen that the stellarity index, which is calculated by SExtractor based on the isophotal areas, peak intensity, and seeing information, separates the point sources (stellarity ∼ 1) and the extended sources (stellarity ∼ 0) with high confidence for the sources with r′ < 22 mag. One more point to be noted in this stellarity distribution is that sources with r′ < 16.5 mag and stellarity > 0.7 are the results of the image saturation. Thus the stellarity index is most relable in the magnitude range of 17 < r′ < 22 mag. 4.1. Source Catalog Considering the source count and the stellarity distribution, we have decided the mag- nitude range of the most reliable sources for the final bright source catalog entry, which is r′ ≤ 23 mag. The number of sources compiled in the final bright source catalog is about 56,000. Table 2 lists a sample of the bright source catalog for reader’s guide and the full source catalog will be available electronically from the online Journal and from an web site http://astro.snu.ac.kr/˜ nhwang/index.files/nep.html. Followings are the short descriptions of data columns in the catalog. Column 1 is the identification number of an optical source. http://astro.snu.ac.kr/~~nhwang/index.files/nep.html – 7 – Column 2 & 3 list, respectively, the J2000.0 right ascension (RA) and declination (DEC) of a source in degrees. The uncertainties of the astrometric solution derived by Elixir are about ±0.5′′. Column 4 ∼ 13 are the AB magnitudes and magnitude errors of a source in the u∗, g′, r′, i′, z′ bands. These are given by MAG AUTO and MAGERR AUTO parameters of SExtractor, which are Kron-like elliptical aperture magnitudes and their errors. Column 14 is the extraction flag of a source given as a sum of powers of 2 by SExtractor. If a source has neighbors (flag = 1) and is blended with another source (flag = 2) and some pixels are saturated (flag = 4), then the final extraction flag given to the source is 7 (= 1 + 4 + 2). Generally, a source with the extraction flag 0 gives the most reliable photometry. For detailed information on this flag, see the SExtractor User’s Manual. Column 15 is the stellarity index of a source calculated from a g′, r′, i′, z′ combined detection image. This index has a value between 1 (point sources) and 0 (extended sources). See Figure 2 for the distribution of this index. Column 16 is the ellipticity of a source calculated by SExtractor using the second order image moments. Column 17 is the FWHM in arcseconds of a source calculated under the assumption that the source has a Gaussian profile. Column 18 is the effective radius or a half-light radius of a source in arcseconds. This value is computed by setting the input parameter PHOT FLUXFRAC = 0.5 and is given by the output parameter FLUX RADIUS. Column 19 is the semi-major axis of a source in arcseconds. Column 20 is the detected isophotal area of a source. This parameter may be used as a measure of the object’s size for the case of extended sources. Column 21 is the field number that the photometric data of a source comes from. The field number 1 represents the NEP-E field and 2 represents the NEP-W field. 4.2. Color-Magnitude and Color-Color Diagrams We have investigated the photometric properties of the NEP sources using several color- magnitude and color-color diagrams. In each diagram, we use the stellarity index given by SExtractor to distinguish between point sources and extended sources: stellarity > 0.95 for point sources (mostly stars) and stellarity < 0.2 for extended sources (mostly galaxies). – 8 – They are found to be statistically very useful tools to separate stars and galaxies brighter than r = 23 mag from all kinds of sources in the catalog. The r′ vs (g′ − r′) color-magnitude diagram (CMD) and (g′ − r′) color histogram in Figure 3 shows two prominent vertical sequences of point sources, i.e., stars at (g′ − r′) ∼ 0.3 (G dwarf stars) and 1.4 (M giant stars), respectively. The extended sources, presumably galaxies, are seen to be concentrated around the peak at (g′ − r′) ∼ 0.8 that corresponds to the (g′ − r′) color of early type galaxies (Fukugita et al. 1995). The extended sources start to dominate at r′ ≈ 22 mag and fainter. Some of these sources may be faint stars that the SExtractor failed to classify as point sources. However, the different (g′ − r′) color histograms of the extended and the point sources suggest that the majority of these faint extended sources are galaxies. Figure 4 displays the r′ vs (u∗− r′) CMD and (u∗− r′) color histogram of sources in the NEP-E field. The CMD shows that there are many galaxies distributed around (u∗ − r′) ∼ 1.0 as shown in the (u∗ − r′) color histogram of extended sources. This value is consistent with the (u∗−r′) color of Scd type galaxies (Fukugita et al. 1995), whereas the (g′−r′) color of most galaxies in Figure 3 is that of early type galaxies. It is also noted that there is a very long redward tail in the (u∗− r′) color distribution of extended sources, reaching nearly (u∗−r′) ≈ 4 or higher. The elliptical galaxies can be as red as (u∗−r′) ∼ 2.8 (Fukugita et al. 1995). Therefore, some of those faint and red sources with (u∗ − r′) > 3.0 could be distant galaxies with redshift z ≥ 0.1. Figures 5 through 7 show the characteristic distribution of point sources and extended sources in three color-color diagrams: (r′ − i′) vs (g′ − r′), (i′ − z′) vs (r′ − i′), and (g′ − z′) vs (u∗ − r′). The most prominent feature in these color-color diagrams (CCD’s) is very distinguishable sequences of point sources, i.e. stars. In Figure 5, the stellar sequence in a flipped ‘L’ shape has a very narrow width of about 0.2 dex in the (r′ − i′) vs (g′ − r′) color space. This tight sequence is also well represented by a straight line in Figure 6. In these diagrams, extended sources are found to populate in a relatively well constrained color space centered at a certain color. It is still clear that we can not possibly separate extended sources from point sources by simply constraining colors in the (r′− i′) vs (g′−r′) and the (i′−z′) vs (r′ − i′) color-color space. However, Figure 7 shows that the (g′ − z′) vs (u∗ − r′) color-color combination enables us to define the two exclusive spaces that are mostly populated by stars and galaxies, respectively. The boundary of these regions can be drawn by combination of three straight lines connecting (u∗ − r′, g′ − z′) = (−1.0, 0.5), (0.5, 0.5), (3.0, 1.7), and (3.0, 5.0). This is a very useful tool for separating faint galaxies from stars to search for distant galaxy clusters or to compute the 2D correlation function of galaxies. – 9 – 4.3. Galaxy Number Counts We have investigated the galaxy number counts using the NEP optical source catalog. To select galaxies efficiently from the photometry catalog, we adopted two different definitions of galaxies: (1) ‘Galaxy I’ sources with stellarity < 0.2 and (2) ‘Galaxy II’ sources that belong to the designated area where the extended sources appeared to occupy in the (u∗ − r′) vs (g′ − z′) CCD as denoted by the dashed line in Figure 7. The results of number counts for Galaxy I and Galaxy II sample, as shown in Figure 8, show a consistent rise from r′ = 17.5 mag up to about r′ = 23 mag and then a steep down turn due to the incompleteness at r′ ≥ 23 mag. From the lower and mid panels of Figure 8, it is clear that Galaxy II sample outnumbers Galaxy I sample in r′ > 23 mag. This difference between number counts is mostly due to the deteriorating reliability of stellarity index in r′ ≥ 23 mag, as shown in the upper panel of Figure 2. Therefore, the Galaxy I sample is likely to lose a large number of faint galaxies compared to the Galaxy II sample. In Figure 8, we also plotted the R-band galaxy number count of Kümmel & Wagner (2001) for comparison after a simple transformation into r′ band. Kümmel & Wagner (2001) derived the number count of extended sources by modeling the number count distribution of point-like sources (star) and then subtracting the extrapolated model count of point-like sources from the detected source count. On the other hand, we defined galaxies as a certain part of all the detected sources that satisfy a given stellarity condition (Galaxy I) or a given two-color criterion (Galaxy II). Nonetheless, it is clear from the plot that the number count by Kümmel & Wagner (2001) is generally consistent with our result except for the differences in the faint magnitude domain where the incompleteness becomes significant. As shown in the mid panel of Figure 8, the logarithmic number counts defined as Log(Count/0.5mag/deg2) also display very similar slopes over the range of r′ = 18 ∼ 22 mag. Simple linear square fits over that magnitude range returned the slope d(LogN)/dm = 0.387 for Galaxy I, 0.408 for Galaxy II, and 0.397 for Kümmel & Wagner (2001) data. The errors of the fitted slopes are about ±0.010 for all cases. Therefore, the slopes of logarithmic galaxy number counts are in good agreement with that of Kümmel & Wagner (2001) within the errors. However, there are some differences in several minor features between the logarithmic profiles of galaxy number count. This is more clearly shown in the upper panel of Figure 8. It is apparent that there are two small dips in Galaxy II data: one at r′ ≈ 17.5 ∼ 18 mag and another smooth one at r′ ≈ 19 ∼ 20 mag. The first dip also appears to exist in Kümmel & Wagner (2001) data but the second smooth and shallow dip could not be identified from Kümmel & Wagner (2001) data in this plot. However, the magnitude of the second dip is roughly coincident with the break point at R ≈ 19 ∼ 20 mag as shown in their – 10 – Figure 4 2. 4.4. Galaxy Clusters We have carried out a galaxy cluster search using our optical photometry catalog. Among numerous galaxy cluster finding methods available, we adopted a simplified version of the ‘C4 cluster finding algorithm’ by Miller et al. (2005) that utilizes a seven-dimensional position and color space. In this study, we use only a three-dimensional color space con- structed based on (g′− r′), (r′− i′), and (i′− z′) colors to select cluster member galaxies and then we investigate the spatial distribution of the selected galaxies to find any overdensity of these galaxies in small regions. Finally, the CMDs of any overdense region are consulted to check whether the red sequence of galaxies is apparent before we identify the region as a galaxy cluster. This approach requires the definition of color ranges spanned by the cluster member galaxies with various redshifts and richness classes before the actual application to the pho- tometric data. We have used the Sloan Digital Sky Survey (SDSS) (York et al. 2000) data for this purpose. The SDSS project provides a huge amount of multiband photometric and spectroscopic data over a large sky area. This enables us to select cluster member galaxies using the spectroscopy information and to reduce the possible contamination by field galax- ies and stars. One more advantage of using the SDSS dataset to define color ranges occupied by cluster member galaxies is that the SDSS filter system is the same as the MegaCam filter system of CFHT except for the u∗ filter (see Section 2 for details). Therefore, we have searched the SDSS database and retrieved photometric and spectroscopic data of nearby galaxy clusters for the calibration of the red sequence colors in the SDSS filter system. 4.4.1. Nearby Galaxy Clusters in SDSS We have searched the SDSS DR5 database (Adelman-McCarthy et al. 2007) and found 101 nearby Abell galaxy clusters whose u′, g′, r′, i′, z′ 3 photometric and spectroscopic data 2Figure 4 of Kümmel & Wagner (2001) is constructed using the galaxy number count made with 0.25 mag bin. But the number count data in a table published in the same paper, which we used for Figure 8, is made with 0.5 mag bin. The use of 0.5 mag bin instead of 0.25 mag bin in the number count is considered to cause the smoothing out the feature. 3Please note that the u′ filter is the Sloan system, which is different from the CFHT u∗ system. See Section 2 for details. – 11 – of member galaxies are available. Figure 9 displays the redshift distribution of these cluster galaxies. The redshift of the selected galaxy clusters runs from 0 to 0.2 with a peak at z ∼ 0.07. After the data retrieval from the SDSS data archive, we have selected member galaxies of each cluster based on the velocity distribution and the spatial separation from the cluster center. The total number of the selected member galaxies for the 101 clusters is about 5,700. From the cluster member galaxies, only the early type galaxies were selected using the ‘fracDev’ and ‘eclass’ parameters provided by the SDSS archive: fracDev > 0.8 and eclass < 0 (for further information on these constraints, see Bernardi et al. 2005). Application of these parameters finally returned about 1,700 early type galaxies in 88 clusters and the number of early type galaxies in each cluster runs from 4 (A2192) to 73 (A2199) depending on the redshift and the richness class. Figure 10 displays the CMDs of the cluster galaxies in the SDSS color system. From these diagrams, it is seen that most early type galaxies in the clusters lie within a well- defined and narrow color range with a width of ≤ 0.2 dex in (g′ − r′)0, (r ′ − i′)0, and (i′ − z′)0. Although the (u ′ − r′)0 vs r 0 CMD shows a rather large dispersion, it still shows a strong concentration at 2.0 ≤ (u′ − r′)0 ≤ 3.0. The colors of these narrow sequences of early type galaxies have been used as indicators of the clusters’ redshifts (Gladders & Yee 2000). We used the SDSS photometry data of member galaxies in four clusters (A2199, A1166, A1349, and A775) to derive the relation between the clusters’ redshift and the (g′−r′)0 color of the sequence as shown in Figure 11. From the linear fit after repeating one sigma clipping twice, we derived a linear and empirical relation between the redshift and the (g′− r′)0 color of the fitted sequences at r′0 = 18 mag (hereafter (g ′ − r′)r18,0) as follows: Redshift [z] = (0.415± 0.044)× (g′ − r′)r18,0 − (0.286± 0.039) (1) Therefore, the galaxies in nearby clusters occupy a certain space in a multi-color parameter space, which can be used to separate galaxies in the clusters from field galaxies and to estimate the approximate redshift of the cluster using the (g′− r′)r18,0 color of the sequence. 4.4.2. Galaxy Cluster Search in the NEP Field To find galaxy clusters based on the results shown in Figure 10 and to estimate the red- shift using Equation 1, the Galactic foreground reddening was corrected for our photometry data using Schlegel et al. (1998), assuming Ar′ ≈ 2.751E(B−V ), E(g ′−r′) ≈ 1.042E(B−V ), E(r′− i′) ≈ 0.665E(B−V ), and E(i′−z′) ≈ 0.607E(B−V ). After some tests on the CFHT data, we have defined color ranges for the cluster galaxies as follows: 0.6 < (g′ − r′)0 < 1.1, 0.1 < (r′ − i′)0 < 0.4, 0.0 < (i − z′)0 < 0.4. We did not use (u − r′)0 color for this color – 12 – space definition because of the unavailability of u∗ band data for the NEP-W field and the general low S/N in the u∗ band in the NEP-E field. According to a test performed after adopting the range 2.0 < (u∗ − r′)0 < 3.0 as the fourth constraining color, the efficiency of finding the cluster galaxies turned out to be comparable to that of another test run using the three color combination of (g′ − r′)0, (r ′ − i′)0, and (i ′ − z′)0. This three color combination approach also enables a homogeneous search of cluster galaxies over our two data fields. Possible cluster galaxies were selected by constraining the colors of galaxies with the predefined parameters of (g′ − r′)0, (r ′ − i′)0, and (i ′ − z′)0. Using their RA and Dec in- formation, we constructed spatial number density maps of the selected galaxies. Then we identified 13 possibly overdense regions in the NEP field. Over these 13 regions, CMDs of galaxies in (g′ − r′)0, (r ′ − i′)0, and (i ′ − z′)0 color were constructed using the source catalog to find any feature that resembles the red sequence of galaxies in a cluster. From this investigation, we have identified two galaxy clusters and one nearby galaxy group, and estimated their redshifts using the (g′− r′)0 colors of the red sequences. The CMDs of those galaxy clusters and the group are presented in Figures 12 − 14. More details about them are discussed below. 4.5. Galaxy Clusters and Groups in X-ray/Radio Source Catalogs Some galaxy clusters or groups were found and reported in the previous studies made by using the X-ray and the radio band data of the NEP region. Henry et al. (1995) found several X-ray-selected groups of galaxies in the NEP region based on RASS data and this work was revised and extended further by Henry et al. (2006). Brinkmann et al. (1999) also found that many X-ray sources in the NEP region have counterparts in the radio bands that were observed with VLA. From the X-ray and radio source catalogs provided by Henry et al. (2006) and Brinkmann et al. (1999), we have found a few galaxy clusters or groups in our data field. Among these clusters and groups, we will discuss two galaxy clusters and a galaxy group that were photometrically identified with our catalog. 4.5.1. NEPX1/VLA 1801.5+6645 This is the richest galaxy cluster found in our data field, which is easily seen in Figure 12. In the upper panel of Figure 12, the CMDs of galaxies show very strong red sequences running from r′0 ≈ 16 mag (magnitude of the third brightest galaxy) in three color domains: (g −r′)0, (r′ − i′)0, and (i ′ − z′)0. Those galaxies belonging to the red sequences are concentrated in a – 13 – very compact region, as shown in the lower panel of Figure 12. The redshift estimated from the red sequence’s (g′ − r′)r18,0 color and Equation 1 is about 0.072 ± 0.037. This cluster was previously discovered by Burg et al. (1992) from ROSAT survey data and named as NEPX1. They estimated the redshift of the cluster to be about 0.09 from the spectroscopic observations, which is in good agreement with our estimate. This cluster is considered as a part of the large supercluster structure that has been reported to exist in the NEP region (Hasinger et al. 1991; Mullis et al. 2001). It was also detected in the radio band from the VLA observations and was suggested as a possible coun- terpart of the X-ray source RXS J180137.7+664526 by Brinkmann et al. (1999). Although Brinkmann et al. (1999) listed this source as a galaxy group, we have reached a conclusion based on the red sequence galaxies in the CMDs that it is a galaxy cluster rather than a galaxy group. 4.5.2. RX J1754.7+6623 Henry et al. (2006) classified this source as a galaxy cluster with redshift z = 0.0879. However, the CMDs shown in Figure 13 suggest that there are two kinds of galaxies: (1) several bright galaxies that form the red sequence running from r′0 ≈ 16 to 18 mag and (2) many faint galaxies in the background. The second component of galaxies are not brighter than r′0 = 19 mag, which is about 1 mag fainter than the faintest galaxy of the brighter component. This indicates that those bright galaxies may happen to be located in front of the faint background galaxies by chance. Therefore, we suggest that these galaxies are members of a galaxy group rather than a galaxy cluster. The red sequence’s (g′ − r′)r18,0 color is about 0.793, corresponding to an estimated redshift of 0.043 ± 0.048. The relative large error may be due to the poorly determined slope of red sequence depends sensitively on several bright galaxies with r′0 < 18 mag. 4.5.3. RX J1757.9+6609 There is a known X-ray source RX J1757.9+6609 at the similar position. It is listed as a type 2 AGN with redshift z = 0.4865 in the catalog of Henry et al. (2006). A type 2 AGN is a narrow emission line (FWHM < 2000 km s−1) object, while a type 1 AGN is a broad emission line (FWHM ≥ 2000 km s−1) object (Gioia et al. 2003). The spatial number density plot (lower panel) of the red sequence galaxies in Figure 14 shows some clustering in the 2D projected space. This point was also noted by Gal et al. (2003) who carried out a cluster – 14 – search using the digitized Second Palomar Observatory Sky Survey (DPOSS) data. They detected an overdensity of galaxies at this location and listed it as a galaxy cluster candidate under the name of NSC J175751+660924, and also estimated its redshift to be z ≈ 0.1663 from their photometric redshift analysis. Our photometric data shows well developed red sequences running from r′0 ≈ 17.5 mag in (g ′− r′)0, (r ′− i′)0, and (i ′− z′)0 colors, confirming that it is a genuine galaxy cluster. The red sequence’s (g′ − r′)r18,0 color is about 0.811 and this leads to the estimated redshift of z = 0.043 ± 0.025. Therefore, there appears to be a galaxy cluster with redshift z < 0.2 and a type 2 AGN with redshift z = 0.4875 in the background of the same projected area. 4.5.4. Other Galaxy Clusters We have identified two galaxy clusters and one galaxy group by applying the simple color cut method to our photometry catalog. However, there are two more galaxy clusters that are listed in the literature in our observation field that we failed to confirm. One is RX J1801.7+6637 (z = 0.57) reported by Bower et al. (1996) and the other is RX J1757.3+6631 (z = 0.6909) from the catalog of Henry et al. (2006). There appear to be some weak hints of clustering around these clusters on the images. But our photometry is not deep enough to identify any red sequence of galaxies in the CMD of the galaxies. 4.6. The Photometric Properties of Other X-ray Sources The photometric properties of various X-ray sources such as stars, AGNs, BL Lac’s could provide valuable information regarding their stellar populations and evolution. Comparison of our photometry catalog with the X-ray source catalog of Henry et al. (2006) returned about 25 possible matches. Based on the classification information of the X-ray sources by Henry et al. (2006), we have investigated the photometric property of each object class using several CMDs and CCDs. Figure 15 shows a (g′ − r′) vs r′ CMD and the three different CCDs of various optical counterparts of X-ray sources. For a guideline in each diagram, the distribution of point sources, which are mostly stars, is also plotted. In Figure 15, it is easily seen that (1) generally, X-ray bright stars do not belong to the well defined stellar sequence in each diagram, and (2) AGNs and BL Lac objects are not readily separated from stars or other kind of objects in these CMDs and CCDs. Most X-ray bright stars that are used in this comparison are very bright (r′ < 15 mag) and are mostly saturated except for one faint giant star as shown in the upper-left panel of Figure 15. This – 15 – may explain why these X-ray bright stars appear to be in different color domains from other generic stars. Even one faint star (marked by a star with a circle in Figure 15) is bluer in (u∗− r′) color than normal stars. Although AGNs and BL Lac’s as a whole do not show any distinct pattern in each color, the AGN1s with stellarity > 0.9, which are relatively free from wrong identification, are found only in the blue domains of (g′ − r′), (r′ − i′), and (u∗ − r′). The (g′ − r′) CMD also shows that any sources with (g′ − r′) < 0.0 and r′ > 15 ∼ 16 mag are very likely to be AGN or BL Lac objects. 5. Summary and Conclusion We have obtained u∗, g′, r′, i′, z′ optical band high resolution images of the two square degrees area centered at the NEP with MegaCam/MegaPrime at CFHT. From the source detection and the photometry using SExtractor, we have compiled about 56,000 sources with r′ ≤ 23 mag, including point and extended sources, into the final optical photometry catalog as listed in Table 2. The use of the color-magnitude diagrams and color-color diagrams revealed strikingly different photometric characteristics of stars and galaxies. We have found that the use of (u∗ − r′) vs (g′ − z′) color enables us to clearly separate galaxies from stars and this separation does not suffer from the uncertainties involved in the morphological classification of faint sources. The galaxy number counts constructed from the galaxies selected based on the (u∗ − r′) vs (g′ − z′) color show a nearly monotonic increase up to about r′ = 23 mag with a slope d(LogN)/dm ≈ 0.40 ± 0.01, which is in agreement with the literature. However, there are some changes in the slope at r′ = 17.5 ∼ 18 mag and r′ = 19 ∼ 20 mag, which needs further studies. Using the SDSS DR5 data of the 101 nearby Abell galaxy clusters with redshift z < 0.2, we have derived a relation between the redshift and the color of the red sequences in the SDSS filter system which is compatible with the CFHT MegaCam filter system. Utilizing the information derived from the nearby galaxy clusters, we have applied a simple color cut method to find galaxy clusters in our data field, which returned two galaxy clusters and one galaxy group. These galaxy clusters and group are also radio and X-ray sources, which were reported by previous studies. We have also estimated the redshift of these galaxy clusters and group using the linear relation between the (g′ − r′)r18,0 color and the redshift. The estimated redshift is in agreement with the known value of 0.09 for galaxies in NEPX1 but it is relatively lower than the spectroscopic redshift in the literature for galaxies in RX J1754.7+6623 and the photometrically derived redshift for RX J1757.9+6609. For RX J1754.7+6623, this underestimation of redshift may be due to the poorly determined slope of red sequence since there are only a few bright galaxies. For RX J1757.9+6609, the galaxy – 16 – cluster seems to be overlayed on the background type 2 AGN with redshift of 0.4875. We have compared our photometry catalog with an X-ray source catalog in the literature to investigate the photometric properties of other X-ray sources. The result of comparison implies that the sources with (g′− r′) < 0.0 could be classified as candidates of AGNs or BL Lac objects. N.H. and T.K. are in part supported by the BK21 program of the Korean Government. This work was in part supported by the ABRL (R14-2002-058-01000-0). REFERENCES Adelman-McCarthy, J., et al. 2007, submitted to ApJS Bernardi, M., Sheth, R. K., Nichol, R. C., Schneider, D. P., & Brinkmann, J. 2005, AJ, 129, Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393 Brinkmann, W., et al. 1999, A&AS, 134, 221 Burg, R., et al. 1992, A&A, 259, L9 Boulade, O., et al. 2003, SPIE, 4841, 72 Bower, R. G., et al. 1996, MNRAS, 281, 59 Fukugita, M., Shimasaku, K., & Ichikawa, T. 1995, PASP, 107, 945 Gal, R. R., et al. 2003, ApJ, 125, 2064 Gal, R. R., et al. 2004, AJ, 128, 3082 Gioia, I. M., et al. 2003, ApJS, 149, 29 Gladders, M. D., & Yee, H. K. C. 2000, AJ, 120, 2148 Hasinger, G., Schmidt, M., & Trümper, J. 1991, A&A, 246, L2 Henry, J. P., et al. 1995, ApJ, 449, 422 Henry, J. P., et al. 2001, ApJ, 553, L113 – 17 – Henry, J. P., et al. 2006, ApJS, 162, 304 Kessler, M. F., et al. 1996, A&A, 315, L27 Kollgaard, R. I., et al. 1994, ApJS, 93, 145 Kümmel, M. W., & Wagner, S. J. 2001, A&A, 370, 384 Magnier, E. A., & Cuillandre, J.-C. 2004, PASP, 116, 449 Matsuhara, H., et al. 2006, PASJ, 58, 673 Miller, C. J., et al. 2005, AJ, 130, 968 Mullis, C. R., et al. 2001, ApJ, 553, L115 Neugebauer, G., et al. 1984, ApJ, 278, L1 Schlegel, D., Finkbeiner, D., & Davis, M., 1998, ApJ, 500, 525 Voges, W., et al. 1999, A&A, 349, 389 Werner, M. W., et al. 2004, ApJS, 154, 1 York, D. G., et al. 2000, AJ, 120, 1579 This preprint was prepared with the AAS LATEX macros v5.2. – 18 – Table 1. Observation Log Field RA DEC Filter Total Exp Time Seeing Depth Observation Date (UT) (hh:mm:ss) (dd:mm:ss) (sec) (arcsec) (AB mag)a (yyyy-mm-dd) NEP-E 18:04:31.51 66:33:38.60 u∗ 1800 × 2 1.13 25.98 2004-09-13/14 g′ 1200 × 2 1.08 26.12 2004-08-22 r′ 1500 × 2 0.99 25.58 2004-08-23 i′ 1500 × 2 0.69 24.70 2004-08-23/09-12 z′ 1800 × 2 0.73 24.03 2004-09-07 NEP-W 17:55:28.49 66:33:38.60 g′ 1200 × 2 1.01 26.12 2004-08-22 r′ 1500 × 2 0.87 25.58 2004-09-19/20 i′ 1500 × 2 0.99 24.85 2004-09-13/14 z′ 1800 × 2 0.85 23.71 2004-09-22 aThe depth of each filter data was measured as 4 σ flux over a circular aperture with a diameter of 1 ′′. Table 2. NEP Optical Source Cataloga ID RA (J2000) Dec (J2000) u∗ err(u∗) g′ err(g′) r′ err(r′) i′ err(i′) z′ err(z′) flag stellarity ellipticity fwhm r eff sma area field [deg] [deg] [mag] [mag] [mag] [mag] [mag] [arcsec] [arcsec] [arcsec] [arcsec2] 3 271.1438293 66.0613251 99.000 99.000 25.438 0.098 22.822 0.020 22.608 0.031 26.193 1.738 0 0.08 0.295 1.611 0.620 0.423 1.677 1 5 271.4050293 66.0613251 99.000 99.000 24.105 0.033 22.895 0.020 22.200 0.019 99.000 99.000 0 0.76 0.042 0.990 0.460 0.347 1.814 1 6 272.0104980 66.0599136 99.000 99.000 99.000 99.000 18.853 0.001 18.484 0.000 99.000 99.000 4 1.00 0.162 0.773 0.326 0.156 0.274 1 17 270.6894836 66.0615005 99.000 99.000 24.819 0.066 22.994 0.025 21.750 0.014 99.000 99.000 0 0.28 0.056 1.373 0.562 0.389 1.814 1 20 270.6847839 66.0614319 99.000 99.000 26.160 0.182 22.875 0.022 21.472 0.010 99.000 99.000 0 0.18 0.133 2.218 0.580 0.398 1.882 1 24 271.0955811 66.0624542 23.651 0.040 23.783 0.026 22.803 0.019 22.544 0.028 99.000 99.000 0 0.76 0.110 1.117 0.531 0.334 1.437 1 25 270.9777527 66.0611649 99.000 99.000 20.694 0.002 19.145 0.001 18.382 0.001 99.000 99.000 0 0.99 0.132 1.025 0.407 0.481 6.640 1 26 270.6968384 66.0618744 23.781 0.050 25.055 0.061 22.911 0.018 22.476 0.022 99.000 99.000 0 0.88 0.129 1.175 0.484 0.318 1.369 1 35 271.8403320 66.0610962 24.433 0.072 24.490 0.041 22.855 0.018 22.677 0.028 99.000 99.000 0 0.88 0.160 1.032 0.468 0.333 1.506 1 37 270.7434998 66.0618744 99.000 99.000 24.161 0.036 22.042 0.011 20.852 0.006 99.000 99.000 0 0.98 0.090 0.995 0.506 0.408 2.772 1 aThe complete version of this table is in the electronic edition of the Journal. The printed edition contains only a sample. – 20 – Fig. 1.— The observation field map around the North Ecliptic Pole (NEP) (dashed line boxes). Squares and asterisks represent, respectively, the known radio (Brinkmann et al. 1999) and X-ray (Henry et al. 2006) sources. Circles with an asterisk represent galaxy groups or clusters classified by Gioia et al. (2003) and cataloged in Henry et al. (2006). The large cross at the center shows the location of NEP. There are about 40 radio and/or X-ray sources found in our two fields of observation, NEP-E (East) and NEP-W (West). – 21 – Fig. 2.— The distribution of the stellarity index (upper panel), the r′ band magnitude error (mid panel), and the source number counts in u∗, g′, r′, i′, z′ bands (lower panel). The number of detected sources reaches its maximum at about 24.3 mag for u∗, 24.0 mag for g′, 23.7 mag for r′, 23.0 mag for i′, and 22.5 mag for z′ band. – 22 – Fig. 3.— Upper panel: (g′ − r′) vs r′ color magnitude diagram (CMD) of NEP sources. Sources selected with r′ ≤ 23 mag and (g′ − r′) color error < 0.1 mag criteria are plotted in different colors: point sources with stellarity > 0.95 in red and extended sources with stellarity < 0.2 in blue. Black dots with r′ < 17 mag mostly represent saturated stars. Lower panel: (g′ − r′) color distribution of those selected sources: point sources in solid line and extended sources in dashed line. Point sources and extended sources show different distributions in (g′ − r′) color: two peaks at (g′ − r′) ≈ 0.3 (G dwarf stars) and 1.4 (M giant stars) for point sources and a single peak at (g′ − r′) ≈ 0.8 for extended sources. – 23 – Fig. 4.— Upper panel: (u∗−r′) vs r′ color magnitude diagram (CMD) of sources in the NEP- E field. Sources selected with r′ ≤ 23 mag and (u∗ − r′) color error < 0.1 mag criteria are plotted in different colors: point sources with stellarity > 0.95 in red and extended sources with stellarity < 0.2 in blue. Lower panel: (u∗ − r′) color distribution of those selected sources: point sources in solid line and extended sources in dashed line. – 24 – Fig. 5.— Panel A: (g′ − r′) vs (r′ − i′) color-color diagram (CCD) of NEP sources. Sources selected with r′ ≤ 23 mag, (g′ − r′) color error < 0.1 mag, and (r′ − i′) color error < 0.1 mag criteria are plotted: point sources with stellarity > 0.95 in crosses and extended sources with stellarity < 0.2 in dots. The color distribution in (g′ − r′) color (Panel B) and (r′ − i′) color (Panel C) of those selected sources: point sources in solid line and extended sources in dashed line. Please note that the (r′− i′) and (g′− r′) color sequence of point sources is very narrow and clear while the feature of extended sources is very diffuse and broad centered at about (g′ − r′) ∼ 0.8 and (r′ − i′) ∼ 0.3. – 25 – Fig. 6.— Panel A: (r′ − i′) vs (i′ − z′) color-color diagram (CCD) of NEP sources. Sources selected with r′ ≤ 23 mag, (r′ − i′) color error < 0.1 mag, and (i′ − z′) color error < 0.1 mag criteria are plotted: point sources with stellarity > 0.95 in crosses and extended sources with stellarity < 0.2 in dots. The color distribution in (r′ − i′) color (Panel B) and (i′ − z′) color (Panel C) of those selected sources: point sources in solid line and extended sources in dashed line. In this plot, point sources are distributed in a narrow straight line running from (r′ − i′, i′ − z′) ≈ (0.1, 0.1) to (1.8, 0.8). – 26 – Fig. 7.— Panel A: (u∗ − r′) vs (g′ − z′) color-color diagram (CCD) of NEP sources in the NEP-E field. Selected sources with r′ ≤ 23 mag, (u∗−r′) color error < 0.1 mag, and (g′−z′) color error < 0.1 mag criteria are plotted: point sources with stellarity > 0.95 in crosses and extended sources with stellarity < 0.2 in dots. The dashed line in Panel A separates point sources from extended sources. See text for details. The color distribution in (u∗ − r′) color (Panel B) and (g′ − z′) color (Panel C) of those selected sources: point sources in solid line and extended sources in dashed line. – 27 – Fig. 8.— The number counts of galaxies in the NEP region, normalized to one square degree area. The Kümmel & Wagner (2001) (KW01) data in R band are also shown in all the three panels for comparison with our data (Galaxy I & II) assuming a simple relation of r′ ≃ R + 0.24. Galaxy I is defined to be sources with stellarity < 0.2 while Galaxy II is selected based on its position on the (u∗ − r′) vs (g′ − z′) CCD. It is clear that the overall shapes and patterns of galaxy number counts are generally in good agreement with Kümmel & Wagner (2001) result. – 28 – Fig. 9.— The redshift distribution of 101 galaxy clusters whose photometric and spectro- scopic data of member galaxies were retrieved from the SDSS DR5 data archive. – 29 – Fig. 10.— The color-magnitude diagrams (CMD) of galaxies in 101 nearby (z < 0.2) galaxy clusters in the SDSS data archive. All the photometric data used in these diagrams are foreground reddening corrected. Early type galaxies are plotted in crosses. Please note the relatively narrow and well-defined red sequence of early type galaxies in each CMD. – 30 – Fig. 11.— The representative (g′ − r′)0 color red sequences of four galaxy clusters: A2199 (inverted triangle, z = 0.0302), A1169 (square, z = 0.0586), A1346 (triangle, z = 0.0975), and A775 (circle, z = 0.1334). The color of the red sequence was derived by fitting the colors of early type galaxies in each cluster marked by the filled symbols while the open symbols were rejected from the fit. The red sequence of a galaxy cluster shifts redward as the corresponding redshift increases. Fig. 12.— The CMDs (the three upper panels), the spatial distribution (lower left), and the number density plot (lower right) of galaxies of NEPX1. In each CMD, only the extended sources are plotted and the magnitude limit adopted for the analysis is indicated with an arrow. The color ranges used to find the cluster galaxies are shown in short dashed lines and the red sequence line fitted in each color is displayed in long dashed lines. The estimated redshift from the (g′ − r′) color of the red sequence is indicated in the head as ‘ZCMR’. The galaxies satisfying the color cut criteria and being located within a solid circle in the lower left panel are represented by filled circles in the CMDs and the spatial distribution plot. The size of the circles in the spatial distribution plot are proportional to luminosity: the larger, the brighter. Galaxies that satisfies only the color cut criteria are plotted in open circles while other galaxies are in open squares. The dashed line in the density plot displays the number density profile of all galaxies around the selected area and the solid line shows the profile of ten times the number density of those selected galaxies and the corresponding errors. – 32 – Fig. 13.— The CMDs (the three upper panels), the spatial distribution (lower left), and the number density plot (lower right) of galaxies of RX J1754.7+6623. There are several bright galaxies in the foreground over the many faint background galaxies in this region. See Figure 12 for the legend. – 33 – Fig. 14.— The CMDs (the three upper panels), the spatial distribution (lower left), and the number density plot (lower right) of galaxies of RX J1757.9+6609. See Figure 12 for the legend. – 34 – Fig. 15.— The photometric properties of X-ray sources. Stellar symbols for stars, triangles for type 1 AGN (AGN1), reversed triangles for type 2 AGN (AGN2), asterisks for BL Lac’s, squares for galaxies, filled circles for galaxy clusters, and dots for point sources from our photometry catalog. The classification information of X-ray sources are from Henry et al. (2006). The distribution of point sources is presented in each CMD or CCD for references. Stars marked by stellar symbols in these diagrams are suspected to suffer from saturation effect except for one faint star represented by a stellar symbol within a circle. Please note that the point-like AGN1 sources (filled triangles) are bluer than (g′ − r′) ∼ 0.5. Introduction Observation Data Reduction Pre-Photometry Processing Source Detection & Photometry Results Source Catalog Color-Magnitude and Color-Color Diagrams Galaxy Number Counts Galaxy Clusters Nearby Galaxy Clusters in SDSS Galaxy Cluster Search in the NEP Field Galaxy Clusters and Groups in X-ray/Radio Source Catalogs NEPX1/VLA 1801.5+6645 RX J1754.7+6623 RX J1757.9+6609 Other Galaxy Clusters The Photometric Properties of Other X-ray Sources Summary and Conclusion

0704.1183

Confirmation of Cylindrical Perfect Invisibility Cloak Using Fourier-Bessel Analysis

Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations Zhichao Ruan∗1,2, Min Yan∗1, Curtis W. Neff1, and Min Qiu1† 1Laboratory of Optics, Photonics and Quantum Electronics, Department of Microelectronics and Applied Physics, Royal Institute of Technology (KTH), Electrum 229, 16440 Kista, Sweden 2Joint Research Center of Photonics of the Royal Institute of Technology (Sweden) and Zhejiang University, Zhejiang University, Yu-Quan, 310027 Hangzhou, PR China Abstract A cylindrical wave expansion method is developed to obtain the scattering field for an ideal two-dimensional cylindrical invisibility cloak. A near-ideal model of the invisibility cloak is set up to solve the boundary problem at the inner boundary of the cloak shell. We confirm that a cloak with the ideal material parameters is a perfect invisibility cloak by systematically studying the change of the scattering coefficients from the near-ideal case to the ideal one. However, due to the slow convergence of the zeroth order scattering coefficients, a tiny perturbation on the cloak would induce a noticeable field scattering and penetration. ∗ These authors contributed equally to this work. † Corresponding author. Electronic address: [email protected]. http://arxiv.org/abs/0704.1183v2 The exciting issue of exotic materials invisible to electromagnetic (EM) waves was dis- cussed in recent works [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Based on a coordinate transformation of Maxwell’s equations, Pendry et al. first proposed an invisibility cloak, which can protect objects inside the cloak from detection [1]: When EM waves pass through the invisibility cloak, the cloak will deflect the waves, guide them around the object, and return them to the original propagation direction without perturbing the exterior field. Numerical methods have been applied to solve the EM problem involving invisibility cloaks [6, 9], and an experi- mental result of the invisibility cloak using metamaterial with simplified material parameters has also recently been reported [7]. Yet, the ideal invisibility cloak has not been confirmed as a perfect cloak, due to the extreme material parameters required (zero or infinity) in the ideal cloak when approaching the inner boundary. Also, numerical methods usually describe the material parameters discretely, which can be computationally intensive in extreme cases. Thus it is preferable to use an analytical or semi-analytical method whenever possible. In this paper, we will study the scattering for an ideal invisibility cloak. We focus our analysis on the 2D cylindrical cloak, because the wave equation can be simplified in comparison with the 3D case, and a 2D invisibility cloak is more feasible to fabricate [7]. Here we take advantage of the cylindrical geometry of the structure and use the cylindrical wave expansion method to study the device semi-analytically. To avoid extreme values (zeros or infinity) of material parameters at the cloak’s inner surface, we introduce a small perturbation into the ideal cloak, and allow the perturbation to approach zero to study the scattering problem for the ideal cloak. Such an asymptotic analysis not only can confirm whether the ideal cloak would be perfectly invisible or not, it also provides hints on how sensitive such a device is to finite perturbations. A sensitivity analysis of the invisibility cloak directly determines the possibility of its application. Our studies show that the cylindrical invisibility cloak is very sensitive to tiny perturbations of the material parameters. First, let’s look at the wave equation inside a cylindrical cloak. According to Ref. [1], a simple transformation r′ = b−a r + a, θ′ = θ, z′ = z (1) can compress space from the cylindrical region 0 < r < b into the annular region a < r′ < b, where a is the inner radius of the cloak, b is the outer radius of the cloak, and r, θ and z (r′, θ′ and z′) are the radial, angular and vertical coordinates in the original (transformed) system, respectively. Following the approach in Ref. [1], the permittivity and permeability tensor components for the cloak shell can be given as εr = µr = , εθ = µθ = εz = µz = )2 r−a and air is assumed for the ambient environment and the interior regions. In the following, the transverse-electric (TE) polarized electromagnetic field is considered (i.e. the electrical field only exists in the z-direction) , however the transverse-magnetic derivation follows in similar manner. Throughout the paper, a exp(−iωt) time dependence is assumed. For the TE-polarized wave, only εz, µr, and µθ are relevant to the following general wave equation governing the Ez field in the cloak’s cylindrical coordinate ) + k20Ez = 0, (3) where k0 is the wave vector of light in vacuum. If we substitute Eq. 2 for εz, µr, and µθ, we + rµθ + εzµθr 2k20Ez + = 0 (4) Equation 4can be solved by a separation of variables Ez = Ψ(r)Θ(θ) and the introduction of a constant l: (r − a)2 + (r − a) )2(r − a)2k20 − l 2]Ψ = 0 (5) + l2Θ = 0. (6) Equation 5 is the lth-order Bessel differential equation, and the general solution of Eq. 6 is exp(ilθ). Therefore, there exists a simple set of solutions to Ez in the cloak shell of the form Fl(k1(r − a)) exp(ilθ) (7) where k1 = k0b/(b − a), Fl is the l-order Bessel function, and l is an integer number as required by the rotational boundary condition. Let us consider the scattering problem in which an arbitrary wave is incident on the cloak. According to the rigorous scattering theory [12], the incident field in the 2D case can be expanded in the cloak’s coordinates with the following expression Einz = αinl Jl(k0r) exp(ilθ), (8) where Jl is the l th-order Bessel function of the first kind. The scattering field can also be expanded as Escz = αscl Hl(k0r) exp(ilθ), (9) where Hl is the l th-order Hankel function of the first kind. FIG. 1: The schematic of a near-ideal invisibility cloak: The distribution of the material parameter is the same as the ideal one shown in Eq. 2, and the outer boundary is still fixed at r = b. However, the actual inner boundary is at r = a+ δ, where δ is a very small positive number. We note that the scattering coefficients cannot be directly obtained for the ideal cloak since εz → 0, µr → 0, and µθ → ∞ when r → a, and the Bessel function of the second kind in Eq. 7 has a singularity at r = a. In order to circumvent this, we introduce a small perturbation to the ideal cloak which we refer to as the near-ideal cloak, see Fig. 1. We expand the inner boundary of the cloak shell slightly, so that it is located at r = a+δ, where δ is a very small positive number. However, the material parameters are still calculated according to Eq. 2 as if the inner boundary is unchanged. The outer boundary remains fixed at r = b. When δ → 0, our model will be equivalent to the ideal cloak. Now the electric-field in each region can be given by (b < r) Ez = αinl Jl(k0r) exp(ilθ) + α l Hl(k0r) exp(ilθ) (a+ δ < r < b) Ez = α1l Jl(k1(r − a)) exp(ilθ) + α lHl(k1(r − a)) exp(ilθ) (r < a+ δ) Ez = α3l Jl(k0r) exp(ilθ) where αil(i = 1, 2, 3) are the expansion coefficients for the resulting field inside the cloak. The tangential fields Ez and Hθ (which can be obtained from Ez), should be continuous across the interfaces at r = a+ δ and r = b; and the orthogonality of exp(ilθ) allows waves in each Bessel order to decouple. Thus, we can have the following four equations: αinl Jl(k0b) + α l Hl(k0b) = α l Jl(k1(b− a)) + α lHl(k1(b− a)) (11a) α1l Jl(k1δ) + α lHl(k1δ) = α l Jl(k0(a+ δ)) (11b) l (k0b) + k0α l(k0b) = µθ(b) α1l J l (k1(b− a)) + µθ(b) l(k1(b− a)) (11c) µθ(a+ δ) α1l J l (k1δ) + µθ(a+ δ) l(k1δ) = k0α l (k0(a+ δ)) (11d) which is a set of linear equations. Thus each order expansion coefficient in each material region can be exactly solved. In turn we can obtain the fields in each region. As a direct result of this set linear equations, we can prove that when δ → 0, αscl = α l → 0 , α1l = α l , and α l → 0 for any α l , i.e., the ideal cloak is a perfect invisibility cloak. Firstly, it can be assumed that |αil(i = sc, 1, 2, 3)| must be finite. Otherwise, the scattering field would be infinite if the incident field has the lth order component. Secondly, due to k1(b−a) = k0b and k1 = k0µθ(b), when δ → 0, Eq. 11a and 11c become (α l )Jl(k0b)+(α l )Hl(k0b) = 0 and (αinl − α l (k0b) + (α l − α l(k0b) = 0, respectively. Since b can be arbitrary and the Bessel functions are not always zeros, αinl = α l and α l = α l must be satisfied. Thirdly, from Eq. 11b, we can obtain the following inequality ∣α2lHl(k1δ) ∣α3l Jl(k0(a+ δ)) ∣α1l Jl(k1δ) ∣ . (12) When δ → 0, the right side of the above inequality approaches a finite value but |Hl(k1δ)| approaches infinity on the left side. Thus, |α2l | must approach zero. Finally, from Eq. 11b, we can also obtain that |α2lH l(k1δ)| ≤ J ′l (k0(a+ δ)) +|α1l J l (k1δ)|. While from Eq. 11d, we have l(k0(a+ δ)) µθ(a+ δ) α1l J l(k1δ) µθ(a+ δ) l(k1δ) . (13) Since µθ(a+ δ) → ∞ and the right side of the above inequality approaches zero when δ → 0, we obtain that |α3l | → 0. Consequently, this argument proves that the scattering field and the field in the interior region of the cloak are zero when δ = 0, i.e., the ideal cloak is a perfect invisibility cloak. FIG. 2: (Color online) (a) Snapshot of the resulting electric-field distribution, (b) the corresponding norm in the vicinity of the cloaked object, and (c) the snapshot of the corresponding scattering field outside the cloak for the near-ideal cloak with δ = 10−5a and when a plane wave is perpendicularly incident on the cloak. The black lines outline the cloak shell. Axis unit: meter. Although we have just confirmed that the ideal cloak can provide perfect invisibility, further study the near-ideal cloak by the above analytical method illuminates how sensitive the parameter δ is to the performance of the cloak. As an example, we use the same material parameters in Ref. [6] where the inner radius of the cloak is a = 0.1m, the outer radius of the cloak is b = 0.2m, and the frequency of the incident plane wave is 2GHz. Similarly, we also consider a plane wave incident on the cloak, where the expansion coefficients in Eq. 8 αinl = i lA exp(−ik0r1 cos(ϕ+ θ1)− ilϕ), (14) where (r1, θ1, 0) is the coordinate of the phase reference point, A is the amplitude of the plane wave, and ϕ is the incident angle [13]. Here the phase reference point is set at r1 = 4a and θ1 = π, the amplitude is A = 1, and the incident angle is ϕ = 0 (i.e. the plane wave propagates from left to right). We use 31 Bessel terms (−15 ≤ l ≤ 15) to calculate the scattering field for the near-ideal cloak with δ = 10−5a. The number of expansion terms is sufficient for convergence of the calculated fields. Figure 2 shows the snapshot of the resulting electric-field distribution (i.e. the real part of the electric-field phasor), and the corresponding norm in the vicinity of the cloaked object. The electric-field distribution clearly demonstrates the cloaking effect of the near-ideal cloak to the incident plane wave. However, the norm of the electric-field (Fig. 2 (b)) reveals that there is still a little bit of the field in the cloak interior and an obvious scattering ripple around the cloak. The amplitude of the resulting electric-field at the center is 0.197. The snapshot of the scattering field (Fig. 2 (c)) shows that it propagates almost isotropically in all angles. Even though the amplitude of the scattering field is much smaller than that of the incident plane wave, the interference of the incident plane wave and the scattering field creates the ripples in the norm (Fig. 2 (b)). Since each order expansion coefficient of the scattering field is only relevant to each order expansion coefficient of the incident field (cf. Eq. 11), we can define the scattering coefficient for each order as cscl = . (15) These coefficients for the field inside the cloak c = αil/α l , i = 1, 2, 3 can also be defined in the same way. To study the ideal cloak, we more δ closer to 0. The amplitude and the phase of these coefficients for 10−8a < δ < 10−2a are shown in Fig. 3, where (a)-(b) and (c)-(d) correspond to the cases of l = 0 and l = 1 , respectively. From Fig. 3, it is clear that c is always equal to 1 for both cases. That is, the incident field propagates into the cloak without any reflection at the outer boundary, which coincides with the explanation of the cloaking effect from the coordination transformation approach FIG. 3: (Color online) The amplitude and phase of the scattering coefficients for the different δ, where (a)-(b) and (c)-(d) correspond to the cases of l = 0 and l = 1 respectively. csc is denoted by the blue point-dashed line, c (the solid black line), c (the red circle-marked), and c green star-marked). [1]. The same behavior for the scattering fields occurs at the outer boundary, where they propagate from inside to outside without any reflection, thus cscl is always equal to c Our computational results also confirm that both cscl and c approach zero when δ → 0. In particular, compared with the case of l = 0, |cscl | and for l = 1 are much smaller, and approach zero more rapidly. This is also observed for the other higher order cases. Thus, in the case of the plane wave incident, where |αinl | is the same for each order, the dominating term of the scattering field outside of the cloak is of the form of the zeroth-order Hankel function of the first kind. Meanwhile, the resulting field in the interior region has a dominating term of J0(k0r) . This explains the near azimuthally invariable distribution of the field in the interior region (see Fig. 2(a)) and the scattering field outside the cloak (see Fig. 2(c)), which has also been mentioned in Ref. [6]. It is worth noting that the zeroth order scattering coefficients csc0 and c 0 decrease ex- tremely slowly with reduced δ, e.g. when δ is decreased from 10−5a to 10−8a, |csc0 | decreased only from 0.175 to 0.099. By utilizing the arbitrary calculation precision of the software MATHEMATICA, we found that the convergence of the limit is so slow that even for δ = 10−99a (i.e. εz ≈ 4 × 10 −99, µr ≈ 10 −99, and µθ ≈ 10 99 at the inner boundary in this case), |csc0 | = 6.973 × 10 −3. Therefore, we conclude that even though an cloak with the ideal material parameters in Ref. [1] is a perfect cloak, a non-ideal invisibility cloak does not provide a good enough cloaking effect due to the slow convergence of |csc0 | and In conclusion, we have used the cylindrical wave expansion method to study the electro- magnetic scattering properties of a 2D invisibility cloak. A near-ideal model of the invisibility cloak is set up to solve the boundary problem at the inner boundary of the cloak shell. By systematically studying the change of the scattering coefficients from the near-ideal case to the ideal one, we confirm that the cloak with the ideal material parameter is a perfect invisibility cloak. But due to the slow convergence of the scattering coefficients, a tiny per- turbation on the cloak would induce a noticeable field scattering and penetration. We also proved that the scattered and penetrated fields are dominated by zeroth-order cylindrical waves. Though our work has focused on the 2D cylindrical cloak, it can be reliably ex- tended to the 3D spherical case. Our method and results are also useful for either designing or detecting this type of the invisibility cloak. This work is supported by the Swedish Foundation for Strategic Research (SSF) through the INGVAR program, the SSF Strategic Research Center in Photonics, and the Swedish Research Council (VR). Z.C.R. acknowledges the partial support from the National Basic Research Program (973) of China under Project No. 2004CB719800. [1] J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006). [2] U. Leonhardt, Science 312, 1777 (2006). [3] A. Alù and N. Engheta, Phys. Rev. E 72, 016623 (2005). [4] D. A. B. Miller, Optics Express 14, 12457 (2006). [5] U. Leonhardt, New J. Phys. 8, 118 (2006). [6] S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, Phys. Rev. E. 74, 036621 (2006). [7] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006). [8] G. W. Milton, M. Briane, and J. R. Willis, New J. Phys. 8, 248 (2006). [9] F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, Opt. Lett. 32, 1069 (2007). [10] W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nature Photonics 1, 224 (2007). [11] H. Chen and C. T. Chan, Appl. Phys. Lett. 90, 241105 (2007). [12] H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981). [13] D. Felbacq, G. Tayeb, and D. Maystre, J. Opt. Soc. Am. A 11, 2526 (1994). References

0704.1184

Sufficiency Criterion for the Validity of the Adiabatic Approximation

Sufficiency Criterion for the Validity of the Adiabatic Approximation D. M. Tong1,2 ∗ , K. Singh1, L. C. Kwek1,3, and C. H. Oh1† 1 Department of Physics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore 2Department of Physics, Shandong Normal University, Jinan 250100, China 3National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 639798, Singapore (Dated: October 29, 2018) Abstract We examine the quantitative condition which has been widely used as a criterion for the adiabatic approximation but was recently found insufficient. Our results indicate that the usual quantitative condition is sufficient for a special class of quantum mechanical systems. For general systems, it may not be sufficient, but it along with additional conditions is sufficient. The usual quantitative condition and the additional conditions constitute a general criterion for the validity of the adiabatic approximation, which is applicable to allN−dimensional quantum systems. Moreover, we illustrate the use of the general quantitative criterion in some physical models. PACS numbers: 03.65.Ta, 03.65.Vf ∗ Electronic address: [email protected] † Electronic address: [email protected] http://arxiv.org/abs/0704.1184v1 The adiabatic theorem reads that if a quantum system with a time-dependent nonde- generate Hamiltonian H(t) is initially in the n-th instantaneous eigenstate of H(0), and if H(t) evolves slowly enough, then the state of the system at time t will remain in the n-th instantaneous eigenstate of H(t) up to a multiplicative phase factor. The theorem is a useful tool [1, 2, 3, 4, 5, 6, 7] but its practical application relies on the criterion of the “slowness” required by the theorem. In the literature, the “slowness” is usually encoded by the quantitative condition, 〈En(t)|Ėm(t)〉 En(t)−Em(t) ≪ 1, m 6= n, t ∈ [0, τ ] (1) where Em(t) and |Em(t)〉 are the eigenvalues and eigenstates of H(t), and τ is the total evolution time. This quantitative condition had been deemed to be a sufficiency criterion, but it was recently found insufficient. In order to resolve the counterexample raised in [8], we showed that fulfilling only the quantitative condition cannot guarantee the validity of the adiabatic approximation[9]. While this explains the counterexample in [8], it raises the obvious question: In what situations would the criterion be sufficient and more importantly how can it be extended to general cases? There have been some attempts to address different aspects of this problem[10, 11, 12, 13], but it remains largely unresolved. In order to resolve the problem, we trace the quantitative condition to its source, and we note that the condition has never been convincingly proven. It looks likely that the condition was first derived from some special quantum systems and it was extended to systems beyond its range of applicability. For instance, in Refs. [3, 4], condition (1) was obtained by assuming a first order approximation and by requiring both Em(t) − En(t) and 〈Em(t)|Ėn(t)〉 to be constants. However, such a proof is not convincing because a first order approximation may be taken as a good approximation of the exact value only if all the higher order corrections are proven to be much smaller. We note that this is not the case here. This has also been pointed out in Refs. [10, 11]. Therefore, even if the condition is sufficient for the special quantum systems, a convincing proof is still necessary. In any case, the sufficiency criterion for general systems is grossly lacking. It should be emphasized that the lack of a sufficiency criterion weakens the applicability of the adiabatic theorem. In the present paper, we address this sufficiency criterion issue. Firstly, we furnish a new proof to show that the quantitative condition is indeed a sufficiency criterion for the adiabatic approximation in the quantum systems which satisfy the requirement of both Em(t)− En(t) and 〈Em(t)|Ėn(t)〉 being constant. Secondly, to extend its validity, we show that the the quantitative condition along with some additional conditions is sufficiency for general systems. Thus, the usual quantitative condition and the additional conditions constitute a general criterion for the adiabatic approximation, which is applicable to all N−dimensional quantum systems. Moreover, we illustrate the use of the general criterion in some physical models. Let us consider an N -dimensional quantum system with the Hamiltonian H(t). The instantaneous nondegenerate eigenvalues and orthonormal eigenstates of H(t), denoted as Em(t) and |Em(t)〉 respectively, are defined by H(t)|Em(t)〉 = Em(t)|Em(t)〉, m = 1, . . . , N. (2) |Em(t)〉 is determined by Eq. (2) up to a phase factor. Hereafter, we choose it such that 〈Em(t)|Ėm(t)〉 = 0[14]. If we assume that the system is initially in the n−th eigenstate |ψ(0)〉 = |En(0)〉, then its state at time t, |ψ(t)〉, is dictated by the Schrödinger equation |ψ(t)〉 = H(t)|ψ(t)〉. (3) In the basis {|Em(t)〉}, |ψ(t)〉 can be expanded as |ψ(t)〉 = cm(t)e ′)dt′ |Em(t)〉, (4) where cm(t) = e ′)dt′〈Em(t)|ψ(t)〉 are the time-dependent coefficients. Substituting it into the Schrödinger equation, we obtain dcm(t) l 6=m 〈Em|Ėl〉e ωmldt cl(t) = 0, (5) which leads to cm(t) = δmn − l 6=m 〈Em|Ėl〉e ωmldt ′)dt′, (6) where Em ≡ Em(t), |Em〉 ≡ |Em(t)〉, ωml ≡ Em(t) − El(t), and m = 1, 2, ..., N . Here, n in Eq. (6) is the index of the initial state |En(0)〉. We want to ascertain the criterion under which the adiabatic approximation is valid, i.e., the condition(s) for which the fidelity F = |〈En(t)|ψ(t)〉| = |cn(t)| ≈ 1. We first discuss the quantum systems for which both ωml and 〈Em|Ėl〉 are constants[15]. In this case, from Eq. (6), we have, by partial integration, cn(t) = 1 + i 〈En|Ėm〉 eiωnmtcm(t)− eiωnmt ċm(t ′)dt′ . (7) Substituting ċm(t) from Eq. (5) into the above equation, we obtain cn(t) = 1 + i 〈En|Ėm〉 eiωnmtcm(t) l 6=m 〈En|Ėm〉 〈Em|Ėl〉 eiωnlt ′)dt′. (8) Noting that |cm(t)| ≤ 1, we have from Eq. (8) 1− |cn(t)| ≤ 〈En|Ėm〉 l 6=m ∣〈Em|Ėl〉 ∣ |Il|  , (9) where Il = iωnlt ′)dt′. Clearly, if the integral Il is bounded by a finite number, the quantitative condition (1) can sufficiently guarantee that 1 − |cn(t)| ≪ 1. We now show that this is indeed the case. To this end, by letting cl(t) = cl(t)e iωnlt and substituting it into Eq. (5), we have dcm(t) − iωnmcm(t) + l 6=m 〈Em|Ėl〉cl(t) = 0. (10) Since Eq. (10) is a system of differential equations, the general solution of cm(t) comprise N special solutions in the form ame iλt, where am and λ are time-independent constants. They are determined by the equations, (ωnm − λ) am + i l 6=m 〈Em|Ėl〉al = 0, (11) where m = 1, 2, ..., N. Solving Eq. (11), we may obtain λ = λ1, λ2, ..., λN , where λj are nonzero real numbers[16]. For each λj, there is a solution cmj(t) = amje iλjt. All the N independent solutions lead to the general solution cm(t) = j=1 pjamje iλjt, where the coefficients pj are determined by the initial conditions cm(0) = δmn. Then, we have |Il| = 0 cl(t ′)dt′ pjamj eiλjt − 1 ∣ ≤ 2 pjamj ∣ , where the latter term is a finite number, independent of time t. This completes the proof that the quantitative condition (1) is a sufficiency criterion for the quantum systems which satisfy the requirement that both ωml and 〈Em|Ėl〉 are constants. We now discuss general quantum systems. Let us return to Eq. (6). We have, by partial integration, cn(t) = 1 + i 〈En|Ėm〉 ωnmdt cm(t) 〈En|Ėm〉 ωnmdt ′)dt′ 〈En|Ėm〉 ωnmdt ċm(t ′)dt′. (12) Substituting Eq. (5) into Eq. (12), we have cn(t) = 1 + i 〈En|Ėm〉 ωnmdt cm(t) 〈En|Ėm〉 ωnmdt ′)dt′ l 6=m 〈En|Ėm〉 〈Em|Ėl〉e ωnldt ′)dt′. (13) In the general case, although it is difficult to estimate exactly the values of the integrals in Eq. (13) as we did in the above special case, it is still possible to obtain bounds on the integrals, which will lead to the sufficiency criterion. Noting that |cm(t)| ≤ 1 and ωnmdt = 1, we 〈En|Ėm〉 ωnmdt ′)dt′ 〈En|Ėm〉 dt′, (14) 〈En|Ėm〉 〈Em|Ėl〉e ωnldt ′)dt′ 〈En|Ėm〉 ∣〈Em|Ėl〉 ∣ dt′. (15) From Eqs. (13), (14), and (15), we obtain 1− |cn(t)| ≤ 〈En|Ėm〉 〈En|Ėm〉 l 6=m 〈En|Ėm〉 ∣〈Em|Ėl〉 ∣ dt′. (16) Since the sums on the right hand side of expression (16) are finite terms, the approximation 1 − |cn(t)| ≪ 1 is guaranteed if each of the terms is small. This is met if the following conditions 〈En(t)|Ėm(t)〉 En(t)−Em(t) ≪ 1, t ∈ [0, τ ], (17) 〈En(t)|Ėm(t)〉 En(t)− Em(t) dt≪ 1, (18) 〈En(t)|Ėm(t)〉 En(t)−Em(t) ∣〈Em(t)|Ėl(t)〉 ∣ dt≪ 1, (19) are satisfied, where m 6= n and τ is the total evolution time for which the adiabatic ap- proximation is valid, t ∈ [0, τ ]. Expression (A) is just the well-known quantitative condition (1), and expressions (B) and (C) are two additional conditions, which set a bound on the total evolution time. Physically, if a quantum system satisfying condition (A) is initially in its n−th eigenstate, it will remain close to its n−th instantaneous eigenstate during the initial short time but it may deviate from its instantaneous eigenstate at a later time. The additional conditions provide a time scale, τ , for which the state remains close to the instan- taneous eigenstate. τ can be obtained after calculating the integrals in Eqs. (18) and (19). In some special cases, the integrals may be easily evaluated. For instance, if En(t)−Em(t) is a monotonic function of t, the integral in Eq. (18) can be carried out. In some other cases, we may not be able to evaluate the integrals analytically. In such instance, we may simplify the conditions by appealing to estimations. In any case, the stronger expressions 〈En(t)|Ėm(t)〉 En(t)−Em(t) τ ≪ 1, (20) 〈En(t)|Ėm(t)〉 En(t)− Em(t) ∣〈Em|Ėl〉 τ ≪ 1, (21) can always cover conditions (B) and (C) respectively, where |f(t)|M means the maximal modulus of f(t) for t ∈ [0, τ ]. As conditions (b) and (c) are stronger than (B) and (C), the latter should be preferentially used when possible. The quantitative condition (A) (i.e.(1)) and the additional conditions (B) and (C) con- stitute a general quantitative criterion for the adiabatic approximation. The general cri- terion can sufficiently guarantee the validity of the approximation and it is applicable to all N−dimensional quantum systems. That is, if a quantum system, initially in the eigen- state |En(0)〉, fulfills the general criterion, it will remain, with high probability, in the n-th instantaneous eigenstate |En(t)〉 up to a phase factor. The fidelity between the approx- imate state and the exact state may be estimated by Eq. (16). It is interesting to use this criterion to reexamine the counterexample furnished with two related Hamiltonians Ha(t) = iU̇(t)U+(t) and Hb(t) = iU̇+(t)U(t) in [8, 9]. Suppose the eigenstate |Eim〉 of H i(t) (i = a, b) has been properly chosen such as 〈Eim|Ė m〉 = 0. We may have the relation, 〈Ebn|Ė m〉 = e (Ean−E 〈Ean|Ė m〉, which result in that condition (B) cannot be satisfied for Hb(t), in general, if it is satisfied for Ha(t). Hence, the counterexample is ruled out from the adiabatic systems. We now apply the general criterion to some quantum systems. Firstly, we specialize the general criterion for the quantum systems in which En(t)−Em(t) is a monotonic function of t. Many interesting adiabatic systems may belong to this class. In this case, we have 〈En|Ėm〉 〈En|Ėm〉 ωnm(τ) ωnm(0) 〈En|Ėm〉 τ. (22) Since | lnωnm(τ)/ωnm(0)| is a finite number and hence the first term is small under condition (A), the additional condition (B) can be written as 〈En|Ėm〉 τ ≪ 1. (23) As we cannot simplify the integral in (19) here, we use the stronger expression (c). Therefore, conditions (A), (B1) and (c) constitute the sufficiency criterion. Since in real physical experiments the total evolution time τ is usually finite, condition (c) is automatically ensured by condition (A). Hence, in this case, we may take (A) and (B1) as the adiabatic criterion. Secondly, we consider a quantum system defined by the parameterized Hamiltonian H(s), where s = t/T, t ∈ [0, T ]. The well-known proofs of adiabatic theorem given in Refs. [5, 6] were carried out by using such a Hamiltonian. We now apply the general criterion to the system. By substituting t = Ts into conditions (A), (B), and (C), we obtain 〈En|Ėm〉 En −Em 〈En(s)|Ėm(s)〉 En(s)− Em(s) 〈En|Ėm〉 En − Em 〈En(s)|Ėm(s)〉 En(s)− Em(s) (En(s)−Em(s)) En(s)− Em(s) 〈En(s)|Ėm(s)〉 En(s)− Em(s) 〈En|Ėm〉 En − Em ∣〈Em|Ėl〉 ∣ dt ≤ 〈En(s)|Ėm(s)〉 En(s)−Em(s) ∣〈Em(s)|Ėl(s)〉 . (24) Since all the terms on the right of the above expressions can be arbitrarily small as T becomes large, conditions (A), (B), and (C) are met if T is large enough. We then arrive at the conclusion that the adiabatic approximation is always valid for quantum systems defined by Hamiltonians of the forms H( t ) with t ∈ [0, T ], as long as T is large enough. This conclusion agrees with the results in [5, 6]. What is special about such Hamiltonians is that the criterion expressed by (A), (B), and (C) can always be satisfied by choosing values of the parameter T . However, once the parameter T is chosen, there is also a bound on total evolution time, τ = T . The adiabatic approximation is valid if t ∈ [0, T ], and it may be invalid if t is larger than T . Finally, we apply the general criterion to a concrete model to understand how the time constraint functions. Consider a spin-half particle in a rotating magnetic field, H(t) = (σx sin θ cosωt + σy sin θ sinωt + σz cos θ). For this model, we have E1 − E2 = ω0, 〈E1|Ė2〉 = − sin θeiωt cos θ. Substituting them into (A), (b), and (c), we have ω sin θ/ω0 ≪ 1 , (ω sin θ/ω0) · ωτ cos θ ≪ 1, and (ω sin θ/ω0) · ωτ sin θ ≪ 1. It shows that, besides the usual condition, there is a time constraint, t ∈ 0, k 1 , where k is a certain number. The time constrain limits the total evolution time to a finite number of rotating periods of the magnetic field. Such a time limit is acceptable in physics, and it is also consistent with the geometric phase consideration. In Ref. [17], it was shown that the geometric phase calculated by using the adiabatic approximation may differ appreciably from its exact value if the evolution time is too large. The difference reads δγ ≃ −ωτ sin θ · ω sin θ 2(ω0+2ω cos θ) . This implies that, in order to guarantee δγ ≪ 1, ωτ must be finite. So, the application of the adiabatic approximation on geometric phase indicates that the time constraint is reasonable. In summary, we have examined the quantitative condition (1). Our results indicate that the usual quantitative condition (1) itself is a sufficiency criterion for the adiabatic approx- imation when it is applied to the quantum systems in which (Em − El) and 〈Em|Ėl〉 are constants. It may not be a sufficiency criterion when it is applied to a general quantum system. To extend its validity to general systems, we have shown that the usual condi- tion (1) and the additional conditions (18) and (19) constitute a general criterion for the adiabatic approximation. The general criterion can sufficiently guarantee the validity of the adiabatic approximation and it is applicable to all N−dimensional quantum systems. We have examined a few examples to illustrate its use. It should be noted that when the sufficiency conditions are used, the eigenstates |Em〉 need to be properly chosen such that 〈Em|Ėm〉 = 0. D. M. Tong acknowledges the useful discussions with B. C. Sanders, K.-P. Marzlin, and L.-A. Wu. This work was supported by NUS Research Grant No. R-144-000-189-305 and NSF of China No. 10675076. Tong also acknowledges the financial support from IQIS at University of Calgary when he visited there. [1] M. Born and V. Fock, Z. Phys. 51, 165(1928). [2] J. Schwinger, Phys. Rev. 51, 648(1937). [3] L. I. Schiff, Quantum Mechanics, McGRAW-Hill Book Co., Inc., New York, (1949). [4] D. Bohm, Quantum Theory, Prentic-Hall, Inc., New York, (1951). [5] T. Kato, J. Phys. Soc. Jap. 5, 435 (1950). [6] A. Messiah, Quantum Mechanics, North-Holland Pub. Co., (1962). [7] L. D. Landau, Zeitschrift 2, 46 (1932); C. Zener, Proc. R. Soc. London A 137, 696 (1932); M. Gell-Mann and F. Low, Phys. Rev. 84, 350 (1951); M.V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984); E. Farhi et al, Science 292, 472(2001); M. S. Sarandy and D. A. Lidar, Phys. Rev. Lett. 95, 250503 (2005); P. Thunstrom, J. Åberg, and E. Sjöqvist 2005 Phys. Rev. A 72, 022328 (2005) [8] K. P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004). [9] D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 95, 110407 (2005). [10] R. Mackenzie, E. Marcotte, and H. Paquette, Phys. Rev. A 73, 042104 (2006). [11] T. Vertesi and R. Englman, Phys. Lett. A 353, 11 (2006). [12] J. Larson and S. Stenholm, Phys. Rev. A 73, 033805 (2006); S. Duki, H. Mathur, and O. Narayan, Phys. Rev. Lett. 97, 128901 (2006); J. Ma et al, Phys. Rev. Lett. 97, 128902 (2006). [13] D. Comparat, quant-ph/0607118 (2006); S. Jansen, M. B. Ruskai, R. Seiler, quant-ph/0603175 (2006); M. Y. Ye et al, quant-ph/0509038 (2005); A. Ambainis and O. Regev, quant-ph/0411152 (2004). [14] If |Em〉 does not satisfy 〈Em|Ėm〉 = 0, then it is replaced by e 〈Em|Ėm〉dt |Em〉. [15] One example of such systems is a spin-half particle in a rotating magnetic field with θ = π/2. See the last page. [16] Since λ can be taken as the eigenvalues of the hermitian matrix defined by Mml = ωnmδml + i〈Em(t)|Ėl(t)〉, they must be real numbers. Besides, λ = 0 is not a solution in general, other- wise, it leads to a trivial proof. http://arxiv.org/abs/quant-ph/0607118 http://arxiv.org/abs/quant-ph/0603175 http://arxiv.org/abs/quant-ph/0509038 http://arxiv.org/abs/quant-ph/0411152 [17] D. M. Tong et al, Phys. Lett. A339,288 (2005). References

0704.1185

Five-dimensional N = 1 AdS superspace: Geometry, off-shell multiplets and dynamics

arXiv:0704.1185 [hep-th] April, 2007 Five-dimensional N = 1 AdS superspace: Geometry, off-shell multiplets and dynamics Sergei M. Kuzenko1 and Gabriele Tartaglino-Mazzucchelli2 School of Physics M013, The University of Western Australia 35 Stirling Highway, Crawley W.A. 6009, Australia Abstract As a step towards formulating projective superspace techniques for supergrav- ity theories with eight supercharges, this work is devoted to field theory in five- dimensional N = 1 anti-de Sitter superspace AdS5|8 = SU(2,2|1)/SO(4, 1) × U(1) which is a maximally symmetric curved background. We develop the differential ge- ometry of AdS5|8 and describe its isometries in terms of Killing supervectors. Vari- ous off-shell supermultiplets in AdS5|8×S2 are defined, and supersymmetric actions are constructed both in harmonic and projective superspace approaches. Several families of supersymmetric theories are presented including nonlinear sigma-models, Chern-Simons theories and vector-tensor dynamical systems. Using a suitable coset representative, we make use of the coset construction to develop an explicit real- ization for one half of the superspace AdS5|8 as a trivial fiber bundle with fibers isomorophic to four-dimensional Minkowski superspace. [email protected] [email protected] http://arxiv.org/abs/0704.1185v3 http://arxiv.org/abs/0704.1185 Contents 1 Introduction 2 2 Covariant derivatives 3 3 Killing supervectors 8 4 Harmonic superspace approach 10 4.1 Analytic multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Harmonic action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 Projective superspace approach 16 5.1 Projective multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 Projective action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Dynamical systems in projective superspace 25 6.1 Projective multiplets revisited . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.2 Projective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Nonlinear sigma-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.4 Vector multiplet and Chern-Simons couplings . . . . . . . . . . . . . . . . 32 6.5 Tensor multiplet and vector-tensor couplings . . . . . . . . . . . . . . . . . 34 7 Coset space realization 35 7.1 Coset representative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 7.2 SO(4,1)×U(1) covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.3 Representation of covariant derivatives . . . . . . . . . . . . . . . . . . . . 40 7.4 Torsion and curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 A 5D Conventions 43 1 Introduction In four-dimensional N = 2 Poincaré supersymmetry, there exist two powerful for- malisms to construct off-shell manifestly supersymmetic actions: harmonic superspace [1, 2] and projective superspace [3, 4, 5, 6]. Both approaches make use of the superspace R4|8 × S2 and its supersymmetric subspaces, which were introduced for the first time by Rosly [7] who built on earlier ideas due to Witten [8]. Both approaches can naturally be extended to the case of d-dimensional supersymmetry with eight supercharges, for d ≤ 6, where the appropriate flat superspace with auxiliary bosonic dimensions is Rd|8 × S2. Specifically, the harmonic superspace formulations were developed in [9, 10] for d = 5, and in [11] for d = 6. The projective superspace formulations were developed in [10] for d = 5, and in [12, 13] for d = 6. In projective superspace, off-shell multiplets are reasonably short and can readily be expressed in terms of 4D N = 1 superfields. The latter property is very appealing from the point of view of brane(-world) models. It is also expected [14, 15] that projective superspace should be relevant in the context of hybrid string theory [16]. For these and similar possible applications, one actually needs projective superspace techniques for supergravity. So far, to the best of our knowledge, the projective superspace approach has been mastered only in the flat case. In harmonic superspace, the prepotential structure of 4D N = 2 supergravity is well understood [17, 2], and similar constructions are clearly applicable in five and six dimen- sions, see [18] for the six-dimensional case. What is still missing here, in our opinion, is a properly incorporated covariant formalism of differential geometry for superfield super- gravity, which should be similar in spirit to the famous Wess-Zumino approach to (the old minimal formulation for) 4D N = 1 supergravity reviewed in [19]. In four-dimensional N = 1 supergravity, it has been recognized for a long time that the most efficient ap- proach to superfield supergravity occurs if one merges together and uses, depending on a concrete application, both the covariant and prepotential techniques [20, 21]. Unlike the purely prepotential approach pursued in [17, 2], this paper is targeted at (making the first step towards) developing covariant superfield techniques for supergravity theories with eight supercharges. Our point of departure is as follows. It is known that all information about off-shell supergravity formulations (including the structure of possible matter multiplets) is encoded in the corresponding algebra of covariant derivatives. We would like to use only this input and try to develop techniques to construct supersymmet- ric actions both in the harmonic and projective settings. In this paper we consider one particular supergravity background – five-dimensional N = 1 anti-de Sitter superspace, AdS5|8, and explicitly develop harmonic and projective formulations in a covariant fashion using only the language of differential geometry. We believe that similar ideas should be applicable for a general supergravity background, as well as in four and six space-time dimensions. In particular, the case of 4D N = 2 anti-de Sitter superspace1 can be treated similarly. This paper is organized as follows. In section 2 we derive the algebra of the covariant derivatives for 5D N = 1 anti-de Sitter superspace by solving the Bianchi identities. In section 3 the isometries of AdS5|8 are realized in terms of Killing supervectors. In section 4 we introduce analytic multiplets over the harmonic superspace AdS5|8×S2 and formulate the harmonic superspace action. Various projective multiplets are defined in section 5, as well as the projective superspace action is formulated. A remarkable feature of this super- symmetric action is that it is uniquely determined by two independent requirements: (i) projective invariance; (ii) invariance under the isometry group SU(2, 2|1). Some impor- tant examples of dynamical systems in the AdS projective superspace are given in section 6. An explicit coset construction for one half of AdS5|8 (Poincaré chart) is elaborated in section 7. Our 5D notation and conventions are collected in Appendix A. 2 Covariant derivatives In this section, we develop the differential geometry of five-dimensional N = 1 anti-de Sitter superspace, AdS5|8. This is a supersymmetric version of spaces of constant curvature and, similar to all symmetric spaces, it can be realized as a coset space, specifically AdS5|8 = SU(2,2|1)/SO(4,1)×U(1). Group-theoretical aspects of AdS5|8 will be discussed in section 7. Let zM̂ = (xm̂, θ i ) be local bosonic (x) and fermionic (θ) coordinates parametrizing AdS5|8, where m̂ = 0, 1, · · · , 4, µ̂ = 1, · · · , 4, and i = 1, 2. The Grassmann variables θµ̂i are assumed to obey a standard pseudo-Majorana reality condition. Since the holonomy group of AdS5|8 is SO(4, 1) × U(1), the superspace covariant derivative D = (Dâ,Diα̂) can be chosen to have the form b̂ĉM J + Ω β̂γ̂ M . (2.1) 1The 4D N = 2 anti-de Sitter superspace was studied in detail in [22] where a manifestly supersym- metric formulation for the off-shell 4D N = 2 anti-de Sitter higher spin supermultiplets [23] was given. A few years later, some formal aspects of this superspace were also discussed in [24]. Here E M̂(z)∂ is the supervielbein, with ∂ = ∂/∂zM̂ , J the Hermitian gen- erator of the group U(1), M the generators of the Lorentz group SO(4, 1), and Φ and Ω b̂ĉ(z) the corresponding connections. The Lorentz generators with vector indices ) and spinor indices (M ) are related to each other by the rule: )α̂β̂M , see Appendix A for more details regarding our 5D notation and conventions. The generators of the holonomy group act on the covariant derivatives as follows: [J,Diα̂] = J ijD , (2.2) ,Diγ̂] = εγ̂α̂Diβ̂ + εγ̂β̂D . (2.3) The Hermitian matrix J ij should be traceless, J i = 0, in order to preserve the pseudo- Majorana condition enjoyed by the covariant derivatives. The latter condition is equiva- lent to the fact that the isotensors2 J ij = εjkJ ik and Jij = εikJ j are symmetric, J ij = J ji, Jij = Jji. The fact that J j is Hermitian, can be seen to be equivalent to (J ij)∗ = −Jij . The algebra of covariant derivatives can be reconstructed if we impose the following two requirements: (i) the torsion tensor is covariantly constant;3 (ii) the group SO(4, 1)× U(1) belongs to the automorphism group. These requirements lead, in particular, to the ansätze: {Diα̂,D } = −2iεijD + xεijε J + f ijM , (2.4) [Dâ,Diα̂] = C ij(Γâ) , (2.5) where = (Γâ) Dâ , (2.6) and x is a constant parameters, f ij = f ji, C ij is a 2×2 matrix. Eq. (2.5) can be rewritten in the equivalent form ,Diγ̂] = −2C ij εγ̂α̂Dj , (2.7) Note that setting x = m = C ij = 0 gives the flat supersymmetry algebra, see. e.g., [10]. The (covariantly) constant parameters x, f ij and C ij in (2.4) and (2.5) turn out to be considerably constrained on general grounds. Firstly, the tensor f ij must be invariant under the action of J , Jf ij = (J ikf kj + J f ik) = 0 ⇐⇒ f ij = mJ ij , (2.8) 2Two-component indices i, j are raised and lowered using the SL(2,C)-invariant antisymmetric tensors εij and εij normalized by ε ikεkj = δ j and ε 12 = ε21 = 1. 3Then, in accordance with Dragon’s theorem [26], the curvature tensor is covariantly constant. with m a constant parameter. Secondly, we should take care of reality conditions such as (Diα̂F )∗ = −(−1)ǫ(F )Dα̂i F ∗ , (2.9) where ǫ(F ) is the Grassmann parity of F . They imply that x = x , m = m . (2.10) and C ij is anti-Hermitian, C† = −C , C = (C ij) . (2.11) Of course, we should also guarantee the fulfillment of the Bianchi identities, and this proves to lead to additional restrictions on the parameters. In particular, the dimension- 3/2 Bianchi identity [Diα̂, {D ,Dkγ̂}] + [D , {Dkγ̂ ,Diα̂}] + [Dkγ̂ , {Diα̂,D }] = 0 (2.12) can be shown to imply C ij = ωJ ij , ω = . (2.13) Imposing the dimension-2 Bianchi identity [Dâ, {Diα̂,D }] + {Diα̂, [D ,Dâ]} − {Dj , [Dâ,Diα̂]} = 0 (2.14) leads, in particular, to [Dâ,Db̂] = − εij(Γb̂) α̂β̂ [Dâ, {Diα̂,D εij(Γb̂) {Diα̂, [D ,Dâ]} − {Dj , [Dâ,Diα̂]} , (2.15) and then [Dâ,Db̂] = mωJ2M , (2.16) where J2 ≡ − 1 J ijJij . (2.17) Another consequence of the dimension-2 Bianchi identity (2.14) is ω = − 1 m . (2.18) As a result, all the parameters in (2.4) and (2.5) have been expressed in terms of ω. With the above conditions taken into account, the remaining dimension-5 Bianchi identity [Dâ, [Db̂,D α̂]] + [Db̂, [D α̂,Dâ]] + [Diα̂, [Dâ,Db̂]] = 0 , (2.19) and dimension-3 Bianchi identity [Dâ, [Db̂,Dĉ]] + [Db̂, [Dĉ,Dâ]] + [Dĉ, [Dâ,Db̂]] = 0 , (2.20) are satisfied identically. Let us summarise the results obtained. The covariant derivatives for AdS5|8 obey the algebra {Diα̂,D } = −2iεijD − 3ωεijε J − 4ωJ ijM , (2.21a) [Dâ,Diβ̂] = ωJ ij(Γâ) , (2.21b) [Dâ,Db̂] = −ω . (2.21c) It is useful to rewrite (2.21b) in the equivalent form ,Diγ̂] = −iωJ ij εγ̂α̂Dj . (2.22) As follows from (2.21c), the bosonic body of the superspace is characterised by a constant negative curvature, and therefore it is AdS5. Indeed, since J j is Hermitian and traceless, we have J ij = J I(σI) j =⇒ J2 = − J ijJij = JIJJ tr(σIσJ) = J IJI > 0 , (2.23) where JI is a real tree-vector, with I = 1, 2, 3, and σI are the Pauli matrices. In section 7, we will give an explicit (coset space) realization of the geometry described. Up to an isomorphism, one can always choose J ij ∝ (σ3)ij , and hence J11 = J22 = 0. Then, it follows from (2.21a–2.21c) that each of the two subsets of covariant derivatives (Dâ,D1α̂) and (Dâ,D α̂) forms a closed algebra, in particular {D1α̂,D } = 0 , (2.24a) [Dâ,D1 ωJ11(Γâ)β̂ γ̂D1γ̂ , (2.24b) [Dâ,Db̂] = −ω . (2.24c) Therefore, one can consistently define covariantly chiral superfields obeying the constraint Φ = 0. Unlike the case of 4D N = 1 anti-de Sitter superspace [27], such multiplets can transform in arbitrary representations of the Lorentz group. In what follows, it will be useful to deal with a different basis for the spinor covariant derivatives. Let us introduce two linearly independent isospinors u+i and u u+iu−i ≡ (u+u−) 6= 0 =⇒ δij = (u+u−) (u+iu−j − u−iu+j ) , (2.25) which do not transform under the action of J , that is J u+i = J u i = 0. Then, defining D±α̂ ≡ Diα̂u±i , (2.26) J++ ≡ J iju+i u+j , J+− ≡ J iju+i u−j , J−− ≡ J iju−i u−j . (2.27) the relations (2.21a) and (2.21b) become } = −4ωJ++M , (2.28a) {D+α̂ ,D−β̂ } = 2(u +u−)iD + 3(u+u−)ωε J − 4ωJ+−M , (2.28b) } = −4ωJ−−M , (2.28c) [Dâ,D+α̂ ] = − 2(u+u−) (Γâ) (J++D− − J+−D+ ) , (2.28d) [Dâ,D−α̂ ] = 2(u+u−) (Γâ) (J−−D+ − J+−D− ) . (2.28e) Eqs. (2.28d) and (2.28e) are equivalent to (u+u−) εγ̂α̂D− (u+u−) εγ̂α̂D+ , (2.29a) ,D−γ̂ ] = (u+u−) εγ̂α̂D− D−α̂ + (u+u−) εγ̂α̂D+ . (2.29b) Under general coordinate and local SO(4,1)×U(1) transfomations, the covariant deriva- tives change as = eτ D e−τ , τ = τ B̂(z)D + i τ(z)J + τ β̂γ̂(z)M . (2.30) This gauge freedom can be used to impose a suitable Wess-Zumino gauge. The latter can be chosen such that Dâ| = ∇â = eâm̂(x) ∂m̂ + b̂ĉ(x)M , (2.31) where U | means the θ independent part of a superfield U(x, θ), U = U(z) = U(x, θ) , U | = U(x, θ = 0) . (2.32) and ∇â stands for the covariant derivatives of anti-de Sitter space, [∇â,∇b̂] = −ω . (2.33) 3 Killing supervectors Similar to the 4D N = 1 case [21], the isometry group SU(2, 2|1) of AdS5|8 is generated by those supervector fields ξÂ(z)E which enjoy the property δξD = −[(ξ + iρJ + Λβ̂γ̂Mβ̂γ̂),DÂ] = 0 , (3.1) where ξ ≡ ξÂD = ξâDâ + ξα̂i Diα̂ = − ξα̂β̂D + ξα̂i Diα̂ , (3.2) for some real scalar ρ(z) and symmetric tensor Λβ̂γ̂(z) = Λγ̂β̂(z). The ξÂ(z)E is called a Killing supervector. The set of all Killing supervectors forms a Lie algebra with respect to the Lie bracket. Given a Killing supervector, it generates a symmetry transformation of matter superfields, which live on AdS5|8, defined as δξχ = −(ξ + i ρJ + Λα̂β̂Mα̂β̂)χ . (3.3) Using the (anti) commutation relations (2.21a) – (2.21c), eq. (3.1) can be seen to be equivalent to Diα̂ξβ̂γ̂ + 2iξiβ̂δ j −Diα̂ξ j + i ρJ α̂ + δ 3ω ξiα̂ + iDiα̂ρ 2ωJ ij(ξ α̂ + ξ α̂) +Diα̂Λβ̂γ̂ , (3.4) and from here we deduce the set of Killing supervector equations Diα̂ξâ = −2i(Γâ)α̂β̂ ξ iβ̂ , (3.5a) J ij −Diα̂ξjβ̂ − i ρJ jεα̂β̂ + δ jΛα̂β̂ , (3.5b) iDiα̂ρ = −3ω ξiα̂ , (3.5c) Diα̂Λβ̂γ̂ = −2ωJ ij(ξ α̂ + ξ α̂) . (3.5d) Note that (3.5b) is equivalent to the following equations Diα̂ξiβ̂ = 2Λα̂β̂ , (3.6a) Diα̂ξj +Djα̂ξiα̂ = 8i J ijρ , (3.6b) (Γâ) α̂β̂(Diα̂ξ ) = −4iω J ijξâ . (3.6c) It is seen that the parameters of U(1) and Lorentz transformations, ρ and Λ , are uniquely expressed in terms of the spinor components of the Killing supervector. As to the vector components ξâ of ξ,which is also uniquely determined in terms of the spinor components of ξ, it obeys the standard Killing equation D(âξ b̂) = 0 . (3.7) To prove (3.7), it suffices to represent Dâ in (3.7) in the form Dâ = i (Γâ)α̂β̂Diα̂Diβ̂ , (3.8) and then make use of relations (3.5a) and (3.6a). As is seen from eqs. (3.5c), (3.6a) and (3.6c), the components of ξ (hence, the Lorentz parameter Λ as well) can be expressed in terms of the scalar parameter ρ as follows: ξiα̂ = − Diα̂ρ , (3.9a) ξâ = − Jij(Γâ) α̂β̂ Diα̂ξ = − 1 3ω2J2 Jij(Γâ) α̂β̂ Diα̂D ρ (3.9b) Diα̂ξiβ̂ = − Diα̂Diβ̂ ρ . (3.9c) This is similar to the situation in 4D N = 2 AdS supersymmetry [23]. We should point out that equation (3.9c) implies 0 = Diα̂ξiα̂ = Diα̂Diα̂ ρ . (3.10) Furthermore, equations (3.5a), (3.5d) and (3.6b) imply Jjk(Γ â)β̂γ̂Diα̂D ξkγ̂ − 2(Γâ)α̂β̂ξ iβ̂ , (3.11a) Diα̂Djβ̂ξ j + 2ωJ ) , (3.11b) (Diα̂Dj +Djα̂Diα̂)ρ+ 8J ijρ . (3.11c) From (3.11b) we also deduce Diα̂Djβ̂ξ j −Diα̂Djγ̂ξ j = 0 =⇒ Diα̂Dβ̂γ̂ρ = 0 , (3.12) and hence ρ = 0 . (3.13) We conclude that ρ is annihilated by the vector covariant derivatives. For later applications, we also observe that the relation Diα̂ξα̂i = 0 and eq. (3.5a) imply ξiβ̂ = iωJ ijξ . (3.14) 4 Harmonic superspace approach In the previous two sections, we have described the differential geometry of five- dimensional N = 1 AdS superspace and its isometries. From now on, we turn to con- structing off-shell supersymmetric theories in AdS5|8. This section is devoted to developing a harmonic superspace approach. To comply with the conventions generally accepted by the harmonic superspace practitioners [1, 2], the isospinors u+ and u− in (2.28a–2.28e) will be chosen to obey the following constraints: − , ui +) ∈ SU(2) , (u+i)∗ = u−i , (u+u−) = 1 . (4.1) As a first step, it is natural to introduce analytic supermultiplets living on harmonic superspace. 4.1 Analytic multiplets We start our analysis with the introduction of O(n) supermultiplets living in AdS5|8. Such a multiplet is described by a completely symmetric superfieldH i1···in(z) = H(i1···in)(z) (with the symmetrization involving a factor of 1/n!) constrained to enjoy the analyticity condition4 Dα̂(i1H i2···in+1)(z) = 0 . (4.2) 4In 4D N = 2 supersymmetry, off-shell superfields H(i1···in)(z) obeying the constrains (i1Hi2···in+1)(z) = D̄α̇ (i1Hi2···in+1)(z) = 0 have a long history. In the presence of an intrinsic cen- tral charge, the cases n = 1 and n = 2 correspond to the Fayet-Sohnius hypermultiplet [28] and the linear multiplet [29] respectively. In the absence of central charge, the case n = 2 corresponds to the tensor multiplet [30]. The case n = 4 was discussed in [31]. The multiplets with n > 2 were introduced in [32], in the projective superspace approach, and then re-discovered in [5]. They were called “O(n) multiplets” in [33]. Their harmonic superspace description was given in [34]. It follows from the algebra of covariant derivatives, that this constraint is consistent provided the superfield is scalar with respect to SO(4,1). If one associates with H i1···in(z) a superfield H(n)(z, u) of harmonic charge n, H(n)(z, u) = u+i1 · · ·u H i1···in(z) , (4.3) the analyticity condition (4.2) can be seen to be equivalent to H(n)(z, u) = 0 , D++H(n)(z, u) = 0 . (4.4) Here D++ is one of the harmonic derivatives (D++, D−−, D0), D++ = u+i , D−− = u−i , D0 = u+i − u−i ∂ [D0, D±±] = ±2D±± , [D++, D−−] = D0 , (4.5) which form a basis in the space of left-invariant vector fields for SU(2). Without imposing the analyticity condition, eq. (4.2), one can consistently define an isotensor superfield F i1···in(z) = F (i1···in)(z) that transforms under the action of the isometry group as follows: i1···in = − ξ + i ρJ F i1···in = −ξF i1···in − i q n ρJ (i1 F i2···in)k , (4.6) where ξ is the Killing supervector, and q is the J-charge of F i1···in. One can associate with F i1···in(z) the harmonic superfield F (n)(z, u) = u+i1 · · ·u F i1···in(z). The latter obeys the algebraic constraint D++F (n) = 0, and its isometry transformation is (n) = −ξF (n) + i qρ(J++D−−F (n) − nJ+−F (n)) , (4.7) where it has been used the fact that u±i are inert under the action of J . It is also worth noting that the Killing supervector can be rewritten as ξ = ξâDâ − ξ+α̂D−α̂ + ξ−α̂D+α̂ . (4.8) It is easy to see that the constraint D++F (n) = 0 is preserved under the isometry trans- formations D++δξF (n) = 0. If the superfield F (n) is constrained to be analytic, D+ F (n) = 0, then the value of its J-charge turns out to be uniquely fixed, and namely q = 1. Therefore, the isometry transformation of the O(n) multiplet is (n) = − ξ + i ρJ H(n) = − ξâDâ − ξ+α̂D−α̂ − i ρ J++D−− − nJ+− H(n) . (4.9) It is not difficult to extend the above consideration to include more general multiplets. Within the harmonic superspace approach [2], one has to deal with superfields of the form Q(n)(z, u), with n an integer, such that (i) Q(n)(z, u) is a smooth function over the group manifold SU(2) parametrized by u = (ui −, ui +); (ii) under harmonic phase transformations u± → exp(±iϕ)u±, the charge of Q(n)(z, u) is equal to n, Q(n)(z, eiϕu+, e−iϕu−) = eniϕQ(n)(z, u+, u−) , ⇐⇒ D0Q(n)(z, u) = nQ(n)(z, u) . Such a superfield can be represented by a convergent Fourier series (for definiteness, we choose n ≥ 0) Q(n)(z, u) = Q(i1···ik+nj1···jk)(z) u+i1 · · ·u u−j1 · · ·u . (4.10) To realise an action of the U(1) generator J on Q(n), we define the component superfields in (4.10) to transform by the law: JQ(i1···ik+nj1···jk)q = q(2k + n)J i2···ik+nj1···jk)r q , (4.11) with the same charge q for all the component superfields. This leads to J Q(n)q (z, u) = q J−−D++ − J++D−− + nJ+− Q(n)q (z, u) . (4.12) The J-charge turns out to be uniquely fixed, q = 1, if Q q is covariantly analytic, q = 0. To summarise, given a covariantly analytic superfield Q(n)(z, u), D+α̂Q(n) = 0 , (4.13) the infinitesimal isometry transformation acts on it as follows: (n) = − ξ + i ρJ ξâDâ − ξ+α̂D−α̂ + i ρ J−−D++ − J++D−− + nJ+− Q(n) . (4.14) Given two covariantly analytic superfields Q(n) and Q(m), their product Q(n)Q(m) is co- variantly analytic and transforms as Q(n+m). In addition, the superfield D++Q(n) can be seen to be covariantly analytic and transform as Q(n+2). 4.2 Harmonic action principle After having introduced various analytic multiplets in AdS5|8 × S2, let us turn to constructing a supersymmetric action. It is worth recalling that in the flat global case (ω = 0), the action principle in 5D harmonic superspace naturally generalizes the original 4D action rule [1, 2] and is given by [10] du (D̂−)4 L(+4) ∣∣∣ , (D̂−)4 = − εα̂β̂γ̂δ̂D− , (4.15) where D− = Diα̂u i , D α̂ are the flat covariant derivatives, and L (+4) is a real analytic Lagrangian of harmonic charge +4, D+ L(+4) = 0. We would like to generalize the flat action to the case of AdS5|8 using the following ansatz: S = S0 + a1S1 + a2S2 d5x e (D̂−)4 + a1 ωJ−−(D̂−)2 + a2 (ωJ−−)2 L(+4) ∣∣ , (4.16) where L(+4) is now covariantly analytic, D+ L(+4) = 0, (D̂−)2 = D−α̂D− , (D̂−)4 = − 1 εα̂β̂γ̂δ̂D− , (4.17) and a1, a2 are two constants to be determined. It is assumed that the above action is evaluated in Wess-Zumino gauge (2.31), using the bar projection (2.32), and as usual e stands for the determinant of the vielbein, e = det(em̂ â), with em̂ âeâ n̂ = δm̂ In accordance with the definition of S, there are several rules for integration by parts which one can use in practice: d5x e duDâQâ| = 0 , (4.18) d5x e duD++Q−−| = d5x e duD−−Q++| = 0 , (4.19) d5x e du J Q(0)| = 0 . (4.20) Here Q(0) is a covariantly analytic superfield of harmonic charge 0. Our aim is to find the constants a1, a2 for which S is invariant under the isometry transformations of AdS5|8. Let us first compute the variation of S0 under infinitesimal isometry transformations. Due to (3.1), we have (D̂−)4L(+4) = (D̂−)4δL(+4) = −(D̂−)4 ξ + i ρJ L(+4) ξ + i ρJ + Λα̂β̂M (D̂−)4L(+4) = − ξ + i ρJ (D̂−)4L(+4) . (4.21) Since L(+4) is covariantly analytic, we obtain δξS0 = − d5x e ξ + i ρJ (D̂−)4L(+4) d5x e ξ+α̂D− (D̂−)4 − ξ−α̂[D+ , (D̂−)4] L(+4) ∣∣∣ . (4.22) Here we have also used eqs. (4.18) and (4.20). To compute D− (D̂−)4 in (4.22), we observe that D− = 0, and then 0 = 5εβ̂γ̂δ̂ρ̂D− D−γ̂ D−δ̂ D = εβ̂γ̂δ̂ρ̂ . (4.23) Moving D− in each term to the left gives (D̂−)4 = ωJ εβ̂γ̂δ̂ρ̂ + 3D− + 2D− . (4.24) This can be further transformed by moving all the Lorentz generators to the right and factors of D− to the left using iteratively the algebra of covariant derivatives. We end up D−α̂ (D̂−)4 = D−α̂ ωJ−−(D̂−)2 + 3 (ωJ−−)2 ωJ−−D−β̂(D̂−)2 − 8 (ωJ−−)2D−β̂ + 1 (ωJ−−)2D−γ̂ M γ̂β̂ . (4.25) The expression in the second line does not contribute when acting on a Lorentz scalar such as L(+4). To compute [D+ , (D̂−)4] in (4.22), we should iteratively use the algebra of covariant derivatives. This is an obvious but tedious procedure. The result is: [D+α̂ , (D̂−)4] = D−β̂(D̂−)2 + 3 ωJD−α̂ (D̂−)2 − ωJ−−iD ω2J−−JD− ωJ+−D− (D̂−)2 − 5ω2J−−J+−D− ωJ−−(D̂−)2D+ ωJ−−D− D+β̂ − 3 ωJ−−D− (ωJ−−)2D+ + ωJ−−iD M β̂γ̂ − 3 ω2J−−JD−β̂M +2ω2J−−J+−D−β̂M ωJ+−D−β̂(D̂−)2M −ω2J−−J+−D−γ̂ M β̂γ̂Mβ̂α̂ . (4.26) Using the relations (4.25) and (4.26), and also the integration by parts identities (4.18) and (4.20), variation (4.22) turns into δξS0 = d5x e du ξ+α̂ ωJ−−D− (D̂−)2 + 3 (ωJ−−)2D− L(+4) d5x e ξ−β̂)D−α̂(D̂−)2 + 3 ω(Jξ−α̂)D− (D̂−)2 ωJ−−(iD ξ−β̂)D−α̂ − 13 ω2J−−(Jξ−α̂)D−α̂ ωJ+−ξ−α̂D−α̂ (D̂−)2 + 5ω2J−−J+−ξ−α̂D−α̂ L(+4) ∣∣∣ . (4.27) Finally, it remains to note Jξ− = J−−ξ+ − J+−ξ− , and also make use of eq. (3.14) projected to the minus-harmonics ξ−β̂ = −5 ω(J−−ξ+ − J+−ξ− ) . (4.28) As a result, the variation of S0 under the isometry transformations takes the final form: δξS0 = d5x e ωJ−−ξ+α̂D− (D̂−)2 − 8 (ωJ−−)2ξ+α̂D− ω2J−−J+−ξ−α̂D− L(+4) ∣∣∣ . (4.29) The next step is to compute the variation of the functional S1 appearing in our action (4.16). Here the procedure is the same as for S0. Varying (D̂−)2L(+4) = −(D̂−)2 ξ + i ρJ L(+4) = − ξ + i ρJ (D̂−)2L(+4) , (4.30) we get δξS1 = d5x e ωJ−−ξ+α̂D− (D̂−)2 − ωJ−−ξ+α̂[D+ , (D̂−)2] L(+4) ∣∣∣ . (4.31) Using the algebra of covariant derivatives gives [D+α̂ , (D̂−)2] = −4iDα̂β̂D −β̂ − 6ωJD−α̂ + 12ωJ+−D−α̂ −2ωJ−−D+ + 8ωJ+−D−β̂M . (4.32) As a result, the variation of S1 is δξS1 = d5x e ωJ−−ξ+α̂D− (D̂−)2 + 4(ωJ−−)2ξ+α̂D− − 16ω2J−−J+−ξ−α̂D−α̂ L(+4) ∣∣∣ . (4.33) It is seen that (4.33) is proportional to (4.29). Therefore, our ansatz (4.16) leads to the unique supersymmetric action: a1 = 2/3 and a2 = 0. The supersymmetric action is d5x e (D̂−)4 + 2 ω J−−(D̂−)2 L(+4) ∣∣∣ , D+α̂L(+4) = 0 . (4.34) This is the main result of this section. By construction, the Lagrangian in (4.34) is a covariantly analytic superfield of har- monic charge +4. It should be also chosen to be real with respect to analyticity preserving conjugation [1] (see also subsection 5.1), and then action (4.34) can be seen to be real. Otherwise, L(+4) is completely arbitrary. Therefore, a great many flat superspace actions [2] can be lifted to the AdS superspace. For instance, an off-shell hypermultiplet can be realized in terms of a covariantly analytic superfield q+(z, u) and its conjugate q̃+(z, u), with respect to the anlyticity preserving conjugation. To describe its dynamics, one can choose L(+4) = −q̃+D++q+ + λ (q̃+q+)2 , (4.35) with λ a coupling constant. 5 Projective superspace approach In the projective superspace approach to d-dimensional theories with eight super- charges, one deals with superfields that live in Md|8 × S2, where Md|8 denotes the conventional superspace, d ≤ 6, and S2 the two-sphere. Such superfields are required to (i) be Grassmann analytic, i.e. to be annihilated by one half of the supercharges; (ii) be holomorphic on an open domain of S2. The latter requirement is equivalently achieved by considering superfields Ψ(n)(z, u+) which are holomorphic functions of a sin- gle isotwsitor u+i ∈ C2−{0}, and have definite degree of homogeneity with respect to u+, Ψ(n)(z, c u+) = cn Ψ(n)(z, u+). The variables u+i can be viewed as homogeneous coordi- nates for CP 1. A second linearly independent isotwistor, u−i, is only required (as a purely auxiliary means, without any intrinsic significance) for constructing a supersymmetric ac- tion which was proposed originally in four dimensions in [3] and then reformulated in [4] in terms of the projective isotwistor u+i. The terminology “isotwistor” is due to [35, 36]. In the flat global case, the 5D N = 1 extension of the 4D N = 2 supersymmetric action [4] is as follows5 [25]: u+i du (u+u−)4 d5x (D̂−)4L++(z, u+) ∣∣ , (5.1) where L++(z, u+) = 0 , L++(z, c u+) = c2 L++(z, u+) , c ∈ C∗ . (5.2) The action is invariant under arbitrary projective transformations of the form − , ui +) → (ui− , ui+)R , R = ∈ GL(2,C) . (5.3) This gauge-like symmetry implies that the action is actually independent of u−i . It can be fixed by imposing, for instance, the gauge u+i ∼ (1, ζ) = ζ i −→ u+i ∼ (−ζ, 1) = ζi , u−i ∼ (0,−1) −→ u−i ∼ (1, 0) , (5.4) in which the action (5.1) reduces to the standard 5D N = 1 projective superspace action [10, 25]. 5.1 Projective multiplets Here we introduce several off-shell projective multiplets that are most interesting from the point of view of model building. By definition, a projective superfield Q(n)(z, u+) lives on the anti-de Sitter superspace and depends parametrically on a non-vanishing isotwistor u+i 6= 0. It is defined to be analytic, Q(n) = 0 , (5.5) and transform by the rule δQ(n) = − ξ + i ρJ Q(n) (5.6) under the isometry group. We specify J to act on Q(n) as follows J Q(n) = −J ++D−−Q(n) − nJ+−Q(n) (u+u−) . (5.7) 5Note that the action given in eq. (B.1) of [25] contains a wrong overal factor of This definition involves an external isotwistor u−i subject to the only requirement (u+u−) 6= 0 . (5.8) Since Q(n) is independent of u−, it is natural to require J Q(n) to be independent of u− as well, that is Q(n) − nJiju+jQ(n) = u+i J Q(n) . (5.9) Contracting this with u+i gives J++u+i Q(n) = nJ++Q(n) . (5.10) Therefore, Q(n) is a homogeneous function of u+ of degree n, Q(n)(z, c u+) = cnQ(n)(z, u+) , c ∈ C∗ . (5.11) The Q(n) will be called a projective superfield of weight n. As is obvious, the complex conjugate of an analytic superfield is not analytic. However, one can introduce a generalized, analyticity-preserving conjugation [7, 1, 3], u+i → ũ+i, which is obtained by composing the complex conjugation, u+i → u+i, with the antipodal map u+i → −u+i . In what follows, it is called “smile-conjugation.” It is thus defined to act on the isotwistor u+ = (u+i) by the rule6 u+ → ũ+ = i σ2 u+ , (u+i) = −u+i , (5.12) with σ2 the second Pauli matrix. Its action on the projective superfields is defined to be Q(n)(u+) → Q̃(n)(u+) ≡ Q̄(n) Q(n) = (−1)nQ(n) . (5.13) It is clear that Q̃(n)(u+) is a homogeneous function of u+ of degree n, that is Q̃(n)(c u+) = cn Q̃(n)(u+), with c ∈ C∗. Due to the identity Q(n) = (−1)ǫ(Q(n)) D+α̂Q̃(n) , (5.14) the smile-conjugation indeed preserves analyticity. It is important to note that, in accordance with (5.13), for an even integer weight, n = 2p, one can consistently define real projective superfields R(2p) with respect to the smile-conjugation: R̃(2p) = R(2p). 6Due to projective invariance, u+i ∼ c u+i, the smile-conjugation could be also defined as u+ → ũ+ = −iσ2 u+, instead of (5.12). Now, let us show that the smile-conjugation is compatible with the superfield transfor- mation law (5.6). To evaluate the smile-conjugate of J Q(n), eq. (5.7), we conventionally define the operation of smile-conjugate for u− = (u−i) to be identical to that we have already chosen for the isotwistor u+, that is u− → ũ− = i σ2 u− , (u−i) = −u−i , (5.15) We should emphasize that such a definition is completely conventional in the sense that the projective superfields are independent of the isotwistor u−. Then it holds D̃−− = D−− , J̃±± = −J±± , J̃+− = −J+− , ˜(u+u−) = (u+u−) . (5.16) This implies J̃ Q(n) = − J Q̃(n) , (5.17) and the smile-conjugate of the transformation law (5.6) is δ̃Q(n) = −(ξ + i ρJ) Q̃(n) = δ Q̃(n) . (5.18) Therefore, the smile-conjugation preserves the superfield transformation laws under the isometry group. As is known, the space CP 1 can be covered by two charts that are defined in terms of u+ = (u+1, u+2) as follows: (i) the north chart on which u+1 6= 0; (ii) the south chart on which u+2 6= 0. As will be described below, the projective action involves the line integral over a closed contour in CP 1, and this contour can be chosen to lie inside one of the coordinate charts. The latter can be chosen to be the north chart, and that is why our local considerations will be mainly restricted to that chart. In the north chart, we can introduce a projective invariant complex coordinate ζ defined as u+i = u+1 (1, ζ), with ζ = u+2/u+1. Since ũ+i = (u+2,−u+1), the smile-conjugation acts as follows: ζ → − 1 . (5.19) The simplest solution to eq. (5.11) is the O(n) multiplet defined by eqs. (4.2) and (4.3). This multiplet is globally defined on CP 1. Allowing for singularities at some points in CP 1 offers the possibility to generate many more interesting supermultiplets. For example, a charged hypermultiplet is described by a weight-one projective superfield Υ+(u+) being holomorphic on CP 1 −{N}, where the North pole is identified with u+i ∼ (0, 1). We can represent Υ+(u+) as Υ+(u+) = u+1Υ+(u+i/u+1) ≡ u+1Υ(ζ) , Υ(z, ζ) = Υk(z)ζ k . (5.20) Its smile-conjugate Υ̃+(u+) is holomorphic onCP 1−{S}, where the South pole is identified with u+i ∼ (1, 0). We can represent Υ̃+(u+) as Υ̃+(u+) = u+2 Υ̃+(u+i/u+2) ≡ u+2 Υ̃(ζ) , Υ̃(z, ζ) = Ῡk(z) (−1)k , (5.21) with Ῡk(z) the complex conjugate of Υk(z). To describe an off-shell vector multiplet, one should use a real weight-zero projective superfield V (u+) being holomorphic on CP 1 − {N ∪ S}. It can be represented as V (z, ζ) = Vk(z)ζ k , V̄k = (−1)kV−k . (5.22) 5.2 Projective action principle Our aim here is to find a generalization of the flat superspace action (5.1) to the case of AdS5|8 superspace. We start with the following ansatz7 S = S0 + β1 S1 + β2 S2 = − 1 u+i du (u+u−)4 d5x e (D̂−)4 + β1ωJ−−(D̂−)2 + β2(ωJ−−)2 L++(z, u+) ∣∣∣ . (5.23) Here L++(z, u+) is a covariantly analytic superfield, D+ L++ = 0, which is homogeneous in u+i of degree +2. The line integral in (5.23) is carried out over a closed contour, γ = {u+i (t)}, in the space of u+ variables. The integrand in (5.23) involves a constant (i.e. time-independent) isotwistor u−i subject to the only condition that u +(t) and u− form a linearly independent basis at each point of the contour γ, that is (u+u−) 6= 0. Our first requirement is that the action (5.23) be invariant under the projective gauge transformations (5.3). First of all, it is obvious that (5.23) is invariant under arbitrary 7An alternative approach to introduce the projective action consists in using a proper generalization of the procedure given in [37]. The latter allows one to derive the projective action as a singular limit of the harmonic action. scale transformations u+i (t) → c(t) u+i (t), with c(t) 6= 0. It is thus only necessary to analyse projective transformations of u− of the form u−i → ũ−i = a(t) u−i + b(t) u+i (t) , a(t) 6= 0 . (5.24) Since both u− and ũ− should be time independent, the coefficients should obey the equa- tions (using the notation f ≡ df(t)/dt, for a function f(t)): a = b (u+u−) b = −b ( (u+u−) . (5.25) As is obvious, the action (5.23) is invariant under arbitrary scale transformations u−i → a(t) u−i , with a 6= 0. Therefore, it only remains to analyse infinitesimal transformations of the form δu−i = b(t)u i , with b(t) obeying the differential equation (5.25). This trans- formation induces the following variations: δD−α̂ = bD+α̂ , δJ−− = 2b J+− . (5.26) Let us start by evaluating the variation of S0. Using the condensed notation dµ++ ≡ − 1 u+i du (u+u−)4 = − 1 (u+u−)4 dt , (5.27) we obtain δS0 = d5x e δ(D̂−)4 α̂β̂γ̂δ̂ dµ++ b d5x e }+ 2D− + {D+ ∣∣∣ . (5.28) Now, making use of the covariant derivatives algebra (2.28a–2.29b) and the identities [J,D+ (u+u−) J+−D+ − J++D− , (5.29a) [J,D− (u+u−) J−−D+ − J+−D− , (5.29b) we can systematically move in (5.28) all space-time derivatives to the left (neglecting total space-time derivatives) and the J operator to the right. This gives δS0 = dµ++ b d5x e ωJ+−(D̂−)2 − 3 ω(u+u−)(D̂−)2J (u+u−)ω2J−−J ∣∣∣ . (5.30) To transform the second and third terms in the square brackets, we should first recall how J acts on the Lagrangian, J L++ = 1 (u+u−) −J++D−−L++ + 2J+−L++ . (5.31) Since L++ is a homogeneous function of degree two, we also have L++ = ( (u+u−) L++ − ( (u+u−) D−−L++ (u+u−) L++ − ( (u+u−) D−−L++ . (5.32) The latter results leads to u+) J L++ = J++ d L++ − 2( (u+u−) J++L++ + 2( (u+u−) J+−L++ . (5.33) One more technical observation, J++ = 2 (u+u−) J++ − 2( (u+u−) J+− , (5.34) allows us to obtain the following identity: (u+u−)3 J L++ = d b J++ (u+u−)3 (u+u−)4 J+−L++ . (5.35) Then (5.30) becomes δS0 = dµ++ b d5x e ωJ+−(D̂−)2 + 22ω2J−−J+− ∣∣∣ . (5.36) Using the same procedure, for δS1 and δS2 we find δS1 = dµ++ b d5x e 2ωJ+−(D̂−)2 + 48ω2J−−J+− ∣∣∣ , (5.37a) δS2 = dµ++ b d5x e 4ω2J−−J+−L++ ∣∣∣ . (5.37b) The relations obtained show that the requirement of projective invariance, δS = δS0 + β1 δS1 + β2 δS2 = 0, uniquely fixes the coefficients in in (5.23) as follows: β1 = 25/24 and β2 = −18. We end up with the projective-invariant action S = − 1 u+i du (u+u−)4 d5x e (D̂−)4 + 25 ωJ−−(D̂−)2 − 18 (ωJ−−)2 ∣∣∣ . (5.38) Now, we are going to demonstrate that (5.38) is supersymmetric, that is this action is invariant under the isometry group of AdS5|8. This requires us to carry out calculations that are very similar to those presented in section 4 for the harmonic case. But there are two technical features being specific for the projective case: (i) unlike the harmonic case, we have (u+u−) 6= 1 in general, and therefore it is necessary to keep track of the factors of (u+u−); (ii) unlike the harmonic superspace identity (4.20), in general we have∮ dµ++J Q−− 6= 0. In all variations involving the U(1) generator J , we will systematically move J ’s to the right to hit the Lagrangian L++, so that eqs. (5.31) and (5.33) can be applied. We start by computing the variation of S0 under the infinitesimal isometry transfor- mation. Making use of (D̂−)4L++ = −(D̂−)4 ξ + i ρJ L++ = − ξ + i ρJ (D̂−)4L++ (5.39) gives δξS0 = d5x e (u+u−) ξ+α̂D− (D̂−)4 − ξ−α̂[D+ , (D̂−)4] − i ρ [J, (D̂−)4] + (D̂−)4J ∣∣∣ . (5.40) To evaluate D− (D̂−)4L++, we note that eq. (4.25) holds even if (u+u−) 6= 1, since in the derivation of (4.25) we only used eq. (2.28c) and the commutation relations of the Lorentz generator M with the covariant derivatives, and both results are clearly not affected by the normalization of (u+u−). Therefore, for the first term on the right of (5.40) we have D−α̂ (D̂−)4L++ = ωJ−−D−α̂ (D̂−)2 + 3 (ωJ−−)2D−α̂ L++ . (5.41) For the operator [D+ , (D̂−)4], which appears in (5.40), we have derived eq. (4.26) in the harmonic case. Now, in evaluating the second term on the right of (5.40), we should take care of the factors of (u+u−), as well as to move the U(1) generator J to the right. This gives , (D̂−)4] = 1 (u+u−)iD D−β̂(D̂−)2 + 3 (u+u−)ωD− (D̂−)2J − 7 ωJ+−D− (D̂−)2 (u+u−)ωJ−−iD D−β̂ − 17 (u+u−)ω2J−−D−α̂ J + ω2J−−J+−D−α̂ + · · · , (5.42) where the dots denote those terms which do not contribute when acting on Lorentz scalar and analytic superfields such as the Lagrangian L++. Inserting (5.42) into δS0, one can get read of the terms with vector covariant derivatives by taking into account the integration by parts rule (4.18) and ξ−β̂ = − 5ω 2(u+u−) (J−−ξ+ − J+−ξ− ) . (5.43) To evaluate the contributions to δS0 which contain JL++, we note that eqs. (5.33) and (5.34) imply (u+u−)4 J L++ = d (J++L++ (u+u−)4 (u+u−)5 J+−L++ . (5.44) The latter observation tells us dµ++ O(u−) J L++ ∣∣∣ = 4 (u+u−) O(u−)L++ ∣∣∣ , (5.45) for any operator O(u−) independent of u+. It follows that d5x e [D+ , (D̂−)4]L++ ∣∣∣ = d5x e (u+u−) ξ+α̂D− (D̂−)2 ωJ−−ξ+α̂D−α̂ + ωJ+− ξ−α̂D−α̂ ∣∣∣ . (5.46) Completely similar considerations, using also D ρ = 0 (3.13), give d5x e ρ[J, (D̂−)4]L++ d5x e ρ J+− (u+u−) 4(D̂−)4 ωJ−−(D̂−)2 − 22 (ωJ−−)2 ∣∣∣ .(5.47) As a result, the variation δξS0 can be represented in the form δξS0 = d5x e (u+u−) ξ+α̂D− (D̂−)2 + 91 (ωJ−−)2 (u+u−) ξ+α̂D− ω2J−−J+− (u+u−) ξ−α̂D− ωJ−−J+− (u+u−) ρ(D̂−)2 − 22 (ωJ −−)2J+− (u+u−) ∣∣∣ . (5.48) The variations δξS1 and δξS2 can be computed by similar means. The results are: δξS1 = d5x e [ ωJ−− (u+u−) ξ+α̂D−α̂ (D̂−)2 + 10 (ωJ−−)2 (u+u−) ξ+α̂D−α̂ − 4 ω 2J−−J+− (u+u−) ξ−α̂D− − 2 ωJ −−J+− (u+u−) ρ(D̂−)2 − 48 (ωJ −−)2J+− (u+u−) ∣∣∣ ,(5.49a) δξS2 = d5x e [ (ωJ−−)2 (u+u−) ξ+α̂D− − 4 (ωJ −−)2J+− (u+u−) ∣∣∣ . (5.49b) Collecting all the results obtained, we conclude δξS = 0 , (5.50) and therefore the action (5.38) is supersymmetric. Actually, it proves to be the only su- persymmetric action in the family (5.23). It is quite remarkable that projective invariance implies supersymmetry and vice versa. 6 Dynamical systems in projective superspace In this section we study in more detail the projective multiplets and then consider sev- eral important supersymmetric theories. To simplify the analysis, it is useful to choose the projective gauge u−2 = 0. Without loss of generality, one can also work in a representation of the algebra in which J11 = J22 = 0, and hence J−− = 0. 6.1 Projective multiplets revisited In each of the two coordinate charts for CP 1, one can describe the projective multiplets by superfields invariant under the projective transformation (5.11). Let us restrict our consideration to the north chart. Given a complex projective multiplet of weight n, Q(n)(u+), it can be equivalently described by a holomorphic function Q[n](ζ) defined as follows: Q(n)(u+i) = (u+1)nQ[n](ζ) , Q[n](ζ) ≡ Q(n)(1, ζ) . (6.1) Here Q[n](ζ) is clearly invariant under (5.11). For the smile-conjugate of Q(n)(u+), we get Q̃(n)(u+i) = (u+2)nQ̃[n](ζ) , Q̃[n](ζ) = Q̄[n](−1/ζ) . (6.2) Given a real projective multiplet R(2p)(u+), with respect to the smile-coinjugation, it can be represented R(2p)(u+) = (iu+1u+2)pR[2p](ζ) , R̃[2p](ζ) ≡ R̃[2p](−1/ζ) = R[2p](ζ) . (6.3) The most general form for Q[n](z, ζ) is Q[n](z, ζ) = (z)ζk . (6.4) In the projective gauge chosen (u−2 = 0, J 11 = J22 = 0), the action of the operator J on our superfield becomes J Q[n](u+) = − 1 (u+u−) J++D−− − nJ+− Q[n](u+) = J12 n− 2u+2 ∂ Q[n](u+) . Then, since the isotwistor u+i is neutral under the action of J , it holds (u+1)n J Q[n](ζ) = J Q(n)(u+) = J12 n(u+1)nQ[n](ζ)− 2u+2 ∂ (u+1)nQ[n](ζ) = (u+1)nJ12 nQ[n](ζ)− 2ζ ∂ Q[n](ζ) and therefore J Q[n](z, ζ) = J11 Q[n](z, ζ) , J Q (z) = (2k − n)J11Q[n]k (z) . (6.5) In the case of a real superfield R(2p)(z, u+) = (iu+1u+2)pR[2p](z, ζ), we have for R[2p] R[2p](z, ζ) = (z)ζk , R̄ = (−1)kR[2p]−k . (6.6) The operator J is represented as follows: J R[2p](z, ζ) = 2J11ζ R[2p](z, ζ) , J R k (z) = 2kJ k (z) . (6.7) Let us analyse the implications of the analyticity condition, D+ Q(n)(u+) = 0. It is useful to change the representation for the projective superfields, Q(n)(u+) → Q[n](ζ). We then have D+ Q[n](ζ) = −u+1(ζD1 )Q[n](ζ), and therefore the analyticity condition is equivalent to Q[n](ζ) = ζD1 Q[n](ζ) . (6.8) For the component superfields Q (z), this implies D2α̂Q k = D k−1 . (6.9) It is natural to think of D1 and D2 as the covariant derivatives associated with two 5D Dirac spinor coordinates, θα̂1 and their conjugates θ 2 . It then follows from (6.8) that the dependence of Q[n](ζ) on θα̂2 is completely determined by the dependence of Q [n](ζ) on θα̂1 . Suppose that the expansion of Q[n](ζ) in powers of ζ terminates from below Q[n](z, ζ) = (z)ζk . (6.10) Then, eq. (6.9) tells us that the two lowest components of Q[n] are constrained as follows: L = 0 , (D̂2)2Q[n]L+1 = 12ω J Q L , (6.11) where (D̂i)2 = Diα̂Diα̂ , i = 1, 2 . (6.12) Therefore, Q L is a five-dimensional chiral superfield, while Q L+1 a complex linear super- field. The union of Q L and Q L+1 forms a 5D analogue of the famous chiral-nonminimal doublet in 4D supersymmetry [38]. Given a real O(2) multiplet H(2)(z, u+), we can represent H [2](z, ζ) in the form H [2](z, ζ) = Φ(z) +G(z)− ζΦ̄(z) , (6.13) where Φ is a five-dimensional chiral superfield, and G a real linear superfield, Φ = 0 , (D̂2)2G = 0 , Ḡ = G . (6.14) If the expansion of Q[n](ζ) in powers of ζ terminates from above, Q[n](z, ζ) = (z)ζk . (6.15) then eq. (6.9) implies that the two highest components of Q[n] are constrained as follows: D1α̂Q L = 0 , (D̂1)2Q[n]L−1 = −12ω J Q L , (6.16) Therefore, Q L is a five-dimensional antichiral superfield, while Q L−1 a complex antilinear superfield. For further analysis, it is useful to switch from the 5D four-component spinor notation to the 4D two-component one by representing Diα̂ = D̄α̇i εijD̄α̇j In such a notation, the algebra of covariant derivatives (2.21a–2.21c) takes the form {Diα,D } = 2εijεαβD5 − 3ωεijεαβJ − 4ωJ ijMαβ , (6.17a) {Diα, D̄ j } = −2iδij(σa)αβ̇Da − 4ωJ ijM β̇α , (6.17b) {D̄α̇i , D̄ j } = −2εijεα̇β̇D5 − 3ωεijεα̇β̇J − 4ωJijM α̇β̇ , (6.17c) [Da,Diα] = − i2ωJ ij(σa)αβ̇D̄ j , [Da, D̄α̇i ] = i2ωJij(σ̃a) α̇βDjβ , (6.17d) [D5,Diα] = 12ωJ jDjα , [D5, D̄α̇i ] = 12ωJ j , (6.17e) [Da,Db] = −ω2J2Mab , [Da,D5] = −ω2J2Ma5 . (6.17f) In the two-component spinor notation, the analyticity condition D+ Q[n](ζ) = 0 is equivalent to D2αQ[n](ζ) = ζ D1αQ[n](ζ) , D̄α̇2Q[n](ζ) = − D̄α̇1Q[n](ζ) . (6.18) For the component superfields Q k (z), this implies = D1αQ k−1 , D̄α̇2Q = −D̄α̇1Q k+1 . (6.19) By analogy with the flat case, these constraints indicate an interesting interpretation. Let us introduce two sets of spinor derivatives, (D1α, D̄ 1 ) and (D2α, D̄ 2 ) which can be viewed as the covariant derivatives corresponding to two different sets of Grassmann variables Θ1 and Θ2. Then, the above constraints imply that the dependence of the projective superfields on Θ2 is uniquely determined in terms of their dependence on Θ1. Unlike the flat case, such an interpretation is somewhat limited in the sense that one can not con- sistently switch off the variables Θ2 (what would be necessary for reducing the multiplets to 4D N = 1 superfields). It follows from the algebra of covariant derivatives, specifically from eq. (6.17d), that [Da,D1α] = − i2ωJ 12(σa)αβ̇D̄ 2 and [Da, D̄α̇1 ] = − i2ωJ 12(σ̃a) α̇βD2β, and therefore the commutation relations mix all the spinor derivatives. This is an important difference between the flat and curved cases. The constraints (6.19) simplify if the series in (6.4) or (6.6) is bounded from below (above). Consider a real O(2n) multiplet H(2n)(z, u+). In accordance with the above general consideration, it can be described by the superfield H [2n](z, ζ) which is defined by H(2n)(z, u+) = (iu+1u+2)nH [2n](z, ζ), and can be represented in the form H [2n](z, ζ) = k (z)ζ k , H̄ k = (−1)kH −k . (6.20) The analyticity constraints (6.19) imply that the two lowest component superfields are constrained by D̄α̇1H −n = 0 , (D̄1)2H [2n]−n+1 = −(4D5 + 6ωJ)H −n = −(4D5 − 12nωJ11)H −n , (6.21) where we have defined (Di)2 ≡ DiαDiα , (D̄i)2 ≡ D̄iα̇D̄α̇i . (6.22) Consider an arctic multiplet of weight n ≥ 0, Υ(n)(u+), defined to be holomorphic on CP 1 − {N}. It can be represented as Υ(n)(z, u+) = (u+1)n Υ[n](z, ζ) , Υ[n](z, ζ) = k (z)ζ k . (6.23) Then the constraints on the two lowest components superfields are D̄α̇1Υ 0 = 0 , (D̄1)2Υ[n]1 = −(4D5 + 6ωJ)Υ 0 = −(4D5 − 6nωJ11)Υ 0 . (6.24) In the flat superspace limit, ω → 0, the constraints (6.21) and (6.24) reduce to those given in [10]. 6.2 Projective action Here we turn to a more detailed analysis of the projective action (5.38). In the pro- jective gauge (u−2 = 0, J 11 = J22 = 0) used throughout this section, we have J−− = 0, and therefore the projective action simplifies S = − 1 u+i du (u+u−)4 d5x e (D̂−)4 L++ ∣∣∣ . (6.25) Of course, the Lagrangian L++ should be real with respect to the smile conjugation, and can be represented as L++(z, u+) = iu+1u+2L(z, ζ) . (6.26) Then, the action turns into S = − 1 d5x e ζ (D̂1)2(D̂1)2L(z, ζ) ∣∣∣ , (6.27) where we have taken into account the fact that {D1 } = 0 in the projective gauge, and also made use of the identity εα̂β̂γ̂δ̂ = εα̂β̂εγ̂δ̂ + εα̂γ̂εδ̂β̂ + εα̂δ̂εβ̂γ̂ . Using the relation (D̂2)2Q[n] = ζ2(D̂1)2Q[n] + 12ωζ JQ[n] , (6.28) we can express action (6.27) in the equivalent forms S = − 1 d5x e (D̂1)2 (D̂2)2 − ζ 12ωJ L(z, ζ) ∣∣∣ , (6.29a) S = − 1 d5x e (D̂2)2 (D̂1)2 + 1 L(z, ζ) ∣∣∣ , (6.29b) where we have used the identities , (D̂2)2] = −18ωJ11D + 4iD D2β̂ + 6ωD2 J − 8ωJ11D2β̂Mα̂β̂ , (6.30) [(D̂1)2, ( ˆ̄D2)2] = 4iDα̂β̂[D1 ] + (6ω[D1 ] + 96ω2J11)J − 8ωJ11[D ]M α̂β̂ . (6.31) Then we can represent the action in the form S = − 1 d5x e (D̂1)2, (D̂2)2 + 24ωJ ζ(D̂1)2 + 1 (D̂2)2 L(z, ζ) ∣∣∣ (6.32) which makes manifest the reality of S with respect to the smile-conjugation. It can be seen from the above relations that there exists a natural “gauge freedom” in the choice of L++. It occurs in the three incarnations: L++ → L++ + Λ++ + Λ̃++ , (6.33) L++ → L++ + i J++ Λ+ Λ̃ , (6.34) L++ → L++ +H++ , (6.35) with Λ++ and Λ arctic multiplets (6.23) of weight +2 and 0, respectively, and H++ a real O(2) multiplet. It is also instructive to express the action in a 4D N = 1 form by switching to the two-component spinor notation D1α̂ = D̄1α̇ D̄α̇2 , D2α̂ = D̄2α̇ −D̄α̇1 . (6.36) Using the analyticity conditions (6.18) we can express D̄α̇2 via D̄α̇1 . As a result our action (6.25) becomes d5x e (D1)2(D̄1)2 − ζωJ11(D1)2 L(z, ζ) ∣∣∣ . (6.37) Using the identities [D̄α̇1 , (D1)2] = 8ωJ11D̄α̇2 − 4i(σ̃a)α̇αDaD1α + 8ωJ11D1αMαα̇ , (6.38) [(D1)2, (D̄1)2] = − 16ωJ11D̄1α̇D̄α̇2 + 16ωJ11D1αD2α + 96ω2J11J +4i(σa) α̇α Da[D1α, D̄1α̇] + 8ωJ11[D1α, D̄1α̇]Mαα̇ , (6.39) the action can also be rewritten in the following form d5x e (D̄1)2(D1)2 + 1(D̄1)2 L(z, ζ) ∣∣∣ , (6.40) or in the manifestly real form d5x e (D1)2, (D̄1)2 1(D1)2 + 1(D̄1)2 L(z, ζ) ∣∣∣ . (6.41) As compared with the flat superspace action [10], the second and third terms on the right of (6.41) are due to the non-vanishing curvature. 6.3 Nonlinear sigma-models We consider a system of interacting artic weight-one multiplets Υ+(z, u+) and their smile-conjugates Υ̃+ described by the Lagrangian L++ = iK(Υ+, Υ̃+) , (6.42) with K(ΦI , Φ̄J̄) a real analytic function. Since L++ = L++(z, u+) is required to be a weight-two projective superfield, the potential K has to respect the following homogeneity condition ( + Φ̄Ī ∂Φ̄Ī K(Φ, Φ̄) = 2K(Φ, Φ̄) . (6.43) For L++ to be real, it is sufficient to require a stronger condition K(Φ, Φ̄) = K(Φ, Φ̄) . (6.44) Such a Lagrangian corresponds to the superconformal sigma-model introduced in [25]. Then, representing Υ+(z, u+) = u+1Υ(z, ζ) and Υ̃+(z, u+) = u+2 Υ̃(z, ζ), we can rewrite the Lagrangian in the form L++(z, u+) = i u+1u+2L(z, ζ) , L = K(Υ, Υ̃) . (6.45) Because of freedom (6.33) in the choice of Lagrangian, we can generalize the above construction by replacing K(ΦI , Φ̄J̄) in (6.42) with K ′(ΦI , Φ̄J̄) = K(ΦI , Φ̄J̄) + Λ(ΦI)− Λ̄(Φ̄J̄) , ΦI ∂ Λ(Φ) = 2Λ(Φ) , (6.46) with Λ(Φ) a holomorphic homogeneous function of degree +2. Then, the homogeneity condition (6.44) turns into K ′(Φ, Φ̄) = K ′(Φ, Φ̄) + Λ(Φ) + Λ̄(Φ̄) . (6.47) We can also consider a system of interacting arctic weight-zero multiplets Υ(z, u+) and their smile-conjugates Υ̃ described by the Lagrangian L++ = i J++K(Υ, Υ̃) , (6.48) with K(ΦI , Φ̄J̄) a real function which is not required to obey any homogeneity condition. Due to the gauge freedom (6.34), the action is invariant under Kähler transformations of the form K(Υ, Υ̃) → K(Υ, Υ̃) +Λ(Υ) + Λ̄(Υ̃) , (6.49) withΛ a holomorphic function. Such dynamical systems generalize the hyperkähler sigma- models on cotangent bundles of Kähler manifolds [39, 40, 41]. 6.4 Vector multiplet and Chern-Simons couplings An Abelian vector mulitplet can be described by a weight-zero real projective superfield V (z, u+) which is required to be holomorphic on CP 1 − {N ∪ S}. V (z, u+) = 0 , V (z, c u+) = V (z, u+) , c ∈ C∗ . (6.50) In the North chart, it is characterized by the series (5.22). It is defined to possess the gauge freedom V → V + λ+ λ̃ , λ(z, ζ) = λk(z)ζ k . (6.51) with λ(z, u+) an arctic multiplet of weight 0. Using considerations similar to those given in subsection 5.2, the field strength (compare with the flat superspace expression [25]) W (z) = − 1 u+i du (u+u−)2 (D̂−)2 − 12ω J−− V (z, u+) (6.52) can be shown to be invariant under the projective transformations (5.3). The field strength turns out to be invariant under the gauge transformations (6.51). In the projective gauge (u−2 = 0, J 11 = J22 = 0), the field strength takes the form W (z) = − 1 dζ (D̂1)2V (z, ζ) , (6.53) compare with the flat superspace result [10]. The AdS transformation law of V , δV = − ξ + i ρJ V , (6.54) can be shown to imply that W transforms as δW = −ξ W (6.55) under the isometry group. The field strength can be shown to obey the Bianchi identity Dγ̂(iDj) W , (6.56) and therefore W = 2ω ε J (ijDk) W , (6.57) compare with the flat superspace case [9, 10]. The Bianchi identity implies that G++(z, u+) = i D+α̂W D+ W (D̂+)2W − 2ω J++W 2 (6.58) is a composite O(2) multiplet, G++ = 0 , G++(z, u+) = Gij(z) u+i u j . (6.59) Let H++(z, u+) be a real O(2) multiplet. Then, similarly to the flat superspace case [10, 25], the supersymmetric action associated with the Lagrangian L++ = V (z, u+)H++(z, u+) (6.60) can be shown to be invariant under the gauge transformations (6.51). Given several Abelian vector multiplets VI(z, u +), where I = 1, . . . , n, the composite superfield (6.58) is generalised to the form: G++IJ = G D+α̂WI D+α̂WJ + W(I (D̂+)2WJ) − 2ω J++WIWJ G++IJ = 0 , G IJ (z, u +) = G IJ(z) u j . (6.61) We then can construct a supersymmetric Chern-Simons action associated with the La- grangian L++CS = cI,JK VI(z, u +)G++JK(z, u +) , cI,JK = cI,KJ , (6.62) for some constant parameters cI,JK (compare with the flat superspace case [10, 25]). In accordance with the above result, the Chern-Simons action is gauge invariant. 6.5 Tensor multiplet and vector-tensor couplings Given several O(2) (or, equivalently, tensor) multiplets H++I (z, u +), a supersymmetric action is generated by the Lagrangian L++ = F(H++I ) , I = 1, . . . , n (6.63) where F(H) is a weakly homogeneous function of first degree in the variables H , ∂F(H) −F(H) = αI HI , (6.64) for some constants α’s.8 Such a Lagrangian occurs in the models for superconformal tensor multiplets in four [42] and five dimensions [25]. One can also consider systems of coupled vector and tensor multiplets described by a Lagrangian of the form L++ = F(H++I ) + VI cI,JK G , (6.65) for some coupling constants κI and cI,JK. 8The projective action principle formulated in subsection 5.2 requires the Lagrangian to be a projective weight-two multiplet. With αI 6= 0 in (6.64), the Lagrangian (6.63) does not have any definite weight, and hence the results of subsection 5.2 are not applicable directly. We plan to discuss the case with αI 6= 0 in more detail somewhere else. 7 Coset space realization In this section we would like to give an explicit realization for the N = 1 AdS5 superge- ometry which we have studied in section 2 using the representation-independent approach. From the group-theoretical point of view, it is known that the N = 1 AdS5 superspace (or simply AdS5|8) can be identified with the coset space SU(2,2|1)/SO(4,1)×U(1). Us- ing the formalism of nonlinear realizations9 [44] (or Cartan’s coset construction), here we introduce a suitable coset representative that makes possible to realize one half of AdS5|8 as a trivial fiber bundle with fibers isomorophic to four-dimensional Minskowski super- space. This realization should be useful if one is interested in having the 4D N = 1 super Poincaré symmetry manifest. However, since it corresponds to one half of AdS5|8 (known as the Poincaré patch [45]), it is not suitable to describe the supersymmetric actions. The analysis of this section builds on the construction given in [46], see also [47] for related issues. Note that we use the superform convenctions of [19]. 7.1 Coset representative As is well known, the supergroup SU(2,2|1) is the four-dimensional N = 1 supercon- formal group. It is generated by Lie-algebra elements of the form (parametrization (7.1) was used in [48, 49]) β −∆δαβ −ibαβ̇ 2ρα −iaα̇β −w̄α̇β̇ + ∆̄δα̇β̇ 2ǭα̇ 2ǫβ 2ρ̄ 2(∆̄−∆)  , (7.1) which satisfy the conditions str X = 0 , BX†B = −X , B = 0 1 0 1 0 0 0 0 −1  . (7.2) The matrix elements in (7.1) correspond to a 4D Lorentz transformation (wα β , w̄α̇ a translation aα̇α, a special conformal transformation bαα̇, a Q–supersymmetry (ǫ α, ǭα̇), an S–supersymmetry (ρα, ρ̄α̇), and a combined scale and U(1)–chiral transformation ∆ = 1 9Many years ago, this formalism was also applied to introduce the 4D N = 1 AdS superspace [43, 27]. The explicit parametrization for the algebra su(2, 2|1), which is given in (7.1), is ideally suited to describe the compactified Minkowski space SU(2,2|1)/(P×C∗), where P denotes the N = 1 super Poincaré group (generated by the parameters (w, w̄, b, ρ, ρ̄) in (7.1)), and C∗ denotes the group of scale and chiral transformations generated by the parameters ∆ and ∆̄ in (7.1). In the case of the coset space SU(2,2|1)/SO(4,1)×U(1), however, this parametrization should be slightly modified. In addition, a re-scaling of some matrix elements is needed in order to incorporate the AdS curvature ω2 into the formalism. As is known, a key role in the coset construction for M = G/H is played by a coset representative S(p) defined to be a smooth mapping S: U → G, for some open domain U ⊂ M, such that S(p)p0 = p, for any point p ∈ U , where p0 ∈ U is a fixed point having H as its isotropy group. On topological grounds, it is not always possible to extend U to the whole coset space M. As a coset representative, S(z), for AdS5|8 = SU(2,2|1)/SO(4,1)×U(1), following mainly [46] we choose S(z) = g(z) · gS · gD 1 0 0 −iωx̃+ 1 2ω 2 θ 0 1 1 2ωηη̄ 2ω 0 1 0 2 η̄ 1 1 0 0 0 0 1  (7.3) ωyδ βα 2ωe ωyηαη̄β̇ 2ω −iωe− 12ωyx̃α̇β+ e − 2iω2x̃α̇γ+ ηγ η̄β̇ + 4ωθ̄α̇η̄β̇ θ̄α̇ − iωx̃α̇γ+ ηγ ωyθβ 2ω + 2ωθγηγ η̄β̇ 1 + 4ωθγηγ where xa± = x a ± iθσaθ̄ denote ordinary 4D N = 1 (anti) chiral bosonic variables. It is worth pointing out that the coset representative g(z) corresponds to the coset P/SO(3, 1) and provides a matrix realization10 for 4D N = 1 Minkowski superspace, with coordinates z = (xa, θα, θ̄α̇). Note that the isotropy group at z = 0 is H = SO(4, 1) × U(1) ∈ SU(2, 2|1) = G, and it is generated by matrices of the form w − i b̃ −w̄ 0 0 0 0 τ 1 0 0 0 − i τ 1 0 0 0 −4i trw = 0 , w̄ = w† , b† = b , τ̄ = τ . (7.4) Setting ω = 1 in (7.3) gives the parametrization used in [46]. 10It is a curious historic fact that the above matirx realization for 4D N = 1 Minkowski superspace was introduced by Akulov and Volkov [50] a year before the official discovery of superspace. Once the coset representative S(p) is chosen, the next step in the coset construction for M = G/H is to compute the Maurer-Cartan one-form S−1dS which proves to encode all the information about the geometry of M. Let G and H be the Lie algebras of G and H , respectively, and G−H be a complement ofH in G such that [G−H,H] ⊂ G−H. Then, the Maurer-Cartan one-form can be uniquely decomposed as S−1dS = S−1dS|G−H+S−1dS|H, where S−1dS|G−H is identified with the vielbein, and S−1dS|H with the connection. In our case, the vielbein E = S−1dS|G−H and the connection Ω = S−1dS|H are: S−1 dS = E+Ω , ωEy δα β − i 2 (Eη)α ωẼα̇β 1 ωEy δ 2 (Ēθ̄) 2 (Eθ) 2 (Ēη̄)β̇ 0  , (7.5) β − i ΩU(1)δα β − i Ω̃α̇β −Ω̄α̇ ΩU(1)δ 0 0 −4i ΩU(1)  . (7.6) The components of the vielbein are given by the one-forms Eαα̇ = eαα̇ e −ωy(1− e2ωyω2η2η̄2) + 2ieωydηαη̄α̇ + 2ieωydη̄α̇ηα +4iωeωydθαη 2η̄α̇ + 4iωe ωydθ̄α̇ηαη̄ 2 , (7.7a) Ey = dy + dθ µ(−2ηµ) + dθ̄µ̇(−2η̄µ̇) , (7.7b) α = dθµ δαµe ωy + em(iωe− σ̃β̇αm ) , (7.7c) α = dηµ δαµe ωy + dθµ δαµ(2ωe ωyη2) + dθ̄µ̇(2ωe ωyη̄µ̇ηα) + em(iω2e ωyη2η̄β̇ σ̃ m ) .(7.7d) The components of the SO(4,1)×U(1) connection read β = dθµ(4ωηαδ µ − 2ωηµδβα) + em(−2iω2ηαη̄β̇σ̃β̇βm − iω2η̄γ̇σ̃γ̇γm ηγδβα) , (7.8a) Ωαα̇ = −eαα̇ ωe−ωy(1 + ω2e2ωyη2η̄2) + dηα(2iωeωyη̄α̇) + dη̄α̇(2iωeωyηα) + dθα(4iω 2eωyη2η̄α̇) + dθ̄α̇(4iω 2eωyηαη̄ 2) , (7.8b) ΩU(1) = dθ µ( 3iωηµ) + dθ̄µ̇(−3iωη̄µ̇) + em(−3ω2η̄µ̇σ̃µ̇µm ηµ) , (7.8c) where em = dxm − i dθµσaµµ̇θ̄µ̇ + i θµσaµµ̇dθ̄µ̇ , (7.9) is the space-time component of the N = 1 flat superspace vielbein [19]. Note that under a group transformation g ∈ SU(2, 2|1) g S(z) = S(g · z) ĥ(z; g) ≡ S ′ ĥ , ĥ(z; g) ∈ H , (7.10) the vielbein E and the connection Ω transform as follows: E′ = ĥE ĥ−1 , Ω′ = ĥΩ ĥ−1 − (dĥ) ĥ−1 . (7.11) It is useful to introduce the inverse EA M of the vielbein supermatrix EM A implic- itly used in the previous equations (EA B = δA B, EM N = δM N ). With the definitions em, dy, dθµ, dθ̄µ̇, dη µ, dη̄µ̇ = EAEA M , (7.12) Ea,Ey, (Eθ) α, (Ēθ̄)α̇, (Eη) α, (Ēη̄)α̇ = εMEM A , (7.13) where em = −1 (σ̃m)α̇αeαα̇ and E a = −1 (σ̃a)α̇αEαα̇, we find eαα̇ = Eαα̇ e ωy(1 + ω2e2ωyη2η̄2) + (Eη)α(−2ie ωyη̄α̇) + (Ēη̄)α̇(−2ie ωyηα) + (Eθ)α(2iωe ωyη̄α̇η 2) + (Ēθ̄)α̇(2iωe ωyηαη̄ 2) , (7.14a) dy = Ey + (Eθ) α( 2e ωyηα) + (Ēθ̄)α̇( 2e ωyη̄α̇) , (7.14b) dθµ = (Eθ) ωyδµα(1− 2ω2e2ωyη2η̄2) + (Eη)α δµα(2ωe ωyη̄2) + (Ēη̄)α̇(−2ωe ωyη̄α̇ηµ) + Ea(−iωeωyη̄ν̇ σ̃ν̇µa ) , (7.14c) dηµ = (Eη) ωyδµα(1− 4ω2e2ωyη2η̄2) + (Eθ)α δµα(−2ωe ωyη2) + (Ēθ̄)α̇(−2ωe ωyη̄α̇ηµ) + Ea(2iω2eωyη̄ν̇ σ̃ 2) . (7.14d) It is also useful to decompose the connection with respect to the curved basis {EA} Ω βα = (Eθ) ωy(4ηαδ γ − 2ηγδβα) + (Eη)γ ωe ωy(4ηαη̄ 2δβγ − 2ηγ η̄2δβα) +Ea ω2eωy(−2iη̄β̇σ̃β̇βa ηα + iη̄γ̇ σ̃γ̇γa ηγδβα) , (7.15a) Ωαα̇ = Eαα̇ ω(−1− 2ω2e2ωyη2η̄2) + (Eη)α(4iωe ωyη̄α̇) + (Ēη̄)α̇(4iωe ωyηα) + (Eθ)α(−4iω2e ωyη̄α̇η 2) + (Ēθ̄)α̇(−4iω2e ωyηαη̄ 2) , (7.15b) ΩU(1) = E a( 3ω2eωyη̄ σ̃β̇βa ηβ) + (Eθ) α( 3iωe ωyηα) + (Ēθ̄)α̇(−3iωe ωyη̄α̇) + (Eη) α( 6iω2e ωyηαη̄ 2) + (Ēη̄)α̇(−6iω2e ωyη̄α̇η2) . (7.15c) 7.2 SO(4,1)×U(1) covariance To better understand the relation between the above coset construction and the AdS5|8 supergeometry of section 2, it is necessary to figure out the precise meaning of the SO(4,1)×U(1) covariance of the vielbein and the connection. We will use several re- sults which are collected in Appendix A and concern the reduction of 5D spinors into 4D ones. First of all, let us recall that choosing g = h ∈ H in relations (7.10, 7.11) gives ĥ = h = const, and the group transformations (7.11) reduce to E′ = hE h−1 , Ω′ = hΩ h−1 , h ∈ SO(4, 1)× U(1) . (7.16) In particular, a 5D Lorentz transformation acts as follows: E′ = ΛEΛ−1 , Ω′ = ΛΩΛ−1 , (7.17) where 0 0 1  , Λα̂β̂ = Λĉd̂(Σ β̂ . (7.18) This transformation law allows us to combine components of the connection into five- dimensional vector and spinor. Explicitly, we can write ωEâ(Γâ)α̂ β̂ 2ω 2 Ēβ̂ 0  , (7.19) Ωâb̂ 0 0 0  + iΩU(1) 0 0 −4  , (7.20) where Eâ = (Ea,E5) = (Ea,Ey) , (7.21a) Eα̂ = (Eη)α (Ēθ̄) , Ēα̂ = α, (Ēη̄)α̇ , (7.21b) Ωâb̂ = (Ωab,Ωa5) , (7.21c) Ωab = −(σab)βαΩαβ + (σ̃ab)β̇ α̇Ω̄α̇β̇ , Ωa5 = −12(σ̃ a)α̇αΩαα̇ . (7.21d) Note that Eâ, Ω and ΩU(1) are real. It follows that E â, Eα̂, Ē α̂, Ω and ΩU(1) transform under the 5D Lorentz group SO(4, 1) respectively as a vector, a Dirac spinor, its Dirac conjugate spinor, an antisymmetric two-tensor and a scalar. Due to (7.21a) we can identify x5 ≡ y . (7.22) Note also that we can combine the two spinors Eα̂ and Ē α̂ into a 5D pseudo-Majorana spinor defined as follows: Eα̂i = (E i ,−Ēiα̇) , (7.23) Eα1 = (Eθ) α , Eα2 = (Eη) α , Ē α̇ = (Ēθ̄)α̇ , Ē α̇ = (Ēη̄)α̇ . (7.24) It remains to consider the transformation properties of the vielbein and the connection under the U(1) part of the isotropy group. In accordance with (7.16), they transform as E′ = ΣEΣ−1 , Ω′ = ΣΩΣ−1 , exp(−1 φ i δ) 0 0 e−  . (7.25) Clearly Ω is invariant under the U(1) transformation, while E transforms as ωEâ(Γâ)α̂ β̂ 2ω eφiEα̂ e−φi Ēβ̂  , (7.26) and hence Eâ is invariant. Note also that (7.25) induces induces the following transfor- mation of Eiα̂: E′α̂i = [exp(−φiJ)]i jEα̂j , Ji j = (σ3)i . (7.27) 7.3 Representation of covariant derivatives With the vielbein and the connection having been introduced, we can now construct the covariant derivatives b̂ĉM bcMbc + Ω b5Mb5 = (Dâ,Diα̂) = (Da, D5, D1α, D̄1α̇, D2α, D̄2α̇) . (7.28) The vector fields E are defined by Eâ, E MDM , ,Dµ, D̄ ∂η̄µ̇ . (7.29) Here the supermatrices E M and EM  have been defined in subsection 7.1. It should be pointed out that (∂m, Dµ, D̄ µ̇) are the 4D N = 1 flat superspace covariant derivatives, + iθ̄µ̇∂µµ̇ and D̄ µ̇ = ∂ ∂θ̄µ̇ + iθµ∂̃ µ̇µ. Furthermore, the connection supefields in D are defined as ΩU(1) = E , Ωâb̂ = EÂ Ω âb̂ . (7.30) It can be shown that the explicit expressions for the covariant derivatives are as follows: Da = eωy(1 + ω2e2ωyη2η̄2)∂a − iωeωy(η̄σ̃a)µDµ − iωeωy(ησa)µ̇D̄µ̇ +2iω2eωyη2(η̄σ̃a) + 2iω2eωyη̄2(ησa)µ̇ ∂η̄µ̇ − 3iω2eωy(ησaη̄)J + ω2eωyηabεbcde(ησcη̄)Mde − ω(1 + 2ω2e2ωyη2η̄2)Ma5 , (7.31a) , (7.31b) D1α = e ωy(1− 2ω2e2ωyη2η̄2)Dα − 2ωe − 2ωe 12ωyηαη̄µ̇ ∂η̄µ̇ +2ηαe − iωe 52ωyη2(σmη̄)α∂m − 3ωe 12ωyηα J + 2ωe ωyηβ(σab)βαMab + 2iω ωyη2(σaη̄)αMa5 , (7.31c) D2α = e− ωy(1− 4ω2e2ωyη2η̄2) ∂ + 2ωe ωyη̄2Dα − 2ωe ωyηαη̄µ̇D̄ µ̇ + ie ωy(σmη̄)α∂m − 6ω2e 32ωyηαη̄2J + 2ω2e ωyηβ η̄2(σab)βαMab − 2iωe ωy(σaη̄)αMa5 , (7.31d) D̄α̇1 = e ωy(1− 2ω2e2ωyη2η̄2)D̄α̇ − 2ωe 12ωyη̄2 ∂ ∂η̄α̇ − 2ωe 12ωyη̄α̇ηµ ∂ +2η̄α̇e − iωe 52ωyη̄2(σ̃mη)α̇∂m ωyη̄α̇ J + 2ωe (σ̃ab)β̇α̇Mab + 2iω ωyη̄2(σ̃aη)α̇Ma5 , (7.31e) D̄α̇2 = e− ωy(1− 4ω2e2ωyη2η̄2) ∂ ∂η̄α̇ + 2ωe ωyη2D̄α̇ − 2ωe 32ωyη̄α̇ηµDµ + ie y(σ̃mη)α̇∂m +6ω2e ωyη̄α̇η2 J + 2ω2e ωyη̄β̇η 2(σ̃ab)β̇α̇Mab − 2iωe ωy(σ̃aη)α̇Ma5 . (7.31f) It is interesting to consider a flat superspace limit, ω → 0, for the covariant derivatives. In this limit, one finds = e−UD eU , U = ηθ + η̄θ̄ , (7.32) where D = (∂â, D α̂) are 5D flat global covariant derivatives, Diα̂ = ∂θα̂i − i (Γb̂) θβ̂i ∂ , (7.33) with θα̂i = (θ i ,−θ̄α̇i) and θαi = (θα, ηα). 7.4 Torsion and curvature Now, we are prepared to demonstrate that the geometry described in the present section reproduces the geometry of AdS5|8 constructed in section 2. We proceed by recalling that, in accordance with the coset construction, the torsion T and curvature R two-forms are defined as follows: T = dE − Ω ∧E − E ∧Ω , R = dΩ − Ω ∧Ω . (7.34) Under group transformations (7.10) they transform covariantly T′ = ĥT ĥ−1 , R′ = ĥR ĥ−1 . (7.35) Keeping in mind the definition E+Ω = G−1dG, we get dE+ dΩ = G−1dG ∧G−1dG = E ∧E + E ∧Ω + Ω ∧ E + Ω ∧Ω , (7.36) from which we obtain dE = (E ∧E)|G−H + E ∧Ω + Ω ∧ E , dΩ = (E ∧E)|H + Ω ∧Ω , (7.37) since (E∧Ω+Ω∧E) ∈ G −H and Ω∧Ω ∈ H. Using the previous formulae we are able to see that the torsion and curvature two-forms are given by simple expressions T = (E ∧E)|G−H , R = (E ∧E)|H . (7.38) Therefore, it remains to compute E ∧E. Direct calculations give E ∧ E = ω2Eâ ∧ Eb̂(Σ β̂ + 4ωE2α̂ ∧Eβ̂1 −iω 2 Eâ ∧ E2γ̂ (Γâ)α̂γ̂ −iω 32 Eγ̂1 ∧ Eb̂ (Γb̂)γ̂ β̂ 4ωE 1 ∧ E2γ̂  , (7.39) and this we should represent as E ∧ E = (E ∧ E)|G−H + (E ∧ E)|H. We end up with ωTâ(Γâ)α̂ β̂ 2ω 2 T2α̂  , (7.40) Râb̂ 0 0 0  + iRU(1) 0 0 −4  , (7.41) where Tâ = ∧Eβ̂j 2i εjk(Γâ) , (7.42) Tα̂i = Eĉ ∧ Eβ̂j ω(σ3) i (Γĉ)β̂ ω(σ3) i (Γb̂)γ̂ , (7.43) Râb̂ = Ed̂ ∧Eĉ ω2(−δâĉ δb̂d̂ + δ ) + Eδ̂l ∧E − 4ωεki(σ3) li (Σâb̂)γ̂δ̂ , (7.44) RU(1) = Eδ̂l ∧ E 3iωεklε . (7.45) Using standard superform definitions [19], we define the components of the torsion and curvature as follows: T = 1 EĈ ∧ EB̂ T  , (7.46) Râb̂ = 1 ED̂ ∧EĈ R âb̂ , RU(1) = ED̂ ∧ EĈ (RU(1))ĈD̂ . (7.47) Now, let us return to the covariant derivatives described in section 2. Their algebra given by eqs. (2.21a–2.21c) can be represented concisely as + i (RU(1))ÂB̂ J + ĉd̂M . (7.48) Comparing (2.21a–2.21c) with eqs. (7.42–7.45), we find that all the components of the torsion and curvature coincide provided J ij = (σ3) j . (7.49) This completes our analysis of the coset construction. Acknowledgements: This work is supported in part by the Australian Research Council and by a UWA research grant. A 5D Conventions Our 5D notation and conventions correspond to [10]. The 5D gamma-matrices Γm̂ = (Γm,Γ5), with m = 0, 1, 2, 3, are defined by {Γm̂ , Γn̂} = −2ηm̂n̂ 1 , (Γm̂)† = Γ0 Γm̂ Γ0 (A.1) are chosen in accordance with (Γm)α̂ 0 (σm)αβ̇ (σ̃m) α̇β 0 , (Γ5)α̂ −i δαβ 0 0 i δα̇ , (A.2) such that Γ0Γ1Γ2Γ3Γ5 = 1. The charge conjugation matrix, C = (ε α̂β̂), and its inverse, C−1 = C† = (ε ) are defined by C Γm̂C −1 = (Γm̂) T , εα̂β̂ = εαβ 0 0 −εα̇β̇ εαβ 0 0 −εα̇β̇ . (A.3) The antisymmetric matrices εα̂β̂ and ε are used to raise and lower the four-component spinor indices. A Dirac spinor, Ψ = (Ψα̂), and its Dirac conjugate, Ψ̄ = (Ψ̄ α̂) = Ψ† Γ0, look like Ψα̂ = , Ψ̄α̂ = (φα , ψ̄α̇) . (A.4) One can now combine Ψ̄α̂ = (φα, ψ̄α̇) and Ψ α̂ = εα̂β̂Ψ = (ψα,−φ̄α̇) into a SU(2) doublet, Ψα̂i = (Ψ i ,−Ψ̄α̇i) , (Ψαi )∗ = Ψ̄α̇i , i = 1, 2 , (A.5) with Ψα1 = φ α and Ψα2 = ψ α. It is understood that the SU(2) indices are raised and lowered by εij and εij , ε 12 = ε21 = 1, in the standard fashion: Ψ α̂i = εijΨα̂j . The Dirac spinor Ψi = (Ψiα̂) satisfies the pseudo-Majorana condition Ψ̄i T = C Ψi. This will be concisely represented as (Ψiα̂) ∗ = Ψα̂i . (A.6) With the definition Σm̂n̂ = −Σn̂m̂ = −14 [Γm̂,Γn̂], the matrices {1,Γm̂,Σm̂n̂} form a basis in the space of 4 × 4 matrices. The matrices ε and (Γm̂)α̂β̂ are antisymmetric, εα̂β̂ (Γm̂)α̂β̂ = 0, while the matrices (Σm̂n̂)α̂β̂ are symmetric. It is useful to write explicitly the 4D reduction of these matrices (Γm)α̂β̂ = 0 −(σm)αβ̇ (σm)β , (Γ5)α̂β̂ = i εαβ 0 0 i εα̇β̇ , (A.7) (Σmn)α̂ (σmn)α 0 (σ̃mn) , (Σm5)α̂ 0 − i (σm)αβ̇ (σ̃m) α̇β 0 , (A.8) (Σmn)α̂β̂ = (σmn)αβ 0 0 −(σ̃mn)α̇β̇ , (Σm5)α̂β̂ = (σm)α (σm)β , (A.9) where (σmn)α β = −1 (σmσ̃n − σnσ̃m)αβ and (σ̃mn)α̇β̇ = −14(σ̃mσn − σ̃nσm) Given a 5-vector V m̂ and an antisymmetric tensor F m̂n̂ = −F n̂m̂, we can equivalently represent them as the bi-spinors V = V m̂ Γm̂ and F = F m̂n̂ Σm̂n̂ with the following symmetry properties , εα̂β̂ V = 0 , F . (A.10) The two equivalent descriptions Vm̂ ↔ Vα̂β̂ and and Fm̂n̂ ↔ Fα̂β̂ are explicitly described as follows: = V m̂ (Γm̂)α̂β̂ , Vm̂ = − (Γm̂) α̂β̂ V F m̂n̂(Σm̂n̂)α̂β̂ , Fm̂n̂ = (Σm̂n̂) α̂β̂ F . (A.11) These results can be easily checked using the identities α̂β̂γ̂δ̂ + εα̂γ̂ εδ̂β̂ + εα̂δ̂ εβ̂γ̂ , εα̂γ̂ εβ̂δ̂ − εα̂δ̂ εβ̂γ̂ = − (Γm̂) (Γm̂)γ̂δ̂ + , (A.12) and therefore α̂β̂γ̂δ̂ (Γm̂) (Γm̂)γ̂δ̂ + , (A.13) with ε α̂β̂γ̂δ̂ the completely antisymmetric fourth-rank tensor. Complex conjugation gives )∗ = −εα̂β̂ , (V )∗ = V α̂β̂ , (F )∗ = F α̂β̂ , (A.14) provided V m̂ and F m̂n̂ are real. References [1] A. S. Galperin, E. A. Ivanov, S. N. Kalitsyn, V. Ogievetsky, E. Sokatchev, “Unconstrained N=2 matter, Yang-Mills and supergravity theories in harmonic superspace,” Class. Quant. Grav. 1, 469 (1984). [2] A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. S. Sokatchev, Harmonic superspace, Cambridge, UK: Univ. Pr. (2001) 306 p. [3] A. Karlhede, U. Lindström and M. Roček, “Self-interacting tensor multiplets in N=2 su- perspace,” Phys. Lett. B 147, 297 (1984). [4] W. Siegel, “Chiral actions for N=2 supersymmetric tensor multiplets” Phys. Lett. B 153 (1985) 51. [5] U. Lindström and M. Roček, “New hyperkähler metrics and new supermultiplets,” Com- mun. Math. Phys. 115, 21 (1988). [6] U. Lindström and M. Roček, “N=2 super Yang-Mills theory in projective superspace,” Commun. Math. Phys. 128, 191 (1990). [7] A. A. Rosly, “Super Yang-Mills constraints as integrability conditions,” in Proceedings of the International Seminar on Group Theoretical Methods in Physics,” (Zvenigorod, USSR, 1982), M. A. Markov (Ed.), Nauka, Moscow, 1983, Vol. 1, p. 263 (in Russian); English translation: in Group Theoretical Methods in Physics,” M. A. Markov, V. I. Man’ko and A. E. Shabad (Eds.), Harwood Academic Publishers, London, Vol. 3, 1987, p. 587. [8] E. Witten, “An interpretation of classical Yang-Mills theory,” Phys. Lett. B 77, 394 (1978). [9] B. Zupnik, “Harmonic superpotentials and symmetries in gauge theories with eight super- charges,” Nucl. Phys. B 554 (1999) 365 [Erratum-ibid. B 644 (2002) 405]. [hep-th/9902038]. [10] S. M. Kuzenko and W. D. Linch, III, “On five-dimensional superspaces,” JHEP 0602, 038 (2006) [hep-th/0507176]. [11] P. S. Howe, K. S. Stelle and P. C. West, “N=1 D = 6 harmonic superspace,” Class. Quant. Grav. 2 (1985) 815; B. M. Zupnik, “Six-dimensional supergauge theories in the harmonic superspace,” Sov. J. Nucl. Phys. 44 (1986) 512. [12] J. Grundberg and U. Lindström, “Actions for linear multiplets in six dimensions,” Class. Quant. Grav. 2, L33 (1985). [13] S. J. Gates, Jr., S. Penati and G. Tartaglino-Mazzucchelli, “6D supersymmetry, projective superspace and 4D, N = 1 superfields,” JHEP 0605, 051 (2006) [hep-th/0508187]; “6D supersymmetric nonlinear sigma-models in 4D, N = 1 superspace,” JHEP 0609, 006 (2006) [hep-th/0604042]. [14] W. Siegel, “Curved extended superspace from Yang-Mills theory a la strings,” Phys. Rev. D 53, 3324 (1996) [hep-th/9510150]. [15] W. D. Linch, III and B. C. Vallilo, “Covariant N = 2 heterotic string in four dimensions,” hep-th/0611105. [16] N. Berkovits, “Covariant quantization of the Green-Schwarz superstring in a Calabi-Yau background,” Nucl. Phys. B 431, 258 (1994) [hep-th/9404162]; N. Berkovits and C. Vafa, “N=4 topological strings,” Nucl. Phys. B 433, 123 (1995) [hep-th/9407190]; N. Berkovits http://arxiv.org/abs/hep-th/9902038 http://arxiv.org/abs/hep-th/0507176 http://arxiv.org/abs/hep-th/0508187 http://arxiv.org/abs/hep-th/0604042 http://arxiv.org/abs/hep-th/9510150 http://arxiv.org/abs/hep-th/0611105 http://arxiv.org/abs/hep-th/9404162 http://arxiv.org/abs/hep-th/9407190 and W. Siegel, “Superspace effective actions for 4D compactifications of heterotic and type II superstrings,” Nucl. Phys. B 462, 213 (1996) [hep-th/9510106]. [17] A. S. Galperin, N. A. Ky and E. Sokatchev, “N=2 supergravity in superspace: Solution to the constraints,” Class. Quant. Grav. 4, 1235 (1987); A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. Sokatchev, “N=2 supergravity in superspace: Different versions and matter couplings,” Class. Quant. Grav. 4, 1255 (1987). [18] E. Sokatchev, “Off-shell six-dimensional supergravity in harmonic superspace,” Class. Quant. Grav. 5, 1459 (1988). [19] J. Wess and J. Bagger, “Supersymmetry and supergravity,” Princeton, USA: Univ. Pr. (1992) 259 p. [20] W. Siegel and S. J. Gates, Jr., “Superfield supergravity,” Nucl. Phys. B 147, 77 (1979); S. J. Gates, Jr., M. T. Grisaru, M. Roček and W. Siegel, Superspace, Or One Thousand and One Lessons in Supersymmetry, Benjamin/Cummings, 1983 [hep-th/0108200]. [21] I. L. Buchbinder and S. M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity or a Walk Through Superspace, IOP, Bristol, 1998. [22] A. Y. Segal and A. G. Sibiryakov, “Explicit N = 2 supersymmetry for higher-spin massless fields in D = 4 AdS superspace,” Int. J. Mod. Phys. A 17, 1207 (2002) [hep-th/9903122]. [23] S. J. Gates, Jr., S. M. Kuzenko and A. G. Sibiryakov, “N = 2 supersymmetry of higher superspin massless theories,” Phys. Lett. B 412, 59 (1997) [hep-th/9609141]; “Towards a unified theory of massless superfields of all superspins,” Phys. Lett. B 394, 343 (1997) [hep-th/9611193]. [24] I. A. Bandos, E. Ivanov, J. Lukierski and D. Sorokin, “On the superconformal flatness of AdS superspaces,” JHEP 0206, 040 (2002) [hep-th/0205104]. [25] S. M. Kuzenko, “On compactified harmonic/projective superspace, 5D superconformal the- ories, and all that,” Nucl. Phys. B 745, 176 (2006) [hep-th/0601177]. [26] N. Dragon, “Torsion and curvature in extended supergravity,” Z. Phys. C 2, 29 (1979). [27] E. A. Ivanov and A. S. Sorin, “Superfield formulation of Osp(1,4) supersymmetry,” J. Phys. A 13, 1159 (1980). [28] M. F. Sohnius, “Supersymmetry and central charges,” Nucl. Phys. B 138 (1978) 109. [29] P. Breitenlohner and M. F. Sohnius, “Superfields, auxiliary fields, and tensor calculus for N=2 extended supergravity,” Nucl. Phys. B 165, 483 (1980). http://arxiv.org/abs/hep-th/9510106 http://arxiv.org/abs/hep-th/0108200 http://arxiv.org/abs/hep-th/9903122 http://arxiv.org/abs/hep-th/9609141 http://arxiv.org/abs/hep-th/9611193 http://arxiv.org/abs/hep-th/0205104 http://arxiv.org/abs/hep-th/0601177 [30] J. Wess, “Supersymmetry and internal symmetry,” Acta Phys. Austriaca 41, 409 (1975); W. Siegel, “Off-shell central charges,” Nucl. Phys. B 173, 51 (1980). [31] M. F. Sohnius, K. S. Stelle and P. C. West, “Representations of extended supersymmetry,” in Superspace and Supergravity, S. W. Hawking and M. Roček (Eds.) Cambridge Unieversity Press, Cambridge, 1981, p. 283. [32] S. V. Ketov, “New self-interaction for N = 2 multiplets in 4d and ultraviolet finiteness of two-dimensional N = 4 sigma-models,” in Proceedings of the International Seminar Group Theory Methods in Physics, (Urmala, USSR, May 1985) Nauka, Moscow, Vol. 1, p. 87; S. V. Ketov and B. B. Lokhvitsky, “Some generalizations of N=2 Yang-Mills matter cou- plings,” Class. Quant. Grav. 4, L137 (1987); S. V. Ketov, B. B. Lokhvitsky and I. V. Tyutin, “Hyperkahler sigma models in extended superspace,” Theor. Math. Phys. 71, 496 (1987). [33] F. Gonzalez-Rey, U. Lindström M. Roček, R. von Unge and S. Wiles, “Feynman rules in N = 2 projective superspace. I: Massless hypermultiplets,” Nucl. Phys. B 516, 426 (1998) [hep-th/9710250]. [34] A. S. Galperin, E. A. Ivanov and V. I. Ogievetsky, “Duality transformations and most general matter self-coupling in N=2 supersymmetry,” Nucl. Phys. B 282, 74 (1987). [35] A. A. Rosly, “Gauge fields in superspace and twistors,” Class. Quant. Grav. 2, 693 (1985). [36] A. A. Rosly and A. S. Schwarz, “Supersymmetry in a space with auxiliary dimensions,” Commun. Math. Phys. 105, 645 (1986). [37] S. M. Kuzenko, “Projective superspace as a double-punctured harmonic superspace,” Int. J. Mod. Phys. A 14 (1999) 1737 [hep-th/9806147]. [38] B. B. Deo and S. J. Gates, Jr., “Comments on nonminimal N=1 scalar multiplets,” Nucl. Phys. B 254, 187 (1985). [39] S. J. Gates, Jr. and S. M. Kuzenko, “The CNM-hypermultiplet nexus,” Nucl. Phys. B 543, 122 (1999) [hep-th/9810137]; “4D N = 2 supersymmetric off-shell sigma models on the cotangent bundles of Kähler manifolds,” Fortsch. Phys. 48, 115 (2000) [hep-th/9903013]. [40] M. Arai and M. Nitta, “Hyper-Kähler sigma models on (co)tangent bundles with SO(n) isometry,” Nucl. Phys. B 745, 208 (2006) [hep-th/0602277]. [41] M. Arai, S. M. Kuzenko and U. Lindström, “Hyperkähler sigma models on cotangent bun- dles of Hermitian symmetric spaces using projective superspace,” JHEP 0702, 100 (2007) [hep-th/0612174]. http://arxiv.org/abs/hep-th/9710250 http://arxiv.org/abs/hep-th/9806147 http://arxiv.org/abs/hep-th/9810137 http://arxiv.org/abs/hep-th/9903013 http://arxiv.org/abs/hep-th/0602277 http://arxiv.org/abs/hep-th/0612174 [42] B. de Wit, M. Roček and S. Vandoren, “Hypermultiplets, hyperkaehler cones and quaternion-Kaehler geometry,” JHEP 0102, 039 (2001) [hep-th/0101161]. [43] B. W. Keck, “An alternative class of supersymmetries,” J. Phys. A 8, 1819 (1975); B. Zu- mino, “Nonlinear realization of supersymmetry in anti de Sitter space,” Nucl. Phys. B 127, 189 (1977). [44] S. Coleman, J. Wess and B. Zumino, “Structure of phenomenological Lagrangians. 1,” Phys. Rev. 177 (1969) 2239; C. G. Callan, S. Coleman, J. Wess and B. Zumino, “Struc- ture of phenomenological Lagrangians. 2,” Phys. Rev. 177 (1969) 2247; C. J. Isham, “A group-theoretic approach to chiral transformations,” Nuovo Cim. 59A (1969) 356; “Metric structures and chiral symmetries,” 61A (1969) 188; A. Salam and J. Strathdee, “Nonlin- ear realizations. 1: The role of Goldstone bosons,” Phys. Rev. 184 (1969) 1750; D. V. Volkov, “Phenomenological Lagrangians,” Sov. J. Part. Nucl. 4, 3 (1973); V. I. Ogievetsky, “Nonlinear realizations of internal and space-time symmetries,” in Proceedings of the Xth Winter School of Theoretical Physics in Karpacz (Wroslaw, 1974), Vol. 1, p. 227. [45] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “Large N field theories, string theory and gravity,” Phys. Rept. 323, 183 (2000) [hep-th/9905111]. [46] S. M. Kuzenko and I. N. McArthur, “Goldstone multiplet for partially broken superconfor- mal symmetry,” Phys. Lett. B 522, 320 (2001) [hep-th/0109183]. [47] S. Bellucci, E. Ivanov and S. Krivonos, “Goldstone superfield actions in AdS(5) back- grounds,” Nucl. Phys. B 672, 123 (2003) [hep-th/0212295]. [48] H. Osborn, “N = 1 superconformal symmetry in four-dimensional quantum field theory,” Annals Phys. 272, 243 (1999) [hep-th/9808041]. [49] S. M. Kuzenko and S. Theisen, “Correlation functions of conserved currents in N = 2 superconformal theory,” Class. Quant. Grav. 17, 665 (2000) [hep-th/9907107]. [50] V. P. Akulov and D. V. Volkov, “Goldstone fields with spin 1/2,” Theor. Math. Phys. 18, 28 (1974). [Teor. Mat. Fiz. 18, 39 (1974)]. http://arxiv.org/abs/hep-th/0101161 http://arxiv.org/abs/hep-th/9905111 http://arxiv.org/abs/hep-th/0109183 http://arxiv.org/abs/hep-th/0212295 http://arxiv.org/abs/hep-th/9808041 http://arxiv.org/abs/hep-th/9907107 Introduction Covariant derivatives Killing supervectors Harmonic superspace approach Analytic multiplets Harmonic action principle Projective superspace approach Projective multiplets Projective action principle Dynamical systems in projective superspace Projective multiplets revisited Projective action Nonlinear sigma-models Vector multiplet and Chern-Simons couplings Tensor multiplet and vector-tensor couplings Coset space realization Coset representative SO(4,1)U(1) covariance Representation of covariant derivatives Torsion and curvature 5D Conventions

0704.1186

Domain wall entropy of the bimodal two-dimensional Ising spin glass

Domain wall entropy of the bimodal two-dimensional Ising spin glass A. Aromsawa and J. Poulter Department of Mathematics, Faculty of Science, Mahidol University, Rama 6 Road, Bangkok 10400, Thailand (Dated: November 15, 2018) We report calculations of the domain wall entropy for the bimodal two-dimensional Ising spin glass in the critical ground state. The L× L system sizes are large with L up to 256. We find that it is possible to fit the variance of the domain wall entropy to a power function of L. However, the quality of the data distributions are unsatisfactory with large L > 96. Consequently, it is not possible to reliably determine the fractal dimension of the domain walls. PACS numbers: 75.10.Hk, 75.10.Nr, 75.40.Mg, 75.60.Ch I. INTRODUCTION The short-range Ising spin glass is still a source of con- troversy in spite of its comparative simplicity. In brief, the exact mechanisms by which widely separated spins influence each other are not clearly understood. The sys- tem Hamiltonian, due to Edwards and Anderson1, is H = − Jijσiσj (1) where the nearest-neighbor exchange interactions Jij are quenched random variables which, for present purposes, take random sign with equal probability. The two partic- ular models of disorder widely studied are the bimodal, or ±J , model where the interactions have fixed magni- tude and the Gaussian model where they are taken from a continuous normal distribution of zero mean. In two di- mensions it is generally accepted that the spin glass only exists at zero temperature2,3, that is the critical temper- ature Tc = 0. Spin correlations in two dimensions for bimodal disor- der are expected to decay algebraically according to [< σ0σR > 2]av ∼ R−η (2) in the ground state. To date, the best estimates from Monte Carlo simulation4 and exact calculation5 seem to agree that the exponent is η = 0.14. For Gaussian dis- order, the exponent is zero since the ground state, apart from a global inversion, is unique. In contrast, bimodal disorder comes with a large ground state degeneracy cor- responding to an entropy of 0.07k per spin6,7,8,9,10,11. In spite of this it has been argued12 that η = 0 in the ther- modynamic limit. Very recently, Hartmann13 has esti- mated η = 0.22 from scaling arguments. This is in fair agreement with a number of previous estimates7,9,14,15,16. Nevertheless, the important issue here is universality; that is whether or not η is positive. Clear evidence of the importance of long range influ- ence in the case of bimodal disorder is apparent in the current controversy regarding the lowest excited state. This issue dates back to the work of Wang and Swendsen7 which proposed that the energy gap should be 2J , not 4J , in the thermodynamic limit. Although some work4,9 has supported the simple naive 4J scenario, it is now be- coming clearer that strong disorder is indeed producing a different result for the infinite system. The energy gap has been often reported11,17,18 as 2J . Further, it has even been suggested, based on a finite size scaling study of the correlation length and the spin glass susceptibility12, that the specific heat should obey a power law. This may indi- cate a feature universal with Gaussian disorder. A good review of the issues involved here is given in Ref. 19. Essentially it appears to be the case that there exist low energy excitations with very long range influence. A standard technique commonly used to investigate the long range response of a spin glass is to introduce a domain wall defect14,20,21. This is done, in two dimen- sions, by drawing a line across the system and reversing the signs of all bonds cut by the line. With Gaussian disorder, the ground and low energy excited states are unique. Consequently, the term domain wall has a clear physical meaning and appears as an optimium fractal path that corresponds to the lowest excited state. The case of bimodal disorder is quite different as a result of the large degeneracy of both the ground and excited states. The degeneracy of the ground state is of the order of exp(0.07L2) for a L × L lattice and that of the first ex- cited state is probably much larger still11. In particular, it is often the case that the defect system is not an ex- cited state of the reference system. Thus, we do not have real domain walls in the same sense as with the case of Gaussian disorder. Nevertheless, we might expect some useful knowledge to be derived from a study of the bi- modal system in response to reversing the signs of bonds along a line as with the Gaussian system. Thinking of the thermodynamic limit, we can discuss this issue in the light of droplet theory22. For the case of a continuous defect distribution, Gaussian for instance, there is a unique ground state and we define a droplet as a region bounded by a closed path (or surface in three dimensions). Below the transition temperature, which is positive in three dimensions at least23, the scaling prop- erties of droplets of various sizes can be investigated by reversing all spins inside the closed path. This creates an excited state of the reference system and it is known that there exist low energy droplet excitations of large spatial extent. The idea here is that a domain wall de- http://arxiv.org/abs/0704.1186v2 fect is much the same as a large droplet excitation in the thermodynamic limit. For bimodal disorder this compar- ison is less clear due to the huge ground state degeneracy. Certain droplets may not represent excitations at all. The most important prediction of droplet theory is that the energy difference Edw between the two systems, with and without the domain wall, can be fitted to a power function of the system size L. We have, for the spin glass, < |Edw| >∼ Lθ (3) where θ is known as the spin-glass stiffness exponent. In three dimensions, this exponent is found to be positive24,25, at about 0.2 for both bimodal and Gaus- sian disorder, showing consistency with the existence of a stable spin glass phase at finite temperature. As a mat- ter of fact the droplet theory22 was originally developed for this type of case where both the critical temperature and stiffness exponent are positive. It is also probable that the Gaussian and bimodal models fit into the same universality class23 with respect to their transitions at fi- nite critical temperature. Degeneracy is not an issue due to the thermal fluctuations. In two dimensions it was not immediately obvious that droplet theory22 is entirely appropriate since the criti- cal temperature is zero and the stiffness exponent is not positive. For Gaussian disorder the stiffness exponent is negative26, θ ≈ −0.28, clearly indicating that the spin glass is unstable at any finite temperature. It is also re- markable that the values of θ obtained from domain wall and droplet calculations are in very good agreement. For bimodal disorder it is generally accepted that the stiffness exponent due to domain wall defects is zero2,27,28. Never- theless the situation for droplets is not clear. Hartmann13 has estimated the droplet stiffness exponent by construct- ing ground and first excited states of three models of bi- modal disorder and found good agreement with Gaussian disorder. Nevertheless, universality is not shown since the correlation function exponent η is not zero. For the case of Gaussian disorder, in two dimensions, it has been suggested29,30 that the domain walls are stochastic Loewner evolution processes31. This theory is able to relate the domain wall fractal dimension df to the stiffness exponent via df = 1 + 4(3 + θ) and the result df ≈ 1.27 agrees well with the literature26. There is also an interesting conjecture due to Fisch32 that θ = ( 6 − 3)/2 exactly. Nevertheless, there is no good reason to believe that Eq. (4) can be used with bimodal disorder, possibly due to the degeneracy of the ground state. The domain wall entropy Sdw is defined in the same manner as Edw. Droplet theory 22 predicts that Sdw should take values with random sign and large variance. In particular, the variance is predicted to scale as < S2dw > − < Sdw >2∼ Ldf (5) which provides a possible means to estimate df . In par- ticular, we can use this to test the appropriateness of droplet theory for the case where both the critical tem- perature and the stiffness exponent tend to zero from above. This is precisely the situation we have for the two-dimensional model with bimodal disorder. Previous estimations of the fractal dimension from di- rect studies of the domain wall entropy have been pub- lished. Saul and Kardar9 predict df ≈ 1.0. Fisch33 ar- gues that df might be an increasing function of |Edw|. The possibility that df = 1.25 in agreement with Eq. (4) is not ruled out. Finally, Lukic et al34 have reported df = 1.03(2). These values should also be compared with those from topological analysis of the ground state. Romá et al35 report df = 1.30(1) while Melchert and Hartmann36 find an interval 1.095(1) ≤ df ≤ 1.395(1). In this article we report calculations of the domain wall entropy on sample sizes that are much larger than any- thing done before. Furthermore, our method is applied at an arbitrarily small temperature and there is no need to extrapolate to the ground state. Our main conclusion is that the domain wall fractal dimension for bimodal disorder, as predicted by droplet theory, is not a well de- fined quantity. The reason for this is that there exists no clear prescription for its estimation if the domain wall entropy distributions are significantly far from normal. A brief overview of the method is given in Sec. II. This is followed by our results in Sec. III and a brief discussion in Sec. IV. II. BACKGROUND The planar Ising model is known to be isomorphic to a system of noninteracting fermions. One particular illustration37 has been adapted by Blackman38 for disor- dered systems. For the square lattice, each site is deco- rated with four fermions. Equivalently, we can decorate each bond with two fermions. For a system of N lattice sites, we have 4N fermions in total. It is useful to think of the two fermions decorating a bond to be placed one on either side. In this way a plaquette (square) is decorated with four fermions; left, right, top and bottom. The partition function for the Ising model on a square lattice with any set of exchange interactions takes the Z = 2N [ cosh(Jij/kT )] (detD) 1/2 (6) where the product is over all nearest neighbor bonds Jij on the N site lattice and D is a skew-symmetric (4N × 4N) matrix. The square root of the determinant of D is also called the Pfaffian. Essentially, it represents the sum over all closed lattice polygons and is equal to the product of all the positive eigenvalues of D. The calculation of the partition function with bimodal disorder has been described in much detail previously8 and a simple summary should suffice here. At zero tem- perature, D is a singular matrix with a set of degener- ate zero eigenvalues exactly equal in number to the total number of frustrated plaquettes. In order to extract the physics of the system, degenerate state perturbation the- ory is applied at an arbitrarily low temperature. The defect eigenvalues occur in pairs and approach zero as some power of exp(−2J/kT ) ǫ = ±1 X exp(−2Jr/kT ) (7) where r is an integer (an order of perturbation theory) and X is a real number that is independent of tempera- ture and depends only on the configuration of frustrated plaquettes. The ground state energy and entropy can be expressed exactly as E = −2NJ + 2J rd (8) S = k lnXd (9) where the sums are over all defect eigenstate pairs. To summarise the perturbation theory, we first write the matrix D exactly as the sum of two terms D = D0 + δD1, where δ = 1 − t with t = tanh(J/kT ). Of course t = 1 and δ = 0 in the ground state. Both of the matrices D0 and D1 are independent of temperature. The matrixD0 has eigenvectors localised inside each frus- trated plaquette; expanded in the basis of the four deco- rating fermions. It is these localised states that form the defect basis for the perturbation theory. The matrix D1 is 2 × 2 block diagonal in the pairs of fermions decorat- ing the bonds (one fermion either side). All degeneracy at first order is lifted by diagonalizing D1 in the defect basis. For example, we can think of just two neighboring frustrated plaquettes. The perturbation theory gives one defect pair with rd = 1 and Xd = 1. In general, the first order calculation will leave some zero eigenvalues. The corresponding eigenvectors of D1 form the basis for second order. We can imagine a sys- tem of two next nearest neighbor frustrated plaquettes. Clearly rd = 2 and Xd = 1 or 2 depending on the ar- rangement. In order to show this we need to use the continuum Green’s function5,8 gc = gc1 + gc2. The ma- trix gc1 is 4×4 block diagonal in the four fermions inside each plaquette and clearly allows us to connect two frus- trated plaquettes, across an unfrustrated plaquette. The second order calculation is performed by diagonalizing D2 = D1gc1D1. The matrix gc2 is, just like D1, 2 × 2 block diagonal in the pairs of fermions decorating bonds. A proof that gc2 is irrelevant for the ground state has been given in Ref. 5. For higher orders we require Green’s functions for states whose degeneracy has already been lifted. We de- fine, for r ≥ 1, Gr = − N(r)∑ | r, i〉(1/ǫir)〈r, i | (10) where | r, i〉 denotes state i (with eigenvalue ǫir) in the set of states whose degeneracy was lifted at order r; there are N(r) of these states. At third order the matrix to be diagonalized is D3 = D2(1 +G1D1)gc1D1 and, generally at arbitrary order Dn = Dn−1(1 + Gn−2Dn−2) · · · (1 + G1D1)gc1D1. The perturbation theory is applied order by order until all degeneracy is lifted. The scheme outlined above allows exact calculations of energy and entropy in the ground state. The method is fully gauge invariant in that it depends only on the number and distribution of frustrated plaquettes. Fur- thermore, there is no requirement to extrapolate to the ground state. The Pfaffian is not calculated at any par- ticular numerical value of the temperature. We believe that this method is the best available for calculating the ground state entropy of large lattices, although matching algorithms are better for the energy2. We have used periodic boundary conditions in one di- mension. The cylindrically wound frustrated patch was nested in an unfrustrated system of infinite extent in the second dimension. In this scheme, the introduction of a domain wall defect is particularly simple. The two pla- quettes at the ends of the defect, one on each side of the patch, change their status; frustrated to unfrustrated or vice versa. For a perfect ferromagnet this gives a domain wall energy proportional to L as required. For a fully frustrated system the defect would make no real differ- ence since the domain wall energy would be independent of L. For the spin glass the domain wall energies are all multiples of 4J since we have L even. The probability that a plaquette is frustrated is expected to be close to 0.5 and it is conceivable that the system with the domain wall could be interchanged with its reference system in another realization of disorder. This is consistent with the prediction of droplet theory that the domain wall entropy has random sign. Since the domain wall entropy is generally a small dif- ference between two larger numbers, we have taken great care with the floating point computations. Although our method is analytically exact, it is subject to numerical propagation error on the computer. Ill-conditioned dis- order realizations were detected by calculating the cor- relation function along a path around the cylinder and repeated in arbitrary precision arithmetic as necessary. III. RESULTS We have calculated the domain wall entropy for bi- modal disorder on L × L lattices where L = 8, 12, 16, 24, 32, 48, 64, 96, 128, 192 and 256. For sizes L ≤ 128, 105 random samples were taken. We also took 4 × 104 for L = 192 and 104 for L = 256. 500 100 50 10 5 FIG. 1: The domain wall energy (in units of J) as a function of system size L. The error bars are two standard deviations of the mean and the curve is a best fit to the form A−BL−p following Ref. 2. To establish the credentials of our boundary scheme, we have calculated the domain wall energy. The data is shown in Fig. 1 where the error bars are two standard deviations of the mean. Following Ref. 2, we have fitted the data to a function A − BL−p and find A = 3.7(2), B = 3.2(1) and p = 0.23(4). We note that the fit is ap- proaching saturation from below. We believe that this is as a consequence of our boundary conditions. The prob- ability of a zero energy (Edw = 0) domain wall is found to decrease with L, contrary to the situation with free boundary conditions in the unwound dimension39. Fur- thermore, since we only use even values of L, the defect energies are all multiples of 4J . The quality of the non- linear fit40 is Q = 0.91. Attempts to fit a power law were not successful. For instance a fit for L ≥ 96 completely missed the point at L = 256. We conclude that our method is reliable although larger system sizes L > 256 are required to conclude more convincingly that θ = 0. The variance of the domain wall entropy Sdw is shown in Fig. 2. A power law fit for 24 ≤ L ≤ 96 predicts that, according to Eq. (5), the fractal dimension of the domain walls is df = 1.090(8) where the quality of the fit is Q = 0.42. A similar fit (Q = 0.30) of < |Sdw| >2 gives 1.080(9) which agrees well, indicating good quality data distributions. A second power law fit for L ≥ 96, also shown in Fig. 2, reveals a significantly higher value df = 1.30(3) with Q = 0.16. However, a fit of < |Sdw| >2 gives only 1.23(2). Although the quality Q = 0.05 is lower, the difference is too large to be disregarded. The reason for 1000 500 100 50 10 5 FIG. 2: The variance of the domain wall entropy as a function of system size L. The error bars are two standard deviations of the mean. Two power law fits are shown; for 24 ≤ L ≤ 96 (dashed line) and L ≥ 96 (dotted line). The powers are respectively 1.090(8) and 1.30(3). this discrepancy must lie in the quality of the distribu- tions for Sdw. For L = 256, for example, the distribu- tion has skewness 0.64 and kurtosis 2.06. Although the mean < Sdw >= 3.53 is still much less than the variance (≈ 350) it reflects a significant sign disparity. The bias most likely arises due to correlations in the distribution of frustrated plaquettes. Incidently, the corresponding dis- tributions for Edw are of excellent quality. For L = 256, the skewness and kurtosis are respectively −0.006 and 0.07. IV. DISCUSSION In summary, the distributions of the domain wall en- tropy for large L, with bimodal disorder in two dimen- sions, are found to deviate significantly from normal. In consequence, even if we assume that droplet theory22 is appropriate, it is unable to prescribe exactly how to get the domain wall fractal dimension. This does not neces- sarily mean that droplet theory is entirely wrong. It just does not give the whole story, only an approximation, for this system; having a large ground state degeneracy, a zero critical temperature and a zero stiffness exponent. Of course, it is quite probable that corrections to scal- ing are large and difficult to manage. This is actually a rather likely scenario in view of the poor results for fitting the ground state energy with cylindrical winding in one dimension28. Scaling corrections are an issue probably related to strong correlations in the distribution of frus- trated plaquettes for large L. All gauge invariant quanti- ties like entropy depend only on the frustrated plaquette distribution; nothing else. The prediction of droplet the- ory that the domain wall entropy is normally distributed with zero, or very small, mean and large variance proba- bly relates to an assumption that a defect system occurs as a reference system in another realization of disorder. This assumption may not be true if the frustrated pla- quette distributions are strongly correlated, as is likely in view of the anamolous behaviour of the degeneracy of the first excitations mentioned earlier. A further scenario is that due to Hartmann13, which proposes that droplet theory is actually inappropriate for estimating the stiff- ness exponent of domain wall defects. If this is correct, it is also unlikely that the fractal dimension of domain walls has anything to do with droplet theory. Neverthe- less, our results do indicate an approximate appropriate- ness for droplet theory in the sense that all the possible estimates for df do not seem unreasonable. Previous studies of the domain wall entropy have worked with much smaller system sizes. Saul and Kardar9 had sizes up to L = 36 and found df ≈ 1.0, while Lukic et al34 used sizes up to L = 50 and fitted < |Sdw| > to find df = 1.03(2). Fisch33 used sizes up to L = 48 and has introduced the idea that the domain wall entropy may be significantly correlated with energy. It is argued that an effective df increases as a function of |Edw| and convergence to the value 1.25 consistent with Eq. (4) is not ruled out. We have tested these predictions and find that, for Edw = 0, the response is in fact much stronger in both the intermediate and large size regimes. For L ≥ 96 we find df = 1.43(4) while, for Edw 6= 0, df = 1.22(3). Also, the probability of finding a disor- der realization with Edw = 0 is under 0.5 for L > 24, much less than 0.75. The cause of these discrepancies is most probably due to boundary conditions and system size. We do not have any evidence from our work that the particular values of Edw are significant for the droplet theory. We also note that a topological analysis of ground states35 predicts df = 1.30(1). Essentially, the technique measured the average length of domain walls. These lengths respond faster than just L since the domain walls bend to avoid the rigid lattice. Nevertheless, only one ground state configuration was studied for each disor- der realization, completely ignoring the entropy issue. A study in a similar vein36 looks at the properties of mini- mum energy domain walls and places the fractal dimen- sion in an interval 1.095(1) ≤ df ≤ 1.395(1). This may agree to some extent with the point that it is not pos- sible, or very difficult, to actually pin down the value of Acknowledgments We would like to thank A. K. Hartmann for an en- lightening correspondence. A. Aromsawa acknowledges support from the Thailand Research Fund in the form of a scholarship from the Royal Golden Jubilee Ph.D. Programme. 1 S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5, 965 (1975). 2 A. K. Hartmann and A. P. Young, Phys. Rev. B 64, 180404(R) (2001). 3 J. Houdayer, Eur. Phys. J. B 22, 479 (2001). 4 H. G. Katzgraber and L. W. Lee, Phys. Rev. B 71, 134404 (2005). 5 J. Poulter and J. A. Blackman, Phys. Rev. B 72, 104422 (2005). 6 H.-F. Cheung and W. L. McMillan, J. Phys. C 16, 7027 (1983). 7 J.-S. Wang and R. H. Swendsen, Phys. Rev. B 38, 4840 (1988). 8 J. A. Blackman and J. Poulter, Phys. Rev. B 44, 4374 (1991). 9 L. Saul and M. Kardar, Phys. Rev. E 48, R3221 (1993); Nucl. Phys. B 432, 641 (1994). 10 J. A. Blackman, J. R. Gonçalves, and J. Poulter, Phys. Rev. E 58, 1502 (1998). 11 J. Lukic, A. Galluccio, E. Marinari, O. C. Martin, and G. Rinaldi, Phys. Rev. Lett. 92, 117202 (2004). 12 T. Jörg, J. Lukic, E. Marinari, and O. C. Martin, Phys. Rev. Lett. 96, 237205 (2006). 13 A. K. Hartmann, arXiv:0704.2748 (2007). 14 A. J. Bray and M. A. Moore, in Heidelberg Colloquium on Glassy Dynamics, edited by J. L. van Hemmen and I. Morgenstern, Lecture Notes in Physics Vol. 275 (Springer- Verlag, Berlin, 1987), p.121. 15 R. N. Bhatt and A. P. Young, Phys. Rev. B 37, 5606 (1988). 16 Y. Ozeki, J. Phys. Soc. Jpn. 59, 3531 (1990). 17 J.-S. Wang, Phys. Rev. E 72, 036706 (2005). 18 H. G. Katzgraber, L. W. Lee, and I. A. Campbell, cond-mat/0510668 (2005). 19 H. G. Katzgraber, L. W. Lee, and I. A. Campbell, Phys. Rev. B 75, 014412 (2007). 20 W. L. McMillan, Phys. Rev. B 29, 4026 (1984); Phys. Rev. B 30, 476 (1984); Phys. Rev. B 31, 340 (1985). 21 A. J. Bray and M. A. Moore, J. Phys. C 17, L463 (1984); Phys. Rev. B 31, 631 (1985); Phys. Rev. Lett. 58, 57 (1987). 22 D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601 (1986); Phys. Rev. B 38, 386 (1988). 23 H. G. Katzgraber, M. Körner, and A. P. Young, Phys. Rev. B 73, 224432 (2006). 24 A. K. Hartmann, Phys. Rev. E 59, 84 (1999). 25 M. Palassini and A. P. Young, J. Phys. Soc. Jpn. 69, 165 (2000). 26 A. K. Hartmann and A. P. Young, Phys. Rev. B 66, 094419 (2002). 27 C. Amoruso, E. Marinari, O. C. Martin, and A. Pagnani, Phys. Rev. Lett. 91, 087201 (2003). 28 I. A. Campbell, A. K. Hartmann, and H. G. Katzgraber, Phys. Rev. B 70, 054429 (2004). 29 C. Amoruso, A. K. Hartmann, M. B. Hastings, and M. A. Moore, Phys. Rev. Lett. 97, 267202 (2006). 30 D. Bernard, P. Le Doussal, and A. A. Middleton, cond-mat/0611433 (2006). http://arxiv.org/abs/0704.2748 http://arxiv.org/abs/cond-mat/0510668 http://arxiv.org/abs/cond-mat/0611433 31 J. Cardy, Ann. Phys. (N.Y.) 318, 81 (2005). 32 R. Fisch, arXiv:0705.0046 (2007). 33 R. Fisch, J. Stat. Phys. 125, 793 (2006). 34 J. Lukic, E. Marinari, O. C. Martin, and S. Sabatini, J. Stat. Mech. L10001 (2006). 35 F. Romá, S. Risau-Gusman, A. J. Ramirez-Pastor, F. Ni- eto, and E. E. Vogel, Phys. Rev. B 75, 020402(R) (2007). 36 O. Melchert and A. K. Hartmann, arXiv:0704.2004 (2007). 37 H. S. Green and C. A. Hurst, Order-Disorder Phenomena (Interscience, London, 1964). 38 J. A. Blackman, Phys. Rev. B 26, 4987 (1982). 39 R. Fisch and A. K. Hartmann, Phys. Rev. B 75, 174415 (2007). 40 W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in Fortran (Cambridge Uni- versity Press, Cambridge, 1992). http://arxiv.org/abs/0705.0046 http://arxiv.org/abs/0704.2004

0704.1187

Magnetic structure of CeRhIn$_{5}$ under magnetic field

Magnetic structure of CeRhIn5 under magnetic field S. Raymond, E. Ressouche, G. Knebel, D. Aoki and J. Flouquet CEA-Grenoble, DRFMC, SPSMS, 38054 Grenoble Cedex 9, France Abstract. The magnetically ordered ground state of CeRhIn5 at ambient pressure and zero magnetic field is an incomensurate helicoidal phase with the propagation vector k=(1/2, 1/2, 0.298) and the magnetic moment in the basal plane of the tetragonal structure. We determined by neutron diffraction the two different magnetically ordered phases of CeRhIn5 evidenced by bulk measurements under applied magnetic field in its basal plane. The low temperature high magnetic phase corresponds to a sine-wave structure of the magnetization being commensurate with k=(1/2, 1/2, 1/4). At high temperature, the phase is incommensurate with k=(1/2, 1/2, 0.298) and a possible small ellipticity. The propagation vector of this phase is the same as the one of the zero-field structure. http://arxiv.org/abs/0704.1187v1 Magnetic structure of CeRhIn5 under magnetic field 2 The interplay between magnetism and superconductivity is one of the most important point of interest in the study of heavy fermion systems [1]. The widely open question concerns the cooperative versus the antagonist nature of the two ground states. The generic phase diagram obtained in heavy fermion systems, high-Tc superconductors and organics is composed of a superconducting region appearing in the vicinity of a vanishing magnetic phase. Because magnetic fluctuations are expected to be strong in this region of the phase diagram, they are invoked to be responsible for the Cooper pair formation [2]. However no definitive symbiosis between experiment and theory firmly establishes this point in the same manner than the classical scenario of phonon mediated conventional superconductivity does. In contrast, the SO(5) theory describes the two phenomena, magnetism and superconductivity, in a unified picture via a superspin order parameter that encompases both states [3]. This leads to a rich variety of possible phase diagrams with yet limited experimental examples to test in detail this theory [4]. To this respect, CeRhIn5 (and the related CeTIn5 compounds with T=Ir, Co [5]) provides a unique experimental case where the Néel (TN ) and the superconducting transition (Tc) temperatures are of the same order (of about 1 K) and can be tuned by pressure (p) or magnetic field (H). The resulting intricated magnetic and superconducting (p, T , H) phase diagram of CeRhIn5 points toward competition between antiferromagnetism and superconductivity [6, 7, 8] : At zero magnetic field, there is a pressure range (1.6-1.9 GPa) where magnetism and superconductivity coexist with TN > Tc. From 2 GPa, where TN ≈ Tc, a pure superconducting phase emerges and antiferromagnetism is suppressed. The application of a magnetic field in this phase induces a spectacular reentrance of the long range magnetic order. Beyond NMR experiments [9], microscopic probes are lacking to address the nature of the magnetically ordered phases in the (p, T , H) phase diagram. In this viewpoint, and as a starting point for further investigations to be performed under pressure, we have investigated the magnetically ordered phases of CeRhIn5 under magnetic field applied in its basal plane at zero pressure. CeRhIn5 crystallises in the tetragonal space group P4/mmm [10]. The sample was obtained by the In self flux method. A rectangular-shaped platelet of width 1 mm normal to the c-axis was cut from this batch, the other dimensions being 4.3 mm along [1, -1, 0] and 2.7 mm along [1, 1, 0]. This geometry is aiming to minimize the strong absorption cross section from In and Rh for the study of the ([1, 1, 0], [0, 0, 1]) scattering plane. The measurements were performed on the two-axis D23-CEA-CRG (Collaborating Research Group) thermal-neutron diffractometer equipped with a lifting detector at the Institut Laue Langevin (ILL), Grenoble. A copper monochromator provides an unpolarized beam with a wavelength of λ=1.276 Å. The sample was mounted in a vertical field 4He flow cryomagnet with the [1, -1, 0] axis along the magnetic field. The (T , H) phase diagram obtained by calorimetry measurements for the field applied perpendicular to the tetragonal axis is shown in Fig.1. It is composed of three magnetically ordered phases (two being induced by the magnetic field) consistently with the data obtained by other goup using calorimetry [11], thermal expansion and Magnetic structure of CeRhIn5 under magnetic field 3 Figure 1. (T , H) phase diagram of CeRhIn5 determined by specific heat for field applied in the basal plane at ambient pressure. It shows three different ordered phases. Open (respect. full) symbols correspond to first (respect. second) order transition. magnetostriction [12]. In the diffraction experiment, we apply the field along [1, -1, 0] and refer to this phase diagram by neglecting the in-plane anisotropy. The magnetic structure at zero field is known to be incommensurate with slightly different propagation vectors reported in the literature, k=(1/2, 1/2, 0.297) [13] or k=(1/2, 1/2, 0.298) [14]. The helicoidal nature of the order, as opposed to a sine-wave modulated structure, is known from the distribution of hyperfine field observed in NQR measurements [15]. In the present experiment, the lattice parameters were obtained from the centering of 18 independent reflections of the crystal and a refinement of the nuclear structure was performed at 1.9 K with 181 Bragg peaks yielding the structural parameters shown in Table 1 and the scale factor for calculation the magnetic structure. These parameters are consistent with the one of the literature [10] as concern the lattice parameters and the fractional coordinate z. The principal mean square atomic deplacements u have typical values of such intermetallic compounds. All refinements were corrected from extinction and absorption with the linear absorption coefficient µ=0.49 mm−1. As far as magnetic scattering is concerned, the measured neutron Bragg intensity after correction for scale factor, extinction, absorption and Lorentz factor, is the square of the component of the magnetic structure factor perpendicular to Q : |FM⊥(Q)|2. In the present case with only one magnetic Ce atom/unit cell at the origin, the magnetic structure factor is : FM(Q) = pf(Q).mk.e −WCe (1) where p ≈ 0.27×10−12 cm is the scattering amplitude at Q=0 for a single magnetic moment of 1 µB, f(Q) is the Ce magnetic form factor, WCe is the Debye-Waller factor of Ce. mk is the Fourier component of the magnetic moment distribution. The magnetic Magnetic structure of CeRhIn5 under magnetic field 4 Table 1. Structural parameters at T = 1.9 K. a = 4.638 Å c = 7.521 Å z 0.30526 (14) uCe 0.0014 (5) Å uRh 0.0006 (4) Å uIn1 0.0018 (5) Å uIn2 0.0015 (4) Å R = 0.0532 structures of interest for the present paper are (i) the collinear sine-wave structure, for which : iΦk (2) and (ii) the non-collinear elliptical structure : (muuk + im iΦk (3) where Ak is the amplitude of the sine-wave, uk and vk are unit vectors, Φk is a phase factor and mu, mvare the component of the magnetic moment along the unit vectors uk and vk. The helicoidal order corresponds to the particular case m u=mv. The obtained propagation vector for the zero field magnetic structure is found to be k=(1/2, 1/2, 0.298) in agreement with the literature. The structure was determined by measuring 16 magnetic peaks and by performing a least square fitting of the helicoidal model. The comparison between the observed intensities and the calculated ones is shown in Table 2 with the given weighted least square factor R. A magnetic moment mI=0.59 (1) µB is found at 1.9 K, a value a little lower than the one found in the literature 0.75 (2) µB at 1.4 K [13]. Given the rather flat temperature evolution of the order parameter between 1.4 and 1.9 K [13], the difference in the magnetic moment determination is not due to the difference in the measurement temperature. We believe that this difference is related to the data treatment, the present work including absorption and exctinction corrections. Figure 2 show Q-scans performed along the c-axis for H = 3 and 5 T (Phase III) with the same scan performed at H = 0 T as a reference (Phase I). The propagation vector is now commensurate being (1/2, 1/2, 1/4). For this phase, 7 magnetic reflections were collected at H = 3 T and T = 1.9 K. The best refinement is obtained for a colinear sine-wave structure (See Table 3) with the moment perpendicular to the field i.e. along [1, 1, 0]. Refinement with an helical structure does not work. For completeness, an elliptic structure was refined and yields, within the error bars, zero component of the magnetic moment along the field and thus confirms the sine-wave refinement. The propagation vector Q=(1/2, 1/2, 1/4) corresponds to a particular case of the sine-wave. For a phase Φk=-π/4 in eq.(2), all the magnetic moments have the same length and the magnetic structure corresponds to the so-called ++ - - structure consisting in up, up, Magnetic structure of CeRhIn5 under magnetic field 5 Table 2. Magnetic refinement with an helicoidal structure at zero field in phase I at T = 1.9 K. The Q vector is the Brillouin zone center +/- the propagation vector k=(1/2, 1/2, 0.298). Q |FM⊥(Q)|2calc |FM⊥(Q)|2obs (1, 1, 0) - 1.10 1.17 (0, 0, 0) + 1.10 1.03 (0, 0 ,1) + 1.52 1.50 (-1, -1, 1) + 1.52 1.39 (0, -1, 1) + 1.52 1.43 (1, 1, 1) - 1.30 1.36 (1, 1, 1) + 0.76 0.46 (0, 0, 2) + 1.54 1.50 (1, 1, 2) - 1.57 1.63 (0, 0, 2) - 1.57 1.53 (1, 0, 2) - 1.57 1.47 (2, 2 ,2) - 0.78 0.57 (1, 1, 3) - 1.47 1.62 (0, 0, 4) + 1.08 1.34 (1, 1, 4) - 1.24 1.14 (1, 1, 5) - 0.97 1.94 R = 0.0696 Table 3. Magnetic refinement with a sine-wave structure in phase III for H = 3 T and T = 1.9 K. Q |FM⊥(Q)|2calc |FM⊥(Q)|2obs (0, 0, 1) + 1.08 1.02 (1, 1, 1) - 0.62 0.65 (0, 0, 2) + 1.40 1.42 (1, 1, 2) - 1.32 1.31 (1, 1, 3) - 1.38 1.30 (0, 0, 4) + 1.08 1.00 (1, 1, 4) - 1.19 1.82 R = 0.0934 down, down sequence of magnetic moment when moving along the c-axis. This structure is favorized at low temperature because it reduces the magnetic entropy. The obtained magnetic amplitude of the sine-wave at 1.9 K is AIII=0.84 (2) µB. For the peculiar ++ - - structure, the magnetic moment mIII is related to the sine wave amplitude by mIII=AIII/ 2. We thus obtain mIII=0.59 µB, the same value than mI . Note that the maximum in plane magnetic moment sustended by the doublet ground state is 0.92 µB as deduced from crystal field spectroscopy [16]. The difference between the paramagnetic moment of the doublet ground state and the saturated ordered moment is often ascribed to the Kondo effect in cerium compounds. Phase II was investigated by performing Q-scans at 3.7 K and 4 T. An example of such a scan along the c-axis is shown on Fig.3a) for Q=(0.5, 0.5, L) with the same Magnetic structure of CeRhIn5 under magnetic field 6 Figure 2. Q-scans performed along the c-axis for H= 0, 3 and 5 T at 1.9 K. Table 4. Magnetic refinement with a sine-wave structure in phase II for H = 4 T and T = 3.6 K. Q |FM⊥(Q)|2calc |FM⊥(Q)|2obs (1, 1, 0)- 0.04 0.07 (0, 0, 1)+ 0.31 0.21 (1, 1, 1)- 0.16 0.19 (1, 1, 2)- 0.36 0.37 R = 0.1902 scan performed at 3.1 K in phase III as a reference. The propagation vector is found to be the same than the helicoidal phase, i.e. k=(1/2, 1/2, 0.298). Figure 3b) shows the temperature variation of the magnetic Bragg peak Q=(0.5, 0.5, 1.298) at 4 T. The difficulty to study this phase is that it exits in a reduced temperature range in the vicinity of the Néel temperature, where magnetic moment is barely developped. As a consequence the magnetic signal is weak. Figure 4 shows the field dependence of the magnetic Bragg peak intensity measured at Q=(1/2, 1/2, 1.298) at 3.6 K. Since the intensity is constant in both phases, this suggests that the propagation vector does not change as a function of field. Because of the weak signal, only 4 magnetic reflections were collected in phase II at 3.6 K and 4 T and the result of a refinement with a sine-wave structure is given in Table 4. For H=4 T and T=3.6 K, the magnetic amplitude is found to be AII=0.44 (2) µB. Refinement with an elliptical phase is slightly better (R=0.1449 instead of R=0.1902) and gives a non zero component along the field m[1,−1,0]=0.12 (5) µB, the component perpendicular to the field being then m[1,1,0]=0.4 µB. We cannot definitivelly conclude on the elliptical nature of this phase given the weak number of collected reflections. In the previous paragraphs, we neglect the possible ferromagnetic component Magnetic structure of CeRhIn5 under magnetic field 7 Figure 3. a) Q-scans performed along the c-axis for H= 4 T at 3.1 and 3.7 K. b) Temperature dependence of the Bragg peak intensity at Q=(0.5, 0.5, 1.295) for H = 4 T. Solid lines are guides for the eyes. Dashed line represents the background. Figure 4. Magnetic field dependence of the Bragg peak intensity at Q=(0.5, 0.5, 1.298) at 3.6 K. The solid line is a guide for the eyes. The dashed line corresponds to the background. along the applied field. The corresponding signal was not observed in the present experiment due to its location on the top of the nuclear peaks. The resulting structure obtained by combining the sine-wave and the ferromagnetic component is a so-called fan structure. The fact that mIII and mI are equal within the error bars indicate that this ferromagnetic component is anyway very weak at least at low temperature. Magnetization measurements performed at 1.3 K in the basal plane give an induced ferromagnetic moment of about 0.08 µB at 5 T [17]. The helicoidal nature of the ordering at zero field is certainly due to the RKKY interactions that allow the conditions Magnetic structure of CeRhIn5 under magnetic field 8 for stabilizing such a state due to their oscillating nature. We invoke RKKY interactions rather than Fermi surface nesting because dHvA experiments suggest the localized nature of the magnetism of CeRhIn5 at ambient pressure [18]. The effect of a magnetic field applied in the plane of an helix is known from a long time and was worked out shortly after the discovery of the helix structure [19]. The resulting sinusoidal oscillating structure or elliptical arrangement depends of the anisotropy and the magnetic field and the details of the complete (T , H) phase diagram depend on the precise Hamiltonian. On general ground and at the mean field level, the possible transition from helix to commensurate structure under field was also predicted in the earlier works for peculiar values of the propagation vectors [20]. A field induced transition to the antiferromagnetic state is expected for k ≈ 1/2 and to the ++ - - structure for k ≈ 1/4, the situation encoutered in the present work. Despite the proximity of the zero field propagation vector to the one of the ++ - - structure, another commensurate structure is reported at zero field for CeRhIn5 based systems with this time k=(1/2, 1/2, 1/2). This antiferromagnetic order occurs in CeRh1−xIrxIn5 (x) [21] and in CeRh0.6Co0.4In5 [22]. Interestingly it is reported to coexist with the incommensurate order and also with the superconducting ground state. On cooling the incommensurate order appears first followed by the commensurate order and the superconducting state. On another hand, it is worthwhile to note that the commensurate order with k=(1/2, 1/2, 1/2) alone is reported for the related CeCoIn5 compound doped with 10 % Cd both in the antiferromagnetic and antiferromagnetic plus superconducting phases [23]. Contrastingly, the occurence of commensurate order is not reported in the diffraction studies performed on CeRhIn5 under pressure. However different groups obtain different results. Either the incommensurate order is reported to change weakly with pressure up to 1.63 GPa [24] or at opposite, the propagation vector changes to k=(1/2, 1/2, 0.396) at 0.1 GPa [14]. This confusing situation asks for new experiments under pressure. The occurence of different commensurate and incommensurate phases in the (T , H , p, x) phase diagram of CeRhIn5 deserves further investigation especially for the interplay between magnetic order and superconductivity. We have determined the two different magnetic ordering states in CeRhIn5 at ambient pressure under magnetic field applied in its basal plane. The low temperature phase is characterized by the commensurate propagation vector k=(1/2, 1/2, 1/4) and a colinear structure with the magnetic moment perpendicular to the field. The saturated magnetic moment of 0.6 µB is the same as the one found in the zero field phase. The high temperature phase is incommensurate with the same propagation vector as the zero field incommensurate helix, k=(1/2, 1/2, 0.298). The structure is colinear at first approximation with an eventual ellipticity of about 1/3. Magnetic structure of CeRhIn5 under magnetic field 9 Acknowledgements We acknowledge M. Zhitomirsky for illuminating discussion concerning helicoidal structures under applied magnetic field. References [1] See e.g. J. Flouquet, Prog. Low. Temp. Phys. 2005 15 Chapter 2. [2] See e.g. N.D. Mathur et al., Nature 1998 394 39. [3] E. Demler, W. Hanke and S.C. Zhang, Rev. Mod. Phys. 2004 76 909. [4] Y. Kitaoka et al., J. Phys. Condens. Matter 2001 13 L79. [5] See e.g. J.D. Thompson et al., J. Magn. Magn. Mat. 2001 226− 230 5. [6] T. Park et al., Nature 2006 440 65. [7] G. Knebel et al., Phys. Rev. B 2006 74 020501(R). [8] G.F. Chen et al., Phys. Rev. Lett. 2006 97 017005. [9] S. Kawasaki et al., Phys. Rev. Lett. 2003 91 137001. [10] E.G. Moschopoulu et al., Applied Physics A 2002 74 Suppl. 5895. [11] A.L. Cornelius et al., Phys. Rev. B 2001 64 144111. [12] V.F. Correa et al., cond-mat/0411359.s [13] W. Bao et al., Phys. Rev. B 2000 62 R14621 and Phys. Rev. B 2003 67 099903 (E). [14] S. Majumdar et al., Phys. Rev. B 2002 66 212502. [15] N.J. Curro et al., Phys. Rev. B 2000 62 R6100. [16] A.D. Christianson et al., Phys. Rev. B 2002 66 193102. [17] T. Takeuchi et al., J. Phys. Soc. Japan 2001 70 877. [18] Y. Onuki et al., Acta Phys. Pol. B 2003 34 667. [19] T. Nagamiya, Solid State Phys. 1967 20 305, New York, Academic Press and references therein. [20] See e.g. Nagamiya et al., Prog. Theor. Phys. 1962 27 1253. [21] A.D. Christianson et al., Phys. Rev. Lett. 2005 95 2117002. [22] M. Yokoyama et al., J. Phys. Soc. Japan 2006 75 103703. [23] M. Nicklas et al., cond-mat/0703703. [24] A. Llobet al al., Phys. Rev. B 2004 69 024403. http://arxiv.org/abs/cond-mat/0411359 http://arxiv.org/abs/cond-mat/0703703

0704.1188

Fermionic Collective Modes in QGP near Critical Temperatures

Fermionic Collective Modes in QGP near Critical Temperatures Yukio Nemoto1,∗) , Masakiyo Kitazawa2,∗∗) , Tomoi Koide3,∗∗∗) and Teiji Kunihiro 1 Department of Physics, Nagoya University, Nagoya 464-8602, Japan 2 RIKEN-BNL Reseach Center, Brookhaven National Laboratory, Upton, NY 11973, USA 3 Instituto de F́ısica, Universidade Federal do Rio de Janeiro, C. P. 68528, 21945-970, Rio de Janeiro, Brasil 4 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502, Japan We investigate the quark spectrum in the quark-gluon plasma phase near color super- conducting (CS) and chiral phase transitions. Owing to the precursory soft modes of the phase transitions, there appear novel excitaion spectra: In the CS transition, the quark mat- ter shows non-Fermi liquid behavior and leads to the pseudogap in the density of states of quarks. In the chiral transition, three collective excitations appear in the quark spectrum. §1. Introduction Recently, the quark-gluon plasma (QGP) just above the chiral and deconfine- ment phase transition is believed to be an unexpectedly strongly interacting system, which is based on the facts that the created matter at RHIC behaves like a perfect fluid and that some hadronic bound states of heavy quarks can survive above the critical temperature (Tc) from Lattice QCD. Because the fundamental degrees of freedom in QGP are quarks and gluons, it is also important to study their properties in such a strongly interacting system. Here, we investigate the quark spectrum just above Tc of the chiral transition at zero density, and the color superconducting (CS) transition around µB = 1 GeV with µB being the baryon number density, focusing on the precursory soft modes of these transitions. It is known that these soft modes exist over a wide range of temperature above Tc owing to a strong coupling nature between quarks.1), 2) In this paper, we show that they affect the quark spectrum significantly in a region just above Tc. §2. Precursory soft modes To describe the quark matter near Tc, we employ the two-flavor Nambu–Jona- Lasinio type interaction with the scalar diquark correlation included, L = ψ̄i/∂ψ + GS [(ψ̄ψ) 2 +(ψ̄iγ5~τψ) 2] +GC(ψ̄iγ5τ2λAψ C)(ψ̄C iγ5τ2λAψ), with ψ C ≡ Cψ̄T and C = ∗) e-mail address: [email protected] ∗∗) e-mail address: [email protected] ∗∗∗) e-mail address: [email protected] †) e-mail address: [email protected] http://arxiv.org/abs/0704.1188v1 2 Y. Nemoto, M. Kitazawa, T. Koide and T. Kunihiro -30-20-10 0 10 20 30 40 50 ω[MeV] k[MeV] S(k,ω) [GeV2] Fig. 1. The dynamic structure factor SC for the diquark mode for µ = 400 MeV and ε ≡ (T − Tc)/Tc = 0.02 (left), and the spectral function ρσ ≡ ρS for the quark-antiquark mode for µ = 0 and ε = 0.1 (right). k and p denote the momentum. iγ2γ0 being the charge conjugation operator. The matrices τ2 and λA (A = 2, 5, 7) are the antisymmetric components of the Pauli and Gell-Mann matrices for the flavor SU(2)f and color SU(3)c, respectively. The coupling constants, GS and GC , and the three-momentum cutoff, Λ, are taken from Refs. 1) and 3). The fluctuations of the diquark (chiral) condensate are described by the diquark (quark-antiquark) Green function in the random phase approximation, DC,S(p, νn) = −[1/2GC,S +QC,S(p, νn)] −1, where the subscript C(S) denotes the diquark (quark- antiquark) sector. νn = 2πnT is the Matsubara frequency for bosons and QC,S(p, νn) is the undressed quark-antiquark (diquark) polarization function at one-loop. To evaluate strengths of the fluctuations, we employ the spectral function, ρ, and the dynamic structure factor, S, given by ρC,S(p, ω) = −(1/π)ImDC,S(p, νn)|iνn=iω+iη and SC,S(p, ω) = ρC,S(p, ω)/(1−e −ω/T ), respectively. SC(p, ω) for the diquark mode and ρS(p, ω) for the quark-antiquark mode near TC are plotted in Fig. 1. One can see that there appear pronounced peaks which denote the precursory soft modes.1), 2) The peak positions of these modes are approximately expressed as ωC ≃ p 2 for the diquark mode, and ωS ≃ ± m∗σ(T ) 2 + p2 for the quark-antiquark mode. A T - dependent ‘mass’ m∗σ(T ) becomes smaller as T approaches Tc, which means the softening at Tc. As will be seen in Sec.3, the difference between ωC and ωS leads to a quite different quark spectrum. §3. Quark spectrum near the phase transitions The effect of the soft modes on the quark spectrum is incorporated in the quark self-energy in the non-selfconsistent way, Σ̃C,S(p, ωn) = −4T (2π)3 DC,S(p − q, ωn − ωm)G0(q, ωm), where G0(q, ωm) is the free quark propagator with ωn = (2n+1)πT being the Matsubara frequency for fermions. The quasi-quark and quasi- antiquark spectral functions, ρ±(p, ω), are obtained from the retarded self-energies, ΣR±(p, ω) = (1/2)Tr[Σ Rγ0Λ±], respectively, i.e. ρ± = −(1/π)Im[ω+µ∓|p|−Σ with the analytic continuation ΣR(p, ω) = Σ̃(p, ωn)|iωn=ω+iη and the projection op- Fermionic Collective Modes in QGP 3 Fig. 2. The quasi-quark spectral function ρ+ above the CS transition for µ = 400 MeV and ε = 0.01 (left), and ρ+ above the chiral transition for µ = 0 and ε = 0.1 (right). k and p denote the momentum. -400 -300 -200 -100 0 100 ω [MeV] ε=0.20 ε=0.05 ε=0.02 ε=0.01 Fig. 3. The density of states of quarks above the CS transition for µ = 400 MeV. The thin dotted curve represents that of free quarks. erators Λ± = (1 ± γ γ · p/|p|)/2. In the following, we show the quark spectral function obtained from the self-energy Σ̃C(Σ̃S) for the CS (chiral) transition. 3.1. Color superconducting transition The quasi-quark spectral function ρ+ for µ = 400 MeV and ε = 0.01 is plotted in the left panel of Fig. 2. We see that the peak has a clear depression around the Fermi energy, ω = 0.4) Owing to the softening of the soft mode near Tc, a quark near the Fermi energy is scattered by the soft mode and create a hole, while a hole can create a quark by absorbing the soft mode. Then, the incident quark and a quark near the Fermi surface make a resonant scattering to form the soft mode and vice versa. This resonant processes induce a virtual mixing between quarks and holes, which leads to the level repulsion of the energy spectrum near the Fermi energy, making the gap-like structure as shown in the left panel of Fig. 2. This behavior, which is quite different from the conventional Fermi liquid, is due to the strong coupling nature between quarks. In fact, we see that the depression is more significant as the diquark coupling GC becomes larger. This depression leads to a depression in the density of states (DOS) of quarks around the Fermi energy, as shown in Fig. 3. Thus, we see a gap-like structure in DOS even above Tc, which we call the pseudogap. 4) The non-Fermi liquid behav- ior is essential for the formation of the pseudogap, which is analogous to high-Tc superconductors, although the origin of the pseudogap of the latter is not known 4 Y. Nemoto, M. Kitazawa, T. Koide and T. Kunihiro precisely. 3.2. Chiral transition The quasi-quark spectral function ρ+ for µ = 0 MeV and ε = 0.1 is plotted in the right panel of Fig. 2. We see a clear three-peak structure at low momentum, which exists even at ε = 0.2.6) Although not shown in the figure, the quasi-antiquark spectrum, ρ−, has also a three-peak structure for a relation, ρ−(p, ω) = ρ+(p,−ω). The mechanism of the appearance of the three-peak structure in ρ+ is as follows: The imaginary part of ΣR+(0, ω) has two peaks at nonzero values of ω, which means that there exist two large damping modes of the quasi-quark there. From a kinematical consideration, we see that one is a collision of a thermally excited antiquark and the quasi-quark creating the soft mode, and the other is a collision of the quasi-quark and the soft mode creating an on-shell quark. Both the processes are interpreted as a Landau damping of the quasi-quark. The point is that the quasi-quark is a mixed state between quarks and ‘antiquark-holes’ which are annihilation of thermally ex- cited antiquarks and have the positive quark number. Then, these damping modes cause a mixing between quarks and antiquark-holes. This mixing mechanism can be described in terms of the resonant scattering as in the case of the color superconduc- tivity, although a crucial difference arises owing to the different nature of the soft modes. We can show that a coupling with the soft mode with a nonzero mass m∗σ(T ) is essential for the appearance of the three-peak structure in the quark spectrum. In fact, we have investigated the quark spectrum in Yukawa models with a massive scalar (pseudo-scalar) and vector (axial-vector) boson of a nonzero mass m, and find that there appears a three-peak structure in the quark spectral function with a collective nature when temperature is compatible with m.7) Because the employed Yukawa models are rather generic, the findings may represent a universal phenomenon for fermions coupled with a massive bosonic excitation with a vanishing or small width. §4. Conclusions We have investigated the quark spectrum near the CS and chiral transitions taking into account the fluctuating soft modes. We have shown that the quark spectrum shows a non-Fermi liquid behavior leading to the pseudogap in the density of states above the CS transition, and a clear three-peak structure above the chiral transition. The gap-like structure in the spectral function near both the transitions can be uderstood in terms of the resonant scattering of each soft mode. References 1) T. Hatsuda and T. Kunihiro, Phys. Rev. Lett. 55 (1985), 158. 2) M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Phys. Rev. D 65 (2002), 091504. 3) T. M. Schwarz, S. P. Klevansky and G. Papp, Phys. Rev. C 60 (1999), 055205. 4) M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Phys. Rev. D 70 (2004), 056003. M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 114 (2005), 205. 5) M. Kitazawa, T. Kunihiro and Y. Nemoto, Phys. Lett. B 631 (2005), 157. 6) M. Kitazawa, T. Kunihiro and Y. Nemoto, Phys. Lett. B 633 (2006), 269. 7) M. Kitazawa, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 117 (2007), 103. Introduction Precursory soft modes Quark spectrum near the phase transitions Color superconducting transition Chiral transition Conclusions

0704.1189

The Kinematics and Dynamics of the Globular Clusters and the Planetary Nebulae of NGC 5128

The Kinematics and Dynamics of the Globular Clusters and Planetary Nebulae of NGC 5128 Kristin A. Woodley Department of Physics & Astronomy, McMaster University, Hamilton ON L8S 4M1, Canada [email protected] William E. Harris Department of Physics & Astronomy, McMaster University, Hamilton ON L8S 4M1, Canada [email protected] Michael A. Beasley Instituto de Astrofisica de Canarias, Calle Vı́a Láctea, s/n E-38200, La Laguna Tenerife, Spain [email protected] Eric W. Peng Herzberg Institute of Astrophysics, 5071 West Saanich Road, Victoria, BC V8V 2X6, Canada [email protected] Terry J. Bridges Department of Physics, Queen’s University, Kingston, ON K7L 3N6, Canada [email protected] Duncan A. Forbes Centre for Astrophysics & Supercomputing, Swinburne University, Hawthorn, VIC 3122, Australia [email protected] Gretchen L. H. Harris http://arxiv.org/abs/0704.1189v2 – 2 – Department of Physics & Astronomy, University of Waterloo, Waterloo ON N2L 3G1, Canada [email protected] ABSTRACT A new kinematic and dynamic study of the halo of the giant elliptical galaxy NGC 5128 is presented. From a spectroscopically confirmed sample of 340 glob- ular clusters and 780 planetary nebulae, the rotation amplitude, rotation axis, velocity dispersion, and total dynamical mass are determined for the halo of NGC 5128. The globular cluster kinematics were searched for both radial de- pendence and metallicity dependence by subdividing the globular cluster sample into 158 metal-rich ([Fe/H]> −1.0) and 178 metal-poor ([Fe/H]< −1.0) globular clusters. Our results show that the kinematics of the metal-rich and metal-poor subpopulations are quite similar: over a projected radius of 0−50 kpc, the mean rotation amplitudes are 47±15 and 31±14 km s−1 for the metal-rich and metal- poor populations, respectively. There is a indication within 0 − 5 kpc that the metal-poor clusters have a lower rotation signal than in the outer regions of the galaxy. The rotation axis shows an interesting twist at 5 kpc, agreeing with the zero-velocity curve presented by Peng and coworkers. Within 5 kpc, both metal-rich and metal-poor populations have a rotation axis nearly parallel to the north-south direction, which is 0o, while beyond 5 kpc the rotation axis twists ∼ 180o. The velocity dispersion displays a steady increase with galactocentric radius for both metallicity populations, with means of 111 ± 6 and 117 ± 6 km s−1 within a projected radius of 15 kpc for the metal-rich and metal-poor pop- ulations; however, the outermost regions suffer from low number statistics and spatial biases. The planetary nebula kinematics are slightly different. Out to a projected radius of 90 kpc from the center of NGC 5128, the planetary nebulae have a higher rotation amplitude of 76±6 km s−1, and a rotation axis of 170±5o east of north, with no significant radial deviation in either determined quantity. The velocity dispersion decreases with galactocentric distance. The total mass of NGC 5128 is found using the tracer mass estimator, described by Evans et al., to determine the mass supported by internal random motions and the spherical component of the Jeans equation to determine the mass supported by rotation. We find a total mass of 1.0 ± 0.2 × 1012 M⊙ from the planetary nebula data ex- tending to a projected radius of 90 kpc. The similar kinematics of the metal-rich and metal-poor globular clusters allow us to combine the two subpopulations – 3 – to determine an independent estimate of the total mass, giving 1.3 ± 0.5 × 1012 M⊙ out to a projected radius of 50 kpc. Lastly, we publish a new and homoge- neous catalog of known globular clusters in NGC 5128. This catalog combines all previous definitive cluster identifications from radial velocity studies and HST imaging studies, as well as 80 new globular clusters with radial velocities from a study of M.A. Beasley et al. (in preparation). Subject headings: galaxies: elliptical and lenticular, cD — galaxies: individual (NGC 5128) — galaxies: kinematics and dynamics — galaxies: star clusters — globular clusters: general — planetary nebulae: general — catalogs 1. Introduction Globular clusters (GCs), as single-age, single-metallicity objects, are excellent tracers of the formation history of their host galaxies, through their dynamics, kinematics, metallicities, and ages. For most galaxies within ≃ 20 Mpc, GCs can be identified through photometry and image morphology, from which follow-up radial velocity studies can be carried out with multi-object spectroscopy on 4 and 8 m class telescopes. The ability to target hundreds of objects in a single field has vastly increased the observed samples of GCs confirmed in many galaxies, providing the necessary basis for detailed kinematic and age studies. Another benefit of using GCs as a kinematic tracer is that they provide a useful, inde- pendent basis for comparison with results from planetary nebulae (PNe). This is particularly true given the current debate surrounding the use of PN velocities and the implications for low dark matter halos. For example, Romanowsky et al. (2003) reported, based on PN ve- locities, that three low-luminosity ellipticals revealed declining velocity dispersion profiles and little or no dark matter. However, subsequent simulations of merger-remnant ellipticals suggested that the radial anisotropy of intermediate-age PNe could give rise to the observed profiles within standard halos of dark matter (Dekel et al. 2005; Mamon & Lokas 2005). Flattening of the galaxy along the line of sight is another possible explanation. One of the ellipticals studied by Romanowsky et al. (2003), NGC 3379, has also been investigated using GC kinematics (Pierce et al. 2006; Bergond et al. 2006); both studies found evidence of a dark matter halo. But the two studies of NGC 3379 suffered from small number statistics. Clearly there is a need to directly compare PN and GC kinematics for the same elliptical with sufficiently large numbers of tracer objects. Previous velocity-based studies of globular cluster systems (GCSs) have shown an in- triguing variety of results in their overall kinematics. Prominent recent examples include the – 4 – following: 1. Côté et al. (2001) performed a kinematic analysis of the GCS in M87 (NGC 4486), the cD galaxy in the Virgo Cluster. With a sample of 280 GCs, they showed that the entire GCS rotates on an axis matching the photometric minor axis of the galaxy, except for the inner metal-poor sample. Inside the onset of the cD envelope, the metal-poor clusters appear to rotate around the major axis of the galaxy instead. This study also found evidence for an increase in velocity dispersion, σv, with radius, due to the larger scale Virgo Cluster mass distribution. They also showed no strong evidence for a difference in σv between the metal-poor and metal-rich groups. Using a Virgo mass model, they investigated anisotropy and found that as a whole, the GCS had isotropy, but considered separately, the metal-poor and metal-rich subpopulations had slight anisotropy. 2. Côté et al. (2003) performed a dynamical analysis for M49 (NGC 4472), the other supergiant member of the Virgo Cluster. Using over 260 GCs, they found that the metal- rich population shows no strong evidence for rotation, while the metal-poor population does rotate about the minor axis of the galaxy. In addition, they found that the metal-poor clusters had an overall higher dispersion than the metal-rich population. 3. Richtler et al. (2004) determined the kinematics for the GCS in NGC 1399, the brightest elliptical in the Fornax Cluster of galaxies. Armed with a sample of over 460 GCs, they found a marginal rotation signal for the entire GC sample and the outer metal-poor sample, while no rotation was seen for the metal-rich subpopulation. Their projected velocity dispersion showed no radial trend within their determined uncertainty, but the metal-poor clusters had a higher dispersion than the metal-rich clusters. The kinematics of the PNe in NGC 1399 have been most recently studied by Saglia et al. (2000) and Napolitano et al. (2002) using a small sample of 37 PNe from Arnoaboldi et al. (1994). These studies do not indicate any significant deviations in velocity dispersion with radius for the PNe, yet interestingly, Napolitano et al. (2002) found stong rotation for the PNe in the inner region of the galaxy. 4. Peng, Ford & Freeman (2004c) studied the GC kinematics of the giant elliptical NGC 5128 from a total of 215 GCs. Their results showed definite rotation signals in both metallicity groups beyond a 5 kpc distance from the center of the galaxy, as well as similar velocity dispersions in both the metal-poor and the metal-rich populations within a 20 kpc projected radius. A similar PN kinematic study in NGC 5128 (Peng, Ford & Freeman 2004b) showed that the PNe population is rotating around a twisted axis that turns just beyond the 5 kpc distance. These few detailed kinematic studies of GCSs in elliptical galaxies, show results that – 5 – appear to differ on a galaxy-by-galaxy basis, without any clear global trends. The age distributions of the GC populations in these same galaxies tend to show a consistent pattern in which the blue, or metal-poor, population is found to be universally old. The red, or metal- rich, population has also been shown to be old in the study of Strader et al. (2005). Their study found that both the metal-poor and metal-rich GCs in a sample of eight galaxies ranging from dwarf to massive ellipticals, all have ages as old as their Galactic GC counterparts. Conversely, in a small sample of studies, the red GC population has also been found to be 2-4 Gyr younger than the metal-poor GC population, and with a wider spread in determined ages (see the studies of Peng, Ford & Freeman 2004c; Puzia et al. 2005, among others), although this is not yet a well-established trend. The GCs in NGC 5128 appear to be old, with an intermediate-age population (Peng, Ford & Freeman 2004c; Beasley et al. 2006). NGC 5128 (Centaurus A), the central giant in the Centaurus group of galaxies at a distance of only ∼ 4 Mpc, is a prime candidate for both kinematic and age studies. Its GCS has a specific frequency of SN ≃ 2.2±0.6 (Harris et al. 2006), toward the low end of the giant elliptical range but about twice as high as in typical disk galaxies. Its optical features show faint isophotal shells located in the halo (e.g. Peng et al. 2002), the prominent dust lane in the inner 5 kpc, the presence of gas, and star formation, all of which suggest that NGC 5128 could be a merger product. Baade & Minkowski (1954) first suggested that NGC 5128 could be the result of a merger between two galaxies, a spiral and an elliptical. This idea was followed by the general formation mechanism of disk-disk mergers proposed by Toomre & Toomre (1972). Bekki, Harris, & Harris (2003) found that the metallicity distribution function of the halo field stars could be reproduced by a gas-free (”dry”) disk-disk merger scenario. Recent numerical simulations by Bekki & Peng (2006) also demonstrate that the PN kinematics observed in NGC 5128 (Peng, Ford & Freeman 2004b) can be reproduced relatively well from a merger of unequal-mass disk galaxies (with one galaxy half the mass of the other) colliding on a highly inclined orbital configuration. Alternatively, much of NGC 5128 could be a ”red and dead” galaxy, passively evolving since its initial formation as a large seed galaxy (Woodley 2006), while undergoing later minor mergers and satellite accretions. Evidence consistent with this scenario is in the halo population of stars in NGC 5128, which have a mean age of 8+3.0 −3.5 Gyr (Rejkuba et al. 2005). Its metal-poor GC ages have also been shown to have ages similar to Milky Way GCs, while the metal-rich population appears younger (Peng, Ford & Freeman 2004c). Our new spectroscopic study (Beasley et al. 2006) suggests that NGC 5128 has a trimodal distribution of cluster ages: ∼ 50% of metal-rich clusters have ages of 6 − 8 Gyr, only a small handful of metal-rich clusters have ages of 1 − 3 Gyr, and a large fraction of both metal-rich and metal-poor clusters have ages of ≃ 12 Gyr. Lastly, the halo kinematics of NGC 5128 have also been recently shown to match the surrounding satellite galaxies in the low-density – 6 – Centaurus group, suggesting that NGC 5128 acts like an inner component to its galaxy group (Woodley 2006). Kinematic and age studies with large number of GCs are thus starting to help disentangle the formation of this giant elliptical. The confirmed GC population in NGC 5128 is now large enough to allow a new kine- matic analysis subdivided to explore radial and metallicity dependence, while avoiding small number statistics in almost all regions of the galaxy. The analysis presented here comple- ments the detailed age distribution study provided by Beasley et al. (2006). The results provide a broader picture of the formation scenario of NGC 5128. The sections of this paper are divided as follows: § 2 contains the full catalog of NGC 5128 GCs with known photometry and radial velocities, § 3 contains the kinematic analysis of GCSs, § 4 contains the kinematic analysis of the PN population, § 5 contains the discussion of the dynamical mass of NGC 5128, and § 6 contains our final discussion, as well as concluding remarks. 2. The Catalog of Globular Clusters in NGC 5128 Finding GCs in NGC 5128 is challenging. This process begins with photometric surveys of the many thousands of objects projected onto the NGC 5128 field (e.g. Rejkuba 2001; Harris et al. 2004); only a few percent of these are the GCs that we seek. This daunting task is made difficult by the galactic latitude of NGC 5128 (b = 19o), which means that many foreground stars are present in the field of NGC 5128. Background galaxies are another major contaminant in the field, forcing the use of search criteria such as magnitude, color, and object morphology to help build the necessary candidate list of GCs. Confirmation of such candidate objects can then be done with spectroscopic radial velocity measurements. The GCs in NGC 5128 have radial velocities in the range vr = 200− 1000 km s −1, while most foreground stars have vr < 200 km s −1. Background galaxies have radial velocities of many thousands of km s−1 and can easily be eliminated (see the recent studies of Peng, Ford, & Freeman 2004a; Woodley, Harris, & Harris 2005). Over the past quarter century, there have been seven distinct radial velocity stud- ies to identify GCs in NGC 5128 (van den Bergh, Hesser, & Harris 1981; Hesser et al. 1984; Hesser, Harris, & Harris 1986; Harris et al. 1992; Peng, Ford, & Freeman 2004a; Woodley, Harris, & Harris 2005; Beasley et al. 2006). Although Harris et al. (1992) was not a radial velocity study it- self, but rather a CCD photometric study of previously confirmed GCs, it does include the GCs determined spectroscopically from Sharples (1988). A recent study with measured GC radial velocities published by Rejkuba et al. (2007) has confirmed two new GCs, HCH15 and – 7 – R122, included in our catalog. Within this catalog are 80 new GCs with radial velocity measurements from Beasley et al. (2006). The combination of these studies now leads to a confirmed population of 342 GCs.1 Also included in our catalog are the new GC candidates from HST STIS imaging from Harris et al. (2002), labelled C100-C106, and from HST ACS imaging from Harris et al. (2006), labelled C111-C179. All these previous studies have their own internal numbering systems, which makes the cluster identifications somewhat confusing at this point. Here we define a new, homogeneous listing combining all this material and with a single numbering system. Our catalog of the GCs of NGC 5128 is given in Table 1. In successive columns, the Table gives the new cluster name in order of increasing right ascension; the previous names of the cluster in the literature; right ascension and declination (J2000); the projected radius from the center of NGC 5128 in arcminutes; the U , B, V , R, and I photometric indices and their measured uncertainties; the C, M , and T1 photometric indices and their uncertainties; the colors U−B, B−V , V −R, V −I, M−T1, C−M , and C−T1; and, lastly, the weighted mean velocity vr and its associated uncertainty from all previous studies. All UBV RI photometry is from the imaging survey described in Peng, Ford, & Freeman (2004a). The CMT1 data are from Harris et al. (2004). The mean velocities are weighted averages with weights on each individual measurement equal to ε−2v where εv is the quoted velocity uncertainty from each study. The uncertainty in the mean velocity is then < εv >= ( ε−2i ) −1/2. There are no individual uncertainties supplied for the velocities for clusters studied by Hesser, Harris, & Harris (1986), but their study reports that the mean velocity uncertainty for clusters with Rgc < 11 ′ is 25 km s−1 and for Rgc > 11 ′ is 44 km s−1. We have adopted these values accordingly for their clusters.2 The study by Harris et al. (1992) also does not report velocity uncertainties; however, these clusters have all been recently measured by Peng, Ford, & Freeman (2004a). The rms 1The confirmed GC list in Woodley (2006) containing 343 GCs has been reduced to 340 based on recent spectroscopic and imaging studies. Object 304867 with a high radial velocity of 305± 56 km s−1 appears to be an M-type star based on the strong molecular bands in its spectrum, see Beasley et al. (2006) for further discussion. Objects pff gc-010 and 114993 are also rejected as GCs because of their starlike appearance under HST ACS imaging; see Harris et al. (2006). However, newly confirmed GCs HCH15 and R122 have now been added (Rejkuba et al. 2007). 2Hesser, Harris, & Harris (1986) report C32 at a distance of Rgc = 10.8 ′, but more recently Peng, Ford, & Freeman (2004a) claim a distance of Rgc = 11.25 ′, so it has an adopted uncertainty of 44 km s−1 in the weighted mean. – 8 – scatter of the Harris et al. (1992) values from theirs was 58 km s−1. This value has been adopted as the velocity uncertainty of the Harris et al. (1992) clusters in the weighted means. The weighted mean velocity of cluster C10 does not include the measured value deter- mined by Peng, Ford, & Freeman (2004a) which is significantly different from other mea- surements. Also, cluster C27 does not include the measurement of vr = 1932 ± 203 km s from Beasley et al. (2006), indicating that this object is a galaxy. We include C27 as a GC, but with caution. In the weighted velocity calculations, the velocities and uncertainties of the 27 GCs from Rejkuba et al. (2007) have been rounded to the nearest whole number, with any velocity uncertainty below 1 km s−1 rounded up to a value of 1. Lastly, the GC pff gc-089 overlaps the previously existing confirmed cluster, C49, within a 0.5” radius; pff gc-089 is therefore removed from the catalog of confirmed GCs. The data in Table 1 provide the basis for the kinematic study presented in this paper. We use them to derive the rotation amplitude, rotation axis and velocity dispersion in the full catalog of clusters, as well as for the metal-poor ([Fe/H] < −1) and metal-rich ([Fe/H] > −1) subpopulations. For this purpose, we define the metallicity of the GCs by transforming the dereddened colors (C − T1)o to [Fe/H] through the standard conversion (Harris & Harris 2002), calibrated through Milky Way cluster data. A foreground reddening value of E(B − V ) = 0.11 for NGC 5128, corresponding to E(C − T1) = 0.22, has been adopted. The division of [Fe/H] = -1 between metal-rich and metal-poor GCs has been shown as a good split between the two metallicity populations from [Fe/H] values converted from C − T1 in Woodley, Harris, & Harris (2005) and Harris et al. (2004) for NGC 5128. If no C and/or T1 values are available for the cluster, it is classified as metal-rich or metal-poor through a transformation from (U − B)o to [Fe/H] from Reed, Harris, & Harris (1994). In Figure 1 we show the spatial distributions of all the GCs from Table 1 (left) and the distribution of the known PNe (right). Both systems are spatially biased to the major axis of the galaxy because this is where most of the GC and PN searches have concentrated. 3. Kinematics of the Globular Cluster System 3.1. Velocity Field For the present discussion we adopt a distance of 3.9 Mpc for NGC 5128. This value is based on four stellar standard candles that each have internal precisions near ±0.2 mag: the PN luminosity function, the tip of the old-halo red giant branch, the long-period variables, – 9 – and the Cepheids (Harris et al. 1999; Rejkuba 2004; Ferrarese et al. 2006). HCH15 and R122 have not been included in our kinematic study, as our study was completed before publication of these velocities. The weighted velocities used in this kine- matic study do not include the most recent 25 velocities of previously known GCs published in Rejkuba et al. (2007). Note that the velocities published in Table 1 do, however, in- clude the Rejkuba et al. (2007) velocities in the quoted final weighted radial velocities for completeness. The velocity distribution of the entire sample of 340 is shown in Figure 2 (top left), binned in 50 km s−1 intervals. A fit with a single Gaussian yields a mean velocity of 546± 7 km s−1, nicely matching the known systemic velocity of 541 ± 7 km s−1 (Hui et al. 1995). There is a slight asymmetry at the low-velocity end that is likely due to contamination by a few metal-poor Milky Way halo stars (also seen in the metal-poor subpopulation in the bottom left panel, which has a mean velocity determined by the Gaussian fit as 532±13 km s−1). Selecting the clusters with radial velocity uncertainties less than 50 km s−1 leaves 226 clusters, plotted in Fig. 2 (top right). The close fit to a single Gaussian is consistent with an isotropic distribution of orbits; the mean velocity is 554 ± 5 km s−1. The metal-rich population, with a mean velocity determined by the Gaussian fit of 565 ± 11 km s−1, is plotted in the bottom right panel, and also shows no strong asymmetries. Looking closer at the metal-poor velocity asymmetry, we note that the 15 metal-poor clusters between 250 and 300 km s−1 (in the region where contamination by Milky Way field stars could occur) are balanced by only two GCs at the high-velocity end on reflec- tion across the systemic velocity. The same velocity regions in the metal-rich population are nearly equally balanced with four clusters between 250 and 300 km s−1 with three clus- ters at the reflected high-velocity range. Interestingly, the four metal-rich clusters between 250 and 300 km s−1 have projected radii > 17 kpc even though the metal-rich popula- tion is more centrally concentrated than the metal-poor (see Peng, Ford & Freeman 2004c; Woodley, Harris, & Harris 2005, among others). The metal-poor clusters between 250 and 300 km s−1, conversely, are more evenly distributed, with five clusters between projected radii of 5 and 10 kpc, five clusters between 10 and 20 kpc, and five clusters beyond 20 kpc from the center of NGC 5128. Some of these low-velocity, metal-poor objects could be fore- ground stars with velocities in the realm of GCs in NGC 5128 (vr & 250 km s −1). However, with only 340 GCs currently confirmed within ∼ 45’ from the center of NGC 5128, out of an estimated ≃ 1500 total clusters within 25’ (Harris et al. 2006), these metal-poor, low- velocity objects could simply be part of a very incomplete GC sample that is also spatially biased. This potential bias is clearly shown in Figure 3, which shows the projected radial – 10 – distribution as a function of azimuthal angle for our GC sample. Beyond 12 kpc, the two ”voids” coincide with the photometric minor axis of the galaxy, attributed at least partly to incomplete cluster surveys in these regions. These objects should, therefore, not be dropped from the GC catalog without further spectroscopic analysis. 3.2. Rotation Amplitude, Rotation Axis, and Velocity Dispersion 3.2.1. Mathematic and Analytic Description We determine the rotation amplitude and axis of the GCS of NGC 5128 from vr(Θ) = vsys + ΩRsin(Θ − Θo) (1) (see Côté et al. 2001; Richtler et al. 2004; Woodley 2006). In Equation 1, vr is the observed radial velocity of the GCs in the system, vsys is the galaxy’s systemic velocity, R is the projected radial distance of each GC from the center of the system assuming a distance of 3.9 Mpc to NGC 5128, and Θ is the projected azimuthal angle of the GC measured in degrees east of north. The systemic velocity of NGC 5128 is held constant at vsys = 541 km s−1 (Hui et al. 1995) for all kinematic calculations. The rotation axis of the GCs, Θo, and the product ΩR, the rotation amplitude of the GCs in the system, are the values obtained from the numerical solution. We use a Marquardt-Levenberg non-linear fitting routine (Press et al. 1992). Eqn. 1 assumes spherical symmetry. While this may be a decent assumption for the inner 12 kpc region (it has a low ellipticity of ∼ 0.2; Peng, Ford & Freeman 2004b), true ellipticities for the outer regions of the system are not well known because of the sample bias (see Fig. 3). Future studies to remove these biases are vital to obtaining a sound kinematic solution for the entire system. Eqn. 1 also assumes that Ω is only a function of the projected radius and that the rotation axis lies in the plane of the sky. It is not entirely clear how these assumptions, discussed thoroughly in Côté et al. (2001), apply to the GC and PN systems of NGC 5128. The Ω we solve for is, therefore, only a lower limit to the true Ω if the true rotation axis is not in the plane of the sky. The projected velocity dispersion is also calculated from the normal condition, σ2v = (vfi − vsys) where N is the number of clusters in the sample, vfi is the GC’s radial velocity after sub- traction of the rotational component determined with Eqn. 1, and σv is the projected velocity dispersion. – 11 – The GCs were assigned individual weights in the sums that combine in quadrature the individual observational uncertainty, εv, in vr and the random velocity component, εrandom, of the GCS. The dominance of the latter is evident by the large dispersion in the GC velocities in the kinematic fitting (see Figure 4). In other words, the clusters have individual weights, ωi = (ε v + ε random) −1; the main purpose of this is to assign a bit more importance to the clusters with more securely measured velocities. This random velocity term dominates in nearly every case, leaving the GCs with very similar base weights in the kinematic fitting. The three kinematic parameters - rotation amplitude, rotation axis, and velocity disper- sion - are determined with three different binning methods. The first involves binning the GCs in radially projected circular annuli from the center of NGC 5128. The chosen bins keep a minimum of 15 clusters in each, ranging as high as 124 clusters. The bins are 0-5, 5-10, 10-15, 15-25, and 25-50 kpc. Also, we include 0-50 kpc to determine the overall kinematics of the system. The second method adopts bins with equal numbers of clusters. The entire population of clusters had nine bins of 38 clusters each, the metal-poor clusters had nine bins of 20 clusters, and the metal-rich clusters had eight bins of 20 clusters. The base weighting is applied to the clusters in both the first and second binning methods. The third method uses an exponential weighting function, outlined in Bergond et al. (2006), to generate a smoothed profile. This method determines each kinematic parameter at the radial position, R, of every GC in the entire sample by exponentially weighting all other GCs surrounding that position based on their radial separation, R− Ri, following wi(R) = −(R − Ri) ]. (3) In Equation 3, wi is the determined weight on each GC in the sample, and σR is the half-width of the window size. For this study, σR is incrementally varied in a linear fashion for the total sample from σR = 1.0 kpc at the radius of the innermost GC in the sample out to σR = 4.5 kpc at the radius of the outermost GC, where the population is lowest. The metal-poor population was given a half-width window of σR = 1.0 − 6.5 kpc, and the metal-rich population was given a half-width window of σR = 2 − 5.3 kpc, again from the innermost to outermost cluster. The progressive radial increase in σR ensured that each point R had roughly equal total weights. – 12 – 3.2.2. Rotation Amplitude of the Globular Cluster System The kinematic parameters were determined for the entire sample of 340 GCs, as well as the subpopulations of 178 metal-poor and 158 metal-rich GCs (four clusters have unknown metallicity). The kinematic results for the entire population of GCs are shown in Table 2, reproduced almost in full from Woodley (2006), while the results for the metal-poor and metal-rich clusters are shown in Tables 3 & 4, respectively. The columns give the radial bin, the mean projected radius in the bin, the radius of the outermost cluster, the number of clusters in the bin, the rotation amplitude, the rotation axis, and the velocity dispersion, with associated uncertainties. These are followed by the mass correction, the pressure-supported mass, the rotationally supported mass, and the total mass in units of solar mass (see § 5 for the mass discussion). The results for the alternate two methods, using an equal number of GCs per bin and the exponentially weighted GCs, are not shown in tabular form but are included in all of the figures. Figure 4 shows the sine fit of Eqn. 1 for the total population and for the metal-poor and metal-rich subpopulations. All three populations show rotation about a similar axis. As discussed in § 3.1, the metal-poor population has more members with low velocities (Vr ≤ 300 km s −1) than the metal-rich population, suggesting possible contamination of Milky Way foreground stars in the sample. Figures 5 & 6 show the rotation amplitude results for the entire population and for the metal-poor and metal-rich subpopulations, respectively. The three kinematic methods, described in Section 3.2.1, appear to agree relatively well for all three populations of clusters. While there appears to be no extreme difference in rotation amplitude between the cluster populations, the metal-poor subpopulation of clusters has lower rotation in the inner 5 kpc of NGC 5128 than the metal-rich subpopulation. The weighted average of the 0-5 kpc radial bin and the innermost equal-numbered bin, shows that the entire population has a rotation amplitude of ΩR = 31 ± 17 km s−1, while the metal-poor population has ΩR = 17 ± 26 km s−1 and the metal-rich population has ΩR = 57±22 km s−1. Peng, Ford & Freeman (2004c) show in their study that the metal-poor population has very little rotation in the central regions, completely consistent with our findings. The rotation amplitude does not appear to differ between the two populations outside of 5 kpc. 3.2.3. Rotation Axis of the Globular Cluster System The results of the rotation axis solutions are shown in Figures 7 & 8, again for the entire population and for the metal-poor and metal-rich subpopulations. The solution for – 13 – Θ0 agrees well for all three kinematic methods and all subgroups. The inner 5 kpc region has a different rotation axis than the outer regions, demonstrated clearly in all three binning methods. The innermost bin yields weighted averages of Θo = 369±24 o, Θo = 25±55 o, and Θo = 352±18 o, all of which are equal within their uncertainties. Beyond 5 kpc, the rotation axes for all three populations are in even closer agreement, with averages of Θo = 189 ± 6 Θo = 199 ± 7 o, and Θo = 196 ± 7 o for the entire population, the metal-poor subpopulation, and the metal-rich subpopulation, respectively. The position angle of the photometric major axis of NGC 5128 is Θ = 35o and 215o east of north and the photometric minor axis is Θ = 119o and 299o east of north (Dufour et al. 1979). It appears the GCS is rotating about an axis similar to the photometric major axis for the full extent of the galaxy, with a possible axial twist or counterrotation within 5 kpc. 3.2.4. Velocity Dispersion of the Globular Cluster System Figures 9 & 10 show the velocity dispersion for the entire population and for the metal- poor and metal-rich subpopulations. Our results for σv show no significant differences be- tween the metallicity subpopulations. All three show a relatively flat velocity dispersion (σv = 119 ± 4, σv = 117 ± 6, and σv = 111 ± 6 km s −1 within 15 kpc of the center of NGC 5128 for the entire population and for the metal-poor and metal-rich subpopulations, re- spectively). These results match the previous study of NGC 5128 by Peng, Ford & Freeman (2004c), whose determined velocity dispersion for the GCs within 20 kpc ranged between 75 and 150 km s−1. At a larger radius, we find that σv then slowly increases to σv > 150 km s−1 towards the outer regions of the halo for all populations. The velocity dispersion of the metal-rich GCs, interestingly, appears higher than that of the metal-poor GCs in the outer regions (although still consistent within the determined uncertainties). In most previous studies, the velocity dispersion of the metal-poor GCs usually appears higher than that of the metal-rich GCs, if there is a notable velocity dispersion difference between the subpopulations (see the studies of Côté et al. 2003; Richtler et al. 2004, as examples). To explore the cause of the distinct rise past 15 kpc a bit further, we have plotted the actual velocity histograms in Figure 11 for the metal-poor and metal-rich subgroups, subdivided further into inner (R < 15 kpc) and outer (R > 15 kpc) regions. In the inner 15 kpc, both samples show histograms strongly peaked near vf = 0 and with at least roughly Gaussian falloff to both high and low velocities. By contrast, the histograms for the outer regions (15−50 kpc) are noticeably flatter, so that the clusters with larger velocity residuals have relatively more importance to the formal value of σv. Nominally, the flatter shape of the velocity distribution would mean that the outer-halo clusters display anisotropy in – 14 – the direction of a bias towards more circular orbits. However, such a conclusion would be premature at this point for two reasons. First, the sample size in the outer regions is still too small to lead to high significance, and a direct comparison between the inner and outer histograms (through a Kolmogorov-Smirnov test) does not show a statistically significant difference between them larger than the 70% level. Second, the outer samples may still be spatially biased in favor of objects along the major axis of the halo, as discussed above, and this bias sets in strongly for R > 12 kpc (see Fig. 3), very near where we have set the radial divisions in this Figure. This type of velocity distribution can also arise from the accretions of satellite galaxies with their own small numbers of GCs (Bekki et al. 2003). We will need to have a larger sample of the outer-halo clusters, and one in which these potential sample biases have been removed, before we can draw any firmer conclusions. However, it needs to be explicitly stated that the outermost point in the kinematic plots for the GCs representing 25 − 50 kpc suffers from very high spatial biases and low number statistics (< 40 GCs) and covers a large radial interval. The rise in velocity dispersion could be driven purely by systematic effects resulting from the radial gradient of the number density of GCs in this outermost bin (Napolitano et al. 2001). 4. Kinematics of the Planetary Nebula System of NGC 5128 NGC 5128 has a large number, 780, of identified PNe with measured radial velocity from the studies of Peng, Ford & Freeman (2004b) and Hui et al. (1995); these PNe are projected out to 90 kpc assuming a distance to NGC 5128 of 3.9 Mpc. Since these are also old objects, it is of obvious interest to compare them with the GCS. The PNe also have the advantage of giving us the best available look at the kinematics of the halo field stars. The PN kinematic results are listed in Table 5, with the same columns as Table 2. The results are shown in Figures 5, 7, & 9 for the rotation amplitude, rotation axis, and velocity dispersion, respectively. The spatial distribution of the known PNe is, like the GCS, biased toward the major axis at large radii (see Fig. 1). Nevertheless, their kinematics closely resemble the GCs. The kinematics of the PN system are very consistent among all three binning methods. The rotation amplitude and rotation axis show little radial trend, while the velocity disper- sion appears relatively flat within the first 15 kpc at σv = 122 ± 7 km s −1 and then slowly decreases to σv ≃ 85 km s −1 at large galactocentric radius. Peng, Ford, & Freeman (2004a) show that the velocity dispersion of the PNe drops from a central value of 140 to 75 km s−1 in the outer regions of the galaxy, consistent with the findings of this study. Their velocity field analysis led to the discovery of a ”zero-velocity curve” located between the photometric – 15 – minor axis, 119±5o east of north (Dufour et al. 1979), and the north-south direction, for the innermost region of the galaxy. Just beyond 5 kpc, the zero-velocity curve turns and follows a straight line at a 7o angle from the photometric major axis, 35o east of north (Dufour et al. 1979) (see Figure 7 of Peng, Ford & Freeman 2004c). This study does not show a strong change in rotation axis for the PNe in the innermost regions of the galaxy. However, it clearly shows in all three GC populations a significant change in the rotation axis just beyond 5 kpc from the center of the galaxy. A change in axis of 5 kpc outward (∼ 180o for the entire population and metal-poor subpopulation and ∼ 160o for the metal-rich subpopulation) has been found, as discussed in § 3.2.3. Similarly, Peng, Ford & Freeman (2004c) show from their sample of 215 clusters that a clear sign of rotation beyond 5 kpc about a misaligned axis appears particularly in their metal-rich subpopulation. The kinematics of the GCs in this study matches the line of zero velocity relatively nicely. Within 5 kpc the rotation axis of the GCS is nearly-parallel to the north- south direction, and beyond 5 kpc the rotation axis is near 200o east of north, which is only ∼ 10o from the zero-velocity curve. However, the velocity field of NGC 5128 is complex and not entirely captured by these approximate solutions. The two-dimensional velocity field shown in Peng, Ford & Freeman (2004b) (see their Figure 7), shows that the photometric major axis (which happens to be very close to our maximum rotation as discussed above) is only 7o from the line of zero velocity. This could lead to a very asymmetric velocity profile that may not be well fit by the sine curve described in Equation 1. Biased kinematics, especially the rotation axis, may develop from the sine fit that could lead to a higher estimated velocity dispersion. Hui et al. (1995) similarly studied the kinematics of the PN system in NGC 5128 with a sample of 433 PNe. They obtain a rotation axis of 344 ± 10o. Our result of 170 ± 5o east of north is consistent with their findings on comparing their sine curve fit of their PN data in their Figure 11 to our corresponding fit shown in Fig. 4 for the GCS, which shares a similar axis to our PN sample (note that in their study, φ = 0o corresponds to our Θ = 305o east of north). Our fits both correspond to a positive rotation amplitude for a rotation axis near 170o east of north and a negative rotation axis near 350o east of north. Therefore, the rotation axis quoted in Hui et al. (1995) of 344 ± 10o corresponds to a negative rotation amplitude of approximately 70 − 75 km s−1 (taken from their Figure 11), nicely matching our result of a positive 76 ± 6 km s−1 about an axis of 170 ± 5o east of north. – 16 – 5. Dynamics of NGC 5128 Both GCs (Côté et al. 2001; Larsen et al. 2002; Evans et al. 2003; Côté et al. 2003; Beasley et al. 2004; Peng, Ford & Freeman 2004c, among others) and PNe (Ciardullo et al. 1993; Hui et al. 1995; Arnaboldi et al. 1998; Peng, Ford & Freeman 2004b, among others) can be used to estimate the total dynamical mass of their host galaxies. A variety of tools are in use including derived mass models, the virial mass estimator (Bahcall & Tremaine 1981), the projected mass estimator (Heisler, Tremaine, & Bahcall 1985), and the tracer mass estimator (Evans et al. 2003). NGC 5128 does not have a large X-ray halo (detected by Kraft et al. 2003; O’Sullivan, Forbes, & Ponman 2001, the latter reporting a measurement of log Lx = 40.10 erg s −1), such as is evident in other giant ellipticals such as M87 (Côté et al. 2001) or NGC 4649 (Bridges et al. 2006). Thus it is difficult to model the dark matter profile of NGC 5128 with a priori constraints. Without such a mass model, we turn to the tracer mass estimator for the dynamical mass determination. The tracer mass estimator has the distinct advantage over the virial and pro- jected mass estimators that the tracer population does not have to follow the dark matter density in the galaxy - an extremely useful feature for stellar subsystems such as GCs and PNe that might, in principle, have significantly different radial distributions (see Evans et al. (2003) for extensive discussion). Below, we determine the mass of NGC 5128 using the tracer populations of GCs and PNe (our mass estimates do not include stellar kinematics in the inner regions). 5.1. Mass Determination The mass contributed by the random internal motion of the galaxy (pressure-supported mass) is determined from the tracer mass estimator as (vfi − vsys) 2Ri (4) where N is the number of objects in the sample and vfi is the radial velocity of the tracer object with the rotation component removed. For an isotropic population of tracer objects, assumed in this study, the value of C is 4(α + γ)(4 − α− γ)(1 − ( rin )(3−γ)) π(3 − γ)(1 − ( rin )(4−α−γ)) where rin and rout are the three-dimensional radii corresponding to the two-dimensional projected radii of the innermost, Rin, and outermost, Rout, tracers in the sample. The – 17 – parameter α is set to zero for an isothermal halo potential in which the system has a flat rotation curve at large distances. Finally, γ is the slope of the volume density distribution, which goes as r−γ, and is found by determining the surface density slope of the sample and deprojecting the slope to three-dimensions. The tracer mass estimator uses a sample of tracer objects defined between rin and rout, yet it is important to emphasize that it determines the total enclosed mass within rout. There is also a contribution to the total mass by the rotational component, as determined in § 3.2.2 for the GCs and § 4 for the PNe. This mass component is determined from the rotational component of the Jeans equation, Routv where Rout is the outermost tracer projected radius in the sample and vmax is the rotation amplitude. Therefore, the total mass of NGC 5128, Mt, is determined by the addition of the mass components supported by rotation, Mr, and random internal motion, Mp, Mt = Mp + Mr. (7) In the determination of the pressure-supported mass, one must estimate values for rin and rout knowing Rin and Rout. Evans et al. (2003) suggest that rin ≃ Rin and rout ≃ Rout for distributions taken over a wide angle. However, in this study the inner and outer radii of the chosen bins are at intermediate radial values within the distribution. Their assumption would therefore lead to an underestimate of the determined mass, since the true rout can be quite a bit larger than the projected Rout. To correct for this contributed uncertainty, distributions of sample tracer populations were generated through Monte Carlo simulations. In the simulations, 340 GCs were randomly placed in a spherically symmetric system extending out to 50 kpc with an r−2 projected density, while 780 PNe were placed in the same environment extending out to 90 kpc. From the generated distributions, the value of C in Eqn. 5 was determined for both the real and projected positions of the tracer populations in each designated radial bin. This correction factor, listed in Tables 3-5 as Mcorr, multiplies the pressure-supported mass from Eqn. 4. The same correction was applied to the full GC sample and the corresponding Mcorr values are listed in Table 1 of Woodley (2006). These values are generally small, but in the worst case they triple Mp. 5.2. Surface Density Profiles In Eqn. 5, the value of γ is determined for the tracer populations by deprojecting the slope of the surface density profile to three-dimensions. Figure 12 shows the surface – 18 – density profiles for the entire, metal-poor, and metal-rich GC populations, along with the PN profile. The populations were binned, following Máız Apellániz & Úbeda (2005), into circular annuli of equal numbers of objects, providing the same statistical weight to each bin (although spatial biases may still affect the GC population in the outer regions along the major axis; see Fig. 3). In the inner 5 kpc of all tracer populations, incompleteness due to the obscuration of the dust lane is evident by the flattening of the surface density profile. The innermost objects were, therefore, excluded from the surface density profile fittings. Outside of 5 kpc, the surface densities fit well to power laws, leading to γ = 3.65 ± 0.17, 3.49 ± 0.34, 3.37 ± 0.30, and 3.47 ± 0.12 for the entire GC population, the metal-poor and metal-rich subpopulations of GCs, and the PNe in NGC 5128, respectively. These are all very similar within their uncertainties. 5.3. Mass Results The similar kinematics we find between the metal-poor and metal-rich subpopulations of GCs in this study strongly justifies the combining of the two populations for the mass determination performed in Woodley (2006). The GC population provides a total mass estimate of (1.3 ± 0.5) × 1012 M⊙ from 340 clusters out to a projected radius of 50 kpc. Removing the GCs in our sample with vr ≤ 300 km s −1, which will remove all possible contamination from foreground stars, discussed in § 3.1, leads to a total mass of (1.0±0.4)× 1012 M⊙. This mass agrees nicely with our mass determined from our entire GC sample. The PN population provides a total mass of (1.0 ± 0.2) × 1012 M⊙ from 780 PNe out to 90 kpc in projected radius, agreeing with the GC value within the uncertainty. We are also able to generate a mass profile of NGC 5128 from the total GC population and the PNe, shown in Figure 13. The tracer mass estimator determines the total enclosed mass for NGC 5128 within the outermost radius of a given tracer sample. It calculates this total mass using a sample of objects defined within the radial range defined by the sample’s inner and outermost radii. It is therefore possible to use a unique set of tracer objects, denoted by the radial bin range, listed in the first column of Tables 2-5, to determine a mass profile from independent mass estimates. The independent binning, leads to sample sizes in the mass determination, in some cases generating higher uncertainties in the total enclosed mass. The most certain mass is the one determined from the full sample of tracers. In the mass determinations above, we have implicity assumed isotropy for the veloc- ity distributions. But the possibility exists that the PNe (for example) might have radial anisotropy which would produce their gradually falling σv(R) curve. Replacing Equation 5 – 19 – in the tracer mass estimator by 16(α + γ − 2β)(4 − α− γ)(1 − ( rin )(3−γ)) π(4 − 3β)(3 − γ)(1 − ( rin )(4−α−γ)) which includes the anisotropy parameter, β, from Evans et al. (2003), we find that the mass estimate from the PNe can be forced to agree with the mass estimate from the GCS for a nominal β = 0.8. For perfect isotropy, β = 0. This would mean roughly 2:1 radial anisotropy for the PNe in the outer halo. However, we find that any β in the wide range of −10 ≤ β ≤ 1 would still keep the two methods in agreement within their internal uncertainties, so we are not yet in a position to tightly constrain any anisotropy. It is possible that the GCs may also have anisotropy; it may therefore be too simplistic to find a range of β for the PNe for which the masses of the PNe and GCs agree. However, the GCs are likelier to be nearly isotropic than the PNe; the GCs are older, ”hotter” subsystems of the halo. In other studies, the isotropy of the GCS orbits has also been shown to be a good assumption from mass profiles of elliptical galaxies using X-ray observations (Côté et al. 2001, 2003; Bridges et al. 2006, among others). Both mass estimates can be compared to previous studies. First, we note the total mass determined from the PN data with that of Peng, Ford & Freeman (2004b). While the rotationally supported mass was determined here with different values of the mean rota- tional velocity, they calculated the pressure supported mass using the identical tracer mass estimator technique with exactly the same PN population. The total mass estimate given by Peng, Ford & Freeman (2004b) is (5.3 ± 0.5) × 1011 M⊙. Subtracting their rotationally supported mass leaves a pressure supported mass of ∼ 3.4×1011 M⊙, quite different from our (8.46±1.72)×1011 M⊙. Recalculating our pressure supported mass estimate with γ = 2.54, which was used in Peng, Ford & Freeman (2004b), we are able to reproduce their mass esti- mate within the uncertainty. The values of γ differ between the two studies simply because the γ used in Peng, Ford & Freeman (2004b) was the inverse of the surface density slope instead of the inverse of the volume density slope. Using the correct value of γ = 3.54, their pressure supported mass estimate would increase to 8.7× 1011M⊙, matching the mass found in this study. Second, we compare our total mass determined using the GC population with that from Peng, Ford & Freeman (2004c). Using 215 GCs out to 40 kpc, they found a pressuresup- ported mass of (3.4±0.8)×1011 M⊙, again much different from our pressure supported mass of (1.26 ± 0.47) × 1012 M⊙ using 340 clusters out to 50 kpc. The large difference can again be attributed to their using γ = 2.72 instead of deprojecting their surface density slope to γ = 3.72. Using the correct value of γ, we find a pressure-supported mass of 7.5 × 1011 M⊙ using the same 215 clusters they used in their study. This corrected estimate is closer to the – 20 – pressure supported mass determined in our study, but it is not necessarily expected to agree with our result, as our sample contains 130 more GCs and uses a slightly different γ that we have independently redetermined. Third, the mass determined by the H I shell study of NGC 5128 by Schiminovich et al. (1994) found a mass of 2×1011 M⊙ within 15 kpc assuming a distance of 3.5 Mpc. With the distance of 3.9 Mpc used in this study, the mass determined in their study would increase to 2.2 × 1011 M⊙, which is 30% smaller than our total mass of 3.89± 0.94× 10 11 M⊙ within 15 kpc. Lastly, we compare our determined mass to a recent study by Samurović (2006). Samurović (2006) determined a total mass of NGC 5128 using GCs, PNe, and an X-ray data tech- nique. The galaxy mass determined from the GC and PN data was obtained using the tracer mass estimator and the spherical Jeans equation, as performed in our study. How- ever, Samurović (2006) used the volume density slopes determined by Peng, Ford & Freeman (2004b) and Peng, Ford & Freeman (2004c), for the PN and GC data, respectively. They ob- tained mass estimates for NGC 5128 similar to those of Peng, Ford & Freeman (2004b) and Peng, Ford & Freeman (2004c), discussed above, using an identical PN sample and slightly increased GC sample. They also included an X-ray-modelling mass estimate for NGC 5128 from which they obtained masses of (7.0± 0.8)× 1011 M⊙ out to 50’ and (11.6± 1.0)× 10 M⊙ out to 80’. This mass estimate is similar to our PNe estimate out to the same radial extent, but the author cautions that it is an overestimate of the true mass of NGC 5128 resulting from a lack of hydrostatic equilibrium in the outer region of the galaxy. We note here that the mass estimates obtained are higher than those from Hui et al. (1995), and Peng, Ford & Freeman (2004b) derived from the PNe using a two-component mass model, as well as Samurović (2006), using an X-ray modelling technique. This dis- crepancy is not fully understood and possibly lies in the assumptions that go into the mass estimators with a spatially biased sample. We intend to pursue this issue further with an upcoming larger sample of GCs with less spatial biases. The mass of NGC 5128 that we find appears to be in the range of other giant elliptical galaxies, such as NGC 1399 (∼ 2 × 1012 M⊙ out to 50 kpc; Richtler et al. 2004), M49 (∼ 2 × 1012 M⊙ out to their kinematically studied radius of 35 kpc; Côté et al. 2003), and M87 (∼ 9 × 1011 M⊙ at 20 kpc, the onset of the projected cD envelope; Côté et al. 2001). Clearly, it is legitimate to say that NGC 5128 is the largest, most massive galaxy in the neighborhood of the Local Group, and one that can be talked about in the same category as these other giants that reside in larger clusters. The sample biases mentioned above in our currently available set of both GCs and PNe – 21 – place limitations on how much we can reasonably interpret the kinematic and dynamic data. We are currently carrying out a set of new spectroscopic programs to increase the tracer sample size and to remove the sample biases, leading to a more complete analysis of the halo velocity field. 6. Discussion and Conclusions Angular momentum is an essential quantity for characterizing the sizes, shapes, and formation of galaxies and is often represented as the dimensionless spin parameter, J |E|1/2 GM5/2 where J is the angular momentum, E is the binding energy, and M is the mass of the galaxy. The spin parameter is representative of a galaxy’s angular momentum compared to the amount of angular momentum needed for pure rotational support: the lower the Λ- value, the less rotation and rotational support within the galaxy. For an elliptical galaxy in gravitational equilibrium, the spin parameter simplifies to Λ ∼ 0.3 < (ΩR/σv) > (Fall 1979), yielding Λ = 0.10 with (ΩR/σv) = 0.33 for the entire population of GCs in NGC 5128. Table 6 shows the spin parameter for four giant galaxies with large GCS kinematic studies, M87, M49, NGC 1399, and NGC 5128. The table columns give the galaxy name, the rotation amplitude, the projected velocity dispersion, and the ratio of the rotation amplitude to the velocity dispersion, followed by the spin parameter. These quantities are shown for the metal-poor and metal-rich populations. What is clearly evident in Table 6 is the strong galaxy to galaxy differences between these four galaxies, already hinted at in § 1. Though the sample is still quite small, no obvious pattern emerges. There is an indication of metal-poor and metal-rich GCSs having similar spin parameters within the same galaxy. M49 is the only galaxy studied here where this may not be the case. Although the metal-poor and metal-rich cluster spin parameters are consistent within the uncertainties, the metal-rich cluster spin parameter of M49 is also consistent with zero. In the monolithic collapse scenario, Peebles (1969) describes the angular momentum within the galaxy as attributed to the tidal torque transferred from neighbouring proto- galaxies during formation. In this scenario, Efstathiou & Jones (1979) found that a spin parameter of Λ = 0.06 for elliptical systems is expected from simulations of the collapse of an isolated protogalactic cloud. But NGC 5128, among many other giant elliptical galaxies, is not in isolation, and therefore not necessarily expected to reproduce such a low spin parameter. Also, the internal rotation axis changes at 5 kpc are not easily explained with – 22 – only the monolithic collapse scenario. In the monolithic collapse model, the inner regions would be expected to have more pronounced rotations. Yet all four of the galaxies with major kinematic studies presented here do not show a higher rotational signal in the inner regions. In fact, for NGC 1399, the outer region (R > 6′) indicates rotation in the metal-poor population that is not evident in the inner regions. Also, a slightly lower rotational signal is present in the inner regions of NGC 5128 for the metal-poor population than in the outer regions. Hierarchical clustering of cold dark matter also relies on angular momentum in a galaxy being produced by gravitational tidal torques during the growth of initial perturbations. Sugerman, Summers, & Kamionkowski (2000) have demonstrated that the tidal torque the- ory predicts an increase in angular momentum during the collapse, and with time, the increase in angular momentum slows. Accretion of satellites and/or merger events is there- fore a possible culprit for moving the angular momentum outward, as major mergers of disks and bulges suggest that angular momentum resides largely in the outer regions of the galaxy (Barnes 1992; Hernquist 1993). Alternatively, Vitvitska et al. (2002) examine the change in spin parameter in a scenario where the angular momentum in a galaxy is built up by mass accretion. Their results show that the spin parameter changes sharply in major merger events in the galaxy and steadily decreases with small satellite accretion events. They also show that the spin parameter for a galaxy with a major merger after a redshift of z = 3 should be notably larger than a galaxy that did not undergo such a major merger. Their study obtains an average of Λ = 0.045 from ΛCDM N -body simulations for galaxies with halo masses of (1.1 − 1.5) × 1012h−1 M⊙ with h = 0.7. NGC 1399 and M49, with their weak rotation signals, are consistent with the model predictions discussed above, whereby their major formation events could have occurred at early times and with perhaps only minor accretions happening since then. However, NGC 5128 and M87 have spin parameters 2-3 times larger than predicted by the model averages. For NGC 5128, this relatively large rotation (which is nearly independent of both metallicity and radius) may, perhaps, be connected with its history within the Centaurus group envi- ronment. The rotation speed and rotation axis for its extended group of satellite galaxies are nearly identical to the NGC 5128 halo (Woodley 2006), much as if the accretion events experienced by the central giant have been taking place preferentially along the main axis of the entire group and in the same orientation. The GCS age distribution discussed by Beasley et al. (2006) and the mean age for the halo field stars (Rejkuba et al. 2005) strongly suggest that a high fraction of the stellar population in NGC 5128 formed long ago, with particularly large bursts between 8 and 12 Gyr. Even if the galaxy underwent a significant – 23 – merger perhaps a few Gyr ago (the traces of which now appear in the halo arcs and shells), the stars in it may already have been old at the time of the merger. Although a very few younger GCs have formed since then, these make up a small minority of what is present, at least for the R > 5 kpc halo outside the bulge region that now contains gas and dust. The situation for M87, with its even larger rotation signature, may require a different sort of individual history. Of the four galaxies compared here, it is at the dynamical center of the richest environment (Virgo), has the most extensive cD-type envelope, and sits within the most massive, extended, and dynamically evolved potential well. A single relatively recent major merger could in principle have caused its present high rotation, but the lack of distinctive tidal features does not necessarily favor such an interpretation and would at least suggest that such a merger should have been with another large elliptical and nondissipative galaxy. Côté et al. (2001) discuss an interpretation - at least partially resembling what we suggest for NGC 5128 - that stellar material ”is gradually infalling onto M87 along the so-called principal axis of the Virgo Cluster.” In conclusion, we have presented a kinematic study of NGC 5128 that makes it now com- parable to recent studies of the other giant galaxies, M87, M49, and NGC 1399. Using 340 GCs (158 metal-rich and 178 metal-poor GCs), we have calculated the rotation amplitude, rotation axis, and velocity dispersion and have searched for radial and metallicity depen- dences. Our findings show that both metallicity populations rotate with little dependence on projected radius, with ΩR = 40±10, 31±14, and 47±15 km s−1 for the total, metal-poor, and metal-rich populations, respectively. Perhaps the inner 5 kpc shows a slower rotation of the metal-poor population, but more clusters would be needed to confirm this finding. The rotation axis is 189±12o, 177±22o, and 202±15o east of north for the total, metal-poor, and metal-rich populations out to a 50 kpc projected radius, assuming the velocity field is best fit by a sine curve. The rotation axis does change at 5 kpc, following the zero-velocity curve proposed by Peng, Ford & Freeman (2004b) or possibly full-on counterrotation. A study with more GCs and lower uncertainties is needed to see what is happening in the innermost 5 kpc of NGC 5128. The velocity dispersion shows a modest increase with galactocentric radius, although the outer regions (especially the metal-rich population) have less reliable statistics; this increase could be driven purely by statistical effects. We find the velocity dis- persion we find 123±5, 117±7, and 129±9 km s−1 for the total, metal-poor, and metal-rich populations, respectively. The PN data are also used to determine the kinematics of the halo of NGC 5128. These show results that are encouragingly similar to those of the GC data, except that no rotation axis change is noted with radius, and a decrease in velocity dispersion is found with radius, possibly indicating a difference in orbital anisotropy compared with the GCs. A – 24 – very similar effect has been noted for the Leo elliptical NGC 3379, although with a much smaller data sample (Romanowsky et al. 2003; Bergond et al. 2006; Pierce et al. 2006). We also determine the total dynamical mass using both the GCs and the PNe by separately calculating the pressure supported mass with the tracer mass estimator and the rotationally supported mass using the spherical component of the Jeans equation. The total mass is (1.3 ± 0.5) × 1012 M⊙ from the GC population out to a projected radius of 50 kpc, or (1.0 ± 0.2) × 1012 M⊙ out to 90 kpc from the PNe. Overall, we have enough evidence to cautiously conclude that a major episode of star formation occurred about 8−10 Gyr ago (corresponding to a redshift z = 1.2 - 1.8) and this may have been when the bulk of the visible galaxy was built. We still do not know just why the most metal-poor clusters show up in such relatively large numbers and appear to have ages of 10 − 12 Gyr, but this is a common issue in all big galaxies. This kinematic study and the age study of Beasley et al. (2006) on the NGC 5128 cluster system indicate that additional spectroscopic studies to build up both the radial velocity database and age distribution can lead to rich dividends. Large GC samples are clearly needed to remove the current sample biases and to fully understand the complex kinematics and history of this giant elliptical galaxy. It seems clear as well that each galaxy needs to be individually studied to fully understand the different galaxy formation histories. We are continuing these studies particularly for NGC 5128, with the eventual aim of at least doubling the total GC sample size in this unique system. Acknowledgements: WEH and GLHH acknowledge financial support from NSERC through operating research grants. DAF thanks the ARC for financial support. – 25 – REFERENCES Arnaboldi, M., Freeman, K.C., Capaccioli, M., & Ford, H. 1994, ESO Messanger, 76, 40 Arnaboldi, M., Freeman, K.C., Gerhard, O., Matthias, M., Kudritzki, R.P. Méndez, R.H., Capaccioli, M., & Ford, H. 1998, ApJ, 507, 759 Baade, W., & Minkowski, R. 1954, ApJ, 119, 215 Bahcall, J.N., & Tremaine, S. 1981, ApJ, 244, 805 Barnes, J.E. 1992, ApJ, 393, 484 Beasley, M.A., Forbes, D.A., Brodie, J.P., & Kissler-Patig, M. 2004, MNRAS, 347, 1150 Beasley, M.A., Bridges, T., Peng, E., Harris, W.E., Harris, G.L.H., Forbes, D.A., & Mackie, G. 2006, in preparation Bekki, K., Harris, W.E., & Harris, G.L.H. 2003, MNRAS, 338, 587 Bekki, K., Forbes, D.A., Beasley, M.A., & Couch, W.J. 2003, MNRAS, 334, 1334 Bekki, K., & Peng, E.W. 2006, MNRAS, 370, 1737 Bergond, G., Zepf, S.E., Romanowsky, A.J., Sharples, R.M., & Rhode, K.L. 2006, A&A, 448, 155 Bridges, T., Gebhardt, K., Sharples, R., Faifer, F.R., Forte, J.C., Beasley, M.A., Zepf, Z.E., Forbes, D.A., Pierce, M. 2006, MNRAS, 373, 157 Ciardullo, R., Jacoby, G.H., & Dejonghe, H.B. 1993, ApJ, 414, 454 Côté, P., McLaughlin, D.E., Hanes, D.A., Bridges, T.J., Geisler, D., Merritt, D., Hesser, J.E., Harris, G.L.H., & Lee, M.G. 2001, ApJ, 559, 828 Côté, P., McLaughlin, D., Cohen, J.G., & Blakesless, J.P. 2003, ApJ, 591, 850 Dekel, A., Stoehr, F., Mamon, G.A., Cox, T.J., Novak, G.S., & Primak, J.R., 2005, Nature, 437, 707 Dufour, R.J., Harvel, C.A., Martins, D.M., Schiffer, F.H., Talent, D.L., Wells, D.C., van den Bergh, S., & Talbot, R.J. 1979, AJ, 84, 284 Efstathiou, G., & Jones, B.J.T. 1979, MNRAS, 186, 133 – 26 – Evans, N.W., Wilkinson, M.I., Perrett, K.M., & Bridges, T.J. 2003, ApJ, 583, 752 Fall, S.M. 1979, Rev. Mod. Phys., 51, 21 Ferrarese, L., Mould, J.R., Stetson, P.B., Tonry, J.L., Blakeslee, J.P., & Ajhar, E.A. 2006, ApJ, in press (astro-ph/0605707) Harris, G.L.H., Geisler, D., Harris, H.C., & Hesser, J.E. 1992, AJ, 104, 613 Harris, G.L.H., Harris, W.E., & Poole, G. 1999, AJ, 117, 855 Harris, W. E., & Harris, G. L. H. 2002, AJ, 123, 3108 Harris, W.E., Harris, G.L.H., Holland, S.T., & McLaughlin, D.E. 2002, AJ, 124, 1435 Harris, G.L.H., Harris, W.E., & Geisler, D. 2004, AJ, 128, 723 Harris, W.E., Harris, G.L.H., Barmby, P., McLaughlin, D.E., & Forbes, D. 2006, AJ, 132, Heisler, J., Tremaine, S., & Bahcall, J.N. 1985, ApJ, 298, 8 Hernquist, I. 1993, ApJ, 409, 548 Hesser, J.E., Harris, H.C., van den Bergh, S., & Harris, G.L.H. 1984, ApJ, 276,491 Hesser, J.E., Harris, H.C., & Harris, G.L.H. 1986, ApJ, 303, L51 Hui, X., Ford, H.C., Freeman, K.C., & Dopita, M.A. 1995, ApJ, 449, 592 Kraft, R.P., Váquez, S.E., Forman, W.R., Jones, C., Murray, S.S., Hardcastle, M.J., Worrall, D.M., Churazov, E. 2003, ApJ, 592, 129 Larsen, S.S., Brodie, J.P., Beasley, M.A., & Forbes, D.A. 2002, AJ, 124, 828 Lee, H., & Worthey, G. 2005, ApJS, 160, 176 Máız Apellániz, J. & Úbeda, L. 2005, ApJ, 629, 7873 Mamon, G.A., & Lokas, E.L. 2005, MNRAS, 362, 95 Napolitano, N.R., Arnaboldi, M., Freeman, K.C., & Capaccioli, M. 2001, A&A, 377, 784 Napolitano, N.R., Arnaboldi, M., & Capaccioli, M. 2002, A&A, 383, 791 O’Sullivan, E., Forbes, D.A., & Ponman, T.J. 2001,MNRAS, 328, 461 http://arxiv.org/abs/astro-ph/0605707 – 27 – Peebles, P.J.E. 1969, ApJ, 155, 393 Peng, E.W., Ford, H.C., Freeman, K.C., & White, R.L. 2002, AJ, 124, 3144 Peng, E.W., Ford, H.C., & Freeman, K.C. 2004, ApJS, 150, 367 Peng, E.W., Ford, H.C., & Freeman, K. 2004, ApJ, 602, 685 Peng, E.W., Ford, H.C., & Freeman, K. 2004, ApJ, 602, 705 Pierce, M., Beasley, M.A., Forbes, D.A., Bridges, T., Gebhardt, K., Faifer, F.R., Forte, J.C., Zepf, S.E., Sharples, R., Hanes, D.A., & Proctor, R. 2006, MNRAS, 366, 1253 Press, W.H., Teukolsky, S.A., Vetterling, W.T., & Flannery, B.P. 1992, Numerical Recipes in Fortran (2nd Ed., Cambridge: Camdridge University Press) Puzia, T.H., Kissler-Patig, M., Thomas, D., Maraston, C., Saglia, R.P., Bender, R., Goud- frooij, P., & Hempel, M. 2005, A&A, 439, 997 Reed, L.G., Harris, G.L.H., & Harris, W.E. 1994, AJ, 107, 555 Rejkuba, M. 2001, A&A, 269, 812 Rejkuba, M. 2004, A&A, 413, 903 Rejkuba, M., Greggio, L., Harris, W.E., Harris, G.L.H., & Peng, E.W. 2005, ApJ, 631, 262 Rejkuba, M., Dubath, P., Minniti, D., & Meylan, G. 2007, A&A, in press (astro-ph/0703385) Richtler, T., Dirsch, B., Gebhardt, K., Geisler, D., Hilker, M., Alonso, M. V., Forte, J. C,. and Grebel, E. K., Infante, L., Larsen, S., Minniti, D., & Rejkuba, M. 2004, AJ, 127, Romanowsky, A.J., Douglas, N.G., Arnaboldi, M., Kuijken, K., Merrifield, M.R., Napolitano, N.R., Capaccioli, M., & Freeman, K.C., 2003, Science, 301, 1696 Saglia, R.P., Kronawitter, A., Gerhard, O., & Bender, R. 2000, AJ, 119, 153 Samurović, S. 2006, Serb. Astron. J., 173, 35 Schiminovich, D., van Gorkom, J.H., van der Hulst, J.M., & Kasow, S. 1994, ApJ, 423, L101 Sharples, R. 1988, IAU Symp. 126: The Harlow-Sharpley Symposium on Globular Cluster Systems in Galaxies, 126, 245 http://arxiv.org/abs/astro-ph/0703385 – 28 – Strader, J., Brodie, J.P., Cenarro, A.J., Beasley, M.A., & Forbes, D.A. 2005, AJ, 130, 1315 Sugerman, B., Summers, F.J., & Kamionkowski, M. 2000, MNRAS, 311, 762 Thomas, D., Maraston, C., & Korn, A. 2004, MNRAS, 351, 19 Toomre, A., & Toomre, J. 1972, ApJ, 178, 623 van den Bergh, S., Hesser, J.E., & Harris, G.L.H. 1981, AJ, 86, 24 Vitvitska, M., Klypin, A.A., Kravtsov, A.V., Wechsler, R.H., Primack, J.R., & Bullock, J.S. 2002, ApJ, 581, 799 Woodley, K.A., Harris, W.E., & Harris, G.L.H. 2005, AJ, 129, 2654 Woodley, K.A. 2006, AJ, 132, 2424 This preprint was prepared with the AAS LATEX macros v5.2. Table 1. The Globular Cluster System Catalogue of NGC 5128 Name Old Name RA Dec. R U B V R I σU σB σV σR σI C M T1 σC σM σT1 U − B B − V V − R V − I M − T1 C − M C − T1 vr (J2000) (J2000) (’) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (km s−1) GC0001 pff gc-028 13 20 01.16 -42 56 51.5 6.46 20.11 19.68 18.75 18.20 17.60 0.07 0.02 0.01 0.01 0.03 19.92 19.04 18.22 0.01 0.01 0.01 0.43 0.93 0.55 1.15 0.82 0.88 1.70 524±165 GC0002 HH-048 13 22 45.36 -43 07 08.8 30.26 - - - - - - - - - - - - - - - - - - - - - - - 856±507 GC0003 HH-080 13 23 38.33 -42 46 22.8 24.84 - - - - - - - - - - 19.45 18.72 17.98 0.02 0.01 0.01 - - - - 0.74 0.72 1.46 497±337 GC0004 HGHH-40/HH-044/C40 13 23 42.37 -43 09 37.8 21.02 19.37 19.01 18.12 17.59 17.03 0.04 0.01 0.01 0.01 0.03 20.12 19.28 18.49 0.02 0.01 0.01 0.35 0.89 0.53 1.10 0.79 0.84 1.63 365±244,5,7 GC0005 HGHH-01/C1 13 23 44.19 -43 11 11.8 21.41 - - - - - - - - - - 18.73 18.00 17.25 0.06 0.03 0.03 - - - - 0.76 0.72 1.48 642±11,5,7,8 GC0006 pff gc-001 13 23 49.62 -43 14 32.0 22.36 20.39 19.91 18.91 18.36 17.60 0.09 0.02 0.01 0.01 0.03 20.22 19.25 18.38 0.02 0.01 0.01 0.47 1.01 0.55 1.31 0.87 0.97 1.84 711±355,7 GC0007 AAT301956 13 23 54.52 -43 20 01.1 25.41 21.22 21.13 20.42 19.98 19.43 0.18 0.06 0.02 0.02 0.04 - - - - - - 0.09 0.71 0.44 0.98 - - - 287±1627 GC0008 HH-099 13 23 56.70 -42 59 59.8 16.66 - - - - - - - - - - 21.72 20.90 20.01 0.09 0.09 0.06 - - - - 0.89 0.82 1.71 798±497 GC0009 AAT101931 13 23 58.58 -42 57 17.0 16.73 20.58 20.57 19.88 19.43 18.96 0.06 0.02 0.01 0.01 0.03 20.60 20.09 19.45 0.01 0.01 - 0.02 0.68 0.45 0.92 0.64 0.51 1.15 590±1447 GC0010 AAT101906 13 23 58.76 -43 01 35.2 16.25 20.45 19.90 18.89 18.28 17.57 0.06 0.02 0.01 0.01 0.03 20.23 19.11 18.28 0.02 0.03 0.02 0.54 1.01 0.61 1.32 0.83 1.12 1.95 511±317 GC0011 pff gc-002 13 23 59.51 -43 17 29.1 22.94 20.70 20.43 19.55 19.02 18.44 0.07 0.02 0.01 0.01 0.03 20.61 19.83 19.07 0.03 0.03 0.01 0.27 0.88 0.53 1.11 0.76 0.78 1.55 653±375,7 GC0012 AAT102120 13 23 59.61 -42 55 19.4 17.11 20.60 20.59 19.92 19.49 19.03 0.06 0.02 0.01 0.01 0.03 20.61 20.13 19.49 0.01 0.01 0.01 0.02 0.67 0.44 0.89 0.64 0.48 1.12 293±837 GC0013 HH-001 13 24 02.67 -42 48 32.2 20.00 - - - - - - - - - - 22.52 21.20 19.89 0.08 0.05 0.03 - - - - 1.31 1.32 2.63 775±747 GC0014 pff gc-003 13 24 03.23 -43 28 13.9 31.17 20.39 20.18 19.31 18.80 18.21 0.15 0.02 0.01 0.01 0.03 20.39 19.64 18.80 0.02 0.02 0.02 0.21 0.87 0.51 1.09 0.84 0.76 1.59 704±225,7 GC0015 pff gc-004 13 24 03.74 -43 35 53.4 37.98 21.12 20.84 20.01 19.49 18.92 0.28 0.03 0.01 0.01 0.03 - - - - - - 0.28 0.83 0.52 1.09 - - - 571±385,7 GC0016 AAT103195 13 24 05.98 -43 03 54.7 15.18 21.57 21.10 20.13 19.58 18.98 0.14 0.03 0.01 0.01 0.03 21.34 20.39 19.50 0.01 0.01 0.01 0.47 0.97 0.56 1.15 0.89 0.95 1.84 277±677 GC0017 AAT30486710 13 24 08.62 -43 16 27.1 21.04 22.12 21.15 19.91 19.20 18.54 0.25 0.03 0.01 0.01 0.03 21.62 20.28 19.18 0.01 - - 0.97 1.24 0.71 1.36 1.10 1.34 2.44 305±567 GC0018 AAT305341 13 24 10.97 -43 12 52.8 18.27 22.13 21.54 20.51 19.88 19.19 0.23 0.04 0.01 0.01 0.03 21.87 20.80 19.90 0.01 0.01 0.01 0.59 1.03 0.63 1.31 0.89 1.07 1.97 440±1187 GC0019 HHH86-28/C28 13 24 18.06 -42 49 01.1 17.57 19.52 19.33 18.50 17.99 17.45 0.03 0.01 0.01 0.01 0.03 19.46 18.75 18.03 0.02 0.01 0.01 0.19 0.83 0.52 1.05 0.72 0.71 1.43 461±223,5,7 GC0020 pff gc-005 13 24 18.92 -43 14 30.1 18.33 21.89 21.31 20.29 19.71 19.07 0.18 0.04 0.01 0.01 0.03 21.63 20.64 19.72 0.02 0.02 0.01 0.58 1.02 0.58 1.22 0.91 1.00 1.91 750±335,7 GC0021 WHH-1/HH-096 13 24 21.40 -43 02 36.8 12.19 19.14 18.85 18.01 17.48 16.98 0.03 0.01 0.01 0.01 0.03 19.11 18.33 17.63 0.02 0.02 0.03 0.29 0.84 0.53 1.03 0.70 0.78 1.48 583±296,7 GC0022 pff gc-006 13 24 23.72 -43 07 52.1 13.48 20.22 20.00 19.22 18.70 18.20 0.05 0.02 0.01 0.01 0.03 20.10 19.46 18.73 0.01 0.01 0.01 0.21 0.78 0.52 1.02 0.73 0.65 1.37 644±285,7 GC0023 WHH-2 13 24 23.98 -42 54 10.7 13.56 20.57 20.45 19.67 19.17 18.65 0.06 0.02 0.01 0.01 0.03 20.54 19.94 19.27 0.03 0.03 0.02 0.12 0.78 0.50 1.02 0.68 0.59 1.27 582±816 GC0024 pff gc-007 13 24 24.15 -42 54 20.6 13.45 21.85 21.16 20.12 19.53 18.82 0.17 0.03 0.01 0.01 0.03 21.54 20.50 19.60 0.02 0.02 0.01 0.69 1.04 0.59 1.30 0.90 1.04 1.94 617±255,7 GC0025 AAT308432 13 24 25.55 -43 21 35.6 23.38 20.98 20.43 19.40 18.80 18.11 0.27 0.02 0.01 0.01 0.03 20.83 19.78 18.84 0.03 0.01 0.02 0.54 1.04 0.60 1.29 0.94 1.05 1.99 835±837 GC0026 C111 13 24 26.97 -43 17 20.0 19.62 - - - - - - - - - - 22.54 22.25 21.36 0.04 0.04 0.04 - - - - 0.89 0.29 1.17 - GC0027 AAT106695 13 24 28.18 -42 53 04.3 13.54 20.95 20.75 19.63 18.96 18.30 0.09 0.03 0.01 0.01 0.03 21.16 20.17 19.19 0.06 0.06 0.04 0.20 1.12 0.67 1.33 0.98 0.99 1.96 835±837 GC0028 AAT106880 13 24 28.44 -42 57 52.9 11.30 21.11 20.94 20.14 19.61 19.11 0.10 0.03 0.01 0.01 0.03 21.06 20.43 19.65 0.01 0.01 0.01 0.18 0.80 0.53 1.03 0.79 0.63 1.41 558±987 GC0029 pff gc-008 13 24 29.20 -43 21 56.5 23.38 21.40 20.93 19.94 19.41 18.75 0.35 0.03 0.01 0.01 0.03 21.24 20.31 19.43 0.02 0.01 0.01 0.47 0.99 0.54 1.19 0.87 0.93 1.81 466±385,7 GC0030 AAT107060 13 24 29.23 -43 08 36.6 13.02 21.81 21.26 20.26 19.63 19.00 0.19 0.04 0.01 0.01 0.03 21.60 20.62 19.69 0.01 0.01 0.01 0.55 1.01 0.63 1.25 0.93 0.99 1.92 600±587 GC0031 AAT107145 13 24 29.73 -43 02 06.5 10.62 21.20 20.93 20.08 19.50 18.97 0.11 0.03 0.01 0.01 0.03 21.16 20.33 19.55 0.03 0.02 0.04 0.27 0.85 0.58 1.11 0.78 0.82 1.60 595±2027 GC0032 pff gc-009 13 24 31.35 -43 11 26.7 14.55 20.82 20.58 19.77 19.25 18.70 0.07 0.02 0.01 0.01 0.03 20.73 20.05 19.26 0.03 0.02 0.01 0.23 0.81 0.52 1.07 0.80 0.68 1.47 683±385,7 GC0033 WHH-3 13 24 32.17 -43 10 56.9 14.10 20.56 20.32 19.49 18.97 18.43 0.06 0.02 0.01 0.01 0.03 20.46 19.76 18.99 0.02 0.02 0.01 0.25 0.83 0.52 1.06 0.76 0.71 1.47 709±546,7 GC0034 C112 13 24 32.66 -43 18 48.8 20.32 - - - - - - - - - - 22.98 22.22 21.31 0.03 0.02 0.02 - - - - 0.91 0.75 1.67 - GC0035 pff gc-0109 13 24 33.09 -43 18 44.8 20.23 20.45 20.36 19.68 19.26 18.84 0.06 0.02 0.01 0.01 0.03 20.45 19.91 19.24 0.01 0.01 0.01 0.09 0.68 0.42 0.85 0.67 0.54 1.21 344±585 GC0036 AAT107977 13 24 34.63 -43 12 50.5 15.18 21.73 21.18 20.18 19.57 18.89 0.17 0.04 0.01 0.01 0.03 21.52 20.54 19.59 0.02 0.02 0.01 0.54 1.01 0.60 1.28 0.95 0.98 1.93 517±1237 GC0037 pff gc-011 13 24 36.87 -43 19 16.2 20.36 20.01 19.82 19.03 18.53 18.00 0.13 0.02 0.01 0.01 0.03 19.95 19.33 18.55 0.01 0.01 0.01 0.18 0.79 0.50 1.04 0.78 0.62 1.40 616±415,7 GC0038 C113 13 24 37.75 -43 16 26.5 17.80 - - - - - - - - - - 20.50 19.87 19.12 0.02 0.02 0.01 - - - - 0.75 0.63 1.38 - GC0039 pff gc-012 13 24 38.77 -43 06 26.6 10.38 21.24 20.51 19.45 18.83 18.15 0.13 0.02 0.01 0.01 0.03 20.92 19.78 18.85 0.01 0.01 0.01 0.73 1.06 0.62 1.30 0.93 1.14 2.07 573±215 GC0040 HGHH-41/C41 13 24 38.98 -43 20 06.4 20.94 20.20 19.59 18.59 17.94 17.32 0.06 0.02 0.01 0.01 0.03 19.95 18.92 17.97 0.02 0.01 0.01 0.61 1.00 0.65 1.27 0.95 1.03 1.98 363±14,5,7,8 GC0041 HGHH-29/C29 13 24 40.39 -43 18 05.3 19.01 19.77 19.15 18.15 17.54 16.89 0.04 0.01 0.01 0.01 0.03 19.46 18.37 17.53 0.04 0.03 0.02 0.62 1.00 0.61 1.26 0.84 1.09 1.92 726±13,5,7,8 GC0042 pff gc-013 13 24 40.42 -43 35 04.9 35.01 20.06 19.83 19.00 18.49 17.98 0.15 0.02 0.01 0.01 0.03 - - - - - - 0.24 0.83 0.51 1.03 - - - 727±315,7 GC0043 C114 13 24 40.48 -42 53 35.3 11.46 - - - - - - - - - - 23.46 22.39 21.42 0.05 0.04 0.03 - - - - 0.97 1.07 2.04 - GC0044 WHH-4/HH-024 13 24 40.60 -43 13 18.1 14.89 20.71 20.12 19.12 18.50 17.85 0.07 0.02 0.01 0.01 0.03 20.46 19.40 18.53 0.03 0.02 0.01 0.60 1.00 0.62 1.27 0.87 1.06 1.93 688±256,7 GC0045 pff gc-014 13 24 41.05 -42 59 48.4 8.62 21.60 21.17 20.23 19.67 18.94 0.26 0.06 0.02 0.02 0.03 21.47 20.56 19.76 0.04 0.03 0.04 0.43 0.94 0.57 1.30 0.80 0.91 1.71 690±345,7 GC0046 pff gc-015 13 24 41.20 -43 01 45.6 8.51 20.27 20.01 19.22 18.74 18.15 0.06 0.02 0.01 0.01 0.03 20.12 19.40 18.70 0.05 0.04 0.05 0.26 0.79 0.47 1.06 0.70 0.72 1.42 533±255,7 GC0047 AAT109380 13 24 43.58 -43 08 43.2 11.05 19.64 19.52 18.80 18.32 17.87 0.04 0.01 0.01 0.01 0.03 19.60 19.02 18.30 0.01 - - 0.12 0.73 0.48 0.93 0.72 0.59 1.31 465±387 GC0048 pff gc-016 13 24 43.60 -42 53 07.3 11.36 21.38 20.87 19.85 19.26 18.58 0.12 0.03 0.01 0.01 0.03 21.19 20.21 19.33 0.02 0.02 0.01 0.51 1.02 0.60 1.27 0.88 0.98 1.85 505±175,7 GC0049 pff gc-017 13 24 43.63 -42 58 16.4 8.54 20.54 20.32 19.51 18.99 18.47 0.07 0.02 0.01 0.01 0.03 20.44 19.78 19.01 0.01 0.01 0.01 0.22 0.81 0.52 1.04 0.77 0.66 1.43 963±505 Table 1—Continued Name Old Name RA Dec. R U B V R I σU σB σV σR σI C M T1 σC σM σT1 U − B B − V V − R V − I M − T1 C − M C − T1 vr (J2000) (J2000) (’) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (km s−1) GC0050 WHH-5 13 24 44.58 -43 02 47.3 8.04 20.47 20.03 19.06 18.49 17.87 0.08 0.02 0.01 0.01 0.03 20.29 19.19 18.90 0.05 0.04 0.04 0.44 0.97 0.57 1.19 0.29 1.09 1.39 671±416 GC0051 HH-054 13 24 45.22 -43 23 19.2 23.49 - - - - - - - - - - 20.92 20.08 19.25 0.05 0.03 0.04 - - - - 0.82 0.84 1.66 998±2507 GC0052 AAT109711 13 24 45.35 -42 59 33.5 7.89 19.88 19.87 19.30 18.90 18.33 0.06 0.02 0.01 0.01 0.03 19.84 19.48 18.91 0.02 0.02 0.04 0.02 0.56 0.40 0.98 0.58 0.36 0.94 276±597 GC0053 AAT109788 13 24 45.78 -43 02 24.5 7.75 21.82 21.02 19.89 19.23 18.56 0.25 0.04 0.01 0.01 0.03 21.39 20.17 19.43 0.02 0.02 0.03 0.80 1.14 0.65 1.33 0.74 1.22 1.96 527±307 GC0054 HGHH-G019/G19 13 24 46.46 -43 04 11.6 8.12 20.16 19.95 19.16 18.64 18.13 0.06 0.02 0.01 0.01 0.03 20.06 19.38 18.64 0.04 0.03 0.03 0.21 0.79 0.52 1.03 0.75 0.67 1.42 710±224,5,7 GC0055 HGHH-09/C9 13 24 46.72 -43 01 18.4 7.48 20.00 19.76 18.95 18.40 17.90 0.05 0.02 0.01 0.01 0.03 19.87 19.13 18.42 0.04 0.04 0.04 0.24 0.81 0.56 1.05 0.70 0.74 1.45 501±214,5 GC0056 pff gc-018 13 24 47.10 -43 06 01.7 8.87 20.90 20.03 18.91 18.25 17.54 0.10 0.02 0.01 0.01 0.03 20.45 19.23 18.28 0.02 0.01 0.01 0.87 1.12 0.66 1.37 0.95 1.22 2.17 534±155,6,7 GC0057 HGHH-G277/G277 13 24 47.37 -42 58 29.8 7.82 20.26 19.96 19.10 18.56 18.02 0.06 0.02 0.01 0.01 0.03 20.14 19.37 18.61 0.01 0.01 0.01 0.30 0.87 0.53 1.08 0.76 0.77 1.53 714±334,5,7 GC0058 WHH-6 13 24 47.37 -42 57 51.2 8.06 21.33 20.75 19.73 19.12 18.44 0.15 0.03 0.01 0.01 0.03 21.08 20.07 19.15 0.01 0.01 0.01 0.58 1.02 0.62 1.29 0.93 1.00 1.93 685±436,7 GC0059 AAT110138 13 24 47.61 -43 10 48.5 12.12 20.87 20.53 19.67 19.11 18.57 0.10 0.03 0.01 0.01 0.03 20.75 19.98 19.16 0.03 0.02 0.01 0.34 0.87 0.56 1.10 0.82 0.77 1.59 358±1377 GC0060 HHH86-10/C10 13 24 48.06 -43 08 14.2 10.13 19.87 19.44 18.49 17.91 17.33 0.05 0.01 0.01 0.01 0.03 19.66 18.75 17.93 0.03 0.03 0.01 0.44 0.95 0.58 1.16 0.81 0.91 1.73 829±222,5,6,7 GC0061 C115 13 24 48.71 -42 52 35.5 11.12 - - - - - - - - - - 20.88 20.23 19.51 0.02 0.02 0.01 - - - - 0.73 0.65 1.37 - GC0062 pff gc-019 13 24 48.97 -42 57 48.4 7.81 20.10 19.87 19.05 18.53 18.00 0.05 0.02 0.01 0.01 0.03 20.00 19.31 18.57 0.03 0.03 0.01 0.23 0.81 0.53 1.05 0.74 0.69 1.43 607±275 GC0063 AAT110410 13 24 49.38 -43 08 17.7 10.00 21.47 21.18 20.34 19.79 19.23 0.16 0.04 0.02 0.01 0.03 21.37 20.66 19.81 0.03 0.02 0.01 0.29 0.84 0.55 1.11 0.85 0.71 1.56 580±817 GC0064 AAT110551 13 24 50.09 -43 07 36.2 9.42 21.43 21.08 20.44 20.04 19.49 0.16 0.04 0.02 0.02 0.03 21.19 20.70 20.03 0.02 0.02 0.01 0.35 0.64 0.41 0.96 0.67 0.49 1.16 615±767 GC0065 pff gc-020 13 24 50.48 -42 59 49.0 6.92 20.57 20.33 19.53 19.00 18.47 0.09 0.03 0.01 0.01 0.03 20.42 19.58 19.68 0.05 0.04 0.04 0.24 0.80 0.53 1.06 -0.11 0.85 0.74 331±475 GC0066 HGHH-42/C42 13 24 50.87 -43 01 22.9 6.72 20.53 19.91 18.86 18.24 17.53 0.09 0.02 0.01 0.01 0.03 20.21 19.11 18.21 0.04 0.03 0.05 0.62 1.05 0.63 1.33 0.90 1.10 2.00 552±124,5,7 GC0067 VHH81-02/C2 13 24 51.49 -43 12 11.1 12.86 19.64 19.33 18.50 18.01 17.42 0.04 0.01 0.01 0.01 0.03 19.48 18.72 17.94 0.04 0.03 0.01 0.31 0.83 0.49 1.07 0.78 0.77 1.55 628±222,5,6,7 GC0068 C100 13 24 51.80 -43 04 33.7 7.33 - - 20.08 - - - - - - - 20.44 19.72 19.03 0.08 0.08 0.10 - - - 1.28 0.69 0.72 1.41 - GC0069 AAT111033 13 24 52.98 -43 11 55.8 12.50 21.17 20.93 20.12 19.66 19.13 0.11 0.03 0.01 0.01 0.03 21.10 20.40 19.64 0.01 0.01 0.01 0.24 0.81 0.46 0.99 0.76 0.70 1.47 302±1667 GC0070 HGHH-G302/G302 13 24 53.29 -43 04 34.8 7.15 20.30 20.01 19.20 18.69 18.16 0.07 0.02 0.01 0.01 0.03 20.18 19.50 18.73 0.02 0.01 0.01 0.28 0.81 0.52 1.04 0.77 0.68 1.45 558±434,5 GC0071 AAT111185 13 24 54.00 -43 04 24.4 6.96 20.92 20.76 19.92 19.41 18.92 0.11 0.03 0.01 0.01 0.03 20.90 20.28 19.37 0.02 0.02 0.02 0.16 0.84 0.51 1.00 0.91 0.62 1.53 466±877 GC0072 pff gc-021 13 24 54.18 -42 54 50.4 8.78 20.21 20.03 19.25 18.75 18.24 0.06 0.02 0.01 0.01 0.03 20.13 19.52 18.81 0.03 0.02 0.01 0.18 0.78 0.50 1.01 0.70 0.62 1.32 594±505 GC0073 pff gc-022 13 24 54.33 -43 03 15.5 6.44 20.72 20.58 19.84 19.37 18.90 0.11 0.03 0.01 0.01 0.03 20.69 20.09 19.37 0.03 0.02 0.02 0.14 0.74 0.48 0.95 0.72 0.60 1.32 619±445,7 GC0074 HHH86-30/C30 13 24 54.35 -42 53 24.8 9.84 18.76 18.24 17.25 16.67 16.02 0.03 0.01 0.01 0.01 0.03 18.47 17.49 16.68 0.03 0.04 0.02 0.52 0.98 0.58 1.23 0.81 0.98 1.79 778±133,5,6,7 GC0075 AAT111296 13 24 54.49 -43 05 34.7 7.50 21.70 20.99 19.96 19.31 18.61 0.23 0.04 0.01 0.01 0.03 21.40 20.28 19.33 0.02 0.01 0.01 0.71 1.03 0.65 1.35 0.96 1.11 2.07 695±457 GC0076 pff gc-023 13 24 54.55 -42 48 58.7 13.59 20.55 20.30 19.44 18.90 18.37 0.06 0.02 0.01 0.01 0.03 20.48 19.74 18.97 0.01 0.01 0.01 0.25 0.86 0.54 1.08 0.77 0.74 1.51 457±315,7 GC0077 HGHH-11/C11 13 24 54.73 -43 01 21.7 6.01 19.55 18.96 17.91 17.26 16.61 0.05 0.01 0.01 0.01 0.03 19.21 17.98 17.20 0.06 0.05 0.06 0.59 1.05 0.66 1.30 0.79 1.23 2.01 753±12,4,5,6,7,8 GC0078 AAT111406 13 24 55.29 -43 03 15.6 6.28 21.84 21.15 20.13 19.50 18.80 0.30 0.05 0.02 0.01 0.03 21.66 20.70 18.80 0.03 0.02 0.01 0.69 1.03 0.63 1.33 1.90 0.96 2.86 669±777 GC0079 C116 13 24 55.46 -43 09 58.5 10.61 - - - - - - - - - - 23.80 22.74 21.84 0.06 0.02 0.03 - - - - 0.90 1.07 1.97 - GC0080 HH-052 13 24 55.71 -43 22 48.4 22.43 - - - - - - - - - - 22.58 21.61 20.42 0.05 0.03 0.03 - - - - 1.19 0.97 2.16 921±1467 GC0081 pff gc-024 13 24 55.71 -43 20 39.1 20.36 21.07 20.73 19.84 19.31 18.73 0.28 0.03 0.01 0.01 0.03 21.00 20.17 19.34 0.02 0.01 0.01 0.34 0.89 0.53 1.11 0.83 0.83 1.66 279±385,7 GC0082 C117 13 24 56.06 -42 54 29.6 8.80 - - - - - - - - - - 21.27 20.18 19.26 0.01 0.02 0.01 - - - - 0.92 1.09 2.01 - GC0083 AAT111563 13 24 56.08 -43 10 16.4 10.79 21.34 21.03 20.45 20.05 19.62 0.15 0.04 0.02 0.02 0.03 21.14 20.73 20.05 0.02 0.01 0.02 0.31 0.58 0.41 0.83 0.68 0.41 1.09 649±1027 GC0084 HGHH-G279 13 24 56.27 -43 03 23.4 6.15 19.97 19.91 19.45 19.17 18.88 0.06 0.02 0.01 0.01 0.03 19.76 19.38 19.83 0.01 0.01 0.01 0.06 0.45 0.29 0.57 -0.45 0.38 -0.07 366±344,5,7 GC0085 C118 13 24 57.17 -43 08 42.6 9.39 - - - - - - - - - - 22.60 21.76 20.67 0.04 0.03 0.02 - - - - 1.08 0.84 1.92 - GC0086 HGHH-31/C31 13 24 57.44 -43 01 08.1 5.52 20.09 19.42 18.38 17.75 17.06 0.08 0.02 0.01 0.01 0.03 19.73 18.59 17.71 0.04 0.03 0.04 0.67 1.04 0.62 1.31 0.88 1.14 2.02 690±183,5,7 GC0087 HGHH-G369 13 24 57.52 -42 59 23.3 5.78 19.82 19.58 18.74 18.22 17.69 0.06 0.02 0.01 0.01 0.03 19.67 18.96 18.24 0.04 0.03 0.05 0.24 0.83 0.53 1.06 0.72 0.71 1.44 512±174,5,6,7 GC0088 pff gc-025 13 24 57.56 -43 05 32.8 7.04 21.50 21.02 20.02 19.39 18.67 0.20 0.04 0.02 0.01 0.03 21.41 20.41 19.45 0.04 0.03 0.02 0.47 1.00 0.63 1.35 0.96 1.00 1.97 923±305 GC0089 C119 13 24 57.69 -42 55 48.4 7.64 - - - - - - - - - - 21.16 21.02 20.67 0.03 0.03 0.02 - - - - 0.35 0.14 0.49 - GC0090 C120 13 24 57.95 -42 52 04.9 10.56 - - - - - - - - - - 23.26 22.34 21.43 0.04 0.04 0.03 - - - - 0.90 0.92 1.83 - GC0091 VHH81-03/C3 13 24 58.21 -42 56 10.0 7.33 19.34 18.73 17.71 17.10 16.44 0.04 0.01 0.01 0.01 0.03 19.02 17.88 17.08 0.04 0.03 0.02 0.61 1.02 0.61 1.26 0.80 1.14 1.94 562±21,3,5,6,7,8 GC0092 C121 13 24 58.42 -43 08 21.2 8.97 - - - - - - - - - - 23.53 23.01 22.45 0.05 0.04 0.09 - - - - 0.56 0.52 1.08 - GC0093 pff gc-026 13 24 58.45 -42 42 53.3 19.02 20.44 20.33 19.52 19.07 18.50 0.08 0.02 0.01 0.01 0.03 20.47 19.82 19.09 0.02 0.01 0.01 0.12 0.81 0.45 1.02 0.73 0.65 1.37 490±675,7 GC0094 C12211 13 24 59.01 -43 08 21.4 8.91 - - - - - - - - - - - - 22.33 - - - - - - - - - - - GC0095 C123 13 24 59.92 -43 09 08.6 9.46 - - - - - - - - - - 21.88 21.10 20.24 0.02 0.01 0.01 - - - - 0.86 0.78 1.64 - GC0096 AAT112158 13 25 00.15 -42 54 09.0 8.61 21.99 21.43 20.43 19.86 19.23 0.25 0.05 0.02 0.01 0.03 21.75 20.78 19.88 0.01 0.01 0.01 0.55 1.00 0.57 1.20 0.90 0.96 1.87 699±437 GC0097 C124 13 25 00.37 -43 10 46.9 10.85 - - - - - - - - - - 23.07 22.36 21.57 0.03 0.02 0.03 - - - - 0.78 0.71 1.50 - GC0098 pff gc-027 13 25 00.64 -43 05 30.3 6.58 21.22 20.75 19.74 19.13 18.45 0.17 0.04 0.01 0.01 0.03 21.05 20.06 19.11 0.01 0.01 0.01 0.47 1.01 0.61 1.29 0.95 0.99 1.94 524±345,7 Table 1—Continued Name Old Name RA Dec. R U B V R I σU σB σV σR σI C M T1 σC σM σT1 U − B B − V V − R V − I M − T1 C − M C − T1 vr (J2000) (J2000) (’) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (km s−1) GC0099 C125 13 25 00.83 -43 11 10.6 11.16 - - - - - - - - - - 22.34 21.55 20.66 0.03 0.02 0.02 - - - - 0.89 0.78 1.67 - GC0100 C126 13 25 00.91 -43 09 14.5 9.45 - - - - - - - - - - 24.02 23.54 22.66 0.04 0.05 0.08 - - - - 0.88 0.48 1.35 - GC0101 C127 13 25 01.32 -43 08 43.4 8.97 - - - - - - - - - - 23.21 22.42 21.62 0.04 0.02 0.03 - - - - 0.80 0.79 1.59 - GC0102 C128 13 25 01.46 -43 08 33.0 8.81 - - - - - - - - - - 23.08 22.04 20.95 0.05 0.03 0.02 - - - - 1.09 1.04 2.13 - GC0103 pff gc-029 13 25 01.60 -42 54 40.9 8.03 19.97 19.75 19.37 19.06 18.71 0.05 0.02 0.01 0.01 0.03 19.72 19.50 19.10 0.03 0.03 0.01 0.22 0.38 0.31 0.65 0.40 0.22 0.62 570±385 GC0104 C129 13 25 01.63 -42 50 51.3 11.34 - - - - - - - - - - 22.59 21.71 20.83 0.04 0.02 0.02 - - - - 0.88 0.88 1.76 - GC0105 pff gc-030 13 25 01.73 -43 00 09.9 4.83 21.68 20.94 19.89 19.26 18.54 0.36 0.06 0.02 0.02 0.03 21.21 19.93 20.43 0.05 0.04 0.02 0.73 1.05 0.64 1.35 -0.50 1.29 0.78 357±295 GC0106 HGHH-04/C4 13 25 01.83 -43 09 25.4 9.52 19.10 18.86 18.04 17.50 16.98 0.03 0.01 0.01 0.01 0.03 18.95 18.24 17.50 0.04 0.03 0.01 0.23 0.82 0.54 1.06 0.74 0.71 1.45 689±161,3,5,6,7 GC0107 C130 13 25 01.86 -42 52 27.8 9.88 - - - - - - - - - - 21.22 20.58 19.85 0.01 0.01 0.01 - - - - 0.73 0.64 1.37 - GC0108 pff gc-031 13 25 02.76 -43 11 21.2 11.17 20.51 20.29 19.48 18.99 18.42 0.07 0.02 0.01 0.01 0.03 20.43 19.75 18.95 0.03 0.02 0.01 0.23 0.81 0.49 1.06 0.80 0.68 1.48 444±335 GC0109 HGHH-G176 13 25 03.13 -42 56 25.1 6.51 20.47 19.94 18.93 18.32 17.67 0.09 0.02 0.01 0.01 0.03 20.24 19.23 18.36 0.01 0.02 0.01 0.54 1.01 0.61 1.26 0.87 1.01 1.88 551±134,5,7 GC0110 HGHH-G066 13 25 03.18 -43 03 02.5 4.85 20.26 19.67 18.68 18.15 17.44 0.09 0.02 0.01 0.01 0.03 19.94 18.96 18.05 0.01 0.01 0.02 0.59 0.99 0.52 1.24 0.91 0.98 1.89 576±84,5 GC0111 pff gc-032 13 25 03.24 -42 57 40.5 5.65 20.92 20.59 19.73 19.18 18.61 0.14 0.04 0.01 0.01 0.03 20.81 20.05 19.22 0.01 0.02 0.01 0.33 0.86 0.55 1.12 0.83 0.76 1.59 648±295,7 GC0112 pff gc-033 13 25 03.34 -43 15 27.4 14.98 21.10 20.49 19.46 18.89 18.21 0.33 0.03 0.01 0.01 0.03 20.83 19.79 18.90 0.02 0.02 0.01 0.61 1.02 0.57 1.25 0.89 1.04 1.93 531±185 GC0113 HHH86-32/C32 13 25 03.37 -42 50 46.2 11.28 20.22 19.50 18.44 17.81 17.15 0.05 0.01 0.01 0.01 0.03 19.86 18.71 17.85 0.03 0.03 0.01 0.72 1.06 0.63 1.29 0.85 1.15 2.01 718±113,5,6,7 GC0114 pff gc-034 13 25 03.37 -43 11 39.6 11.41 20.98 20.70 19.91 19.35 18.84 0.09 0.03 0.01 0.01 0.03 20.89 20.19 19.38 0.03 0.02 0.01 0.28 0.80 0.56 1.06 0.81 0.69 1.50 605±465 GC0115 C131 13 25 03.67 -42 51 21.7 10.72 - - - - - - - - - - 21.57 20.98 20.20 0.02 0.02 0.01 - - - - 0.78 0.59 1.36 - GC0116 AAT112752 13 25 04.12 -43 00 19.6 4.37 20.51 20.16 19.34 18.88 18.44 0.15 0.04 0.02 0.01 0.03 20.35 19.58 18.87 0.02 0.02 0.03 0.35 0.82 0.46 0.90 0.72 0.76 1.48 679±827 GC0117 pff gc-035 13 25 04.48 -43 10 48.4 10.54 21.32 20.73 19.71 19.07 18.44 0.14 0.03 0.01 0.01 0.03 21.13 20.07 19.13 0.03 0.02 0.01 0.59 1.02 0.64 1.27 0.94 1.06 2.00 627±225,7 GC0118 AAT112964 13 25 04.61 -43 07 21.7 7.50 20.45 20.39 19.61 19.13 18.68 0.08 0.03 0.01 0.01 0.03 20.47 19.90 19.11 0.01 0.01 0.01 0.06 0.78 0.49 0.94 0.78 0.57 1.35 456±1187 GC0119 HGHH-43/C43 13 25 04.81 -43 09 38.8 9.47 19.74 19.45 18.60 18.07 17.53 0.04 0.01 0.01 0.01 0.03 19.59 18.85 18.07 0.03 0.02 0.01 0.29 0.85 0.53 1.07 0.78 0.74 1.52 518±254,5,7 GC0120 WHH-7 13 25 05.02 -42 57 15.0 5.68 18.96 18.39 17.41 16.81 16.19 0.04 0.01 0.01 0.01 0.03 18.69 17.60 16.83 0.02 0.03 0.01 0.57 0.98 0.60 1.22 0.77 1.09 1.86 722±206 GC0121 HGHH-G035 13 25 05.29 -42 58 05.8 5.09 21.02 20.59 19.62 19.01 18.41 0.17 0.04 0.01 0.01 0.03 20.87 19.95 19.08 0.01 0.02 0.01 0.43 0.97 0.61 1.21 0.87 0.92 1.80 776±264,5 GC0122 pff gc-036 13 25 05.46 -43 14 02.6 13.52 20.09 19.83 19.04 18.52 17.91 0.05 0.02 0.01 0.01 0.03 19.96 19.27 18.49 0.02 0.02 0.01 0.26 0.79 0.52 1.13 0.78 0.69 1.47 666±305,7 GC0123 HGHH-12/C12/R281 13 25 05.72 -43 10 30.7 10.18 - - - - - - - - - - 19.34 18.23 17.36 0.04 0.03 0.02 - - - - 0.87 1.11 1.98 440±12,5,8 GC0124 HGHH-G342 13 25 05.83 -42 59 00.6 4.52 19.65 19.14 18.18 17.57 16.96 0.08 0.02 0.01 0.01 0.03 - - - - - - 0.51 0.96 0.61 1.22 - - - 553±214,5,6,7 GC0125 HHH86-13/C13 13 25 06.25 -43 15 11.6 14.58 20.02 19.60 18.68 18.13 17.51 0.04 0.01 0.01 0.01 0.03 19.82 18.90 18.12 0.03 0.03 0.02 0.41 0.93 0.55 1.16 0.78 0.92 1.70 601±125,6,7 GC0126 R276 13 25 07.33 -43 08 29.6 8.23 21.93 21.62 20.78 20.26 19.73 0.26 0.07 0.02 0.02 0.04 21.88 21.16 20.32 0.03 0.03 0.02 0.31 0.83 0.52 1.05 0.84 0.71 1.55 550±285 GC0127 AAT113428 13 25 07.33 -43 06 20.6 6.38 21.25 20.92 20.04 19.49 18.93 0.16 0.04 0.01 0.01 0.03 21.14 20.36 19.51 0.02 0.01 0.01 0.33 0.88 0.54 1.11 0.85 0.78 1.63 657±677 GC0128 pff gc-037 13 25 07.48 -43 12 29.4 11.93 21.60 21.29 20.42 19.91 19.33 0.55 0.05 0.02 0.01 0.03 21.46 20.73 19.90 0.02 0.01 0.01 0.31 0.87 0.50 1.08 0.83 0.73 1.56 554±235 GC0129 WHH-8 13 25 07.62 -43 01 15.2 3.66 19.83 19.16 18.12 17.46 16.82 0.13 0.03 0.01 0.01 0.03 19.48 18.32 17.45 0.03 0.03 0.04 0.67 1.04 0.66 1.30 0.87 1.16 2.03 690±326 GC0130 WHH-9 13 25 08.51 -43 02 57.4 3.93 20.59 19.95 18.91 18.37 17.64 0.17 0.03 0.01 0.01 0.03 20.28 19.17 18.30 0.05 0.04 0.05 0.64 1.04 0.54 1.27 0.87 1.11 1.98 315±1006 GC0131 C132 13 25 08.79 -43 09 09.6 8.72 - - - - - - - - - - 20.38 19.69 18.90 0.02 0.02 0.01 - - - - 0.79 0.69 1.48 - GC0132 R271 13 25 08.81 -43 09 09.5 8.72 20.49 20.23 19.42 18.91 18.39 0.08 0.02 0.01 0.01 0.03 20.38 19.69 18.90 0.02 0.02 0.01 0.26 0.80 0.52 1.03 0.79 0.69 1.48 436±455 GC0133 pff gc-038 13 25 08.82 -43 04 14.9 4.63 20.83 20.29 19.29 18.69 18.03 0.17 0.04 0.01 0.01 0.03 20.58 19.57 18.71 0.03 0.03 0.03 0.54 1.00 0.61 1.26 0.87 1.01 1.88 431±195 GC0134 pff gc-039 13 25 09.10 -42 24 00.9 37.29 20.01 19.77 18.95 18.44 17.91 0.17 0.02 0.01 0.01 0.03 19.92 19.22 18.50 0.01 0.01 0.01 0.23 0.82 0.51 1.04 0.72 0.70 1.42 387±275,7 GC0135 K-029 13 25 09.19 -42 58 59.2 4.00 19.13 18.65 17.71 17.13 16.55 0.08 0.02 0.01 0.01 0.03 - - - - - - 0.48 0.94 0.58 1.16 - - - 677±447 GC0136 pff gc-040 13 25 09.54 -42 55 18.5 6.71 20.09 19.89 19.12 18.60 18.11 0.06 0.02 0.01 0.01 0.03 20.00 19.37 18.66 0.02 0.02 0.01 0.20 0.76 0.52 1.01 0.71 0.62 1.34 504±355 GC0137 HGHH-G329 13 25 10.20 -43 02 06.7 3.33 21.73 20.89 19.71 19.07 18.38 0.69 0.11 0.03 0.02 0.03 21.34 20.04 19.07 0.02 0.01 0.03 0.84 1.18 0.64 1.33 0.97 1.29 2.26 502±164,5,7 GC0138 K-033 13 25 10.25 -42 55 09.5 6.78 21.07 20.47 19.46 18.90 18.21 0.13 0.03 0.01 0.01 0.03 20.78 19.77 18.86 0.01 0.01 0.01 0.60 1.01 0.57 1.25 0.91 1.02 1.93 582±337 GC0139 K-034 13 25 10.27 -42 53 33.1 8.23 19.33 18.79 17.80 17.26 16.55 0.04 0.01 0.01 0.01 0.03 19.14 18.03 17.25 0.03 0.03 0.02 0.55 0.99 0.54 1.25 0.79 1.11 1.89 464±427 GC0140 HHH86-14/C14 13 25 10.49 -42 44 52.6 16.57 19.23 18.84 17.94 17.40 16.79 0.03 0.01 0.01 0.01 0.03 19.06 18.16 17.41 0.01 0.02 0.01 0.39 0.90 0.54 1.15 0.75 0.90 1.65 705±102,5,6,7 GC0141 AAT113992 13 25 10.51 -43 03 24.0 3.85 21.95 21.39 20.40 19.79 19.19 0.58 0.11 0.03 0.03 0.04 21.77 20.78 19.82 0.03 0.03 0.01 0.57 0.99 0.61 1.20 0.97 0.99 1.95 648±587 GC0142 C13313 13 25 11.05 -43 01 32.3 3.05 - - - - - - - - - - - - 19.30 - - - - - - - - - - - GC0143 HGHH-G348 13 25 11.10 -42 58 03.0 4.32 19.90 19.66 18.92 18.46 18.03 0.10 0.03 0.01 0.01 0.03 19.80 19.15 18.45 0.01 0.01 - 0.24 0.74 0.46 0.89 0.70 0.65 1.35 416±304,5,7 GC0144 pff gc-041 13 25 11.17 -43 03 09.6 3.62 20.57 20.32 19.51 18.99 18.48 0.19 0.05 0.02 0.02 0.03 20.48 19.77 19.01 0.02 0.02 0.02 0.24 0.81 0.52 1.03 0.76 0.71 1.47 456±295,7 GC0145 HGHH-G327 13 25 11.98 -43 04 19.3 4.27 21.46 20.83 19.78 19.16 18.56 0.34 0.06 0.02 0.02 0.03 21.16 20.11 19.19 0.01 0.01 0.01 0.63 1.05 0.61 1.22 0.92 1.05 1.97 608±194,5,7 GC0146 HGHH-G081 13 25 12.11 -42 57 25.2 4.68 19.90 19.52 18.62 18.05 17.49 0.08 0.02 0.01 0.01 0.03 19.72 18.88 18.10 0.03 0.03 0.01 0.38 0.90 0.57 1.13 0.77 0.85 1.62 536±434,5 GC0147 pff gc-042 13 25 12.21 -43 16 33.9 15.67 19.91 19.74 18.97 18.47 17.91 0.12 0.02 0.01 0.01 0.03 19.82 19.19 18.47 0.02 0.02 0.01 0.17 0.78 0.50 1.05 0.72 0.63 1.35 627±235,6,7 Table 1—Continued Name Old Name RA Dec. R U B V R I σU σB σV σR σI C M T1 σC σM σT1 U − B B − V V − R V − I M − T1 C − M C − T1 vr (J2000) (J2000) (’) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (km s−1) GC0148 AAT114302 13 25 12.34 -42 58 07.7 4.11 21.87 21.39 20.44 19.90 19.35 0.53 0.11 0.03 0.03 0.04 21.64 20.76 19.97 0.02 0.01 0.01 0.47 0.95 0.55 1.09 0.79 0.88 1.67 754±1437 GC0149 pff gc-043 13 25 12.45 -43 14 07.4 13.27 20.51 20.22 19.40 18.89 18.33 0.20 0.02 0.01 0.01 0.03 20.38 19.67 18.90 0.03 0.02 0.01 0.29 0.82 0.52 1.07 0.77 0.70 1.48 527±335,7 GC0150 WHH-10 13 25 12.84 -42 56 59.8 4.95 20.62 20.36 19.57 19.07 18.48 0.12 0.03 0.01 0.01 0.03 20.43 19.90 19.10 0.03 0.02 0.01 0.26 0.79 0.50 1.09 0.80 0.54 1.33 664±1416 GC0151 R261 13 25 12.90 -43 07 59.1 7.35 19.65 19.15 18.20 17.60 17.03 0.05 0.01 0.01 0.01 0.03 19.42 18.49 17.62 0.02 0.01 0.01 0.51 0.94 0.60 1.17 0.88 0.93 1.81 615±45,7,8 GC0152 pff gc-044 13 25 13.19 -43 16 35.6 15.67 20.29 20.11 19.37 18.87 18.33 0.16 0.02 0.01 0.01 0.03 20.19 19.59 18.87 0.02 0.02 0.01 0.17 0.74 0.50 1.04 0.72 0.60 1.32 568±535 GC0153 C134 13 25 13.20 -43 02 31.3 2.97 - - - - - - - - - - 22.45 21.64 20.71 0.07 0.05 0.07 - - - - 0.93 0.80 1.74 - GC0154 pff gc-045 13 25 13.31 -42 52 12.4 9.31 21.57 20.86 19.82 19.21 18.53 0.19 0.04 0.01 0.01 0.03 21.38 20.32 19.40 0.07 0.06 0.04 0.71 1.04 0.62 1.29 0.93 1.06 1.99 474±235 GC0155 R259 13 25 13.88 -43 07 32.5 6.87 22.13 21.70 20.70 20.11 19.48 0.37 0.08 0.02 0.02 0.03 21.96 21.07 20.10 0.02 0.01 0.01 0.43 1.00 0.60 1.23 0.96 0.89 1.85 628±485 GC0156 HGHH-G271 13 25 13.95 -42 57 42.6 4.25 19.45 19.36 18.72 18.32 17.90 0.07 0.02 0.01 0.01 0.03 19.40 18.92 18.30 0.01 0.01 - 0.09 0.63 0.40 0.82 0.62 0.47 1.09 353±274,5,6,7 GC0157 C13513 13 25 14.07 -43 00 51.8 2.49 - - - - - - - - - - - - 19.90 - - - - - - - - - - - GC0158 C136 13 25 14.07 -43 03 35.0 3.47 - - - - - - - - - - 22.48 21.50 21.19 0.04 0.02 0.02 - - - - 0.30 0.99 1.29 - GC0159 WHH-11/K-051 13 25 14.24 -43 07 23.5 6.71 21.26 20.63 19.55 18.90 18.21 0.18 0.04 0.01 0.01 0.03 21.02 19.95 18.95 0.03 0.02 0.01 0.64 1.07 0.65 1.34 1.00 1.06 2.07 582±306,7 GC0160 pff gc-046 13 25 14.83 -43 41 10.6 40.10 19.91 19.72 18.93 18.43 17.95 0.11 0.02 0.01 0.01 0.03 19.87 19.14 18.37 0.01 - 0.01 0.20 0.79 0.50 0.98 0.77 0.73 1.50 532±215,7 GC0161 pff gc-047 13 25 15.12 -42 50 30.4 10.88 20.92 20.72 19.87 19.37 18.87 0.09 0.03 0.01 0.01 0.03 20.87 20.22 19.41 0.02 0.02 0.01 0.20 0.85 0.50 0.99 0.81 0.65 1.46 717±555 GC0162 AAT114769 13 25 15.12 -42 57 45.7 4.08 21.20 21.00 20.25 19.76 19.25 0.30 0.09 0.03 0.03 0.04 21.11 20.52 19.80 0.02 0.02 0.01 0.20 0.75 0.49 1.00 0.71 0.59 1.30 650±1697 GC0163 R257 13 25 15.24 -43 08 39.2 7.84 22.29 21.77 20.78 20.18 19.58 0.35 0.07 0.02 0.02 0.03 22.02 21.09 20.16 0.02 0.02 0.02 0.51 1.00 0.60 1.20 0.92 0.93 1.85 339±515 GC0164 pff gc-048 13 25 15.79 -42 49 15.1 12.09 20.92 20.67 19.85 19.30 18.78 0.09 0.03 0.01 0.01 0.03 20.85 20.13 19.41 0.02 0.02 0.02 0.25 0.82 0.55 1.07 0.71 0.72 1.44 535±535,7 GC0165 AAT114913 13 25 15.93 -43 06 03.3 5.35 21.69 21.33 20.39 19.88 19.29 0.34 0.08 0.02 0.02 0.04 21.60 20.72 19.86 0.02 0.02 0.02 0.36 0.94 0.52 1.10 0.86 0.88 1.74 563±637 GC0166 pff gc-049 13 25 16.06 -43 05 06.5 4.49 20.74 20.45 19.59 19.05 18.50 0.15 0.04 0.01 0.01 0.03 20.64 19.85 19.07 0.03 0.03 0.03 0.29 0.86 0.54 1.09 0.78 0.79 1.57 674±325 GC0167 C13713 13 25 16.06 -43 02 19.3 2.42 - - - - - - - - - - - - 19.04 - - - - - - - - - - - GC0168 VHH81-05/C5 13 25 16.12 -42 52 58.2 8.44 18.70 18.49 17.68 17.13 16.68 0.03 0.01 0.01 0.01 0.03 - - - - - - 0.21 0.81 0.55 1.00 - - - 555±31,2,3,5,7,8 GC0169 HCH01 13 25 16.22 -42 59 43.4 2.52 18.69 18.36 17.47 16.94 16.40 0.52 0.15 0.04 0.03 0.04 - - - - - - 0.33 0.89 0.53 1.07 - - - 649±455,7 GC0170 HHH86-33/C33 13 25 16.26 -42 50 53.3 10.47 19.58 19.34 18.50 18.02 17.50 0.04 0.01 0.01 0.01 0.03 - - - - - - 0.24 0.84 0.48 1.00 - - - 596±183,5,6,7 GC0171 AAT1149939 13 25 16.44 -43 03 33.1 3.16 22.49 21.50 20.44 19.82 19.32 1.41 0.19 .382 0.308 0.224 21.95 20.81 19.83 0.02 0.02 0.03 0.99 1.06 0.62 1.12 0.98 1.14 2.12 352±1367 GC0172 HCH02 13 25 16.69 -43 02 08.7 2.23 19.27 18.87 17.93 17.37 16.76 0.98 0.26 0.07 0.05 0.05 - - - - - - 0.40 0.94 0.56 1.17 - - - 300±25,7,8 GC0173 pff gc-050 13 25 16.73 -42 50 18.4 11.02 21.63 21.28 20.39 19.86 19.27 0.16 0.04 0.01 0.01 0.03 - - - - - - 0.35 0.89 0.54 1.12 - - - 688±705 GC0174 C138 13 25 16.91 -43 03 08.0 2.79 - - - - - - - - - - 21.53 20.79 19.97 0.02 0.02 0.03 - - - - 0.81 0.74 1.55 - GC0175 R254 13 25 16.96 -43 09 28.0 8.54 20.52 20.26 19.43 18.91 18.33 0.09 0.03 0.01 0.01 0.03 20.42 19.65 18.91 0.04 0.03 0.03 0.27 0.82 0.52 1.10 0.73 0.77 1.51 561±275,7 GC0176 C139 13 25 17.06 -43 02 44.6 2.50 - - - - - - - - - - 20.73 19.62 18.86 0.03 0.02 0.04 - - - - 0.76 1.11 1.87 - GC0177 HGHH-G219 13 25 17.31 -42 58 46.6 3.03 20.69 19.92 18.84 18.20 17.55 0.43 0.08 0.02 0.02 0.03 - - - - - - 0.77 1.09 0.63 1.29 - - - 535±244,5,7 GC0178 R253 13 25 17.33 -43 08 39.0 7.74 20.75 20.49 19.67 19.14 18.64 0.09 0.03 0.01 0.01 0.03 20.68 20.00 19.47 0.03 0.03 0.05 0.26 0.81 0.53 1.04 0.53 0.69 1.22 486±525,7 GC0179 C140 13 25 17.42 -43 03 25.2 2.94 - - - - - - - - - - 21.40 20.65 19.83 0.03 0.02 0.03 - - - - 0.82 0.75 1.57 - GC0180 C141 13 25 18.14 -43 02 50.9 2.43 - - - - - - - - - - 22.23 21.55 20.95 0.06 0.05 0.06 - - - - 0.60 0.68 1.29 - GC0181 WHH-12 13 25 18.27 -42 53 04.8 8.25 20.36 20.00 19.07 18.47 17.97 0.07 0.02 0.01 0.01 0.03 20.23 19.35 18.71 0.04 0.04 0.01 0.36 0.93 0.60 1.10 0.64 0.89 1.52 558±996 GC0182 AAT115339 13 25 18.44 -43 04 09.8 3.45 21.19 20.91 20.11 19.53 19.01 0.39 0.10 0.03 0.03 0.04 21.11 20.38 19.56 0.02 0.01 0.02 0.27 0.81 0.58 1.09 0.81 0.73 1.55 618±1087 GC0183 C142 13 25 18.50 -43 01 16.4 1.67 - - - - - - - - - - - - 17.64 - - - - - - - - - - - GC0184 AAT320656 13 25 19.13 -43 12 03.8 11.03 20.61 20.50 19.81 19.35 18.89 0.08 0.03 0.01 0.01 0.03 20.57 20.01 19.35 0.02 0.01 0.02 0.10 0.70 0.46 0.92 0.66 0.56 1.22 453±2337 GC0185 HGHH-G331 13 25 19.50 -43 02 28.4 1.99 20.00 19.60 18.92 18.47 17.93 0.55 0.14 0.05 0.04 0.05 - - - - - - 0.40 0.68 0.44 0.99 - - - 371±224,5,7 GC0186 AAT115561 13 25 19.83 -42 58 27.2 3.05 22.13 21.47 20.50 19.92 19.29 1.44 0.26 0.07 0.06 0.06 - 20.98 20.15 1.00 0.04 0.02 0.66 0.98 0.57 1.20 0.83 - - 514±467 GC0187 R247 13 25 19.99 -43 07 44.1 6.73 21.86 21.20 20.11 19.40 18.62 0.28 0.05 0.02 0.01 0.03 21.73 20.65 19.75 0.05 0.04 0.04 0.66 1.10 0.70 1.49 0.90 1.08 1.98 662±485 GC0188 AAT115605 13 25 20.44 -42 54 08.5 7.13 21.44 21.22 20.47 20.00 19.50 0.18 0.05 0.02 0.02 0.03 21.36 20.76 20.11 0.01 0.03 0.01 0.22 0.75 0.47 0.97 0.66 0.60 1.25 713±1317 GC0189 AAT115679 13 25 20.72 -43 06 35.9 5.60 21.93 21.49 20.44 19.80 19.15 0.40 0.09 0.02 0.02 0.03 21.71 20.79 19.84 0.02 0.02 0.03 0.44 1.05 0.64 1.30 0.95 0.92 1.87 511±1657 GC0190 WHH-13/HH-090 13 25 21.29 -42 49 17.7 11.91 20.75 20.22 19.20 18.61 17.95 0.08 0.02 0.01 0.01 0.03 20.59 19.52 18.74 0.03 0.03 0.02 0.53 1.02 0.59 1.25 0.78 1.08 1.85 444±336,7 GC0191 AAT321194 13 25 21.32 -43 23 59.3 22.87 21.46 21.03 20.06 19.48 18.82 0.38 0.03 0.01 0.01 0.03 21.33 20.42 19.53 0.03 0.02 0.01 0.43 0.97 0.58 1.24 0.90 0.90 1.80 254±1697 GC0192 HGHH-06/C6 13 25 22.19 -43 02 45.6 1.89 18.65 18.17 17.21 16.61 16.03 0.24 0.06 0.02 0.01 0.03 - - - - - - 0.48 0.96 0.60 1.18 - - - 855±22,4,5,6,7,8 GC0193 pff gc-051 13 25 22.35 -43 15 00.1 13.89 21.15 20.73 19.79 19.20 18.61 0.10 0.02 0.01 0.01 0.03 21.00 20.10 19.22 0.03 0.03 0.02 0.42 0.94 0.59 1.18 0.88 0.90 1.78 468±395,7 GC0194 C143 13 25 23.20 -43 03 12.9 2.22 - - - - - - - - - - 21.87 21.35 20.52 0.05 0.04 0.06 - - - - 0.82 0.52 1.34 - GC0195 AAT116025 13 25 23.46 -42 53 26.2 7.75 21.23 20.88 20.00 19.48 18.85 0.14 0.04 0.01 0.01 0.03 21.16 20.32 19.62 0.04 0.05 0.01 0.35 0.88 0.51 1.15 0.70 0.84 1.54 545±647 GC0196 AAT116220 13 25 24.40 -43 07 58.9 6.86 20.76 20.39 19.47 18.91 18.36 0.11 0.03 0.01 0.01 0.03 20.62 19.72 18.92 0.03 0.03 0.03 0.37 0.92 0.56 1.11 0.80 0.90 1.70 524±417 Table 1—Continued Name Old Name RA Dec. R U B V R I σU σB σV σR σI C M T1 σC σM σT1 U − B B − V V − R V − I M − T1 C − M C − T1 vr (J2000) (J2000) (’) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (km s−1) GC0197 AAT116385 13 25 25.39 -42 58 21.5 2.82 21.71 21.23 20.26 19.59 18.99 1.22 0.26 0.07 0.05 0.06 21.58 20.68 19.94 0.02 0.02 0.01 0.48 0.97 0.66 1.27 0.74 0.90 1.64 550±407 GC0198 WHH-14 13 25 25.49 -42 56 31.2 4.64 20.99 20.83 20.06 19.56 19.05 0.18 0.05 0.02 0.02 0.03 20.98 20.36 19.68 0.02 0.03 0.01 0.15 0.77 0.50 1.01 0.68 0.62 1.30 461±756,7 GC0199 AAT204119 13 25 25.70 -42 37 40.9 23.47 20.59 20.49 19.81 19.35 18.93 0.18 0.02 0.01 0.01 0.03 20.60 20.04 19.42 0.01 - - 0.10 0.67 0.47 0.88 0.62 0.56 1.18 404±747 GC0200 pff gc-052 13 25 25.75 -43 05 16.5 4.14 20.88 20.70 19.88 19.39 18.86 0.19 0.06 0.02 0.02 0.03 20.86 20.17 19.42 0.03 0.03 0.03 0.18 0.82 0.48 1.01 0.75 0.69 1.44 462±525 GC0201 HGHH-46/C46 13 25 25.97 -43 03 25.7 2.30 19.74 19.71 19.10 18.74 18.30 0.37 0.12 0.04 0.04 0.05 19.77 19.30 18.64 0.01 0.02 0.22 0.03 0.61 0.36 0.80 0.66 0.47 1.13 508±194,5,7 GC0202 C144 13 25 26.28 -43 04 38.5 3.50 - - - - - - - - - - 23.22 22.36 21.96 0.06 0.05 0.05 - - - - 0.41 0.85 1.26 - GC0203 AAT116531 13 25 26.75 -43 08 53.4 7.74 20.92 20.65 19.83 19.34 18.83 0.10 0.03 0.01 0.01 0.03 20.77 19.95 19.94 0.07 0.06 0.02 0.27 0.82 0.49 1.00 0.01 0.82 0.83 267±717 GC0204 WHH-15 13 25 26.78 -42 52 39.9 8.48 20.73 20.56 19.81 19.36 18.87 0.09 0.03 0.01 0.01 0.03 20.68 20.80 19.53 0.03 0.02 0.01 0.16 0.75 0.46 0.94 1.27 -0.12 1.15 513±536,7 GC0205 R235 13 25 26.82 -43 09 40.5 8.53 20.55 20.30 19.53 19.02 18.51 0.08 0.02 0.01 0.01 0.03 20.42 19.73 19.01 0.04 0.03 0.04 0.25 0.77 0.51 1.03 0.72 0.69 1.41 498±285,7 GC0206 WHH-16/K-102 13 25 27.97 -43 04 02.2 2.89 20.90 20.23 19.18 18.56 17.90 0.56 0.11 0.03 0.02 0.04 20.64 19.52 18.60 0.04 0.03 0.03 0.66 1.06 0.61 1.28 0.93 1.11 2.04 661±476,7 GC0207 HCH13 13 25 28.69 -43 02 55.0 1.78 21.32 20.82 19.89 19.30 18.63 2.53 0.55 0.15 0.12 0.12 - - - - - - 0.50 0.93 0.59 1.26 - - - 641±245,7 GC0208 C145 13 25 28.81 -43 04 21.6 3.22 - - - - - - - - - - 19.32 18.54 17.81 0.01 0.01 0.02 - - - - 0.73 0.78 1.51 - GC0209 WHH-17 13 25 29.25 -42 57 47.1 3.38 19.68 19.39 18.57 18.06 17.48 0.15 0.04 0.02 0.01 0.03 19.54 18.77 18.13 0.02 0.03 - 0.29 0.82 0.51 1.09 0.65 0.76 1.41 619±456,7 GC0210 AAT116969 13 25 29.41 -42 53 25.6 7.73 21.55 21.28 20.46 20.10 19.48 0.21 0.06 0.02 0.02 0.03 21.45 20.77 20.10 0.03 0.02 0.01 0.27 0.82 0.35 0.98 0.68 0.68 1.36 446±617 GC0211 HGHH-G169 13 25 29.43 -42 58 09.9 3.00 21.39 20.58 19.49 18.92 18.12 0.81 0.13 0.03 0.03 0.04 21.00 19.80 18.94 0.01 0.02 0.01 0.81 1.10 0.57 1.37 0.86 1.21 2.07 643±244,5,7 GC0212 pff gc-053 13 25 29.62 -42 54 44.5 6.42 20.91 20.29 19.26 18.73 17.98 0.14 0.03 0.01 0.01 0.03 20.65 19.54 18.78 0.03 0.04 0.01 0.62 1.03 0.53 1.29 0.76 1.11 1.86 439±205 GC0213 R229 13 25 29.74 -43 11 42.8 10.57 20.70 20.49 19.72 19.20 18.69 0.08 0.02 0.01 0.01 0.03 20.62 19.94 19.22 0.04 0.03 0.04 0.21 0.77 0.52 1.03 0.73 0.68 1.41 517±665 GC0214 HCH15 13 25 29.80 -43 00 07.0 1.10 19.27 18.87 17.93 17.37 16.76 0.98 0.26 0.07 0.05 0.05 - - - - - - 0.04 0.94 0.56 1.17 - - - 519±18 GC0215 C146 13 25 29.87 -43 05 09.2 4.03 - - - - - - - - - - 21.71 20.78 19.94 0.03 0.02 0.02 - - - - 0.84 0.93 1.77 - GC0216 WHH-18 13 25 30.07 -42 56 46.9 4.39 19.85 19.36 18.36 17.85 17.16 0.09 0.02 0.01 0.01 0.03 19.63 18.60 17.88 0.01 0.03 - 0.50 0.99 0.51 1.20 0.72 1.03 1.75 752±256,7 GC0217 pff gc-054 13 25 30.28 -43 41 53.6 40.75 20.65 20.60 19.99 19.57 19.09 0.19 0.03 0.01 0.01 0.03 - - - - - - 0.05 0.62 0.41 0.89 - - - 297±405,7 GC0218 HCH16 13 25 30.29 -42 59 34.8 1.64 - - - - - - - - - - - - - - - - - - - - - - - 458±25,7,8 GC0219 HHH86-15/C15/R226 13 25 30.41 -43 11 49.6 10.69 20.11 19.56 18.56 17.94 17.35 0.06 0.02 0.01 0.01 0.03 19.80 18.68 17.92 0.05 0.05 0.05 0.55 1.00 0.62 1.21 0.76 1.12 1.88 644±15,6,7,8 GC0220 C147 13 25 30.65 -43 03 47.1 2.70 - - - - - - - - - - 21.58 20.88 20.05 0.05 0.05 0.03 - - - - 0.83 0.69 1.52 - GC0221 pff gc-055 13 25 30.72 -42 48 13.4 12.94 20.87 20.68 19.92 19.44 18.97 0.08 0.03 0.01 0.01 0.03 20.83 20.20 19.54 0.02 0.02 0.02 0.19 0.76 0.48 0.96 0.66 0.62 1.28 485±475,7 GC0222 AAT205071 13 25 30.74 -42 30 16.0 30.89 19.65 19.61 18.86 18.37 17.96 0.09 0.01 0.01 0.01 0.03 19.68 19.10 18.50 0.02 0.02 - 0.04 0.74 0.49 0.90 0.60 0.58 1.18 288±617 GC0223 WHH-19 13 25 31.03 -42 50 14.9 10.92 18.47 18.16 17.29 16.78 16.24 0.02 0.01 0.01 0.01 0.03 18.08 17.67 16.91 0.06 0.06 0.02 0.31 0.87 0.51 1.05 0.76 0.42 1.18 451±406,7 GC0224 AAT117287 13 25 31.08 -43 04 17.0 3.20 22.20 21.44 20.36 19.82 19.13 1.22 0.20 0.05 0.04 0.05 21.82 20.56 20.45 0.04 0.03 0.03 0.76 1.08 0.54 1.23 0.11 1.27 1.37 554±607 GC0225 HGHH-G292 13 25 31.48 -42 58 08.3 3.09 21.77 21.05 19.97 19.33 18.64 1.04 0.18 0.05 0.03 0.04 21.48 20.85 19.53 0.01 0.02 0.01 0.72 1.08 0.64 1.34 1.33 0.63 1.95 655±434,5 GC0226 HCH18 13 25 31.60 -43 00 02.8 1.32 21.06 20.99 19.93 19.20 18.43 2.42 0.76 0.19 0.13 0.12 - - - - - - 0.07 1.06 0.73 1.50 - - - 455±15,7,8 GC0227 AAT117322 13 25 31.73 -42 55 15.7 5.93 21.14 20.91 20.08 19.55 19.07 0.19 0.05 0.02 0.02 0.03 21.07 20.38 19.75 0.04 0.05 0.01 0.22 0.84 0.52 1.01 0.63 0.69 1.32 636±1337 GC0228 HGHH-44/C44 13 25 31.73 -43 19 22.6 18.25 19.76 19.50 18.69 18.14 17.66 0.04 0.01 0.01 0.01 0.03 19.59 18.85 18.15 0.03 0.02 0.01 0.26 0.80 0.55 1.03 0.70 0.74 1.44 505±14,5,6,7,8 GC0229 C148 13 25 31.75 -43 05 46.0 4.68 - - - - - - - - - - 21.25 20.79 20.21 0.07 0.06 0.05 - - - - 0.58 0.46 1.04 - GC0230 R224/C149 13 25 32.33 -43 07 17.1 6.20 21.07 20.90 20.15 19.64 19.19 0.15 0.04 0.02 0.02 0.03 21.02 20.42 19.68 0.03 0.02 0.02 0.16 0.76 0.50 0.96 0.74 0.60 1.34 389±455 GC0231 HGHH-G359 13 25 32.42 -42 58 50.2 2.47 20.44 19.86 18.86 18.24 17.64 0.45 0.10 0.03 0.02 0.04 20.13 19.09 18.33 0.01 0.01 - 0.58 1.00 0.62 1.22 0.76 1.04 1.80 489±344,5,7 GC0232 pff gc-056 13 25 32.80 -42 56 24.4 4.83 19.62 19.43 18.64 18.15 17.64 0.06 0.02 0.01 0.01 0.03 19.51 18.82 18.22 0.03 0.04 - 0.19 0.79 0.48 1.00 0.60 0.69 1.29 306±275,6 GC0233 R223 13 25 32.80 -43 07 02.2 5.97 20.08 19.67 18.77 18.23 17.64 0.07 0.02 0.01 0.01 0.03 19.89 18.98 18.19 0.03 0.03 0.03 0.41 0.91 0.54 1.13 0.79 0.92 1.71 776±15,7,8 GC0234 K-131 13 25 32.88 -43 04 29.2 3.48 21.23 20.44 19.37 18.77 18.12 0.38 0.07 0.02 0.02 0.03 20.84 19.65 18.74 0.03 0.02 0.03 0.79 1.08 0.59 1.24 0.91 1.20 2.10 639±447 GC0235 pff gc-057 13 25 33.17 -42 59 03.2 2.33 22.21 21.38 20.35 19.75 18.95 2.11 0.37 0.09 0.07 0.07 21.67 20.75 19.86 0.03 0.02 0.01 0.82 1.04 0.60 1.40 0.90 0.91 1.81 515±125 GC0236 C15013 13 25 33.82 -43 02 49.6 2.03 - - - - - - - - - - - - 19.78 - - - - - - - - - - - GC0237 C151 13 25 33.93 -43 03 51.4 2.94 - - - - - - - - - - 22.27 20.90 19.95 0.03 0.01 0.03 - - - - 0.95 1.37 2.32 - GC0238 MAGJM-11 13 25 33.94 -42 59 39.4 1.89 19.80 19.64 18.87 18.44 17.83 0.80 0.25 0.07 0.06 0.07 - - - - - - 0.16 0.77 0.43 1.04 - - - 444±175,7 GC0239 HGHH-G206 13 25 34.10 -42 59 00.7 2.44 20.68 20.09 19.10 18.58 17.91 0.56 0.12 0.03 0.03 0.04 20.37 19.36 18.61 0.02 0.03 0.01 0.58 0.99 0.52 1.19 0.75 1.00 1.75 600±244,5 GC0240 HGHH-45/C45 13 25 34.25 -42 56 59.1 4.33 19.99 19.81 19.03 18.60 18.05 0.10 0.03 0.01 0.01 0.03 19.91 19.17 18.67 0.08 0.07 0.02 0.18 0.78 0.43 0.98 0.50 0.75 1.25 612±344,5,7 GC0241 WHH-20 13 25 34.36 -42 51 05.9 10.12 20.30 19.93 19.01 18.52 17.90 0.06 0.02 0.01 0.01 0.03 20.11 19.25 18.59 0.04 0.05 0.01 0.37 0.92 0.49 1.10 0.66 0.86 1.52 259±336,7 GC0242 C15213 13 25 34.64 -43 03 16.4 2.48 - - - - - - - - - - - - 17.82 - - - - - - - - - - - GC0243 C15313 13 25 34.64 -43 03 27.8 2.65 - - - - - - - - - - - - 18.23 - - - - - - - - - - - GC0244 HCH21 13 25 34.65 -43 03 27.7 2.65 - - - - - - - - - - - - - - - - - - - - - - - 663±25,7,8 GC0245 C15413 13 25 34.71 -43 03 30.2 2.69 - - - - - - - - - - - - 19.58 - - - - - - - - - - - Table 1—Continued Name Old Name RA Dec. R U B V R I σU σB σV σR σI C M T1 σC σM σT1 U − B B − V V − R V − I M − T1 C − M C − T1 vr (J2000) (J2000) (’) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (km s−1) GC0246 HHH86-16/C16 13 25 35.00 -42 36 05.0 25.10 20.01 19.57 18.59 18.00 17.41 0.12 0.01 0.01 0.01 0.03 19.85 18.83 18.12 0.03 0.03 0.01 0.44 0.98 0.59 1.19 0.70 1.02 1.73 538±162,4,7 GC0247 pff gc-058 13 25 35.12 -42 56 45.3 4.60 19.67 19.47 18.68 18.19 17.70 0.07 0.02 0.01 0.01 0.03 19.55 18.85 18.29 0.03 0.03 0.01 0.20 0.78 0.49 0.99 0.56 0.70 1.26 365±225,6,7 GC0248 K-144 13 25 35.16 -42 53 01.0 8.25 22.10 21.31 20.25 19.63 18.94 0.33 0.05 0.02 0.01 0.03 21.72 20.44 19.75 0.01 0.01 0.01 0.79 1.06 0.62 1.31 0.69 1.28 1.97 593±437 GC0249 WHH-21 13 25 35.22 -43 12 01.5 10.97 19.64 19.60 19.00 18.63 18.25 0.04 0.01 0.01 0.01 0.03 19.61 19.15 18.62 0.02 0.01 0.02 0.04 0.60 0.37 0.75 0.53 0.46 0.99 243±616 GC0250 WHH-22 13 25 35.31 -43 05 29.0 4.56 - - - - - - - - - - 19.66 18.80 18.03 0.04 0.04 0.04 - - - - 0.77 0.86 1.63 492±1266 GC0251 MAGJM-08 13 25 35.50 -42 59 35.2 2.12 20.76 20.18 19.17 18.68 17.94 1.15 0.24 0.06 0.05 0.05 - - - - - - 0.58 1.01 0.49 1.22 - - - 701±285,7 GC0252 R215 13 25 35.64 -43 08 36.8 7.61 22.06 21.68 20.80 20.27 19.70 0.27 0.06 0.02 0.02 0.03 21.98 21.18 20.35 0.03 0.03 0.04 0.38 0.87 0.53 1.11 0.83 0.79 1.63 543±575,7 GC0253 R213 13 25 35.93 -43 07 27.9 6.50 22.70 22.03 20.92 20.35 19.62 0.56 0.10 0.03 0.02 0.03 22.34 21.26 20.34 0.03 0.01 0.02 0.67 1.11 0.58 1.30 0.92 1.08 2.00 532±525,7 GC0254 AAT117899 13 25 36.05 -42 53 40.3 7.63 21.70 21.17 20.17 19.62 18.97 0.23 0.05 0.02 0.01 0.03 21.51 20.51 19.70 0.03 0.03 0.01 0.53 1.01 0.54 1.20 0.81 1.01 1.81 609±1167 GC0255 C155 13 25 36.47 -43 08 03.5 7.10 - - - - - - - - - - 22.98 22.21 21.36 0.06 0.05 0.05 - - - - 0.85 0.77 1.63 - GC0256 MAGJM-06 13 25 36.69 -42 59 59.2 2.02 - - - - - - - - - - - - - - - - - - - - - - - 443±155,7 GC0257 AAT118198 13 25 37.47 -43 05 44.9 4.94 21.49 20.68 19.64 18.98 18.29 0.31 0.05 0.02 0.01 0.03 21.14 19.94 19.03 0.04 0.02 0.03 0.81 1.04 0.66 1.35 0.91 1.20 2.11 575±367 GC0258 AAT118314 13 25 38.03 -43 16 59.2 15.95 20.69 20.62 19.89 19.45 18.98 0.08 0.03 0.01 0.01 0.03 20.67 20.13 19.42 0.01 0.01 0.01 0.08 0.72 0.44 0.92 0.71 0.53 1.24 257±597 GC0259 R209 13 25 38.13 -43 13 02.2 12.04 21.92 21.51 20.62 20.05 19.43 0.20 0.05 0.02 0.01 0.03 21.91 20.95 20.13 0.04 0.04 0.04 0.41 0.90 0.57 1.19 0.83 0.96 1.79 558±405,7 GC0260 C156 13 25 38.43 -43 05 02.6 4.37 - - - - - - - - - - 19.76 18.55 17.65 0.02 0.02 0.03 - - - - 0.90 1.21 2.11 - GC0261 C15712 13 25 38.45 -43 03 28.9 3.06 - - - - - - - - - - - - 19.21 - - - - - - - - - - - GC0262 HGHH-G268 13 25 38.61 -42 59 19.6 2.71 20.23 19.83 18.93 18.36 17.79 0.34 0.08 0.03 0.02 0.04 20.05 19.15 18.44 0.02 0.02 - 0.40 0.91 0.56 1.13 0.71 0.91 1.62 436±434,5 GC0263 C158 13 25 38.76 -43 05 34.5 4.88 - - - - - - - - - - 21.67 20.86 20.06 0.05 0.04 0.05 - - - - 0.80 0.81 1.61 - GC0264 C159 13 25 39.17 -43 04 33.8 4.02 - - - - - - - - - - 21.91 20.82 19.93 0.02 0.02 0.02 - - - - 0.90 1.09 1.99 - GC0265 MAGJM-01 13 25 39.33 -43 00 48.8 2.17 19.90 19.52 18.61 18.06 17.50 0.32 0.08 0.02 0.02 0.03 - - - - - - 0.37 0.91 0.55 1.11 - - - 645±365,7 GC0266 pff gc-059 13 25 39.65 -43 04 01.4 3.62 21.53 20.97 19.96 19.35 18.73 0.55 0.11 0.03 0.02 0.04 21.42 20.33 19.48 0.05 0.04 0.04 0.56 1.01 0.61 1.23 0.86 1.09 1.94 525±275,7 GC0267 HGHH-17/C17 13 25 39.73 -42 55 59.2 5.62 18.82 18.51 17.63 17.10 16.57 0.04 0.01 0.01 0.01 0.03 18.61 17.72 17.19 0.04 0.05 0.01 0.32 0.88 0.53 1.06 0.53 0.89 1.42 782±22,5,6,7,8 GC0268 HHH86-18/C18/K-163 13 25 39.88 -43 05 01.9 4.49 18.79 18.42 17.53 16.97 16.43 0.04 0.01 0.01 0.01 0.03 18.49 17.51 16.89 0.05 0.04 0.05 0.38 0.89 0.56 1.10 0.62 0.99 1.60 480±22,5,6,7,8 GC0269 C16012 13 25 40.09 -43 03 07.1 3.01 - - - - - - - - - - - - 19.99 - - - - - - - - - - - GC0270 C101 13 25 40.47 -42 56 02.7 5.65 - - 20.34 - - - - - - - 21.84 20.90 20.08 0.03 0.04 0.01 - - - - 0.82 0.94 1.76 - GC0271 R204/C161 13 25 40.52 -43 07 17.9 6.59 21.58 20.84 19.80 19.16 18.41 0.21 0.04 0.01 0.01 0.03 21.21 20.15 19.26 0.04 0.03 0.03 0.74 1.05 0.63 1.39 0.89 1.06 1.95 425±325 GC0272 HGHH-34/C34 13 25 40.61 -43 21 13.6 20.22 19.37 19.01 18.12 17.59 17.03 0.04 0.01 0.01 0.01 0.03 19.64 18.54 17.76 0.04 0.03 0.02 0.35 0.89 0.53 1.10 0.77 1.10 1.87 669±183,5,7 GC0273 C162 13 25 40.87 -43 05 00.4 4.56 - - - - - - - - - - 21.95 21.41 20.91 0.06 0.05 0.01 - - - - 0.51 0.54 1.05 - GC0274 R203 13 25 40.90 -43 08 16.0 7.52 20.66 20.56 19.84 19.36 18.88 0.09 0.03 0.01 0.01 0.03 20.62 20.08 19.38 0.04 0.03 0.03 0.09 0.73 0.48 0.96 0.70 0.54 1.24 455±355,7 GC0275 AAT118874 13 25 41.36 -42 58 08.9 3.91 21.55 21.02 19.99 19.43 18.81 0.44 0.09 0.03 0.02 0.04 21.49 20.30 19.57 0.05 0.02 - 0.53 1.03 0.56 1.18 0.73 1.19 1.92 570±827 GC0276 C163 13 25 41.63 -43 03 45.8 3.66 - - - - - - - - - - 21.98 21.25 20.43 0.05 0.04 0.04 - - - - 0.81 0.73 1.55 - GC0277 R202 13 25 42.00 -43 10 42.2 9.91 20.34 20.08 19.26 18.72 18.22 0.06 0.02 0.01 0.01 0.03 20.35 19.74 18.94 0.07 0.03 0.06 0.27 0.81 0.54 1.05 0.80 0.61 1.41 286±455,7 GC0278 C16414 13 25 42.09 -43 03 19.5 3.43 - - - - - - - - - - - - 19.60 - - - - - - - - - - - GC0279 HGHH-G370 13 25 42.25 -42 59 17.0 3.26 19.61 19.29 18.39 17.86 17.32 0.17 0.05 0.02 0.01 0.03 19.45 18.59 17.93 0.01 0.03 0.01 0.32 0.90 0.54 1.08 0.66 0.86 1.52 507±344,5,7 GC0280 pff gc-060 13 25 42.43 -42 59 02.6 3.43 20.11 19.74 18.81 18.24 17.66 0.20 0.05 0.02 0.01 0.03 19.96 19.02 18.33 0.02 0.03 0.01 0.37 0.93 0.57 1.15 0.69 0.95 1.63 890±195,6 GC0281 AAT119058 13 25 42.53 -43 03 41.5 3.73 21.85 21.42 20.39 19.77 19.18 0.60 0.13 0.04 0.03 0.04 21.69 20.74 19.86 0.04 0.03 0.04 0.43 1.03 0.62 1.21 0.88 0.95 1.83 385±877 GC0282 pff gc-061 13 25 42.62 -42 45 10.9 16.20 21.59 21.15 20.25 19.69 19.12 0.14 0.03 0.01 0.01 0.03 21.42 20.55 19.75 0.01 0.01 0.01 0.44 0.90 0.55 1.13 0.80 0.87 1.67 500±485,7 GC0283 pff gc-062 13 25 43.23 -42 58 37.4 3.81 21.13 20.46 19.42 18.83 18.18 0.34 0.06 0.02 0.02 0.03 20.79 19.71 18.92 0.01 0.01 0.01 0.67 1.04 0.59 1.24 0.80 1.08 1.88 697±375 GC0284 HGHH-19/C19 13 25 43.40 -43 07 22.8 6.87 19.37 19.01 18.12 17.59 17.03 0.04 0.01 0.01 0.01 0.03 18.96 17.83 18.64 0.05 0.04 0.01 0.35 0.89 0.53 1.10 -0.80 1.13 0.32 632±103,4,5,6,7 GC0285 C165 13 25 43.43 -43 04 56.5 4.77 - - - - - - - - - - 20.10 18.98 18.17 0.05 0.04 0.05 - - - - 0.81 1.12 1.93 - GC0286 pff gc-063 13 25 43.80 -43 07 54.9 7.39 20.66 20.55 19.83 19.37 18.89 0.08 0.03 0.01 0.01 0.03 20.63 20.13 19.45 0.04 0.03 0.03 0.11 0.72 0.46 0.94 0.67 0.50 1.18 554±755 GC0287 pff gc-064 13 25 43.90 -42 50 42.7 10.85 21.16 20.67 19.66 19.07 18.41 0.11 0.03 0.01 0.01 0.03 20.96 19.97 19.20 0.02 0.03 0.01 0.50 1.01 0.59 1.25 0.76 0.99 1.76 560±245,6,7 GC0288 HGHH-35/C35 13 25 44.21 -42 58 59.4 3.72 19.92 19.51 18.58 18.01 17.43 0.13 0.03 0.01 0.01 0.03 19.74 18.79 18.11 0.02 0.02 0.01 0.41 0.93 0.57 1.14 0.68 0.94 1.63 544±133,4,5 GC0289 C16611 13 25 44.90 -43 04 21.1 4.50 - - - - - - - - - - - - 20.72 - - - - - - - - - - - GC0290 AAT208065 13 25 45.77 -42 34 18.0 27.05 21.79 21.45 20.09 19.40 18.71 0.49 0.05 0.01 0.01 0.03 22.06 20.70 19.66 0.04 0.03 0.02 0.33 1.36 0.69 1.39 1.04 1.36 2.40 836±417 GC0291 WHH-23 13 25 45.90 -42 57 20.2 5.07 19.26 19.12 18.37 17.89 17.41 0.05 0.02 0.01 0.01 0.03 19.13 18.50 18.00 0.03 0.05 0.01 0.14 0.75 0.48 0.96 0.50 0.64 1.14 286±636 GC0292 C16713 13 25 45.97 -43 06 45.4 6.54 - - - - - - - - - - - - 20.41 - - - - - - - - - - - GC0293 WHH-24 13 25 46.00 -42 56 53.0 5.43 20.64 20.43 19.63 19.12 18.62 0.12 0.04 0.01 0.01 0.03 20.54 19.91 19.25 0.02 0.03 0.01 0.21 0.80 0.51 1.01 0.66 0.63 1.29 566±486 GC0294 AAT119596 13 25 46.06 -43 08 24.5 8.01 21.19 20.69 19.70 19.10 18.51 0.12 0.03 0.01 0.01 0.03 - - - - - - 0.50 1.00 0.60 1.19 - - - 573±537 Table 1—Continued Name Old Name RA Dec. R U B V R I σU σB σV σR σI C M T1 σC σM σT1 U − B B − V V − R V − I M − T1 C − M C − T1 vr (J2000) (J2000) (’) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (km s−1) GC0295 HGHH-G378 13 25 46.13 -43 01 22.9 3.39 21.30 20.59 19.53 18.86 18.22 0.55 0.10 0.03 0.02 0.03 - - - - - - 0.71 1.06 0.67 1.31 - - - 487±314,5,7 GC0296 AAT208206 13 25 46.49 -42 34 54.1 26.47 20.08 20.10 19.47 19.07 18.69 0.12 0.02 0.01 0.01 0.03 20.13 19.73 19.19 0.01 0.01 0.01 -0.02 0.63 0.40 0.78 0.55 0.40 0.94 293±1157 GC0297 HGHH-G284 13 25 46.59 -42 57 03.0 5.37 21.47 20.89 19.87 19.28 18.63 0.26 0.05 0.02 0.01 0.03 21.26 20.20 19.41 0.01 0.01 0.01 0.57 1.03 0.58 1.24 0.79 1.06 1.85 479±124,5,7 GC0298 AAT119697 13 25 46.68 -42 53 48.6 8.12 21.61 21.06 20.05 19.49 18.87 0.23 0.05 0.02 0.01 0.03 21.42 20.42 19.65 0.02 0.02 0.01 0.54 1.01 0.57 1.18 0.77 1.01 1.78 633±1087 GC0299 pff gc-065 13 25 46.92 -43 08 06.6 7.81 20.60 20.17 19.23 18.66 18.08 0.08 0.02 0.01 0.01 0.03 20.43 19.55 18.71 0.01 0.01 0.01 0.42 0.94 0.57 1.15 0.83 0.88 1.72 450±445 GC0300 HGHH-G204 13 25 46.99 -43 02 05.4 3.66 19.35 19.12 18.33 17.83 17.31 0.12 0.04 0.01 0.01 0.03 - - - - - - 0.23 0.79 0.51 1.02 - - - 690±184,5,7 GC0301 pff gc-066 13 25 47.14 -43 06 08.8 6.14 21.21 20.82 19.92 19.37 18.80 0.15 0.04 0.01 0.01 0.03 21.07 20.21 19.44 0.01 0.01 0.01 0.39 0.91 0.55 1.11 0.77 0.86 1.63 530±445,7 GC0302 HGHH-G357 13 25 47.78 -43 00 43.4 3.71 20.87 20.55 19.71 19.24 18.71 0.26 0.07 0.02 0.02 0.04 20.71 19.96 19.31 0.01 0.02 0.01 0.33 0.83 0.47 1.00 0.66 0.75 1.41 664±314,5 GC0303 AAT119894 13 25 47.92 -42 55 52.4 6.45 21.45 21.30 20.25 19.59 18.92 0.23 0.07 0.02 0.02 0.03 21.68 20.75 19.85 0.05 0.04 0.03 0.15 1.05 0.66 1.33 0.90 0.93 1.84 448±907 GC0304 C16815 13 25 48.46 -43 07 12.5 7.16 - - - - - - - - - - - - 19.71 - - - - - - - - - - - GC0305 HGHH-G251 13 25 48.54 -42 57 41.2 5.16 20.50 19.93 18.93 18.33 17.69 0.13 0.03 0.01 0.01 0.03 20.23 19.22 18.46 0.02 0.01 0.01 0.57 1.00 0.60 1.24 0.76 1.01 1.77 574±274,5,7 GC0306 pff gc-067 13 25 48.77 -43 11 38.7 11.19 20.51 20.35 19.60 19.08 18.60 0.07 0.02 0.01 0.01 0.03 20.47 19.87 19.14 0.01 0.01 0.01 0.16 0.75 0.52 1.00 0.74 0.60 1.33 632±435,7 GC0307 pff gc-068 13 25 49.27 -43 02 20.4 4.13 20.73 20.41 19.55 18.99 18.48 0.20 0.05 0.02 0.02 0.03 20.65 20.17 19.13 0.03 0.04 0.01 0.32 0.85 0.56 1.08 1.04 0.48 1.52 393±265,7 GC0308 HGHH-20/C20 13 25 49.69 -42 54 49.3 7.50 19.18 18.83 17.95 17.43 16.88 0.04 0.01 0.01 0.01 0.03 19.01 18.19 17.54 0.03 0.02 0.01 0.35 0.88 0.53 1.07 0.65 0.82 1.47 744±92,5,6,7 GC0309 AAT120259 13 25 49.73 -43 05 04.7 5.64 21.95 21.16 20.12 19.51 18.80 0.35 0.06 0.02 0.01 0.03 21.57 20.48 19.56 0.01 0.01 0.01 0.79 1.04 0.62 1.32 0.92 1.09 2.01 477±547 GC0310 HGHH-48/C48 13 25 49.82 -42 50 15.3 11.62 19.63 19.46 18.67 18.16 17.67 0.04 0.01 0.01 0.01 0.03 19.56 18.92 18.28 0.02 0.02 0.01 0.17 0.79 0.51 1.01 0.64 0.64 1.28 547±194,5,6,7 GC0311 pff gc-069 13 25 49.93 -42 40 08.2 21.40 21.55 20.90 19.87 19.26 18.62 0.41 0.03 0.01 0.01 0.03 21.29 20.24 19.39 0.01 0.01 0.01 0.64 1.03 0.61 1.25 0.85 1.05 1.90 518±315,7 GC0312 HGHH-47/C47 13 25 49.95 -42 52 09.4 9.87 19.80 19.55 18.69 18.18 17.65 0.05 0.02 0.01 0.01 0.03 19.70 18.97 18.30 0.03 0.02 0.01 0.26 0.86 0.51 1.04 0.67 0.73 1.40 589±224,5,6 GC0313 AAT120336 13 25 50.22 -43 06 08.5 6.48 21.66 21.14 20.17 19.57 18.96 0.23 0.05 0.02 0.01 0.03 21.48 20.50 19.67 0.02 0.01 0.01 0.52 0.97 0.60 1.21 0.83 0.97 1.81 452±677 GC0314 WHH-25 13 25 50.34 -43 04 08.2 5.12 21.33 20.93 19.97 19.42 18.83 0.22 0.05 0.02 0.01 0.03 21.19 20.27 19.57 0.02 0.02 0.01 0.40 0.95 0.55 1.14 0.71 0.92 1.62 525±506,7 GC0315 AAT120355 13 25 50.37 -43 00 32.6 4.20 21.26 21.14 20.42 20.02 19.58 0.28 0.09 0.03 0.03 0.04 21.32 20.75 20.17 0.03 0.03 0.01 0.12 0.72 0.39 0.84 0.59 0.56 1.15 548±777 GC0316 pff gc-070 13 25 50.40 -42 58 02.3 5.20 20.39 20.23 19.42 18.94 18.45 0.12 0.04 0.01 0.01 0.03 20.35 19.67 19.06 0.01 0.01 0.01 0.16 0.81 0.48 0.97 0.61 0.68 1.29 556±495 GC0317 C169 13 25 51.01 -42 55 36.3 7.00 - - - - - - - - - - 21.60 21.00 20.39 0.03 0.03 0.02 - - - - 0.62 0.60 1.21 - GC0318 AAT120515 13 25 51.31 -42 59 29.4 4.64 21.11 20.81 19.92 19.38 18.81 0.21 0.06 0.02 0.02 0.03 21.04 20.24 19.46 0.04 0.04 0.01 0.30 0.90 0.53 1.11 0.77 0.81 1.58 461±1417 GC0319 pff gc-071 13 25 51.54 -42 59 46.8 4.58 20.79 20.59 19.82 19.34 18.83 0.16 0.05 0.02 0.02 0.03 20.71 20.07 19.45 0.01 0.01 0.01 0.20 0.77 0.48 0.99 0.62 0.64 1.27 475±415 GC0320 C102 13 25 52.07 -42 59 14.4 4.87 - - 21.43 - - - - - - - 22.71 21.87 21.04 0.02 0.02 0.02 - - - - 0.83 0.84 1.67 - GC0321 pff gc-072 13 25 52.14 -42 58 30.2 5.20 20.68 20.45 19.62 19.12 18.54 0.15 0.04 0.02 0.01 0.03 20.64 19.92 19.25 0.02 0.02 0.01 0.23 0.83 0.51 1.08 0.67 0.72 1.39 504±225 GC0322 HGHH-21/C21 13 25 52.74 -43 05 46.4 6.52 19.16 18.76 17.87 17.32 16.77 0.04 0.01 0.01 0.01 0.03 18.97 18.17 17.40 0.03 0.02 0.01 0.40 0.89 0.55 1.11 0.77 0.80 1.58 462±23,5,6,7,8 GC0323 pff gc-073 13 25 52.78 -42 58 41.7 5.21 21.36 20.94 19.97 19.38 18.74 0.25 0.06 0.02 0.02 0.03 21.25 20.29 19.50 0.02 0.01 0.01 0.42 0.98 0.58 1.22 0.79 0.97 1.76 401±295,6,7 GC0324 HGHH-G256 13 25 52.88 -43 02 00.0 4.70 20.69 20.18 19.00 18.35 17.65 0.18 0.04 0.01 0.01 0.03 20.50 19.31 18.44 0.03 0.04 0.01 0.51 1.17 0.65 1.36 0.87 1.20 2.06 495±184,5,7 GC0325 AAT209412 13 25 53.30 -42 30 52.7 30.63 21.94 21.23 20.22 19.60 18.97 0.52 0.04 0.01 0.01 0.03 21.69 20.64 19.76 0.01 0.01 0.01 0.71 1.01 0.62 1.25 0.88 1.05 1.92 998±1357 GC0326 pff gc-074 13 25 53.37 -42 51 12.4 11.00 21.16 20.83 20.00 19.48 18.90 0.11 0.03 0.01 0.01 0.03 21.04 20.27 19.56 0.01 0.01 0.01 0.33 0.83 0.52 1.10 0.71 0.77 1.48 471±245,7 GC0327 pff gc-075 13 25 53.50 -43 03 56.6 5.50 20.96 20.45 19.48 18.90 18.28 0.14 0.03 0.01 0.01 0.03 20.78 19.76 19.01 0.02 0.03 0.01 0.51 0.97 0.58 1.20 0.75 1.02 1.77 748±185,6,7 GC0328 HGHH-22/C22 13 25 53.57 -42 59 07.6 5.16 19.39 19.06 18.15 17.62 17.07 0.05 0.02 0.01 0.01 0.03 19.21 18.34 17.70 0.02 0.03 - 0.34 0.90 0.54 1.09 0.64 0.87 1.52 578±12,4,5,7,8 GC0329 pff gc-076 13 25 53.75 -43 19 48.6 19.26 20.67 20.06 19.07 18.41 17.78 0.08 0.02 0.01 0.01 0.03 20.44 19.44 18.52 0.02 0.01 0.01 0.62 0.99 0.66 1.29 0.92 1.00 1.92 368±155,6,7 GC0330 AAT120976 13 25 54.28 -42 56 20.6 6.84 21.51 21.23 20.38 19.84 19.27 0.23 0.06 0.02 0.02 0.03 21.50 20.73 19.99 0.03 0.02 0.02 0.28 0.85 0.54 1.11 0.74 0.77 1.51 595±697 GC0331 AAT328533 13 25 54.39 -43 18 40.1 18.19 21.79 21.26 20.31 19.74 19.09 0.16 0.04 0.01 0.01 0.03 21.62 20.64 19.77 0.01 0.01 0.01 0.53 0.95 0.58 1.23 0.87 0.98 1.85 577±897 GC0332 HGHH-23/C23 13 25 54.58 -42 59 25.4 5.22 18.92 18.29 17.22 16.62 15.95 0.04 0.01 0.01 0.01 0.03 18.59 17.44 16.69 0.01 0.03 - 0.63 1.07 0.60 1.28 0.75 1.15 1.90 674±12,3,5,7,8 GC0333 C103 13 25 54.98 -42 59 15.4 5.36 - - 18.88 - - - - - - - 20.43 20.44 18.44 0.01 0.02 0.02 - - - - 2.00 -0.01 1.99 - GC0334 C170 13 25 56.11 -42 56 12.9 7.17 - - - - - - - - - - 23.31 22.83 22.16 0.05 0.04 0.06 - - - - 0.67 0.48 1.15 - GC0335 AAT121367 13 25 56.26 -43 01 32.9 5.25 21.06 20.92 20.24 19.82 19.39 0.25 0.08 0.03 0.03 0.04 21.01 20.45 19.85 0.01 0.01 0.01 0.13 0.68 0.42 0.85 0.60 0.56 1.15 438±807 GC0336 WHH-26 13 25 56.59 -42 51 46.6 10.76 20.50 20.02 19.05 18.43 17.81 0.07 0.02 0.01 0.01 0.03 20.33 19.36 18.56 0.02 0.01 0.01 0.48 0.97 0.62 1.24 0.79 0.97 1.76 412±366 GC0337 AAT329209 13 25 57.28 -43 41 09.0 40.37 20.87 20.70 19.87 19.40 18.90 0.21 0.02 0.01 0.01 0.03 20.78 20.12 19.35 0.01 - 0.01 0.17 0.82 0.47 0.97 0.77 0.66 1.43 601±657 GC0338 C171 13 25 57.78 -42 55 36.1 7.82 - - - - - - - - - - 22.25 21.45 20.67 0.04 0.03 0.02 - - - - 0.78 0.79 1.58 - GC0339 C172 13 25 57.95 -42 53 04.3 9.79 - - - - - - - - - - 22.15 21.45 20.91 0.02 0.01 0.01 - - - - 0.55 0.70 1.24 - GC0340 pff gc-077 13 25 58.15 -42 31 38.2 30.03 20.57 20.32 19.50 18.96 18.47 0.16 0.02 0.01 0.01 0.03 20.55 19.83 19.09 0.02 0.01 0.01 0.24 0.83 0.53 1.03 0.74 0.71 1.45 675±415 GC0341 pff gc-078 13 25 58.47 -43 08 06.3 8.96 20.93 20.34 19.30 18.69 18.02 0.10 0.02 0.01 0.01 0.03 20.70 19.68 18.75 0.02 0.01 0.01 0.59 1.03 0.62 1.28 0.93 1.02 1.95 545±185,7 GC0342 HGHH-G143 13 25 58.69 -43 07 11.0 8.29 20.31 20.02 19.21 18.68 18.18 0.06 0.02 0.01 0.01 0.03 20.22 19.52 18.77 0.02 0.01 0.01 0.29 0.81 0.53 1.03 0.75 0.70 1.45 503±384,5,7 GC0343 pff gc-079 13 25 58.91 -42 53 18.9 9.70 20.74 20.43 19.58 19.06 18.52 0.10 0.03 0.01 0.01 0.03 20.62 19.86 19.14 0.02 0.01 0.01 0.30 0.86 0.51 1.06 0.72 0.77 1.48 410±215,7 Table 1—Continued Name Old Name RA Dec. R U B V R I σU σB σV σR σI C M T1 σC σM σT1 U − B B − V V − R V − I M − T1 C − M C − T1 vr (J2000) (J2000) (’) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (km s−1) GC0344 AAT121826/C104 13 25 59.49 -42 55 30.8 8.10 20.91 20.65 19.81 19.28 18.73 0.13 0.03 0.01 0.01 0.03 20.81 20.15 19.40 0.02 0.02 0.01 0.26 0.83 0.53 1.08 0.74 0.66 1.40 448±987 GC0345 pff gc-080 13 25 59.55 -42 32 39.4 29.08 21.10 20.80 19.96 19.40 18.84 0.25 0.03 0.01 0.01 0.03 21.07 20.32 19.56 0.02 0.01 0.01 0.29 0.85 0.56 1.11 0.75 0.75 1.51 598±565,7 GC0346 C173 13 25 59.57 -42 55 01.5 8.46 - - - - - - - - - - 23.15 22.00 21.06 0.04 0.03 0.02 - - - - 0.94 1.15 2.09 - GC0347 C174 13 25 59.63 -42 55 15.7 8.30 - - - - - - - - - - 23.24 22.42 21.42 0.04 0.04 0.03 - - - - 0.99 0.82 1.81 - GC0348 pff gc-081 13 26 00.15 -42 49 00.7 13.51 20.43 20.17 19.34 18.80 18.28 0.07 0.02 0.01 0.01 0.03 20.36 19.61 18.93 0.02 0.01 0.01 0.26 0.83 0.53 1.05 0.69 0.75 1.43 304±315,7 GC0349 K-217 13 26 00.81 -43 09 40.1 10.46 21.63 21.09 20.09 19.54 18.91 0.16 0.03 0.01 0.01 0.03 21.42 20.48 19.57 0.01 0.01 0.01 0.54 0.99 0.56 1.19 0.91 0.94 1.85 315±1577 GC0350 C175 13 26 00.93 -42 58 28.9 6.65 - - - - - - - - - - 23.51 22.53 21.84 0.06 0.04 0.03 - - - - 0.69 0.97 1.67 - GC0351 pff gc-082 13 26 00.98 -42 22 03.4 39.56 20.83 20.17 19.16 18.58 17.89 0.21 0.02 0.01 0.01 0.03 20.51 19.53 18.69 0.02 0.01 0.01 0.66 1.01 0.58 1.27 0.84 0.98 1.82 573±195,7 GC0352 AAT122146 13 26 01.00 -43 06 55.3 8.40 21.14 20.90 20.05 19.53 19.04 0.11 0.03 0.01 0.01 0.03 21.09 20.38 19.58 0.01 0.01 0.01 0.24 0.85 0.52 1.01 0.80 0.71 1.51 517±997 GC0353 HGHH-G221 13 26 01.11 -42 55 13.5 8.52 20.87 20.24 19.30 18.75 18.13 0.10 0.02 0.01 0.01 0.03 20.54 19.60 18.83 0.01 0.01 0.01 0.62 0.94 0.55 1.16 0.77 0.94 1.72 390±154,5 GC0354 AAT329848 13 26 01.29 -43 34 15.5 33.68 19.44 19.19 18.39 17.89 17.37 0.08 0.01 0.01 0.01 0.03 - - - - - - 0.25 0.79 0.50 1.03 - - - 558±467 GC0355 pff gc-083 13 26 01.83 -42 58 15.0 6.89 21.23 20.80 19.86 19.30 18.72 0.16 0.04 0.01 0.01 0.03 21.12 20.21 19.44 0.02 0.02 0.01 0.42 0.95 0.55 1.13 0.77 0.91 1.68 458±315,6,7 GC0356 pff gc-084 13 26 02.25 -43 08 55.6 10.03 21.31 20.80 19.84 19.24 18.63 0.12 0.03 0.01 0.01 0.03 21.13 20.19 19.32 0.02 0.01 0.01 0.51 0.97 0.59 1.21 0.87 0.93 1.80 458±385,7 GC0357 AAT122445 13 26 02.43 -43 00 11.7 6.43 21.13 20.99 20.28 19.81 19.42 0.15 0.05 0.02 0.02 0.03 21.08 20.50 19.92 0.01 0.02 0.01 0.14 0.71 0.47 0.86 0.58 0.58 1.16 342±987 GC0358 C176 13 26 02.79 -42 57 05.0 7.61 - - - - - - - - - - 22.35 21.92 21.30 0.02 0.02 0.03 - - - - 0.62 0.43 1.05 - GC0359 HGHH-25/C25 13 26 02.85 -42 56 57.0 7.69 20.17 19.56 18.49 17.85 17.17 0.07 0.02 0.01 0.01 0.03 19.92 18.83 17.97 0.03 0.02 0.01 0.61 1.07 0.64 1.32 0.86 1.09 1.95 703±93,5,7 GC0360 AAT122526 13 26 02.90 -43 05 43.0 7.90 21.89 21.35 20.36 19.73 19.05 0.24 0.05 0.02 0.01 0.03 21.72 20.72 19.81 0.02 0.01 0.01 0.54 1.00 0.62 1.31 0.90 1.00 1.90 506±507 GC0361 C177 13 26 03.20 -42 54 30.1 9.30 - - - - - - - - - - 22.95 22.11 21.09 0.11 0.07 0.05 - - - - 1.02 0.84 1.87 - GC0362 C178 13 26 03.85 -42 56 45.3 7.95 - - - - - - - - - - 23.02 22.32 21.53 0.04 0.04 0.03 - - - - 0.79 0.69 1.48 - GC0363 HGHH-G293/G293 13 26 04.20 -42 55 44.7 8.60 20.11 19.90 19.10 18.61 18.11 0.06 0.02 0.01 0.01 0.03 20.04 19.36 18.69 0.01 0.01 0.01 0.21 0.80 0.49 0.99 0.67 0.69 1.35 581±154,5,7 GC0364 AAT122808 13 26 04.61 -43 09 10.2 10.49 21.05 20.87 20.12 19.66 19.18 0.10 0.03 0.01 0.01 0.03 21.00 20.45 19.69 0.01 0.01 0.01 0.18 0.75 0.46 0.94 0.76 0.55 1.31 264±1317 GC0365 AAT122794 13 26 04.69 -42 47 35.1 15.16 21.89 21.15 20.00 19.29 18.69 0.21 0.04 0.01 0.01 0.03 21.60 20.37 19.38 0.01 0.01 0.01 0.74 1.15 0.71 1.31 0.98 1.23 2.22 336±1617 GC0366 C105 13 26 05.12 -42 55 37.0 8.80 - - 22.01 - - - - - - - 23.68 22.58 21.76 0.07 0.04 0.04 - - - - 0.82 1.10 1.92 - GC0367 HGHH-07/C7 13 26 05.41 -42 56 32.4 8.30 18.38 18.03 17.17 16.65 16.08 0.03 0.01 0.01 0.01 0.03 18.18 17.33 16.64 0.03 0.03 0.01 0.35 0.86 0.53 1.09 0.68 0.85 1.53 595±11,2,3,5,6,7,8 GC0368 C106 13 26 06.15 -42 56 45.4 8.32 - - 21.28 - - - - - - - 22.54 21.46 20.66 0.03 0.01 0.01 - - - - 0.80 1.08 1.88 - GC0369 pff gc-085 13 26 06.42 -43 00 38.1 7.11 20.39 20.20 19.43 18.99 18.47 0.07 0.02 0.01 0.01 0.03 20.29 19.64 19.03 0.02 0.02 0.01 0.20 0.76 0.44 0.96 0.61 0.65 1.26 548±315,6,7 GC0370 pff gc-086 13 26 06.55 -43 06 14.5 8.75 21.48 20.93 19.95 19.37 18.76 0.16 0.03 0.01 0.01 0.03 21.29 20.32 19.45 0.02 0.01 0.01 0.56 0.97 0.59 1.20 0.87 0.97 1.84 440±265,7 GC0371 pff gc-087 13 26 06.87 -42 33 17.3 28.77 20.26 19.96 19.11 18.56 18.05 0.14 0.02 0.01 0.01 0.03 20.17 19.42 18.70 0.02 0.01 0.01 0.30 0.86 0.55 1.05 0.73 0.75 1.47 830±295,7 GC0372 HGHH-G170 13 26 06.93 -42 57 35.1 8.02 20.78 20.21 19.22 18.65 17.97 0.11 0.02 0.01 0.01 0.03 20.57 19.56 18.73 0.02 0.02 0.01 0.57 0.99 0.57 1.25 0.83 1.01 1.84 636±274,5,7 GC0373 AAT123188 13 26 06.94 -43 07 52.6 9.85 21.70 21.30 20.40 19.85 19.27 0.17 0.04 0.01 0.01 0.03 21.57 20.71 19.94 0.02 0.01 0.01 0.40 0.90 0.56 1.13 0.77 0.86 1.63 364±567 GC0374 HGHH-36/C36/R113 13 26 07.73 -42 52 00.3 11.72 19.42 19.19 18.35 17.81 17.33 0.04 0.01 0.01 0.01 0.03 19.32 18.61 17.94 0.02 0.01 0.01 0.23 0.84 0.54 1.03 0.67 0.71 1.38 703±13,5,6,7,8 GC0375 AAT123453 13 26 08.38 -42 59 18.9 7.67 21.29 20.95 20.09 19.55 19.03 0.17 0.04 0.02 0.01 0.03 21.20 20.40 19.67 0.03 0.02 0.01 0.34 0.86 0.54 1.06 0.73 0.80 1.53 257±1547 GC0376 pff gc-088 13 26 08.86 -43 01 21.4 7.54 19.99 19.79 18.99 18.46 18.00 0.05 0.02 0.01 0.01 0.03 19.86 19.14 18.59 0.03 0.03 0.01 0.20 0.80 0.53 0.99 0.55 0.72 1.27 554±295,7 GC0377 AAT123656 13 26 09.61 -43 07 05.9 9.71 21.77 21.13 20.08 19.46 18.79 0.19 0.04 0.01 0.01 0.03 21.57 20.51 19.59 0.03 0.02 0.02 0.64 1.05 0.63 1.29 0.92 1.07 1.99 380±937 GC0378 R111 13 26 09.71 -42 50 29.5 13.14 22.07 21.57 20.59 19.91 19.09 0.28 0.06 0.02 0.02 0.03 21.96 21.03 20.20 0.03 0.03 0.02 0.51 0.97 0.68 1.50 0.82 0.94 1.76 717±485,7 GC0379 C179 13 26 09.87 -42 56 36.0 8.96 - - - - - - - - - - 22.13 21.71 21.04 0.03 0.03 0.01 - - - - 0.67 0.42 1.09 - GC0380 HGHH-37/C37/R116 13 26 10.58 -42 53 42.7 10.81 19.86 19.38 18.43 17.87 17.26 0.04 0.01 0.01 0.01 0.03 19.65 18.72 17.96 0.01 0.01 0.01 0.48 0.95 0.56 1.17 0.76 0.93 1.69 612±13,5,6,7,8 GC0381 WHH-27 13 26 12.82 -43 09 09.2 11.51 20.27 19.68 18.66 18.06 17.42 0.05 0.01 0.01 0.01 0.03 20.02 19.03 18.13 0.02 0.01 0.01 0.59 1.02 0.60 1.24 0.90 0.99 1.89 545±606 GC0382 WHH-28 13 26 14.18 -43 08 30.4 11.25 20.21 19.98 19.17 18.65 18.14 0.05 0.02 0.01 0.01 0.03 20.13 19.48 18.71 0.02 0.01 0.01 0.23 0.81 0.52 1.03 0.77 0.65 1.41 506±1236 GC0383 HHH86-26/C26 13 26 15.27 -42 48 29.4 15.36 19.99 19.26 18.13 17.47 16.77 0.05 0.01 0.01 0.01 0.03 19.66 18.45 17.59 0.04 0.03 0.02 0.73 1.13 0.66 1.35 0.86 1.21 2.07 377±142,5,6,7 GC0384 R122 13 26 15.95 -42 55 00.5 10.76 19.02 18.78 18.02 17.58 17.14 0.03 0.01 0.01 0.01 0.03 - - - - - - 0.24 0.76 0.44 0.88 - - - 588±28 GC0385 AAT125079 13 26 17.29 -43 06 39.3 10.62 21.13 20.90 20.09 19.59 19.06 0.10 0.03 0.01 0.01 0.03 21.07 20.41 19.64 0.02 0.01 0.01 0.23 0.81 0.51 1.03 0.77 0.66 1.43 513±1837 GC0386 C50/K-233 13 26 19.66 -43 03 18.6 9.76 20.09 19.68 18.74 18.17 17.57 0.05 0.02 0.01 0.01 0.03 19.91 18.92 18.28 0.04 0.04 0.01 0.41 0.93 0.58 1.17 0.64 1.00 1.63 615±586,7 GC0387 C49/pff gc-089 13 26 20.20 -43 10 35.7 13.48 19.90 19.73 18.95 18.45 17.95 0.12 0.02 0.01 0.01 0.03 19.85 19.30 18.53 0.03 0.02 0.01 0.17 0.78 0.50 1.00 0.77 0.56 1.32 538±305,6,7 GC0388 pff gc-090 13 26 20.53 -43 03 18.5 9.91 20.90 20.34 19.33 18.74 18.10 0.10 0.02 0.01 0.01 0.03 20.70 19.61 18.85 0.03 0.03 0.01 0.56 1.00 0.60 1.24 0.75 1.09 1.85 486±265,7 GC0389 AAT215171 13 26 20.66 -42 38 32.0 24.60 21.03 20.63 19.69 19.11 18.53 0.27 0.03 0.01 0.01 0.03 20.97 20.06 19.26 0.02 0.02 0.01 0.40 0.94 0.58 1.15 0.80 0.90 1.70 527±487 GC0390 R107 13 26 21.11 -42 48 41.1 15.84 21.53 20.82 19.75 19.13 18.48 0.16 0.03 0.01 0.01 0.03 21.23 20.16 19.30 0.05 0.03 0.02 0.71 1.07 0.62 1.27 0.86 1.06 1.92 405±285,7 GC0391 pff gc-091 13 26 21.14 -43 42 24.6 42.41 20.42 20.13 19.29 18.77 18.21 0.15 0.02 0.01 0.01 0.03 20.30 19.56 18.72 0.01 0.01 0.01 0.29 0.84 0.52 1.08 0.84 0.74 1.58 623±375,7 GC0392 pff gc-092 13 26 21.31 -42 57 19.1 10.53 21.36 20.86 19.92 19.35 18.76 0.14 0.03 0.01 0.01 0.03 21.18 20.27 19.46 0.02 0.02 0.01 0.50 0.93 0.57 1.16 0.81 0.92 1.73 462±275,7 Table 1—Continued Name Old Name RA Dec. R U B V R I σU σB σV σR σI C M T1 σC σM σT1 U − B B − V V − R V − I M − T1 C − M C − T1 vr (J2000) (J2000) (’) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (km s−1) GC0393 R117 13 26 21.99 -42 53 45.5 12.38 20.52 20.41 19.68 19.23 18.75 0.07 0.02 0.01 0.01 0.03 20.50 19.98 19.32 0.02 0.02 0.01 0.10 0.73 0.45 0.93 0.66 0.53 1.18 484±265,7 GC0394 R118 13 26 22.01 -42 54 26.5 11.99 20.73 20.60 19.87 19.38 18.93 0.08 0.03 0.01 0.01 0.03 20.72 20.19 19.50 0.03 0.02 0.01 0.13 0.73 0.49 0.95 0.70 0.52 1.22 440±735 GC0395 WHH-29 13 26 22.08 -43 09 10.7 12.79 20.97 20.70 19.82 19.28 18.73 0.08 0.02 0.01 0.01 0.03 20.92 20.17 19.38 0.03 0.02 0.01 0.27 0.88 0.54 1.09 0.79 0.75 1.54 505±786 GC0396 pff gc-093 13 26 22.65 -42 46 49.8 17.50 20.58 20.39 19.62 19.13 18.66 0.08 0.03 0.01 0.01 0.03 20.52 19.91 19.23 0.02 0.01 0.01 0.20 0.77 0.48 0.96 0.69 0.61 1.30 576±425,7 GC0397 WHH-30 13 26 23.60 -43 03 43.9 10.55 20.56 20.25 19.38 18.83 18.30 0.10 0.03 0.01 0.01 0.03 20.45 19.62 18.96 0.03 0.03 0.01 0.31 0.87 0.55 1.08 0.65 0.83 1.49 470±666 GC0398 pff gc-094 13 26 23.66 -43 00 45.6 10.25 20.99 20.77 19.97 19.49 18.93 0.09 0.03 0.01 0.01 0.03 20.92 20.24 19.57 0.03 0.02 0.01 0.22 0.80 0.47 1.04 0.67 0.68 1.35 334±645,7 GC0399 HHH86-38/C38/R123 13 26 23.78 -42 54 01.1 12.50 19.67 19.30 18.41 17.87 17.32 0.04 0.01 0.01 0.01 0.03 19.51 18.68 17.96 0.02 0.02 0.01 0.37 0.89 0.54 1.09 0.72 0.83 1.54 405±13,5,6,8 GC0400 HGHH-51 13 26 23.86 -42 47 17.1 17.26 19.63 19.34 18.47 17.93 17.38 0.05 0.02 0.01 0.01 0.03 - - - - - - 0.29 0.87 0.54 1.09 - - - 343±434,5 GC0401 AAT335187 13 26 23.95 -43 17 44.4 19.53 21.19 21.11 20.46 20.04 19.60 0.10 0.03 0.01 0.01 0.03 21.20 20.72 20.08 0.01 0.01 0.01 0.08 0.65 0.42 0.86 0.64 0.49 1.12 277±1587 GC0402 pff gc-095 13 26 25.50 -42 57 06.2 11.33 21.13 20.60 19.60 19.02 18.40 0.11 0.02 0.01 0.01 0.03 20.97 19.97 19.17 0.04 0.03 0.02 0.53 1.00 0.58 1.20 0.80 0.99 1.79 405±185,7 GC0403 R124 13 26 28.87 -42 52 36.4 14.08 20.32 20.17 19.42 18.96 18.43 0.05 0.02 0.01 0.01 0.03 20.28 19.69 19.02 0.02 0.02 0.01 0.15 0.75 0.46 0.99 0.68 0.58 1.26 541±355,7 GC0404 pff gc-096 13 26 30.29 -42 34 41.7 28.83 21.24 20.73 19.73 19.10 18.42 0.31 0.03 0.01 0.01 0.03 21.08 20.09 19.24 0.01 0.01 0.01 0.52 1.00 0.63 1.30 0.85 0.99 1.83 532±255,7 GC0405 R105 13 26 33.55 -42 51 00.9 15.74 21.28 20.78 19.86 19.28 18.70 0.11 0.03 0.01 0.01 0.03 21.18 20.25 19.46 0.02 0.01 0.01 0.50 0.92 0.58 1.15 0.79 0.93 1.72 534±345,7 GC0406 HGHH-27/C27 13 26 37.99 -42 45 49.9 20.00 19.60 19.39 18.60 18.11 17.62 0.03 0.01 0.01 0.01 0.03 - - - - - - 0.21 0.78 0.49 0.98 - - - 492±372,5 GC0407 WHH-31 13 26 41.43 -43 11 25.0 16.96 20.76 20.44 19.50 18.94 18.36 0.07 0.02 0.01 0.01 0.03 20.67 19.89 19.05 0.03 0.02 0.02 0.32 0.94 0.56 1.14 0.85 0.77 1.62 573±676 GC0408 HHH86-39/C39 13 26 42.03 -43 07 44.8 15.12 18.72 18.33 17.43 16.91 16.42 0.02 0.01 0.01 0.01 0.03 18.57 17.73 16.92 0.01 0.01 - 0.39 0.89 0.53 1.01 0.81 0.83 1.65 271±203,5,6,7 GC0409 pff gc-097 13 26 45.40 -43 26 34.1 29.13 19.75 19.51 18.72 18.23 17.76 0.10 0.01 0.01 0.01 0.03 19.66 19.04 18.28 0.02 0.01 0.01 0.24 0.79 0.49 0.96 0.76 0.62 1.38 599±305,7 GC0410 HH-017 13 26 49.31 -43 04 57.8 15.41 23.21 22.54 20.80 19.87 19.06 0.67 0.11 0.02 0.01 0.03 23.32 21.63 20.13 0.05 0.03 0.02 0.67 1.75 0.93 1.73 1.50 1.68 3.19 839±637 GC0411 pff gc-098 13 26 53.94 -43 19 17.7 24.05 19.54 19.17 18.28 17.73 17.16 0.09 0.01 0.01 0.01 0.03 - - - - - - 0.37 0.89 0.55 1.12 - - - 631±185,7 GC0412 AAT222977 13 26 58.91 -42 38 53.4 27.82 21.18 21.02 20.34 19.84 19.38 0.28 0.03 0.01 0.01 0.03 21.16 20.64 20.02 0.02 0.01 0.01 0.17 0.68 0.50 0.96 0.62 0.52 1.14 655±1127 GC0413 HH-060 13 26 59.78 -42 55 26.5 17.79 - - - - - - - - - - 23.41 21.69 20.14 0.05 0.04 0.02 - - - - 1.55 1.72 3.26 811±327 GC0414 pff gc-099 13 26 59.82 -42 32 40.5 33.09 20.96 20.72 19.94 19.40 18.92 0.23 0.03 0.01 0.01 0.03 20.92 20.29 19.59 0.02 0.01 0.02 0.25 0.78 0.53 1.02 0.70 0.63 1.33 425±345,7 GC0415 pff gc-100 13 27 03.41 -42 27 17.2 38.12 19.92 19.37 18.41 17.81 17.17 0.11 0.01 0.01 0.01 0.03 - - - - - - 0.56 0.96 0.60 1.24 - - - 513±105,7 GC0416 pff gc-101 13 27 21.56 -42 38 41.4 30.63 19.89 19.51 18.72 18.20 17.74 0.10 0.01 0.01 0.01 0.03 19.74 19.02 18.39 0.01 0.01 - 0.38 0.79 0.51 0.97 0.63 0.72 1.35 263±165,7 GC0417 pff gc-102 13 28 18.45 -42 33 12.5 41.90 20.38 20.11 19.29 18.76 18.24 0.14 0.02 0.01 0.01 0.03 20.28 19.59 18.94 0.04 0.04 0.01 0.27 0.83 0.52 1.04 0.65 0.70 1.34 428±405,7 1Radial velocity measured by van den Bergh, Hesser, & Harris (1981). 2Radial velocity measured by Hesser et al. (1984). 3Radial velocity measured by Hesser, Harris, & Harris (1986). 4Radial velocity measured by Harris et al. (1992). 5Radial velocity measured by Peng, Ford, & Freeman (2004a). 6Radial velocity measured by Woodley, Harris, & Harris (2005). 7Radial velocity measured by Beasley et al. (2006). 8Radial velocity measured by Rejkuba et al. (2007). 9These objects appear starlike under HST/ACS imaging (Harris et al. 2006). 10This object appears to be a M-type star based on the strong molecular bands in its spectrum (Beasley et al. 2006). 11Faint globular (Harris et al. 2006). 12ocated in the inner bulge region which is very crowded (Harris et al. 2006). 13A very compact globular or star (Harris et al. 2006). 14A possible globular, but slightly elliptical (Harris et al. 2006). 15A globular next to a very bright star (Harris et al. 2006). – 39 – Table 2. Kinematic and Dynamic Solutions for the Globular Cluster System of NGC 51281 Radial Bin Rmean Rout N ΩR Θo σv Mcorr Mp Mr Mt (kpc) (kpc) (kpc) (km s−1) (o E of N) (km s−1) (×1010M⊙) (×10 10M⊙) (×10 10M⊙) 0-50 12.9 48.8 340 40±10 189±12 123±5 1.0 125.8±46.52 3.1±1.3 128.9±46.5 0-5 3.64 4.96 54 24±21 334±59 120±12 - - - - 5-10 7.65 9.96 124 43±15 195±20 112±8 2.3 37.4±6.4 0.4±0.3 37.8±6.4 10-15 12.4 14.9 68 83±25 195±12 105±10 1.6 36.5±9.3 2.4±1.5 38.9±9.4 15-25 19.0 24.3 56 35±26 184±34 147±16 1.3 89.5±27.4 0.7±1.0 90.2±27.4 25-50 34.7 48.8 39 96±45 169±17 148±21 1.0 184.7±84.5 10.4±9.8 195.1±85.1 1This table is reproduced from Woodley (2006) for completeness. 2The Tracer Mass estimator determines the total mass enclosed within the outermost radius using a sample defined in the range of the inner and outer radial bins. Table 3. Kinematic and Dynamic Solutions for the Metal-Poor Globular Cluster System of NGC 5128 Radial Bin Rmean Rout N ΩR Θo σv Mcorr Mp Mr Mt (kpc) (kpc) (kpc) (km s−1) (o E of N) (km s−1) (×1010M⊙) (×10 10M⊙) (×10 10M⊙) 0-50 13.8 48.1 178 31±14 177±22 117±7 1.0 94.5±58.5 1.1±1.0 95.6 ±58.6 0-5 3.84 4.96 22 16±38 373±120 99±19 - - - - 5-10 7.76 9.96 66 30±21 223±39 116±12 2.1 38.9±12.4 0.2±0.3 39.1±12.4 10-15 12.7 14.8 41 97±38 199±13 102±15 1.5 31.4±15.1 3.3±2.5 34.6±15.3 15-25 18.8 24.3 26 45±31 139±49 112±20 1.3 59.7±32.7 1.2±1.6 60.8±32.7 25-50 36.7 48.1 23 61±43 137±50 141±24 1.0 122.4±82.9 4.2±5.9 126.7±83.1 Table 4. Kinematic and Dynamic Solutions for the Metal-Rich Globular Cluster System of NGC 5128 Radial Bin Rmean Rout N ΩR Θo σv Mcorr Mp Mr Mt (kpc) (kpc) (kpc) (km s−1) (o E of N) (km s−1) (×1010M⊙) (×10 10M⊙) (×10 10M⊙) 0-50 12.0 48.8 158 47±15 202±15 129±9 1.0 116.5±73.8 4.3±2.3 120.8±73.9 0-5 3.61 4.98 29 44±27 308±47 134±23 - - - - 5-10 7.52 9.93 58 65±22 180±19 105±11 1.5 17.4±7.7 1.0±0.7 18.3±7.7 10-15 12.0 14.9 27 74±34 186±22 108±17 1.2 23.8±11.3 1.9±1.8 25.7±13.3 15-25 19.3 24.3 30 50±36 223±32 168±24 1.1 72.1±46.2 1.2±1.7 73.3±46.3 25-50 31.7 48.8 15 102±96 191±21 146±65 1.0 219.1±174.6 11.8±22.2 230.9±176.0 – 40 – Table 5. Kinematic and Dynamic Solutions for the Planetary Nebula System of NGC Radial Bin Rmean Rout N ΩR Θo σv Mcorr Mp Mr Mt (kpc) (kpc) (kpc) (km s−1) (o E of N) (km s−1) (×1010M⊙) (×10 10M⊙) (×10 10M⊙) 0-90 14.1 88.2 780 76±6 170±5 118±13 1.0 84.6±17.2 11.9±1.9 96.5±17.3 0-5 3.3 4.99 184 75±16 179±11 131±7 - - - - 5-10 7.6 9.98 211 82±12 177±8 120±6 3.0 43.9±3.6 1.6±0.5 45.4±3.6 10-15 11.8 14.8 138 76±15 171±10 118±7 2.0 42.3±5.0 2.0±0.8 44.2±5.1 15-20 17.4 20.0 71 96±24 169±12 116±10 1.7 47.7±5.6 4.3±2.1 52.0±6.9 20-30 24.6 30.0 87 76±17 157±17 108±8 1.5 53.2±8.4 4.0±1.8 57.2±8.6 30-40 34.6 39.8 50 44±15 132±38 87±9 1.3 40.7±7.4 1.8±1.2 42.3±7.5 40-80 48.7 71.2 36 61±45 183±22 85±11 1.0 48.6±11.5 6.2±9.1 54.8±14.7 Table 6. Spin Parameters for Giant Elliptical Galaxies Galaxy ΩRMP σv,MP (ΩR/σv)MP ΛMP ΩRMR σv,MR (ΩR/σv)MR ΛMR (km s−1) (km s−1) (km s−1) (km s−1) NGC 5128 31±14 117±7 0.26±0.12 0.08±0.04 47±15 129±9 0.36±0.11 0.11±0.03 M878,9 186+58 397+36 0.47+0.13 −0.11 0.14 +0.04 −0.03 155 365+38 0.43+0.14 −0.12 0.13 +0.04 −0.04 M499 93+69 342+33 0.27+0.19 −0.11 0.08 +0.06 −0.03 -26 265+34 0.10+0.27 −0.25 0.03 +0.08 −0.08 NGC 139910 15±26 291±14 0.05±0.09 0.02±0.03 7±24 255±13 0.03±0.09 0.01±0.03 8Kinematic data taken from Côté et al. (2001). 9Kinematic data taken from Côté et al. (2003). 10Kinematic data taken from Richtler et al. (2004). – 41 – Fig. 1.— Positions of the GCS (left) totaling 340 objects and the PN system (right) totaling 780 objects in NGC 5128. The PN system has more than double the objects and extends ∼ 40 kpc further out from the center of NGC 5128. Both systems are spatially biased to the major axis of the galaxy with a position angle of 35o east of north. – 42 – Fig. 2.— Velocity histograms fit with a Gaussian (mean velocity and standard deviation in km s−1) for the samples. Top left, the entire GC (546.4 ± 7.3, 128.8 ± 7.3); top right, GCs with a radial velocity uncertainty < 50 km s−1 (554.8 ± 5.3, 126.2 ± 7.3); bottom left, all metal-poor GCs (532.5 ± 12.7, 123.8 ± 12.8); bottom right, all metal-rich GCs (565.5 ± 10.5, 140.3 ± 10.7). – 43 – Fig. 3.— Projected radial distribution as a function of projected angular position of all known GCs in NGC 5128. The angular distribution of metal-poor (squares), metal-rich (crosses), and unknown metallicity (circles) subpopulations of GCs clearly shows the observational bias in GC searches along the photometric minor axis (Θ = 119 and 299o east of north, Dufour et al. 1979) beyond 15 kpc. – 44 – Fig. 4.— Sine curve fit for the GCs in NGC 5128 (circles) with a fixed systemic velocity of vsys = 541 km s −1, for 0-50 kpc from the center of NGC 5128. The top panel shows all 340 GCs with rotation amplitude ΩR = 40± 10 km s−1 and rotation axis Θo = 189± 12 o east of north. The middle panel shows the 178 metal-poor clusters with ΩR = 31 ± 14 km s−1 and Θo = 177 ± 22 o east of north, and the bottom panel shows the 158 metal-rich clusters with ΩR = 47± 15 km s−1 and Θo = 202± 15 o east of north. The squares represent the weighted velocities in 72o bins. – 45 – Fig. 5.— Rotation amplitude as a function of projected galactocentric radius from the center of NGC 5128, including all known GCs (left) and the 780 PNe (right). The data are binned in radial bins (squares) of 0-5, 5-10, 10-15 15-25, and 25-50 kpc for the GCs or 0-5, 5-10, 10-15, 15-20, 20-30, 30-40, and 40-80 kpc for the PNe, in equal number of 38 GCs (filled circles) or 60 PNe, and exponentially weighted GCs or PNe with varying bin width (solid line) and 35% uncertainties of the weighted data (dashed lines). The radial distribution of data is shown at the bottom (open circles). – 46 – Fig. 6.— Same as Fig. 5, but for the metal-poor subpopulation of GCs (left) and the metal- rich subpopulation of GCs (right). The equal numbered bins (filled circles) consist of 20 metal-poor or metal-rich GCs. – 47 – Fig. 7.— Rotation axis as a function of projected galactocentric radius from the center of NGC 5128, including all known GCs (left) and the 780 PNe (right). The symbols are the same as in Fig. 5, but the uncertainty of the weighted data (dashed lines) is 25%. – 48 – Fig. 8.— Same as Fig. 7, but for the metal-poor (left) and metal-rich (right) subpopulations of GCs. – 49 – Fig. 9.— Velocity dispersion as a function of projected galactocentric radius from the center of NGC 5128 for all known GCs (left) and the 780 PNe (right). The symbols are the same as in Fig. 7, but the uncertainties of the weighted data (dashed lines) are determined by Eqn 2. – 50 – Fig. 10.— Same as Fig. 9, but for the metal-poor (left) and metal-rich (right) subpopulations of GCs. – 51 – Fig. 11.— Velocity histograms for the GCs, subdivided by both metallicity and radius, binned in 50 km s−1 intervals. Here vf is the residual velocity after subtraction of the systemic velocity of 541 km s−1 and the overall rotation component for either the metal-poor or metal-rich subpopulation, determined from Eqn. 1. The histograms for the metal-poor clusters (left panels) include 129 clusters between 0 and 15 kpc and 49 clusters between 15 and 50 kpc, while the histograms for the metal-rich clusters (right panels) include 114 clusters between 0 and 15 kpc and 44 clusters between 15 and 50 kpc. – 52 – Fig. 12.— Surface density of known GCs fit with a powerlaw for the entire population (solid line), the metal-poor subpopulation (dotted line), and the metal-rich subpopulation (short- dashed line) in NGC 5128 with the best-fit linear coefficients shown. The radial distribution of the entire GC population is shown as filled squares, yet clusters with projected radii < 5 kpc are shown as open squares and are not included in the powerlaw fits. Also overplotted is the surface density of the PN system (long-dashed line) fit from all PNe beyond 5 kpc with the best-fit linear coefficients also shown. – 53 – Fig. 13.— Total mass profile for the radially binned estimates of the GC system (filled squares) from Woodley (2006) and the PNe (filled triangles). The GC total mass from 0 to 50 kpc, shown as the open square, is 1.3 ± 0.5 × 1012M⊙, and the PN total mass from 0 to 90 kpc, shown as the open triangle, is 1.0± 0.2× 1012M⊙, agreeing within the uncertainties. Introduction The Catalog of Globular Clusters in NGC 5128 Kinematics of the Globular Cluster System Velocity Field Rotation Amplitude, Rotation Axis, and Velocity Dispersion Mathematic and Analytic Description Rotation Amplitude of the Globular Cluster System Rotation Axis of the Globular Cluster System Velocity Dispersion of the Globular Cluster System Kinematics of the Planetary Nebula System of NGC 5128 Dynamics of NGC 5128 Mass Determination Surface Density Profiles Mass Results Discussion and Conclusions

0704.1190

Topological Properties of Phase Singularities in Wave Fields

Topological Properties of Phase Singularities in Wave Fields Yi-Shi Duan, Ji-Rong Ren, and Tao Zhu∗ Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, P. R. China (Dated: November 21, 2018) Phase singularities as topological objects of wave fields appear in a variety of physical, chemical, and biological scenarios. In this paper, by making use of the φ-mapping topological current theory, we study the topological properties of the phase singularities in two and three dimensional space in details. The topological inner structure of the phase singularities are obtained, and the topological charge of the phase singularities are expressed by the topological numbers: Hopf indices and Brouwer degrees. Furthermore, the topological invariant of the closed and knotted phase singularities in three dimensional space are also discussed in details. PACS numbers: 02. 40. Xx, 03. 65. Vf, 05. 45. Yv I. INTRODUCTION In physics, the solutions of wave equations in two and three dimensional space often possess phase singularities[1, 2]. At the point of phase singularity, the phase of the wave is undefined and wave intensity van- ishes. This phase singularity is a common property to all waves, it exists in many area of physical, chemical, and biological scenarios, such as the quantized vortices in superfluid or superconductor systems[3, 4], the vor- tices phase singularities in Bose-Einstein condensates[5], the streamlines or singularities in quantum mechanical wavefunctions[6, 7], the optical vortices in optical wave systems[8, 9, 10, 11, 12, 13], the vortex filaments in chem- ical reaction and molecular diffusion [15, 16, 17], and the phase singularities in biological systems[17, 18]. In recent years, the phase singularities have drawn great interest because it is of importance for understanding fundamental wave physics and have many important applications, and a great deal of works on the phase singularities in wave fields have been done by many physicists[1, 2, 12, 13]. Phase singularities are generically points in two dimen- sional space and form a network of lines in three di- mensional space. Around the phase singularity points in two dimensions or phase singularity lines in three dimensions, the phase changes by 2π times an inte- ger. This integer is the so-called topological charge (the strength of the singularity) of the phase singularity ob- jects (points or lines). In three dimensional space, a im- portant case is that the phase singularity lines are closed and knotted curves, these knot-like configurations exist in a variety of physical scenarios, including Bose-Einstein condensations[14], chemical reaction and molecular diffu- sion systems[15, 16, 17], optical wave systems[19, 20] and field theory[21, 22]. The phase singularities (points, lines, or knotted) are topological objects, and have a very rich topological properties, so it is necessary to use the topo- logical method to study the topology of these objects. ∗Corresponding author. Email : [email protected] In this paper, we have given a generic feature of the topological properties of the phase singularities in wave fields. By making use of the φ-mapping topological current theory[23], we study the topological current of the phase singularity objects (points in two dimensions and lines in three dimensions), and the topological inner structure of these topological objects are obtained. For the case that the phase singularity lines in three dimen- sional space are closed and knotted curves, we discussed the topological invariant of these knotted family in de- tails. This paper is arranged as follows. In Sec.II, we construct a topological current density of the phase sin- gularity points in two dimensions, this topological cur- rent density don’t vanish only when the phase singular- ity points exist, the topological charge of phase singu- larity points are expressed by the topological quantum numbers,the Hopf indices and Brouwer degrees of the φ- mapping. In Sec.III is the topology of the phase singu- larity lines in three dimensions. In Sec.IV, we introduce a important topological invariant to describe the phase singularity lines when they are linked and knotted, it is just the sum of all the self-linking and all the linking numbers of the knot family. In Sec.V is our concluding remarks. II. TOPOLOGICAL STRUCTURE OF PHASE SINGULARITIES IN TWO DIMENSIONS Let us study a complex scalar field ψ(~x, t), the wave fields function ψ(~x, t) is maps from space (of either two or three dimensions) to the complex numbers, so ψ : R2, R3 → C. The complex scalar field ψ(~x, t) is time dependent as well as space dependent, and it can be writ- ten as ψ(~x, t) = ‖ψ‖ eiθ = φ1(~x, t) + iφ2(~x, t), (1) where φ1(~x, t) and φ2(~x, t) can be regarded as complex representation of a two-dimensional vector field ~φ = (φ1, φ2) over two or three dimensional space, ‖ψ‖ =√ ψ∗ψ is the modulus of ψ, and θ is the phase factor. In this section, we will restrict our attention to the phase http://arxiv.org/abs/0704.1190v1 singularities in two dimensional space, and labeled points ~x = (x1, x2) in cartesian coordinates. The phase singu- larities in three dimensional space will be discussed in Sec.III and Sec.IV. It is known that the current density ~J associated with ψ(~x, t) is defined as ~J = Im(ψ∗∇ψ) = ‖ψ‖2~v, (2) the ~v is the velocity field. From the expressions in Eq.(1) and Eq.(2), the velocity field ~v can be rewritten as (ψ∗∇ψ −∇ψ∗ψ) = ∇θ, (3) it becomes the gradient of the phase factor θ. The vor- ticity field ~ω of the velocity field ~v is defined as ∇× ~v. (4) Obviously, in two dimensional space the ~ω only has z- component, i.e., ~ω = ω · ~ez. Form Eq.(3),we directly obtain a trivial curl-free result: ~ω = 1 ∇×∇θ = 0. But in topology, because of the existence of phase singularities in the wave fields ψ, the vorticity ~ω does not vanish[4]. So in the following discussions, we will study that what the exact expression for ~ω is in topology. Introducing the unit vector na = φa/‖φ‖(a = 1, 2;nana = 1), and defining a topological current in (2+1) dimensions space-time, ǫijkǫab∂jn b, i = 0, 1, 2. (5) Obviously, the topological current is identically con- served, i = 0. (6) By making use of the definition of the unit vector na, one can reexpressed the velocity field ~v as ~v = ǫabn a∇nb, (7) and the vorticity ~ω is ω = j0 = ǫjkǫab∂jn b j, k = 1, 2, (8) it is just the time component of the topological cur- rent ji. According to the φ-mapping topological current theory[23], one can prove that ji = δ2(~φ)Di( ), (9) where εijkǫab∂jφ b (10) is the Jacobian vector, in which the time component D0(φ/x) is the usual two dimensional Jacobian D(φ/x). The expression of the topological current in Eq.(9) shows that the topological current ji does not vanish only at the zero points of ~φ, i.e., the phase singularities of the wave fields ψ. According to the Eq.(8) and Eq.(9), the vorticity can be expressed ω = j0 = δ2(~φ)D( ), (11) obviously, ω also does not vanish only when the phase singularities of the wave fields ψ exist. The topological current ji and vorticity ω paly an important role in the topological property of the phase singularities, so it is necessary to study the zero points of ~φ to determine the non-zero solutions of ji. According to the implicit function theory[24], while the regular condition Di(φ/x) 6= 0 is satisfied, the general solutions of φ1(x1, x2, t) = 0, φ2(x1, x2, t) = 0, (12) can be expressed as x1 = x1k(s, t), x 2 = x2k(s, t), k = 1, 2, . . . , N, (13) which represent N zero points ~zk (k=1, 2, . . . , N) where ψ(~r, t) = 0 in two-dimensional space. These zero points solutions are just the so-called phase singularity points of the wave fields ψ, they represent the zeroes of the real part and imaginary part of the complex scalar wave fields: Re(ψ) = φ1 = 0 and Im(ψ) = φ2 = 0. In the theory of δ function of ~φ(~r), one can prove that in 2-dimensional space[25] δ2(~φ) = |D(φ/x)~zk | δ2(~x− ~zk), (14) where the positive integer βk is called the Hopf index of map x→ φ[12, 13]. The meaning of βk is that when the point ~zk covers the neighborhood Σk of zero ~zk once, the vector field ~φ covers the corresponding region βk times. By substituting Eq.(14) into Eq.(9), one can obtain that 2(~x− ~zk) , (15) whereWk = βkηk is the winding number of ~φ around zero point ~zk, and ηk = sgn(D(φ/x)) = ±1 is the Brouwer degrees of φ-mapping. The sign of Brouwer degrees are very important, for the case of vortices phase singulari- ties, the ηk = +1 corresponds to the vortex, and ηk = −1 corresponds to the antivortex. The topological charge (the strength of phase singular- ities) of the phase singularities point ~zk is defined by the Gauss map ~n : ∂Σk → S1[26]: n∗(ǫabn adnb) (16) where n∗ is the pullback of Gauss map n, ∂Σk is the boundary of a neighborhood Σk of point ~zk and Σk ∩ Σm = ∅ for Σm is the neighborhood of another arbitrary zero point ~zm. In topology it means that, when the point ~zk covers ∂Σk once, the unit vector ~n will cover S 1, or ~φ covers the corresponding region Wk times, which is a topological invariant. By using the Stokes’ theorem and in term of Eq.(8), one can obtain that jk∂jn ωd2x = βkηk. (17) It is clear that the topological charges densities of phase singularity points is just the non-vanishing vorticity ω of the velocity field ~v. Using the notion j = (ω,~j), Eq.(6) can be rewrite as +∇ ·~j = 0, (18) this expression tells us that the topological charges of the phase singularity points are conserved. This is the only topological property of the complex wave fields. In this section, we obtain the topological inner struc- ture of the phase singularity points in two dimensional space, the solutions of the Eq.(13) possess N isolated phase singularity points of which the kth point possesses topological charge Wk = βkηk. III. TOPOLOGICAL CURRENT OF PHASE SINGULARITIES IN THREE DIMENSIONS In this section,we will study the phase singularities in three dimensions space, some results which discussed in the above section in two dimensions space are also useful here. Differently with in two dimensions, the point is labeled ~x = (x1, x2, x3) and the vorticity ~ω have its all three components. In three dimensional space, the ~ω can be reexpressed as ǫijkǫab∂jn b, (19) it is clearly that the vorticity ~ω is just the topological current of phase singularities in three dimensions space. Using the φ-mapping theory[23], the topological cur- rent ~ω is rewritten as ωi = δ2(~φ)Di( ), (20) we can see from this expression that the vorticity ~ω is non-vanishing only if ~φ = 0, i.e., the existence of the phase singularities, so it is necessary to study these zero solutions of ~φ. In three dimensions space, these solutions are some isolate zero lines, which are the so-called phase singularity lines in three dimensions space. Under the regular condition Di(φ/x) 6= 0, the general solutions of φ1(x1, x2, x3, t) = 0, φ2(x1, x2, x3, t) = 0 (21) can be expressed as x1 = x1k(s, t), x 2 = x2k(s, t), x 3 = x3k(s, t), (22) which represent the world surfaces of N moving iso- lated singular strings Lk with string parameter s (k = 1, 2, · · · , N). These singular strings solutions are just the phase singularities solutions in three dimensions space. In δ-function theory[25], one can obtain in three di- mensions space δ2(~φ) = δ3(~x − ~xk()s) ds, (23) where )|Σk = ǫjkǫmn and Σk is the kth planar element transverse to Lk with local coordinates (u1, u2). The βk is the Hopf index of φ mapping, which means that when ~x covers the neigh- borhood of the zero point ~xk(s) once, the vector field φ covers the corresponding region in φ space βk times. Meanwhile the direction vector of Lk is given by |xk = Di(φ/x) D(φ/u) |xk . (24) Then from Eq.(23) and Eq.(24) one can obtain the inner structure of ωi: δ3(~x− ~xk(s))ds, (25) whereWk = βkηk is the winding number of ~φ around Lk, with ηk = sgnD(φ/u)|~xk = ±1 being the Brouwer degree of φ mapping. The sign of Brouwer degrees are very im- portant, for the case of vortices phase singularities, the ηk = +1 corresponds to the vortex, and ηk = −1 corre- sponds to the antivortex. Hence the topological charge of the phase singularity line Lk is[21] ωidσi =Wk. (26) The results in this section show us the topological inner structure of the topological current density ωi. The topo- logical charge of the kth phase singularity line in three dimensions can be expressed by the topological numbers: Qk =Wk = βkηk. IV. TOPOLOGICAL ASPECT ON KNOTTED PHASE SINGULARITY LINES An important case is that the phase singularity lines in three dimensions space from closed and knotted curves. Topology has play a very important role in understanding these knot configurations, so it is necessary to study the topology in the knotted phase singularity lines. In order to do that, we define a the helicity integral[27] ~v · ∇ × ~vd3x, (27) this is an important topological knot invariant and it measures the linking of the phase singularity lines. From the Eq.(4), the helicity integral can be changed as ~v · ~ωd3x. (28) Substituting Eq. (25) into Eq. (28), one can obtain ~v · d~x, (29) when these phase singularities are closed and knotted lines, i.e., a family of knots ξk(k = 1, 2, . . . , N), Eq. (29) becomes ~v · d~x. (30) It is well known that many important topological num- bers are related to a knot family such as the self-linking number and Gauss linking number. In order to discuss these topological numbers of knotted phase singularity lines, we define Gauss mapping: ~m : S1 × S1 → S2, (31) where ~m is a unit vector ~m(~x, ~y) = ~y − ~x |~y − ~x| , (32) where ~x and ~y are two points, respectively, on the knots ξk and ξl (in particular, when ~x and ~y are the same point on the same knot ξ , ~n is just the unit tangent vector ~T of ξ at ~x ). Therefore, when ~x and ~y , respectively, cover the closed curves ξk and ξl once, ~n becomes the section of sphere bundle S2. So, on this S2 we can define the two-dimensional unit vector ~e = ~e(~x, ~y). ~e, ~m are normal to each other, i.e. , ~e1 · ~e2 = ~e1 · ~m = ~e2 · ~m = 0, ~e1 · ~e1 = ~e2 · ~e2 = ~m · ~m = 1. (33) In fact, the velocity field ~v can be decomposed in terms of this two-dimensional unit vector ~e: vi = ǫabe b − ∂iθ, where θ is a phase factor[4]. Since one can see from the expression ~ω = 1 ∇ × ~v that the (∂iθ) term does not contribute to the integral H , vi can in fact be expressed vi = ǫabe b. (34) Substituting it into Eq.(14), one can obtain a(~x, ~y)∂ie b(~x, ~y)dxi. (35) Noticing the symmetry between the points ~x and ~y in Eq.(32), Eq.(35) should be reexpressed as k,l=1 ǫab∂ie bdxi ∧ dyj . (36) In this expression there are three cases: (1) ξk and ξl are two different phase singularities (ξk 6= ξl), and ~x and ~y are therefore two different points (~x 6= ~y); (2) ξk and ξl are the same phase singularities (ξk = ξl), but ~x and ~y are two different points (~x 6= ~y); (3) ξk and ξl are the same phase singularities (ξk = ξl), and ~x and ~y are the same points (~x = ~y). Thus, Eq.(36) can be written as three terms: k=1(k=l, ~x 6=~y) ǫab∂ie bdxi ∧ dyj k,l=1(k 6=l) ǫab∂ie bdxi ∧ dyj .(37) By making use of the relation ǫab∂ie b = 1 ~m · (∂i ~m× ∂j ~m)[28], the Eq.(37) is just k=1(~x 6=~y) ~m∗(dS) k,l=1(k 6=l) ~m∗(dS), (38) where ~m∗(dS) = ~m ·(∂i ~m×∂j ~m)dxi∧dyj(~x 6= ~y) denotes the pullback of the S2 surface element. In the following we will investigate the three terms in the Eq.(38) in detail. Firstly, the first term of Eq.(38) is just related to the writhing number[29] Wr(ξk) of ξk Wr(ξk) = ~m∗(dS). (39) For the second term, one can prove that it is related to the twisting number Tw(ξk) of ξk bdxi = (~T × ~V ) · d~V = Tw(ξk), (40) where ~T is the unit tangent vector of knot ξk at ~x (~m = ~T when ~x = ~y) and ~V is defined as ea = ǫabV b(~V ⊥ ~T ,~e = ~T × ~V ). In terms of the White formula[30] SL(ξk) =Wr(ξk) + Tw(ξk), (41) we see that the first and the second terms of Eq.(38) just compose the self-linking numbers of knots. Secondly, for the third term, one can prove that ~m∗(dS) (xk − yk) ‖~x− ~y‖3 = Lk(ξk, ξl) (k 6= l), (42) where Lk(ξk, ξl) is the Gauss linking number between ξk and ξl[29]. Therefore, from Eqs.(39), (40), (41) and (42), we obtain the important result: W 2kSL(ξk) + k,l=1(k 6=l) WkWlLk(ξk, ξl). (43) This precise expression just reveals the relationship be- tween H and the self-linking and the linking numbers of the phase singularity knots family[29]. Since the self- linking and the linking numbers are both the invari- ant characteristic numbers of the phase singularity knots family in topology, H is an important topological invari- ant required to describe the linked phase singularities in wave fields. V. CONCLUSION In the present study, by making use of the φ-mapping topological current theory, the topology of the phase sin- gularity in wave fields is studied. Firstly, we obtain the inner structure of the phase singularities in two and three dimensional space. The phase singularity objects have been found at the every zero point of the wave func- tion ψ under the condition that the Jacobian determi- nateDi(φ/x) 6= 0. One shows that the topological charge (strength of the phase singularity) of the phase singular- ity objects are determined by the topological numbers: Hopf indices and Brouwer degrees. Secondly, we have studied the topological invariant of the knotted phase sin- gularity lines in three dimensional space in details. This topological invariant can be expressed as the sum of all the self-linking and all the linking numbers of the knotted phase singularity lines. Finally, it should be pointed out that in the present pa- per, when we discussed the topological properties of the phase singularities, the condition Di(φ/x) 6= 0 must be satisfied. Now the question is coming, when this condi- tion fails, what will happen about the phase singularity objects? The answer is related to the evolution of the phase singularity objects[31]. The topological object will generate, annihilate, split, or merge when the condition fails , and these dynamics properties of the phase singu- larity objects will be discussed in further work. Acknowledgments This work was supported by the National Natural Sci- ence Foundation of China. [1] L. M. Pismen, Vortices in Nolinear fields, (Clarendon Press, Oxford, 1999). [2] J. Adachi, G. Ishikawa, arxiv: math/0608721. [3] R. J. Donnelly, Quantized Vortices in Helium II (Cam- bridge University Press, Cambridge, England, 1991). [4] Y. S. Duan, X. Liu and P. M. Zhang, J. Phys: Condens. Mather 14, 7941 (2002). [5] S. Inouye, S. Gupta, T. Rosenband, A. P. Chikkatur, A. Görlitz, T. L. Gustavson, A. E. Leanhardt, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 87, 080402 (2001). [6] J. O. Hirschfelder, A. C. Christoph, and W. E. Palke, J. Chem. Phys. 61, 5456 (1974). [7] J. Riess, Phys. Rev. D 2, 647 (1970); J. Phys. A: Math. Gen. 20, 5179 (1987). [8] A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Optical Vortices and Vortex Solitons, Progress in Optics Vol. 47 (North-Holland, Amsterdam, 2005). [9] J. F. Nye, Natural focusing and fine struture of light (Institute of Physics Publishing, 1999). [10] M. V. Berry and M. R. Dennis, Proc. R. Soc. A 456, 2059 (2000). [11] M. V. Berry and J. F. Nye, Proc. R. Soc. A 336, 165 (1974). [12] Y. S. Duan, G. Jia and H. Zhang, J. Phys. A: Math. Gen. 32, 4943 (1999). [13] Y. S. Duan, H. Zhang and G. Jia, Phys. Letts. A 253, 57 (1999). [14] E. Babaev, L. D. Faddeev, and A. J. Niemi, Phys. Rev. B 65, 100512 (2002); E. Babaev, Phys. Rev. Lett. 88, 177002 (2002). [15] A. T. Winfree and S. H. Strogatz, Physica D 8, 35 (1983); http://arxiv.org/abs/math/0608721 Physica D 9, 65 (1983); Physica D 9, 333 (1983); Physica D 13, 221 (1984); Physica D 17, 109 (1985). [16] A. T. Winfree, Physica D 84, 126 (1995). [17] A. T. Winfree, Nature 371, 233 (1994); Science 266, 1003 (1994). [18] A. T. Winfree, The Geometry of Biological Time (Springer, 1980). [19] M. V. Berry and M. R. Dennis, Proc. R. Soc. A 457, 2251 (2001). [20] J. Leach, M. R. Dennis, J. Courtial and M. J. Padgett, Nature 432, 165 (2004), New. J. Phys. 7, 55 (2005). [21] Y. S. Duan, X. Liu, and L. B. Fu, Phys. Rev. D 67, 085022 (2003). [22] F. Faddeev and A. J. Niemi, Nature 387, 58 (1997). [23] Y. S. Duan, SLAC-PUB-3301, 1984; Y. S. Duan, H. Zhang, G. Jia, Phys. Letts. A 253, 57 (1999). [24] E. Goursat, A Course in Mathematical Analysis, trans- lated by E. R. Hedrick (Dover, New York, 1904), Vol.I. [25] J. A. Schouten, Tensor Analysis for Physicists (Claren- don Press, Oxford, 1951). [26] Y. S. Duan, S. Li and G. H. Yang, Nucl. Phys. B 514, 705 (1998). [27] H. Aref and I. Zawadzki, Nature 354, 50 (1991). [28] N. D. Mermin, T. L. Ho, Phys. Rev. Lett. 36,594 (1976). [29] E. Witten, Commun. Math. Phys. 121, 351 (1989); A. M. Polyakov, Mod. Phys. Lett. A 3, 325 (1988). [30] D. Rolfsen, Knots and Links (Publish or Perish, Berkeley, CA, 1976). [31] L. B. Fu, Y. S. Duan, and H. Zhang, Phys. Rev. D 61, 045004 (2000).

0704.1191

Five Intermediate-Period Planets from the N2K Sample

to appear in ApJ Five Intermediate-Period Planets from the N2K Sample 1,2 Debra A. Fischer3, Steven S. Vogt4, Geoffrey W. Marcy5, R. Paul Butler6, Bun’ei Sato7, Gregory W. Henry8, Sarah Robinson4, Gregory Laughlin4, Shigeru Ida7, Eri Toyota9 Masashi Omiya10 Peter Driscoll11, Genya Takeda12, Jason T. Wright5, John A. Johnson5 [email protected] ABSTRACT We report the detection of five Jovian mass planets orbiting high metallicity stars. Four of these stars were first observed as part of the N2K program and exhibited low RMS velocity scatter after three consecutive observations. How- ever, follow-up observations over the last three years now reveal the presence of 1Based on observations obtained at the W. M. Keck Observatory, which is operated by the University of California and the California Institute of Technology. Keck time has been granted by NOAO and NASA. 2Based on observations obtained at the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan 3Department of Physics & Astronomy, San Francisco State University, San Francisco, CA 94132; fis- [email protected] 4UCO/Lick Observatory, University of California at Santa Cruz, Santa Cruz, CA 95064 5Department of Astronomy, University of California, Berkeley, CA USA 94720 6Department of Terrestrial Magnetism, Carnegie Institute of Washington DC, 5241 Broad Branch Rd. NW, Washington DC, USA 20015-1305 7Tokyo Institute of Technology, 2-12-1 Okayama, Meguro-ku, Tokyo 152-8550, Japan 8Center of Excellence in Information Systems, Tennessee State University, 3500 John A. Merritt Blvd., Box 9501, Nashville, TN 37209 9Department of Earth and Planetary Sciences, Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan 10Department of Physics, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan 11Department of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, MD 21218 12Department of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208 http://arxiv.org/abs/0704.1191v2 – 2 – longer period planets with orbital periods ranging from 21 days to a few years. HD 11506 is a G0V star with a planet of M sin i = 4.74 MJUP in a 3.85 year orbit. HD 17156 is a G0V star with a 3.12 MJUP planet in a 21.2 day orbit. The eccentricity of this orbit is 0.67, one of the highest known for a planet with a relatively short period. The orbital period for this planet places it in a region of parameter space where relatively few planets have been detected. HD 125612 is a G3V star with a planet of M sin i = 3.5 MJUP in a 1.4 year orbit. HD 170469 is a G5IV star with a planet of M sin i = 0.67 MJUP in a 3.13 year orbit. HD 231701 is an F8V star with planet of 1.08 MJUP in a 142 day orbit. All of these stars have supersolar metallicity. Three of the five stars were observed photometrically but showed no evidence of brightness variability. A transit search conducted for HD 17156 was negative but covered only 25% of the search space and so is not conclusive. Subject headings: planetary systems – stars: individual (HD 11506, HD 17156, HD 125612, HD 170469, HD 231701) 1. Introduction Ongoing Doppler surveys of stars closer than 150 pc have detected more than 200 exoplanets (Butler et al. 2006, Wright et al. 2007). This ensemble of exoplanets exhibit a diverse range of statistical characteristics (Marcy et al. 2005). Notably, the mass distribution of exoplanets falls exponentially for masses greater than one Jupiter-mass. In addition, there is a statistical pile-up of planets in orbits of just a few days, a paucity of planets with periods between 10 - 100 days, and a rising number of gas giant planets found at separations greater than 1 AU. Orbital eccentricities span a surprising range from 0 - 0.93, although 92% of planet eccentricities are less than 0.6 and even for planets with periods longer than five days (i.e., not tidally circularized) the median exoplanet eccentricity is 0.26. It has also been shown that planet formation is tied to the chemical composition of the host star. There is a few percent probability of finding a gas giant planet around a solar metallicity star, while planet occurrence rises dramatically to ∼ 25% for stars with three times the heavy metal composition of the Sun (Santos et al. 2005, Fischer & Valenti 2005). Ida & Lin (2004) have accounted for this metallicity correlation within the context of core accretion. The statistical characteristics of exoplanets serve as tracers of planet formation and mi- gration histories. The planet-metallicity correlation indicates initial high metallicity in the – 3 – protoplanetary disk which in turn may be correlated with a higher surface density of solid particles in the midplane of the disk that enhances core accretion. Orbital eccentricities and the proximity of gas giant plants to their host stars are remnant signatures of gravitational interactions that drive orbital migration. The architecture of multi-planet systems, some- times locked in resonances, adds to our understanding of evolution of the protoplanetary disk. The N2K program (Fischer et al. 2005) is a survey of metal-rich stars, designed to identify short period planets. These planets are geometrically endowed with a higher transit probability; transit events provide a rare opportunity to derive information about the planet density, internal structure and the atmosphere (Burrows et al. 2007, Charbonneau 2006, Sato et al. 2005). Because short-period planets can be flagged with just a few observations, the N2K program only requires three Doppler measurements to screen each star. However, an increased occurrence of planets is correlated with high host star metallicity at all detected separations. Therefore, additional observations were obtained for the highest metallicity stars to check for longer period planets. This extended program has detected six new intermediate period planets: HD 5319 and HD 75898 (Robinson et al. 2007), and four of the five planets presented here: HD 11506, HD 17156, HD 125612, HD 231701. 2. HD 11506 2.1. Stellar Characteristics HD 11506 is classified as a G0 star with V=7.51. The Hipparcos catalog (ESA 1997) lists B-V = 0.607 with a parallax of 18.58 milliarcseconds, corresponding to a distance of 53.8 pc. The distance and apparent magnitude set the absolute visual magnitude as MV =3.85 and the stellar bolometric luminosity as 2.29 L⊙ , including a bolometric correction of -0.033 (VandenBerg & Clem 2003) based on effective temperature, surface gravity and metallicity of the star. High resolution spectroscopic analysis described in Valenti & Fischer (2005) yields Teff = 6058 ± 51K, log g = 4.32 ± 0.08, v sin i = 5.0 ± 0.5 km s −1 , and [Fe/H]= 0.31 ± 0.03 dex. HD 11506 is about 0.65 magnitudes above the main sequence. We expect the star to be located about 0.3 magnitudes above the main sequence because of high stellar metallicity, so this star appears to be slightly evolved by a few tenths of a magnitude, and is likely just beginning to transition onto the subgiant branch. The stellar radius is calculated to be 1.3 R⊙ using L = 4πR 2σT 4. We have also run a fine grid of evolutionary tracks (described in Takeda et al. 2007), tuned to the uniform spectroscopic analysis of Valenti & Fischer (2005) and based on the Yale Stellar Evolution – 4 – Code. This analysis provides posterior probability distributions for stellar mass, radius, gravity and ages. Based on these evolutionary tracks, we derive a stellar mass of 1.19 M⊙, a radius of 1.38 R⊙, and an age of 5.4 Gyr. As a measure of the formal uncertainties, the lower and upper 95% credibility intervals from a Bayesian posterior probability distribution are provided in parentheses in Table 1 for these values. The Ca II H & K lines (Figure 1) show that HD 11506 is chromospherically inactive. We measure SHK , the core emission in the Ca II H & K lines relative to the continuum, for all of our stars. Based on nineteen Keck observations, we measure an average SHK = 0.156 for HD 11506. The ratio of flux from SHK to the bolometric stellar flux is designated as logR′HK and gives the best diagnostic of chromospheric activity. The average logR −4.99 for this star and we derive an activity-based rotational period (Noyes et al. 1984), PROT = 12.6d, and an activity-based age of 5.4 Gyr, in excellent agreement with the age from evolutionary tracks. The activity, spectral type and evolutionary stage of the star allow us to estimate an additional source of astrophysical noise, or stellar jitter for the velocities of each of the stars on our Doppler survey (Wright 2005). We monitored the brightness of HD 11506 with the T10 0.8 m automatic photometric telescope (APT) at Fairborn Observatory (Henry 1999, Eaton, Henry & Fekel 2003). The T10 APT measures the brightness of program stars relative to nearby constant comparison stars with a typical precision of 0.0015–0.0020 mag for a single measurement. For HD 11506, we obtained 102 b and y measurements spanning 451 days between 2004 October and 2006 January. The standard deviation of a single observation from the mean was 0.0023 mag, our upper limit to possible photometric variability in HD 11506. A periodogram analysis found no significant periodicity between 1 and 225 days, so our photometry confirms the star’s low chromospheric activity. The stellar parameters are summarized in Table 1. 2.2. Doppler Observations and Keplerian Fit Doppler observations were made at the Keck telescope using HIRES (Vogt et al. 1994) with an iodine cell to model the instrumental profile and to provide the wavelength scale (Butler et al. 1996). An exposure meter maintains a constant signal-to-noise ratio of about 200 in our spectra, yielding a mean radial velocity precision of 2.75 m s−1 for HD 11506. We obtained a total of 26 Doppler measurements. The observation dates, radial velocities and measurement uncertainties for the radial velocities are listed in Table 2 and plotted in Figure 2. In addition to velocity errors arising from our measurement uncertainties (including – 5 – photon shot noise), the star itself can have pulsations, cool spots or granular convective flows that contribute non-dynamical velocity noise. These astrophysical sources of noise are termed jitter and we empirically estimate stellar jitter based on the spectral type and chromospheric activity of the star, following Wright (2005). For purposes of fitting a Keplerian model, the stellar jitter is added in quadrature to the formal instrumental errors, however the estimated jitter is never included in the measurement uncertainties for the tabulated radial velocity sets. The periodogram of the radial velocities shows a strong, broad peak in the power spec- trum at about 1270 days with an associated false alarm probability (FAP) < 0.0001. The FAP associated with the periodogram tests whether scrambled velocities yield power that exceeds the observed, unscrambled velocities. A high FAP suggests that the signal is not significant or could have been caused by a window function in the data. Using a Monte Carlo simulation, one thousand data sets of noise were generated by randomly drawing (with re- placement) sets of actual stellar velocities. The fraction of trials with maximum periodogram power that exceeds the observed value from the initial unscrambled data set defines the FAP (Cumming 2004). For each of the radial velocity data sets in this paper, a Levenberg-Marquardt fitting algorithm was used to model the radial velocities with a theoretical Keplerian orbital curve. There are six orbital parameters derived in the fit: orbital period (P), time of periastron passage (TP ), eccentricity (e), the orientation of the orbit (or line of apsides) (ω), the semi- velocity amplitude (K) and the residual center of mass radial velocity (after subtracting a median radial velocity). Uncertainties in the orbital parameters are determined with a bootstrap Monte Carlo analysis. First, a best-fit Keplerian model is obtained. Then, for each of 100 trials, the theoretical best fit is subtracted from the observed radial velocities. The residual velocities are then scrambled (with replacement) and added back to the theoretical best fit velocities and a new trial Keplerian fit is then obtained. The standard deviation of each orbital parameter for the 100 Monte Carlo trials was adopted as the parameter uncertainty. The best fit Keplerian model gives an orbital period of 1405 ± 45 d, semi-velocity amplitude of 80 ± 3 m s−1and orbital eccentricity of 0.3 ± 0.1. The RMS to this fit is 10.8 m s−1. Based on the chromospheric activity of this star, we estimated a jitter of 2 m s−1 (Wright 2005). When this jitter is added in quadrature with the error bars listed in Table 2, χ2ν = 3.2. While the large amplitude Doppler variation is clear, the χ2ν fit is worse than usual, suggesting that our velocity errors may be underestimated or that additional low amplitude dynamical velocities are present. A periodogram of the residual velocities to the single Keplerian fit show several peaks with similar power. For example, we – 6 – can fit a second planet with a period of 170 days with a significant reduction in the residual velocity RMS and an improvement in χ2ν , however this is not yet a unique double planet fit; additional data are required to better evaluate the possible second signal. Using the stellar mass of 1.19 M⊙ derived from evolutionary tracks, we findM sin i= 4.74 MJUP and a semi-major axis of 2.48 AU. At the distance of this star, this physical separation corresponds to an angular separation of α = 0.′′04. The Keplerian orbital solution is listed in Table 3 and the best-fit Keplerian model is plotted in Figure 2. 3. HD 17156 3.1. Stellar Characteristics HD 17156 is a listed as a G5 star in the SIMBAD database and the Hipparcos catalog. However, this spectral type seems at odds with other data for the star. The visual magnitude is V = 8.17, B− V = 0.59, and the Hipparcos parallax (ESA 1997) is 12.78 milliarcseconds, corresponding to a distance of 78.24 pc. The bolometric correction -0.039 (VandenBerg & Clem 2003) and absolute visual magnitude, MV=3.70, imply a bolometric stellar luminosity of 2.6 L⊙ . Spectroscopic analysis yields Teff = 6079 ± 56K, log g = 4.29 ± 0.06, v sin i = 2.6 ± 0.5 km s−1 , and [Fe/H]= 0.24 ± 0.03. The B−V color and the effective temperature are independent measurements that are consistent with each other. Together with the absolute magnitude and position on the H-R diagram, the spectral type for this star is more likely to be G0 and the star is just beginning to evolve off the main sequence. The stellar mass, from evolutionary models described by Takeda et al. (2007), is 1.2 M⊙, and the age is 5.7 Gyr. The stellar radius from evolutionary models is 1.47 R⊙, and agrees with the value we derive using the observed luminosity and the Stefan-Boltzmann relation. The absence of Ca II H & K emission (Figure 1) demonstrates low chromospheric ac- tivity. Taking the average of 25 observations, we measure SHK = 0.15 and logR HK = -5.04 and derive a rotational period, PROT = 12.8 d, with an estimated stellar age of 6.4± 2Gyr, which compares favorably with the age derived above from stellar evolution tracks. We obtained 241 photometric measurements with the T12 APT spanning 179 days be- tween 2006 September and 2007 March. The standard deviation of the observations from their mean was 0.0024 mag, the upper limit to photometric variability in the star. Peri- odogram analysis revealed no significant periodicity between 1 and 100 days. In particular, a least-squares sine fit of the observations on the 21.22-day radial velocity period resulted – 7 – in a photometric amplitude of only 0.00039± 0.00023 mag, providing further evidence that the radial velocity variations in HD 17156 are not due to chromospheric activity. The stellar characteristics, including our assessment of photometric variability, are summarized in Table 3.2. Doppler Observations and Keplerian Fit We initially obtained eight Doppler observations of HD 17156 using the High Dispersion Spectrometer (Noguchi et al. 2002) at the Subaru Telescope in 2004 and 2005. For the first observing runs, the iodine absorption cell was located behind the entrance slit of the spectrometer (Kambe et al. 2002, Sato et al. 2002, Sato et al. 2005). The box holding the I2 cell included a window with a lens to maintain constant focal length inside the spectrometer. This eliminated the need to adjust the collimator position when moving the I2 cell in and out of the light path (i.e., when taking program and template observations). However, the lens introduced a different wavelength dispersion for program observations relative to the template observation. Modeling of those early data is still ongoing, however, standard stars, known to have constant radial velocities show RMS scatter greater than 15 m s−1, with larger run-to-run velocity offsets for Doppler observations obtained with that setup. The Subaru N2K program was awarded ten nights of “intensive” time in summer 2006 and in December 2006. Before the intensive time allocation, the iodine cell was moved in front of the slit, eliminating the change in wavelength dispersion between template and program observations. With this new setup, the RMS scatter decreased, ranging from 4 - 12 m s−1 in a set of four RV standard stars. HD 17156 had exhibited large radial velocity variations in 2004 - 2005 at Subaru. Follow- up observations at Keck confirmed velocity variations, so the star was observed on nine consecutive nights at Subaru from 8 December to 16 December, 2006. Setup StdI2b was used to cover the wavelength region of 3500–6100 Å with a mosaic of two CCDs. The slit width of 0′′.6 was used to give a reciprocal resolution (λ/∆λ) of 60000. We obtained a typical signal-to-noise ratio of S/N ∼ 150 pixel−1 at 5500 Å with exposure times of about 120 seconds. Because of the larger systematic errors for observations taken before summer 2006 (with the iodine cell behind the slit), only the nine radial velocities from December 2006 are listed in Table 4. To account for the intrinsic RMS velocity scatter in standard stars, 5 m s−1 was added in quadrature to the nine Subaru observations in Table 4. After HD 17156 was flagged as an N2K candidate at Subaru, it was added to the N2K planet search program at Keck. We obtained 24 radial velocity measurements at the – 8 – Keck Observatory with an average internal velocity precision of 1.6 m s−1. Observation dates, radial velocities and uncertainties for 33 observations are listed in Table 4. The last column designates the source of the observations as “K” (Keck Observatory) or “S” (Subaru Observatory). The periodogram of the radial velocity data shows a strong narrow peak at 21.1 days with a FAP less than 0.0001 (for 10000 Monte Carlo trials). When combining the Subaru and Keck velocities, we first determined a velocity differ- ence of about 130 m s−1 between two observations taken at Subaru and Keck on the same night (JD 2454083.9). With that initial guess, we included a velocity offset as a free pa- rameter and found that an offset of 116.0 m s−1 produced a minimum χ2ν . That offset was added to the Subaru velocities listed in Table 4. The best fit Keplerian model for the combined Subaru and Keck data sets yields an orbital period of 21.2 ± 0.3 d, semi-velocity amplitude, K = 275± 15 m s−1, and orbital eccentricity, e = 0.67± 0.08. The RMS to the fit is 3.97 m s−1. Adding jitter of 3 m s−1 (expected for this star) in quadrature with the actual single-measurement errors gives χ2ν = 1.04 for this Keplerian fit. Adopting a stellar mass of 1.2 M⊙, we derive M sin i = 3.12 MJUP and a semi-major axis of 0.15 AU. The Keplerian orbital solution is summarized in Table 3. The phase-folded plot of the Doppler measurements for Keck and Subaru observations are shown in the left plot of Figure 3 and include 3 m s−1 jitter. Keck observations are represented by diamonds and the Subaru observations are shown as filled circles. Because the high eccentricity is unusual, we examined the Keplerian fit for the Keck data alone, shown in the right plot Figure 3. The Keck data have poor phase coverage near periastron, and yield a Keplerian fit with lower amplitude and lower eccentricity. The Subaru observations map periastron passage and help to model the eccentricity of the orbit. 3.3. Transit Search The 21.22 day period of the companion to HD 17156 is by far the shortest planetary orbital period in this paper. The orbital semi-major axis of 0.15 AU and the stellar radius of 1.47 R⊙ lead to an a priori transit probability of 7% (Seagroves et al. 2003). Therefore, we used our 241 brightness measurements to conduct a preliminary transit search. The orbital parameters in Table 3 constrain the predicted times of transit to about ± 0.3 days, which is slightly greater than the 0.25-day duration of a central transit. We performed our transit search, using a technique similar to the one described by Laughlin (2000), over all orbital phases for periods between 20 and 23 days. The search was negative but was able to cover effectively only 25% of the period-phase search space corresponding to the uncertainties in – 9 – the orbital parameters. Thus, our photometric data do not preclude the possibility of transits in HD 17156. 4. HD 125612 4.1. Stellar Characteristics HD 125612 is a G3V main sequence star with V=8.31, B-V = 0.628, and Hipparcos parallax (ESA 1997) of 18.93 corresponding to a distance of 52.82 pc and absolute visual magnitude, MV =4.69. Spectroscopic analysis yields Teff = 5897 ± 40K, log g = 4.45 ± 0.05, v sin i = 2.1 ± 0.5 km s−1 , and [Fe/H]= 0.24 ± 0.03 dex. The bolometric correction is -0.061, giving a stellar luminosity of 1.08 L⊙ . The luminosity and Teff imply a stellar radius of 1.0 R⊙. Within uncertainties, this agrees well with the value of 1.05 R⊙ determined from stellar evolutionary tracks. We also derive a stellar mass of 1.1 M⊙ from stellar evolution models and an age of 2.1 Gyr. Figure 1 shows the Ca H line for HD 125612; the lack of emission indicates low chromo- spheric activity for this star. Taking the mean of 18 observations, we measure SHK = 0.178 and logR′HK = -4.85, and derive PROT = 10.5 d, and a stellar age of 3.3 ± 2 Gyr (which compares well with the age of 2.1 Gyr from stellar evolution tracks). Stellar parameters are summarized in Table 1. 4.2. Doppler Observations and Keplerian Fit We obtained 19 Keck velocity measurements for HD 125612 with a typical uncertainty of 2.2 m s−1. Observation dates, radial velocities and instrumental uncertainties in the radial velocities are listed in Table 5. A periodogram of the velocities shows a strong broad peak at about 500 days. The best fit Keplerian model is plotted in Figure 4 and yields a period of 510 ± 14 d, with semi-velocity amplitude 90.7 ± 8 m s−1, orbital eccentricity, 0.38 ± 0.05, and a linear trend of 0.037 meters per day. Adopting a stellar mass of 1.1 M⊙, we derive M sin i = 3.5 MJUP and semi-major axis of 1.2 AU (angular separation, α = 0 ′′.023). The Keplerian orbital solution is listed in Table 3 and the RV data are plotted with the best-fit Keplerian model (solid line) in Figure 4. The RMS to the Keplerian fit shown in Figure 4 is 10.7 m s−1. The velocity jitter for this star is expected to be about 2 m s−1. Therefore, the residual RMS is several times – 10 – the typical error bar, consistent with the poor χ2ν statistic of 3.56. A periodogram of the residuals to a 510-day planet fit shows power near 3.5 d. However, there are several other peaks of nearly comparable height, showing that other orbital solutions may give similar improvements. Thus, while we could fit the residuals with a second Keplerian, the FAP of the peak does not yet meet our standards of statistical significance, and more data are required for follow up. 5. HD 170469 5.1. Stellar Characteristics HD 170469 is a G5 subgiant star with visual magnitude V=8.21, B-V = 0.677, and Hipparcos parallax (ESA 1997) of 15.39 milliarcseconds, corresponding to a distance of 64.97 pc. The absolute visual magnitude of the star is MV=4.14. The bolometric correction is -0.072 providing a bolometric stellar luminosity of 1.6 L⊙ and (with Teff ) stellar radius of 1.2 R⊙ calculated from the luminosity. Evolutionary tracks provide a stellar mass estimate of 1.14 M⊙ and stellar radius of 1.22 R⊙ and age of 6.7 Gyr. Our spectroscopic analysis gives Teff = 5810 ± 44K, log g = 4.32 ± 0.06, v sin i = 1.7 ± 0.5 km s −1 , and [Fe/H]= 0.30 ± 0.03 dex. The Ca H & K lines (Figure 1) indicate low chromospheric activity. Taking the mean of thirteen observations, we measure SHK = 0.145 and logR HK = -5.06 and derive a rotational period, PROT = 13.0 d and an activity-calibrated age (Noyes et al. 1984) of 7± 2 Gyr. We obtained 215 brightness measurements with the T10 APT spanning 630 days be- tween 2005 March and 2006 November. The standard deviation of the observations was 0.0018 mag, the upper limit to photometric variability in HD 170469. A periodogram anal- ysis found no significant periodicity between 1 and 315 days, confirming the star’s low chro- mospheric activity. The stellar characteristics are summarized in Table 6. 5.2. Doppler Observations and Keplerian Fit We obtained 35 Keck velocities for HD 170469 with a mean velocity precision of 1.6 m s−1. Observation dates, radial velocities and instrumental uncertainties in the radial velocities are listed in Table 7. A periodogram of the velocities yields very strong power at about 1100 days with a FAP less than 0.0001. The best fit Keplerian model gives an orbital period of 1145 ± 18 d, semi-velocity – 11 – amplitude of 12.0 ± 1.9 m s−1 and orbital eccentricity, 0.11 ± 0.08. The RMS to the fit is 4.18 m s−1 with χ2ν = 1.59, including the estimated astrophysical jitter of 2.0 m s Adopting a stellar mass of 1.14 M⊙, we derive M sin i = 0.67 MJUP and a semi-major axis of 2 AU (α = 0.′′03). The Keplerian orbital parameters are listed in Table 8 and the RV data are plotted with the best-fit Keplerian model (solid line) in Figure 5. 6. HD 231701 6.1. Stellar Characteristics HD 231701 is an F8V star with V = 8.97, B − V = 0.539, and Hipparcos parallax (ESA 1997) of 9.22 milliarcseconds, corresponding to a distance of 108.4 pc. The absolute visual magnitude is MV =3.79, so this star is beginning to evolve onto the subgiant branch. Spectroscopic analysis yields Teff = 6208 ± 44K, log g = 4.33 ± 0.06, v sin i = 4.0 ± 0.5 km s−1 , and [Fe/H]= 0.07 ± 0.03 dex. The bolometric correction is -0.037 and bolometric luminosity is 2.4 L⊙ . The luminosity and effective temperature yield a stellar radius of 1.36 R⊙ . Modeling the stellar evolutionary tracks, we derive a stellar mass of 1.14 M⊙, radius of 1.35 R⊙, and age of 4.9 Gyr. The Ca H&K lines (Figure 1) show that the star has low chromospheric activity. We measure SHK = 0.159 and logR HK = -5.0 and derive a rotational period, PROT = 12.2 d and a stellar age of 5.6± 2 Gyr. Stellar parameters are listed in Table 6. 6.2. Doppler Observations and Keplerian Fit We obtained 17 Keck observations of HD 231701 with mean internal errors of 3.2 m s−1. Observation dates, radial velocities and measurement uncertainties in the radial velocities are listed in Table 9. The periodogram of this data set has a FAP of 0.006 for a period near 140 days. The best fit Keplerian model has an orbital period of 141.6 ± 2.8 d, with semi-velocity amplitude 39 ± 3.5 m s−1 and orbital eccentricity, 0.1 ± 0.06. The RMS to this fit is 5.9 m s−1. The expected astrophysical jitter for this star is 2.2 m s−1. Adding this jitter in quadrature with the error bars listed in Table 9 yields χ2ν = 1.46 for this Keplerian fit. Adopting the stellar mass of 1.14 M⊙, we derive M sin i = 1.08 MJUP and semi-major axis of 0.53 AU. The Keplerian orbital solution is listed in Table 8 and the phased RV data is plotted with the best-fit Keplerian model (solid line) in Figure 6. – 12 – 6.3. Discussion Here, we present the detection of 5 new exoplanets detected with Doppler observa- tions. For each of the Keplerian models, we also carried out a Markov Chain Monte Carlo (MCMC) analysis to better estimate the orbital parameters and their uncertainties following the algorithm described by Ford (2003). Unlike the Levenberg-Marquardt algorithm that we generally use to determine a best fit Keplerian orbit, the MCMC analysis provides the full posterior probability density distribution for each parameter. This approach is particularly useful for data sets where the Levenberg-Marquardt algorithm can minimize χ2ν with a model that fits a sparse data set. The MCMC algorithm explores a wider range of param- eter space because it is not driven solely by χ2ν minimization. However, MCMC does not explore an exhaustive range of parameter space. For example, solutions with very different orbital periods might be missed. For each of the models presented here, we began with the input parameters found with Levenberg Marquardt fitting and confirmed that the orbital elements were recovered with strongly peaked probability distributions using MCMC. HD 170469 is a star on the regular planet search at Keck that has a planet of M sin i = 0.66 MJUP in a ∼3 yr orbit with eccentricity 0.23. The host star is metal-rich with [Fe/H] = 0.3. The remaining four exoplanets were initially part of the N2K program at Keck. The N2K program targets metal-rich stars for rapid identification of short-period planets. The first three radial velocity measurements for the stars presented here had RMS scatter less than 5 m s−1(except HD 17156, with initial RMS scatter of 34 m s−1 ), so these stars were not candidates for short period planets. However, a follow-up program to obtain Doppler observations on N2K-vetted high metallicity stars with low chromospheric activity and low RMS velocity scatter has detected the presence of these longer period planets. HD 11506 b is a fairly massive planet, with M sin i = 4.74 MJUP and a semi-major axis of 2.5 AU. This could well constitute the outer edge of a habitable zone location for putative rocky moons orbiting the planet, depending on atmospheric properties of any moons. The host star has a luminosity that is 2.3 times that of the Sun. The eccentricity of this system is 0.3, so the temperature at the top of the planet atmosphere would change by about 50K between apastron and periastron. HD 17156 b has a mass of M sin i = 3.12 MJUP and an orbital period of 21.2 days, placing it in the so-called period valley between 10 and 100 days (Udry et al. 2003), where a relatively small fraction of exoplanets have been detected. We derive a substantial orbital eccentricity of 0.67 for HD 17156 b. At this proximity to the subgiant host star, the planet moves between 0.05 and 0.25 AU, experiencing temperature changes of a few hundred degrees between periastron and apastron. It is possible that these thermal changes could be observed with sensitive IR flux measurements from space, even though the planet is not known to – 13 – transit its host star. The distribution of orbital eccentricities for known exoplanets is shown in Figure 7. An upper envelope in the distribution of eccentricities rises steeply from periods of a few days to reach the maximum observed eccentricities (for HD 80606 and HD 20782) at periods of 100 - 1000 days. Although an orbital eccentricity of 0.67 seems remarkable for HD 17156 b, given its orbital period of just 21.2 days, the eccentricity still falls along the upper edge of the observed eccentricity distribution. HD 125612 b has M sin i = 3.5 MJUP with a semi-major axis of 1.2 AU. This planet has an eccentricity of 0.38. The planet is carried from 0.47 AU at periastron, where the temperature at the top of the atmosphere is about 300 K, to about 2.1 AU where the temperature falls below the freezing point of water to about 200 K. A single planet model does not appear to adequately describe the velocities of HD 125612 because the RMS to that fit is 10.7 m s−1, yet the star is chromospherically quiet and slowly rotating, with v sin i = 2 km s−1. This star may well have an additional planet orbiting in a relatively short period. Velocity variations in HD 231701 have been modeled as a planet with M sin i = 1.08 MJUP with a semi-major axis of 0.53 AU and orbital eccentricity of about 0.1. The MCMC probability distributions for HD 231701 are consistent with this Keplerian model, but al- low for eccentricity solutions that extend to zero. This analysis alerts us that more RV measurements should be taken to better constrain the orbital eccentricity of this system. We have now obtained three or more Doppler observations for 423 stars at Keck Ob- servatory as part of the N2K program. Spectral synthesis modeling has been carried out for all of these stars, and we plot the percentage of stars with detected planets in each 0.1 dex metallicity bin in Figure 8. Superimposed on this plot is the planet detectability curve from Fischer & Valenti (2005). A planet probability can be assigned based on the stellar metal- licity. Integrating planet probabilities we expect 27± 5 exoplanets with masses greater than 1 MJUP and orbital periods shorter than 4 years. Fourteen, or about half of the expected planets in the sample have now been detected. We gratefully acknowledge the dedication and support of the Keck Observatory staff, in particular Grant Hill for support with HIRES. We thank Akito Tajitsu and Tae-Soo Pyo for their expertise and support of the Subaru HDS observations. DAF acknowledges sup- port from NASA grant NNG05G164G and from Research Corporation. SSV acknowledges support from NSF AST-0307493. BS is supported by Grants-in-Aid for Scientific Research (No. 17740106) from the Japan Society for the Promotion of Science (JSPS). We thank the Michelson Science Center for travel support through the KPDA program. We thank the NASA and UC Telescope assignment committees for generous allocations of telescope – 14 – time. The authors extend thanks to those of Hawaiian ancestry on whose sacred mountain of Mauna Kea we are privileged to be guests. Without their kind hospitality, the Keck observations presented here would not have been possible. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and of NASA’s Astrophysics Data System Bibliographic Services and is made possible by the generous support of Sun Microsystems, NASA, and the NSF. – 15 – REFERENCES Butler, R. P., Wright, J. T., Marcy, G. W., Fischer, D. A., Vogt, S. S., Tinney, C. G., Jones, H. R. A., Carter, B. D., Johnson, J. A., McCarthy, C., Penny, A. J. 2006 ApJ 646, Butler, R. P., Marcy, G. W., Williams, E., McCarthy, C., Dosanjh, P., Vogt, S. S. 1996, PASP, 108, 500 Burrows, A., Hubeny, I., Budaj, J., Hubbard, W. 2007, ApJ, in press Charbonneau, D., Brown, T. M., Burrows, A., Laughlin, G. 2006, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, and K. Keil (University of Arizona Press) Cumming, A. 2004, MNRAS, 354, 1165 Eaton, J. A., Henry, G. W., Fekel, F. C. 2003, in The Future of Small Telescopes in the New Millennium, Volume II–The Telescopes We Use, ed. T. D. Oswalt (Dordrecht: Kluwer), 189 ESA 1997, The Hipparcos and Tycho Catalogs. ESA-SP 1200 Fischer, D. A. et al. 2005, ApJ, 629, 481 Fischer, D. A., Valenti, J. A. 2005, ApJ 622, 1102 Ford, E. 2003, ApJ, 129, 1706 Henry, G. W. 1999, PASP, 111, 845 Laughlin, G. 2000, ApJ, 545, 1064 Ida, S., Lin, D. N. C. 2004, ApJ 616, 567 Kambe, E. et al. 2002, PASJ, 54, 873 Marcy, G. W., Butler, R. P., Fischer, D. A., Vogt, S. S., Wright, J. T., Tinney, C. G., Jones, H. R. A. 2005, PThPS, 158, 24 Noguchi, K. et al. . 2002, PASJ, 54, 855 Noyes, R. W., Hartmann, L., Baliunas, S. L., Duncan, D. K., Vaughan, A. H. 1984, ApJ, 279, 763 Robinson, S. E., Laughlin, G., Vogt, S. S., Fischer, D., Butler, R. P., Marcy, G. W., Takeda, G., Driscoll, P. 2007, ApJ in press Santos, N. C., Israelian, G., Mayor, M., Bento, J. P., Almeida, P. C., Sousa, S. G., Ecuvillon, A. 2005, A & A, 437, 1127 Sato, B., Kambe, E., Takeda, Y., Izumiura, H., Ando, H. 2002, PASJ, 54 873 Sato, B. et al. 2005, ApJ, 633, 465 Seagroves, S., Harker, J., Laughlin, G., Lacy, J., & Castellano, T. 2003, PASP, 115, 1355 Takeda, G., Ford, E. B., Sills, A., Rasio, F., Fischer, D. A., Valenti, J. A. 2007, ApJS, 168, – 16 – Udry, S., Mayor, M., Clausen, J. V., Freyhammer, L. M., Helt, B. E., Lovis, C., Naef, D., Olsen, E. H., Pepe, F., Queloz, D., Santos, N. C. 2003, A & A, 407, 679 Valenti, J. A., Fischer, D. A. 2005, ApJS, 159, 141 VandenBerg, D. & Clem, J. 2003, AJ, 126, 778 Vogt, S. S. et al. 1994, SPIE, 2198, 362. Wright, J. T. 2005, PASP, 117, 657 Wright, J. T., Marcy, G. W., Fischer, D. A., Butler, R. P., Vogt, S. S., Tinney, C. G., Jones, H. R. A., Carter, B. D., Johnson, J. A., McCarthy, C., Apps, K. 2007, ApJ, 657, 533 This preprint was prepared with the AAS LATEX macros v5.2. – 17 – Fig. 1.— The Ca H line for HD 11506 and 17156 are plotted in the left panel, with the same wavelength segment of the Sun shown for comparison. HD 125612, HD 170469 and HD 231701 are plotted in the right panel. All of these stars have low chromospheric activity based on our measurement of line core emission relative to the continuum. – 18 – 2004.0 2005.0 2006.0 2007.0 2008.0 Time (Years) Keck Obs P = 3.85 yr K = 80.3 ms e = 0.30 RMS = 10.8 ms Fig. 2.— Radial velocities for HD 11506. The velocity error bars have been augmented by adding 2 m s−1 in quadrature to the single measurement precision listed in Table 2. This gives χ2ν = 3.2 for the Keplerian fit. With a stellar mass of 1.19 M⊙, we derive a planet mass of M sin i = 4.74 MJUP and semi-major axis, arel = 2.48 AU. – 19 – 0.0 0.5 1.0 Orbital Phase P = 21.22 day K = 275. m s e = 0.67 RMS = 3.98 m s 0.0 0.5 1.0 Orbital Phase P = 21.22 day K = 147. m s e = 0.43 RMS = 3.67 m s Fig. 3.— (left) Radial velocities for HD 17156 from Keck Observatory (diamonds) and Subaru Observatory (filled circles) have 3 m s−1 added in quadrature to the uncertainties listed in Table 4 to account for expected photospheric jitter. Adopting a stellar mass of 1.2 M⊙ we derive a planet mass, M sin i = 3.12 MJUP and semi-major axis for the orbit, arel = 0.15 AU. (right) The plot on the right shows the Keck velocities only. Although the phase coverage misses periastron the Keck velocites alone confirm high eccentricity in HD 17156 b. – 20 – 2004.5 2005.0 2005.5 2006.0 2006.5 2007.0 2007.5 Time (Years) Keck Obs P = 1.39 yr K = 90.7 ms e = 0.38 RMS = 10.7 ms Fig. 4.— Radial velocities for HD 125612. The velocity measurements have 2 m s−1 added to their error bars listed in Table 5 to account for the level of astrophysical noise (jitter) we expect from the star. With the assumed stellar mass of 1.1 M⊙, we derive a planet mass, M sin i = 3.5 MJUP and semi-major axis of 1.2 AU. This Keplerian model still has a high RMS and χ2ν of 3.56, suggesting the possible presence of an additional planet. – 21 – 2002 2004 2006 2008 Time (Years) P = 3.13 yr K = 12.0 ms e = 0.11 RMS = 4.18 ms Fig. 5.— Keck radial velocities for HD 170469 include 2.0 m s−1 added in quadrature with the internal error bars listed in Table 7. With the added jitter, the Keplerian fit has χ2ν = 1.59. The assumed stellar mass of 1.14 M⊙ yields a planet mass of M sin i = 0.67 MJUP and semi-major axis of about 2 AU. – 22 – 0.0 0.5 1.0 Orbital Phase P = 141.6 day K = 39.0 m s e = 0.10 RMS = 5.90 m s Fig. 6.— Phase-folded radial velocities for HD 231701 include stellar jitter of 2.2 m s−1added to the errors listed in Table 9, giving χ2ν = 1.46. The stellar mass of 1.14 M⊙ implies a planet mass of M sin i = 1.08 MJUP and orbital radius of 0.53 AU. – 23 – Fig. 7.— Orbital eccentricity distribution for exoplanets. A rising envelope defines the distribution for planets with periods between 2 and 100 days. The distribution peaks for periods between 100 - 1000 days. The arrow points to the dot representing HD 17156 b. With an orbital period of 21 days and eccentricity of 0.67, HD 17156 b still fits within the envelope of this eccentricity distribution. – 24 – Fig. 8.— Stars with at least 4 Doppler observations that were observed at Keck as part of the N2K program were been binned according to metallicity. In each 0.1 dex metallicity bin, the percentage of stars with detected planets is plotted and Poisson error bars are shown. Superimposed on this histogram is the curve of planet occurrence as a function of metallicity from Fischer & Valenti (2005). Most stars on the N2K program have only 4-5 observations, however the metallicity correlation is still emerging and is supported in this sample. – 25 – Table 1. Stellar Parameters Parameter HD11506 HD17156 HD125612 V 7.51 8.17 8.31 MV 3.85 3.70 4.69 B-V 0.607 0.590 0.628 Spectral Type G0V G0V G3V Distance (pc) 53.82 78.24 52.82 Lbol/L⊙ 2.29 2.6 1.08 [Fe/H] 0.31 (0.03) 0.24 (0.03) 0.24 (0.03) Teff (K) 6058 (51) 6079 (56) 5897 (40) v sin i (km s−1) 5.0 (0.50) 2.6 (0.50) 2.1 (0.50) log g 4.32 (0.08) 4.29 (0.06) 4.45 (0.05) MSTAR (M⊙ ) a (1.1) 1.19 (1.29) (1.1) 1.2 (1.3) (1.04) 1.1 (1.17) RSTAR (R⊙ ) a (1.25) 1.38 (1.53) (1.3) 1.47 (1.6) (0.99) 1.05 (1.13) Age (Gyr)a (3.9) 5.4 (7.0) (3.8) 5.7 (7.0) (0.16) 2.1 (5.6) SHK 0.156 0.15 0.178 logR′HK -4.99 -5.04 -4.85 PROT (d) 12.6 d 12.8 d 10.5 d σphot (mag) 0.0023 0.0024 · · · aStellar masses, radii and ages are derived from evolutionary tracks – 26 – Table 2. Radial Velocities for HD 11506 JD RV Uncertainties -2440000. (m s−1 ) (m s−1 ) 13014.73505 -6.67 2.94 13015.73893 -6.58 2.89 13016.74089 -18.73 2.98 13191.12201 -43.89 3.56 13207.10116 -71.30 3.17 13208.08401 -56.92 3.41 13368.83778 -80.87 2.26 13369.75897 -80.35 2.18 13370.73242 -81.24 2.17 13397.73009 -75.66 2.36 13750.73807 59.16 2.67 13775.72853 71.94 2.77 13776.70435 74.90 2.57 13777.72528 76.27 2.93 13778.71858 78.70 2.73 13779.74737 71.93 2.77 13926.12744 73.68 2.66 13933.09065 75.37 2.62 13959.13935 80.47 2.46 13961.12421 81.24 2.73 13981.98256 74.91 3.12 14023.97438 44.32 3.07 14083.84327 26.82 2.44 14085.92115 38.66 2.54 14129.74332 56.29 2.48 14286.11838 28.66 3.01 – 27 – – 28 – Table 3. Orbital Parameters Parameter HD 11506 HD 17156 HD 125612 P (d) 1405 (45) 21.2 (0.3) 510 (14) Tp (JD) 13603 (102) 13738.529 (0.5) 13228.3 (12) ω (deg) 262 (19) 121 (11) 21 (9) ecc 0.3 (0.1) 0.67 (0.08) 0.38 (0.05) K1 (m s −1) 80 (3) 275 (15) 90.7 (8) dv/dt (m s−1per day) · · · · · · 0.037 arel (AU) 2.48 0.15 1.2 a1 sin i (AU) 0.0099 0.00039 0.0039 f1(m) (M⊙) 6.53e-08 1.83e-08 3.09e-08 M sin i (MJup) 4.74 3.12 3.5 Nobs 26 33 19 RMS (m s−1 ) 10.8 3.97 10.7 Jitter (m s−1 ) 2 3 2 Reduced χ2ν 3.2 1.17 3.56 FAP (periodogram) < 0.0001 < 0.0001 0.0003 – 29 – Table 4. Radial Velocities for HD 17156 JD RV Uncertainties Observatory -2440000. (m s−1 ) (m s−1 ) 13746.75596 88.49 1.70 K 13748.79814 138.15 1.73 K 13749.79476 151.35 1.67 K 13750.80160 169.65 1.76 K 13775.77821 235.17 1.83 K 13776.80791 253.80 1.81 K 13779.82897 239.14 1.64 K 13959.13219 97.33 1.55 K 13962.07028 152.64 1.51 K 13963.10604 165.48 1.68 K 13964.13118 194.69 1.70 K 13982.03231 132.55 1.20 K 13983.08575 146.35 1.70 K 13983.99480 166.11 1.32 K 13985.00847 187.22 1.57 K 14023.95206 114.39 1.77 K 14047.95773 166.01 1.74 K 14078.01162 -116.60 5.14 S 14078.92501 -261.56 5.18 S 14079.91371 -164.10 5.23 S 14080.98093 -89.57 5.13 S 14081.89406 -44.08 5.15 S 14082.86071 2.39 5.14 S 14083.88445 37.62 5.16 S 14083.90314 32.76 1.33 K 14084.82860 63.22 1.63 K 14085.82560 86.67 5.20 S 14085.86537 84.28 1.56 K 14086.87960 99.01 5.20 S – 30 – Table 4—Continued JD RV Uncertainties Observatory -2440000. (m s−1 ) (m s−1 ) 14129.92513 113.30 1.43 K 14130.73019 133.66 1.32 K 14131.85485 151.11 1.77 K 14138.76720 261.56 1.37 K – 31 – Table 5. Radial Velocities for HD 125612 JD RV Uncertainties -2440000. (m s−1 ) (m s−1 ) 13190.83262 52.80 2.85 13197.83363 61.40 2.66 13198.85557 60.49 2.59 13199.83792 44.35 2.70 13604.75480 -60.07 2.04 13754.12824 45.68 1.83 13776.15380 17.80 2.03 13777.11940 21.71 2.16 13838.01492 -77.76 2.35 13841.94894 -86.49 2.52 13927.79007 -90.28 2.01 13961.75080 -79.07 2.01 13962.74127 -80.81 1.83 13981.72862 -83.16 1.90 13983.74037 -81.92 2.11 13984.72815 -80.88 2.03 14130.13392 -22.62 2.18 14139.12383 1.31 1.94 14251.82778 90.28 2.39 – 32 – Table 6. Stellar Parameters Parameter HD 170469 HD 231701 V 8.21 8.97 MV 4.14 3.79 B-V 0.677 0.539 Spectral Type G5IV F8V Distance (pc) 64.97 108.4 Lbol/L⊙ 1.6 2.4 [Fe/H] 0.30 (0.03) 0.07 (0.03) Teff (K) 5810 (44) 6208 (44) v sin i (km s−1) 1.7 (0.5) 4 (0.50) log g 4.32 (0.06) 4.33 (0.06) MSTAR (M⊙ ) a (1.05) 1.14 (1.16) (1.08) 1.14 (1.22) RSTAR (R⊙ ) a (1.15) 1.22 (1.3) (1.16) 1.35 (1.55) Age (Gyr)a (5.0) 6.7 (7.8) (3.5) 4.9 (6.2) SHK 0.145 0.159 logR′HK -5.06 -5.00 PROT (d) 13.0 d 12.2 d σphot (mag) 0.0018 · · · aStellar masses, radii and ages are derived from evolu- tionary tracks – 33 – Table 7. Radial Velocities for HD 170469 JD RV Uncertainties -2440000. (m s−1 ) (m s−1 ) 11705.96808 6.04 1.51 11793.81330 -0.05 1.39 12008.04881 -10.21 1.50 12099.03294 -14.05 1.59 12162.76894 -15.48 1.42 12364.13287 -7.07 1.67 12390.12499 -2.46 1.63 12391.12567 2.80 1.76 12445.93867 -12.12 1.72 12515.82777 14.53 1.96 12535.75539 3.45 1.53 12536.74191 0.06 1.50 12537.82520 1.72 1.50 12538.74254 0.83 1.30 12539.75501 4.11 1.51 12572.69435 6.59 1.60 12573.69333 5.76 1.34 12574.70725 10.11 1.45 12575.69822 2.35 1.40 12778.04455 15.48 1.96 12804.05044 7.80 1.56 12848.92274 5.06 2.35 13180.90825 -14.03 1.55 13181.89752 -12.91 1.57 13548.99248 -2.83 1.54 13603.80234 4.11 1.50 13842.01212 9.79 1.59 13932.96654 14.41 1.49 13960.91798 10.12 1.42 – 34 – Table 7—Continued JD RV Uncertainties -2440000. (m s−1 ) (m s−1 ) 13961.83115 11.29 1.54 13981.82421 7.85 1.27 13982.77494 5.33 1.26 13983.76067 5.96 1.33 13984.83377 7.38 1.22 14250.01196 -4.92 1.57 – 35 – Table 8. Orbital Parameters Parameter HD 170469 HD 231701 P (d) 1145 (18) 141.6 (2.8) Tp (JD) 11669.0 (21) 13180.0 (4.2) ω (deg) 34 (19) 46 (24) ecc 0.11 (0.08) 0.10 (0.08) K1 (m s −1 ) 12.0 (1.9) 64 (8) dv/dt (m s−1per day) · · · · · · arel (AU) 2.1 0.53 a1 sin i (AU) 0.00126 0.0005 f1(m) (M⊙) 2.03e-10 8.6e-10 M sin i (MJup) 0.67 1.08 Nobs 35 17 RMS (m s−1 ) 4.18 5.90 Jitter (m s−1 ) 2.0 2.22 Reduced χ2ν 1.59 1.46 FAP (periodogram) < 0.0001 0.006 – 36 – Table 9. Radial Velocities for HD 231701 JD RV Uncertainties -2440000. (m s−1 ) (m s−1 ) 13190.98047 1.98 3.41 13198.03808 -2.44 4.00 13199.02309 -11.19 3.74 13199.95689 -16.16 3.62 13603.87134 40.18 3.32 13928.04498 -38.57 2.90 13931.08095 -32.84 2.93 13932.02123 -36.55 3.38 13961.87814 -31.29 3.34 13981.76292 5.66 3.18 13983.77988 5.49 3.11 14023.71873 25.79 3.53 14083.69519 -39.18 2.43 14085.70009 -37.48 2.83 14217.13510 -40.18 2.99 14250.07410 -15.89 2.90 14286.00169 31.17 2.86 Introduction HD 11506 Stellar Characteristics Doppler Observations and Keplerian Fit HD 17156 Stellar Characteristics Doppler Observations and Keplerian Fit Transit Search HD 125612 Stellar Characteristics Doppler Observations and Keplerian Fit HD 170469 Stellar Characteristics Doppler Observations and Keplerian Fit HD 231701 Stellar Characteristics Doppler Observations and Keplerian Fit Discussion

0704.1195

On the Kaehler rank of compact complex surfaces

ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES MATEI TOMA Abstract. Harvey and Lawson introduced the Kähler rank and com- puted it in connection to the cone of positive exact currents of bidimen- sion (1, 1) for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of sur- faces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions (C2, 0) → (C2, 0). Harvey et Lawson ont introduit et calculé le rang de Kähler en re- lation avec le cône des courants positifs fermés de bidimension (1, 1) pour beaucoup de classes de surfaces complexes compactes. Dans ce travail nous étendons ces calculs à la seule classe de surfaces connues et qui n’avait pas été considérée par eux, celle des surfaces de Kato. Notre outil principal est la réduction à la dynamique des contractions holomorphes (C2, 0) → (C2, 0) associées. 1. Introduction In [8] Harvey and Lawson give a characterisation of Kählerianity for compact complex surfaces in terms of existence (or rather non-existence) of closed positive currents which are (1, 1)-components of a boundary. The authors also investigate and describe the cones formed by such currents for many types of non-Kähler surfaces: elliptic, Hopf, Inoue. Later Lamari proved that every non-Kähler surface admits non-trivial positive d-exact currents of bidimension (1, 1); cf. [10]. In order to estimate the degree of non- Kählerianity of a compact complex surface smooth positive d-exact currents are considered in [8] and the Kähler rank is defined as follows: the Kähler rank is two if the surface admits some Kähler metric, one if it admits some positive d-exact (1, 1)-form with some supplementary property and zero in Date: October 30, 2018. I wish to thank the Max-Planck-Institut für Mathematik in Bonn and the University of Osnabrück for their hospitality and for financial support during the preparation of this paper. Furthemore I thank Charles Favre and Karl Oeljeklaus for useful discussions and the referee for his remarks which improved the presentation of the paper. Keywords: compact complex surface, global spherical shell, closed positive current, iter- ation of polynomial maps. Mots-clé: surface complexe compacte, coquille sphérique globale, courant positif fermé, itération des applications polynômiales AMS Classification (2000): 32J15, 32H50 . http://arxiv.org/abs/0704.1195v2 2 MATEI TOMA the remaining case; see the more precise Definition 2. Unfortunately it is not clear whether the Kähler rank is a bimeromorphic invariant. We consider in this paper instead a bimeromorphic invariant which we call the modified Kähler rank and which we define to be two if the surface is Kähler, one if the cone of positive d-exact currents of bidimension (1, 1) is larger than a half-line and zero if this cone is a half-line. The two notions agree in the cases considered in [8]. Our main result is the computation of the cones of positive exact (1, 1)-currents for Kato surfaces. These are surfaces whose minimal models have positive second Betti number and ad- mit a global spherical shell (see Definition 9) and are the only “known” compact complex surfaces not considered in [8]. (We refer the reader to [1] for the general theory of compact complex surfaces.) It turns out that the modified Kähler rank does not coincide with the Kähler rank in general. In order to perform our computations we reduce ourselves to the investigation of plurisubharmonic functions with a certain invariance property with re- spect to polynomial automorphisms of C2 associated to Kato surfaces. As a corollary we obtain Theorem 1. (a) Every positive d-exact (1, 1)-current on a Kato surface, and more generally on any “known” non-Kählerian compact complex sur- face, is a foliated current for some holomorphic foliation of the surface. (b) All positive d-closed (1, 1)-currents on the “known” non-Kählerian com- pact complex surfaces excepting on parabolic Inoue surfaces are foliated cur- rents for some holomorphic foliations. See Section 3 for definitions. 2. The Kähler rank We let X be a compact complex surface and denote by Pbdy = Pbdy(X), P∞bdy = P bdy(X) the cones of positive currents of bidimension (1, 1) which are boundaries (i.e. are d-exact), respectively smooth such currents. These objects have been first studied in [8]. In [10] it was shown that X is Kähler if and only if Pbdy(X) is trivial. It is easy to see that on a non-Kähler surface every positive d-closed differential form of type (1, 1) is d-exact, cf. [10]. The following definition of Harvey and Lawson gives a measure of non-Kählerianity by looking at the positive closed differential(1, 1)-forms on X . It is roughly speaking the largest generic rank of a positive closed (1, 1)-form on X . Definition 2. Let B(X) = {x ∈ X | ∃φ ∈ P∞bdy(X) φx 6= 0}. The Kähler rank of X is defined to be two if X admits a Kähler metric. If X is non- Kähler the Kähler rank is set to be one when B(X) contains a non-trivial Zariski open subset of X and zero otherwise. It is not known whether the Kähler rank is a bimeromorphic invariant. We propose also the following: ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 3 Definition 3. The modified Kähler rank of X is defined to be two when X admits no non-trivial positive exact current of bidimension (1, 1), zero when it admits exactly one such current up to a multiplicative constant and one otherwise. One can show easily that the modified Kähler rank is a bimeromorphic invariant by taking push-down and pull-back of currents through blowing up maps. See the proof of Proposition 7 for a more precise description. For elliptic surfaces, primary Hopf surfaces and Inoue surfaces one sees that the Kähler rank and the modified Kähler rank coincide using the precise description of Pbdy given in [8] in these cases. Proposition 4. Let T be a positive exact current of bidimension (1, 1) on the compact complex surface X. Then there is a representation ρ : π1(X) → (R,+) and a plurisubharmonic function u on the universal cover X̃ of X such that T = ddcu and u ◦ g = u+ ρ(g) for all g ∈ π1(X). The function u can be chosen to be smooth if T is smooth. Proof. One has b1 = dimCH 1(X,C) = dimCH 1(X,OX)+dimCH 0(X,Ω1X) = h0,1 + h1,0, cf [1]. Denoting the sheaf of closed differential (1, 0)-forms by dOX and looking at the long exact cohomology sequence of 0 → CX → OX → dOX → 0 one gets an exact sequence 0 → H0(dOX) → H 1(CX) → H 1(OX). By the above equality follows now the surjectivity of the natural map H1(CX) → H 1(OX). This is given by mapping a de Rham cohomology class [β] of a differential form β = β1,0+β0,1 onto the Dolbeault cohomology class of β0,1. Let now T be a positive exact current of bidimension (1, 1) on X . Then T = dS with S = S1,0+ S0,1, S1,0, S0,1 currents of order zero and bidegree (1, 0) and (0, 1) respectively, and S1,0 = S̄0,1. Since ∂̄S0,1 = 0, S0,1 represents a cohomology class in H1(OX) and let β = β 1,0 + β0,1 be a closed differential form with [β0,1] = [S0,1] in H1(OX) and U a current of degree 0 on X with S0,1 = β0,1 + ∂̄U . The lift β̃ of β to the universal cover X̃ is d-exact and let f be a smooth function on X̃ with df = β. In particular ∂̄f = β0,1. This implies S0,1 = ∂̄(f + U) and T = dS = d(∂̄(f + U) + ∂(f̄ + Ū)) = i∂∂̄(2ℑm(f + U)). Moreover for g ∈ π1(X) we have d(f ◦ g − f) = 0 hence f ◦ g − f must be constant. Set ρ(g) = 2ℑm(f ◦g−f). The current 2ℑm(f +U) is associated to a plurisubharmonic function u on X̃ . Since u − 2ℑmf = 2ℑmU comes from X we see that u has the desired automorphy behaviour with respect to the action of π1(X). It is clear that u can be chosen to be smooth when T is smooth. � 4 MATEI TOMA Definition 5. We say that an effective reduced divisor C = C1 + ... + Cn on X is a cycle of rational curves if n ≥ 1, C1,..., Cn are rational curves and either n = 1 and C1 has a node or n > 1, all components C1,..., Cn are smooth and the dual graph of C is cyclic. Corollary 6. For a compact complex surface X with a cycle of rational curves C and b1(X) = 1 the Kähler rank is zero. Proof. Under the above hypotheses the natural map Z ∼= π1(C) → π1(X) is an isomorphism by a Theorem of Nakamura, [13]. Let g be a generator of π1(X). If the Kähler rank of X were one, we would get a smooth non- constant plurisubharmonic function u on the universal cover X̃ satisfying u ◦ g = u+ c for some constant c ∈ R by Proposition 4. The inverse image C̃ of the cycle of rational curves C of X is an infinite chain of rational curves on X̃ . Since u is smooth and constant on each link of this chain we would get c = 0 hence u would descend to X . But here u must be constant contradicting our assumptions. � Proposition 7. The modified Kähler rank is a bimeromorphic invariant. Proof. It is known that for smooth surfaces the property of being Kähler is invariant under bimeromorphic transformations. Let now X be non-Kähler and X ′ the blown-up surface at a point x ∈ X . Any plurisubharmonic function u on the universal cover X̃ may be lifted to X̃ ′. Conversely let u′ be a plurisubharmonic function on X̃ ′. It will be constant on the exceptional divisors of X̃ ′ coming from X ′ → X . Thus u′ is the pull-back of a plurisubharmonic function on X̃ , which satisfies the same invariance condition as u with respect to the action of π1(X) ∼= π1(X This gives a bijective correspondence between Pbdy(X) and Pbdy(X ′), hence the invariance of the modified Kähler rank. � From the above argument it also follows that a counterexample to the bimeromorphic invariance of the Kähler rank could only be given by some non-elliptic non-Kähler minimal surface X of Kähler rank zero admitting a continuous plurisubharmonic function u on the universal cover X̃ which is smooth outside a discrete subset of X̃ and exhibits the automorphy be- haviour from Proposition 4. From [8] we gather that this is not the case of Hopf surfaces or Inoue surfaces and as we shall see it is not going to be the case of Kato surfaces either. Finally let us mention the following description of the cone P (X) of positive d-closed (1, 1)-currents: ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 5 Remark 8. Let C1, ..., Cn be the irreducible curves of negative self- intersection on a non-Kählerian surface X and [C1], ..., [Cn] the corre- sponding currents of integration. Then P (X) = Pbdy(X) + R≥0[Ci]. Proof. Let us denote by Fj the irreducible curves ofX of zero self-intersection. Since X is non-algebraic it will admit no curve of positive self-intersection. Thus the Siu decomposition of a positive closed current T takes the form: ai[Ci] + bj [Fj ] +R, where ai, bj ∈ R and R is a positive current whose Lelong level sets are finite. By [11] Prop. 4.3 and its proof R is nef. Since X is non-Kählerian it follows that R must be exact, cf. [11] Thm. 7.1. The integration currents [Fj] are also exact, since F j = 0 and the intersection form on H (X) is negative definite for a non-Kählerian surface. � 3. Kato surfaces We recall here some facts on Kato surfaces which will be needed in the sequel. We refer the reader to [9], [3], [4], [5], [6] for more details. Definition 9. A global spherical shell on a compact complex surface X is the image Σ of the sphere S3 by a holomorphic embedding of a neighbourhood of S3 from C2 into X such that X \ Σ is connected. We call X a Kato surface if X admits a global spherical shell and the second Betti number of a minimal model of X is positive. The other minimal surfaces admitting a global spherical shell are the Hopf surfaces. Their second Betti number is zero. The notion of global spherical shell and a construction method for sur- faces with global spherical shells were introduced by Ma. Kato in [9]. An important analytic object associated to this construction method was intro- duced and studied by Dloussky in [3]. It is the germ of a holomorphic map (C2, 0) → (C2, 0). We shortly recall the facts now. Take the unit ball B in C2 around the origin and blow-up the origin. Choose a point P1 on the exceptional curve C1 thus obtained and blow it up again. Continue by blowing up a point on the last created exceptional curve. Af- ter n blow-ups one considers the blowing down map π : B′ → B. The exceptional divisor on B′ is a tree of n smooth rational curves. The only (−1)-curve among them is the last created curve Cn. Choose a point Pn on Cn, a biholomorphic map σ : B̄ → σ(B̄) onto a small compact neighbour- hood of Pn in B ′ and glue the two components of the boundary of B′ \σ(B) by means of σ ◦π. In this way one obtains a minimal compact complex sur- face X . One can show that the image of S3 through σ is a global spherical shell on X and that b2(X) = n. Thus X is a minimal Kato surface. 6 MATEI TOMA The images of the exceptional curves are the only (compact) rational curves on X . They form an effective reduced divisor which we denote by D. De- pending on the structure of D one subdivides the class of minimal Kato surfaces into: (1) Enoki surfaces, when D is a cycle of rational curves and D is homo- logically trivial, (2) intermediate surfaces, when D consists of a cycle of rational curves and of at least one further rational curve attached to the cycle, (3) Inoue-Hirzebruch surfaces, when D consists of one or two cycles of rational curves and D is not homologically trivial. In particular Kato surfaces admit cycles of rational curves and therefore their Kähler rank is zero. In the case of Enoki surfaces a further curve might appear. In such a case the curve will be elliptic and the surface is called a parabolic Inoue surface. By definition the Dloussky germ associated to X is the germ of the map f := π ◦ σ : B → B around the origin. It can be shown that the conjugacy class of this germ determines the isomorphy class of X . We shall denote by X(f) a Kato surface associated to such a germ of holomorphic map f : (C2, 0) → (C2, 0). One can relate certain analytic objects on X to germs of objects on (C2, 0) which are invariant under f as follows. One recovers first the universal cover X̃ ofX from the above construction by considering an infinite number of copies (Ai)i∈Z of B̄ ′ \ σ(B) and by gluing for all i ∈ Z the pseudoconvex component of the border ∂Ai−1 of Ai−1 to the pseudoconcave component of ∂Ai by means of σ ◦ π again. If one glues in this way only the copies (Ai)i≤0 and then caps the pseudoconcave end of A0 by a copy of B using σ, one obtains a non-compact complex surface X̂ , a holomorphic map p : X̃ → X̂ , a (−1)-curve Ĉ on X̂ and a point Ô on Ĉ such that p extends the identity map of ∪i≤0Ai, Ĉ is the isomorphic image of a rational curve C of X̃ through p and p restricts to an isomorphism X̃ \ p−1(Ô) → X̂ \ {Ô}. In fact p−1(Ô) is the union of the infinitely many rational curves appearing after C on X̃ in the “order of creation”, cf. [3], Prop. 3.4. Thus p : X̃ → X̂ can be seen as a blowing down of the infinitely many exceptional curves in p−1(Ô). The generator g of π1(X) mapping Ai to Ai+1 induces a holomorphic map ĝ : X̂ → X̂ with p ◦ g = ĝ ◦ p. One sees that Ô is fixed by ĝ, that ĝ(Ĉ) = Ô, and that the germ of ĝ at Ô is the same as the germ of π ◦ σ : B → B at the origin. Let now u be any plurisubharmonic function on X̃ . The restriction of u to X̂ \ {Ô} extends to a plurisubharmonic function û on X̂ , [7]. It is clear that û ◦ ĝ = û − c in case u ◦ g = u − c for some c ∈ R. Conversely one gets a plurisubharmonic function u = û ◦ p starting from û. In fact, since the germ of ĝ around Ô is contracting, it is enough to have only a germ of û around Ô satisfying û ◦ ĝ = û − c in order to recover u on X̃ with the property u ◦ g = u− c. ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 7 According to [3], [4], [5], [6] we get the following three normal forms for representatives of conjugacy classes of Dloussky germs: (1) in the case of Enoki surfaces f(z, w) = (αz, wzs +Q(z)), where α ∈ ∆∗ = ∆ \ {0}, s ≥ 1 and Q is a complex polynomial of degree at most s and with Q(0) = 0; we have denoted by ∆ the unit disc in C; (2) in the case of intermediate surfaces f(z, w) = (zp, λwzs +Q(z)), where p ≥ 2, s ≥ 1, λ ∈ C∗ and Q(z) = m=1 amz m + az p−1 is a complex polynomial with gcd{p, m | am 6= 0} = 1 and a = 0 if (p− 1) ∤ s or λ 6= 1; (3) in the case of Inoue-Hirzebruch surfaces f(z, w) = (zawb, zcwd), where the matrix is a product of b2(X) matrices of the form with at least one factor of the first kind. In the rest of this paper we shall determine the germs of plurisubharmonic functions around the origin of C2 satisfying u ◦ f = u − c for some fixed c ∈ R>0 and each type of germ f as above. 4. The main results Theorem 10. (a) Let f : C2 → C2, f(z, w) = (αz, wzs +Q(z)), with α ∈ ∆∗, s ≥ 1 and Q is a complex polynomial of degree at most s and with Q(0) = 0. Let u : C2 → [−∞,∞[, u(z, w) = log |z|. Then up to some additive constant, u is the only plurisubharmonic function on C2 which satisfies u ◦ f = u+ log |α|. (b) On an Enoki surface the integration current on the cycle of rational curves is the only positive exact current of bidimension (1, 1) up to multi- plicative constants. Proof. We start by proving part (b) of the theorem. Let C = C1 + ... + Cn be the cycle of rational curves on the Enoki surface X . Since C is homologically trivial the current of integration [C] along C is a positive d-exact current of bidimension (1, 1). We denote by E the elliptic curve on X in case it exists. Let T be an arbitrary positive exact current of bidimension (1, 1) on X and mCi := inf{ν(T, x) |x ∈ Ci}, mC = minmCi , mE its generic Lelong numbers 8 MATEI TOMA along Ci, C and E respectively. We denote by χA the characteristic function of a subset A of X . The Siu decomposition of T has the form T = χET + χCT + χX\(C∪E)T = mE [E] + mCi [Ci] + χX\(C∪E)T = = mE [E] + (mCi −mC)[Ci] +mC [C] + χX\(C∪E)T, see [2] 6.18, 3.2.4. Since C is the only homologically trivial effective reduced non-trivial divisor X we get as in Remark 8 that χX\(C∪E)T is exact and T = mC [C] + χX\(C∪E)T. Hence we may replace T by χX\(C∪E)T which is positive, d-exact, has van- ishing generic Lelong number along C and is the trivial extension toX of the restriction of T to X \(C∪E). After normalisation we obtain a correspond- ing plurisubharmonic function v on C2 satisfying v ◦ f = v + log |α|. We denote the current ddcv on C2 by T again. Consequently we have f ∗T = T . Moreover, since C corresponds to the axis A = {z = 0} ⊂ C2, the generic Lelong number mA of T along A must vanish. Since f(A) = {0} and in general ν(f ∗T, x) ≥ ν(T, f(x)) (cf. [12]) it follows that ν(T, 0) = 0. The differential form dz defines a holomorphic foliation on C2 which is invariant by f , since f ∗(dz) = αdz. Claim: T is a foliation current for this foliation. This means that for any test function φ ∈ C∞c (C 2) one has T (φdz∧dz̄) = 0. As before T is the trivial extension to C2 of its restriction to C2 \ A. It is therefore enough to check that T (φdz ∧ dz̄) = 0 for test functions φ with support in C2 \ A. We have |T (φdz ∧ dz̄)| = |f ∗T (φdz ∧ dz̄)| = |T ((f−1)∗(φdz ∧ dz̄))| = = |T ( φ ◦ f−1 dz ∧ dz̄)| ≤ max |φ| 2σT (f(Suppφ)), where σT = T ∧ (dz ∧ dz̄ + dw ∧ dw̄) denotes the trace measure of T . Iterating we obtain (1) |T (φdz ∧ dz̄)| ≤ max |φ| |α|2n 2σT (f n(Supp φ)), for any n ∈ N. We now need to estimate how large fn(B(0, R)) is. Take C1 = max{ |Q(z)| | z ∈ ∆̄}. We denote by P (R1, R2) the bidisc of radii R1, R2 centered at the origin of C2. Then f(P (1, R2) ⊂ P (|α|, R2 + C1), f 2(P (1, R2) ⊂ P (|α| 2, |α|(R2 + C1) + |α|C1) ⊂ ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 9 ⊂ P (|α|2, |α|(R2 + C1) + ( 1− |α| − 1)C1), f 3(P (1, R2) ⊂ P (|α| 3, |α|3(R2 + C1) + 1− |α| |C1) ⊂ ⊂ P (|α|3, |α|3(R2 + C1) + ( 1− |α| − 1)C1), f 4(P (1, R2) ⊂ P (|α| 4, |α|6(R2 + C1) + 1− |α| |C1) ⊂ ⊂ P (|α|4, |α|6(R2 + C1) + ( 1− |α| − 1)C1), and further fn(P (1, R2) ⊂ P (|α| n, |α| n(n−1) 2 (R2 + C1) + |α|n−1 1− |α| |C1). From this and from (1) it follows that there is a constant C2 such that |T (φdz ∧ dz̄)| ≤ C2max |φ| σT (B(0, |α| |α|2n for all n sufficiently large. But σT (B(0, |α| |α|2n = πν(T, 0) = 0, and the claim follows. Using the claim it can be easily shown that T is invariant under translations in the w-direction, hence T = pr∗1S, where S is a positive current on C of dimension zero and pr1 : C 2 → C denotes the first projection. Set f1 : C → C, f1(z) = αz. Then pr1 ◦ f = f1 ◦ pr1, hence f 1 (S) = S. The current S is of the form µidz ∧ dz̄, where µ is a positive measure on C. Denote by ∆(r) the disc of radius r around the origin of C. The invariance property of S implies µ(∆(r)) = µ(∆(|α|r)) = µ(∆(|α|nr)) for all n ∈ N and r ∈ R>0. This entails that S is supported at the origin of C and hence that T = 0, since T is not carried by A. We now turn to part (a) of the theorem. Let v be a plurisubharmonic function on C2 satisfying v ◦ f = f + log |α|. By part (b) of the theorem we see that ddcv ≤ ddcu or ddcu ≤ ddcv. But then u− v or v − u would give a plurisubharmonic function on X which has to be constant. � Notation For α ∈ R>0 we denote by Kα the set of continuous α-periodic functions ψ : R → R which fulfil the inequality −ψ′′ + ψ′ + 1 ≥ 0 in generalised sense. 10 MATEI TOMA Notice that Kα is infinite dimensional: for any α-periodic smooth function φ : R → R and for any small enough factor ǫ ∈ R>0 one has ǫφ ∈ Kα. Theorem 11. (a) Let f : ∆× C → ∆× C, f(z, w) = (zp, λwzs +Q(z)), where p ≥ 2, s ≥ 1, λ ∈ C∗ and Q(z) = m=1 amz m + az p−1 is a complex polynomial with gcd{p, m | am 6= 0} = 1 and a = 0 if (p − 1) ∤ s or λ 6= 1. The plurisubharmonic functions u : ∆×C → [−∞,∞[ which satisfy u ◦ f = u− log p are precisely the functions of the form u(z, w) = − log(− log |z|)− ψ(log(− log |z|)) for ψ ∈ Klog p. (b) The cone of positive exact currents of bidimension (1, 1) on the inter- mediate surface X(f) corresponds bijectively to the cone of currents of the form cddcu on ∆×C for u as above and c ∈ R>0. In particular, the modified Kähler rank of X(f) is one. Proof. There are no homologically trivial divisors on X(f) this time (ex- cepting 0 of course). As in the case of Enoki surfaces we reduce ourselves to the investigation of closed positive currents T on ∆ × C with vanishing Lelong number at the origin and satisfying f ∗T = T . Moreover we may again suppose that T is the extension to ∆×C of its restriction to ∆∗ ×C. The differential form dz defines a holomorphic foliation on ∆× C which is invariant under f . We start again by showing that T is a foliation current for this foliation. For this it is enough to check that the measure idz ∧ dz̄ ∧ T vanishes on ∆∗ × C. We use the invariance of T by f again. Take 0 < r1 < r2 < 1, r ′ > 0, D := (∆(r2) \ ∆(r1)) × ∆(r ′), D′ := f−n(fn(D)) and A(r1, r2) = A(r1, r2, r ′) := (idz ∧ dz̄ ∧ T )(D) the measure of the set D. We have of course D′ ⊃ D. Since f : ∆∗ × C → ∆∗ × C is p to 1 we get (idz ∧ dz̄ ∧ T )(fn(D)) = pn(i|z|2(p n−1)dz ∧ dz̄ ∧ T )(D′) ≥ ≥ pn(i|z|2p dz ∧ dz̄ ∧ T )(D′) ≥ pnr 1 A(r1, r2). Hence (2) A(r1, r2) ≤ n i(∂∂̄(|z|2 + |w|2p) ∧ T )(fn(D)) As before we need to estimate the width of fn(D). Let C1 = max{ |Q(z)| | z ∈ ∆̄} and C2 = max(1, |λ|). For 0 < r < 1, 0 < r ′ we see that f(P (r, r′)) ⊂ P (rp, rC2(r )) ⊂ P (rp, rC2(r (1− r)C2 ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 11 f 2(P (r, r′)) ⊂ P (rp , rp+1C22r rp−1C22C1 (1− r)C2 ) ⊂ P (rp , rp−1C22(r (1− r)C2 f 3(P (r, r′)) ⊂ P (rp p−1C22(r (1− r)C2 ⊂ P (rp 2+p−1C32r 2−1C32C1 (1− r)C2 For sufficiently large n we thus obtain fn(P (r, r′)) ⊂ {|z|2 + |w|2p < 2C1r which in combination with the inequality (2) gives A(r1, r2) ≤ n 2i(∂∂̄(|z|2 + |w|2p) ∧ T )({|z|2 + |w|2p < 2C1r But the factor 2i(∂∂̄(|z|2 + |w|2p) ∧ T )({|z|2 + |w|2p < 2C1r converges to 4C1ν(T, φ, 0), where ν(T, φ, 0) is the Lelong number with re- spect to φ = log(|z|2 + |w|2p). This Lelong number vanishes by the Com- parison Theorem for Lelong numbers [2], since the usual Lelong number vanishes. So for any ǫ > 0 we can find some N ∈ N such that A(r1, r2) ≤ for all n ≥ N . Moreover this inequality holds also for smaller r1 and r2. δ = ( and look at the division (r1, δr1, δ 2r1, ..., δ 2pnr1 = r2) of the interval [r1, r2]. We have seen that A(δir1, δ i+1r1) ≤ ǫ = 1 ǫ hence A(r1, r2) = i=1A(δ i−1r1, δ ir1) ≤ 2 ǫ, which proves that T is a foliation current. As in the proof of Theorem 10, T is invariant under translations in the w- direction, hence T = pr∗1S, where S is a positive current on ∆ of dimension zero and pr1 : ∆ × C → ∆ is the first projection. On the other hand T = ddcu, where u is a plurisubharmonic function on ∆ × C with u ◦ f = u− log p. The restriction of u to a general leaf {z = const.} of our foliation is a harmonic function in w. We shall show that ∂u = 0, the case of ∂u being similar. Since T is a foliation current one has that ∂ ∂z̄∂w = 0 as distributions. In particular ∂u = 0 is invariant by translations in the z-directions. The invariance condition by translations in w implies that (3) λzs (f(z, w)) = (z, w). 12 MATEI TOMA Take now a small non-zero solution z′ of the equation (4) w = λzsw +Q(z) for fixed small w, w 6= 0. Using the invariance in the z-direction together with (3) and (4) we obtain (z, w) = (z′, w) = λ(z′)s ((z′)p, w) = λ(z′)s (z, w), hence ∂u (z, w) = 0 for small w. But for large w the vanishing holds as well since we can iterate on (3). Thus u is constant on the leaves {z = const.} and therefore descends to a subharmonic function u on ∆ fulfilling the condition u(zp) = u(z) − log p for all z ∈ ∆. Take now z ∈ ∆ and a pn-th root of unity θ. The invariance condition implies u(z) = u(zp ) + n log p = u(θz) hence each value u(z) is attained on a dense subset of the circle {|z| = const.}. By semi-continuity it follows that u depends only on r = |z|. Let v = u|[0,1[. By the maximum principle and the invariance condition on u one sees that v is strictly increasing. Using this, the semi-continuity of u and the definition of subharmonicity (cf. [2] 1.4.13.b) one infers that v is continuous. The Laplace operator takes the form ∂ in polar coordinates (r, φ). We thus reduce ourselves to the search of continuous functions v : [0, 1[→ [−∞,∞[ which satisfy (5) v(rp) = v(r)− log p (6) v′′(r) + v′(r) ≥ 0. Let h :] − ∞,∞] → [0, 1[, h(t) = exp(− exp t) and ψ = −v ◦ h − h. The conditions (5) and (6) translate into ψ(t+ log p) = ψ(t), −ψ′′ + ψ′ + 1 ≥ 0 and thus ψ ∈ Klog p and the theorem is proved. � For the case of Inoue-Hirzebruch surfaces we need some preparations. Let f : C2 → C2, f(z, w) = (zawb, zcwd), and denote by A the matrix We suppose that A is a product of matrices of the form with at least one factor of the first kind. We have det(A) = ±1 and by Lemma 2.2 from [4], trace(A) > 2 unless A = or A = ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 13 any case one sees that the eigenvalues λ1, λ2 of A t are real, irrational and one of them is larger than 1. We set λ = λ1 > 1. Let (α, β) = (α1, β1), (α2, β2) be eigenvectors associated to λ1 and λ2 respectively. An easy computation shows that αβ = α1β1 > 0, α2β2 < 0. We choose α, β, α2 ∈ R>0 and set φ = φ1 = α log |z|+ β log |w|, φ2 = α2 log |z|+ β2 log |w|, U := {φ < 0} ⊂ C2. Then φi ◦ f = λiφi for i = 1, 2. Now we can state Theorem 12. (a) For A, λ, α, β as above let φ = α log |z| + β log |w|, U = φ−1([−∞, 0[), f : U → U , f(z, w) = (zawb, zcwd). Then the plurisubharmonic functions u on U which satisfy u ◦ f = u− log λ are precisely the functions of the form u = − log(−φ)− ψ(log(−φ)), for ψ ∈ Klog p. (b) The cone of positive exact currents of bidimension (1, 1) on the Inoue- Hirzebruch surface X(f) corresponds bijectively to the cone of currents of the form cddcu on U for u as above and c ∈ R>0. In particular, the modified Kähler rank of X(f) is one. Proof. As in the proof of Theorem 11 we reduce ourselves to the investi- gation of currents T on U of the form T = ddcu with u plurisubharmonic on U and such that u ◦ f = u − log λ and ν(T, 0) = 0. Moreover we may suppose that T is the extension to U of its restriction to U ∩ (C∗ × C∗). There is again a holomorphic foliation on U , this time singular, which will be shown to be compatible with T . On U∩(C∗×C∗) this foliation is induced by the smooth positive (1, 1)-form ddcv where v := − log(−φ). As before it is enough to show that the measure ddcv∧T is zero on any compact subset of U ∩ (C∗ × C∗). For a better visualisation of the present situation we introduce the map E : C × C → C∗ × C∗, E(ζ, ω) = (exp ζ, expω). We denote the linear automorphism induced by A on C2 again by A and get the relations f ◦ E = E ◦ A, φ ◦ E(ζ, ω) = αiℜeζ + βiℜeω, i = 1, 2. In particular E−1(U) = {αℜeζ + βℜeω < 0}. We also see that U is covered by compact subsets of the formD = D(c1, δ, c2) := φ−11 ([−c1δ,−c1]) ∩ φ 2 ([−c2, c2]) for c1, c2 ∈ R>0, δ ∈ [1,∞[. Since both T and ddcv are invariant by f we have A(c1, δ) = A(c1, δ, c2) := (dd cv ∧ T )(D(c1, δ, c2)) = = (ddcv ∧ T )fn(D(c1, δ, c2))) = (dd cv ∧ T )(D(λnc1, δ, λ −nc2)). We need to estimate ddcv on fn(D). On D we have 14 MATEI TOMA −c1δ ≤ α1 log |z|+ β1 log |w| ≤ −c1, −c2 ≤ α2 log |z| + β2 log |w| ≤ c2, hence c1δβ2 − c2β1 α2β1 − α1β2 ) ≤ |z| ≤ exp( c1β2 + c2β1 α2β1 − α1β2 −c1δα2 − c2α1 α2β1 − α1β2 ) ≤ |w| ≤ exp( −c1α2 + c2α1 α2β1 − α1β2 Therefore we get on fn(D) λnc1δβ2 − λ −nc2β1 α2β1 − α1β2 ) ≤ |z| ≤ exp( λnc1β2 + λ −nc2β1 α2β1 − α1β2 −λnc1δα2 − λ −nc2α1 α2β1 − α1β2 ) ≤ |w| ≤ exp( −λnc1α2 + λ −nc2α1 α2β1 − α1β2 hence fn(D) ⊂ B(0, rn), where r2n = exp(2 λnc1β2 + λ −nc2β1 α2β1 − α1β2 ) + exp(2 −λnc1α2 + λ −nc2α1 α2β1 − α1β2 Now on D again i∂∂̄v = i∂φ ∧ ∂̄φ i(α dz + β dw ) ∧ (α dz̄ + β dw̄ min{|z|2, |w|2 | (z, w) ∈ D} i(dz ∧ dz̄ + dw ∧ dw̄) where C1 is a constant not depending on D. This implies A(c1, δ) ≤ min{φ2(z, w)|z|2, φ2(z, w)|w|2 | (z, w) ∈ fn(D)} σT (B(0, r As in the previous cases σT (B(0,r converges to zero, therefore for any ǫ > 0 we get C1σT (B(0,r < ǫ as soon as n is large enough. Setting 2c1 max{α1,−β2} α2β1−α1β2 + 1 we get by our estimates r2n ≤ exp(−λ nC2) and min{|z|2, |w|2 | (z, w) ∈ fn(D)} ≥ exp(−λnC2δ), hence A(c1, δ) ≤ exp(λnC2(δ − 1)) Set m = ⌊λ2n⌋ and consider the division (1, exp 1 , exp 2 , ..., exp m = e) of the interval [1, e]. For any subinterval our estimates hold with eC2 instead of C2, hence A(c1, e, c2) = A(c1 exp , exp , c2) ≤ ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 15 exp(λnC2(exp − 1)) ≤ exp(C2e exp 1 and the last term converges to exp(C2e) showing that A(c1, e, c2) = 0. As in the proof of Theorem 11 we next check that u is constant on the leaves of the foliation given by α dz + β dw on U ∩ (C∗ × C∗). Indeed, for V := u ◦ E one has V ◦ A = V − log λ on {αℜeζ + βℜeω < 0} and ddcV is a foliation current for the foliation given by αdζ + βdω. We consider a linear change of coordinates which diagonalizes A on C2 leading to A(ξ, τ) = (λξ, λ−1τ). Keeping the notation V for V after this coordinate change we notice that V restricted to the leaves {ξ = const.} is harmonic and that ∂V are invariant by translations in the ξ-direction. As in the case of intermediate surfaces we get (ξ, τ). Iterating this relation and using the continuity of ∂V on {ξ = const.} we obtain τ) = λn (ξ, τ) which forces ∂V to vanish. Similarly ∂V = 0. Thus V is constant on the leaves {ξ = const.}. Returning now to the coordinates (ζ, ω) on C2 we shall show that V only depends on ℜeζ and ℜeω. The function V is doubly periodic in ℑmζ , ℑmω since V = u ◦ E. Fix ℜeζ and ℜeω and look at a leaf {αζ + βω = const.}. Under this restriction the imaginary parts must satisfy some rela- tion αℑmζ + βℑmω = const. which describes a dense subset in the “torus of the imaginary parts” R2/(2πZ)2 since α/β is irrational. On this subset V is constant and by semi-continuity it must be constant on the whole torus. Switching once more to coordinates (ξ, τ) we see that V depends on ℜeξ alone. We have reduced ourselves in this way to the search of continuous functions v : R<0 → R subject to the conditions v(λt) = v(t)− log λ, ∀t ∈ R<0, v′′ ≥ 0. Remark that the continuity of v can be deduced as in Theorem 11 after first restricting u to the line {z = w}. Take now h : R → R<0, h(s) = − exp(s) and ψ = −v ◦ h − h. Then the conditions on v translate into ψ ∈ Klog λ. Thus u = − log(−φ)−ψ(log(−φ)) and the proof is finished. � Finally, the first part of Theorem 1 is a direct consequence of the description of Pbdy(X) given in [8] and in the above Theorems. For the second part it suffices to apply Remark 8 and to notice that the only case of a curve on a “known” non-Kählerian surface, which is not invariant under a holomorphic foliation, is that of the elliptic curve on a parabolic Inoue surface. 16 MATEI TOMA References [1] Barth W., Hulek K., Peters C., Van de Ven A.: Compact complex sur- faces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Berlin, Springer-Verlag, (2004). [2] Demailly J.P.: Complex analytic and algebraic geometry, http://www-fourier.ujf-grenoble.fr/∼demailly/books.html. [3] Dloussky G.: Structure des surfaces de Kato, Mémoire de la S.M.F 112.n◦14 (1984). [4] Dloussky G.: Sur la classification des germes d’applications holomorphes con- tractantes, Math. Ann. 280, (1988), 649-661. [5] Dloussky G., Oeljeklaus K.: Vector fields and foliations on surfaces of class VII0, Ann. Inst. Fourier 49, (1999), 1503-1545. [6] Favre, Ch.: Classification of 2-dimensional contracting rigid germs, Jour. Math. Pures Appl. 79 (2000), 475-514. [7] Grauert, H., Remmert, R.: Plurisubharmonische Funktionen in komplexen Räumen, Math. Z. 65 (1956), 175-194. [8] Harvey, R., Lawson, H. B.: An intrinsic characterization of Khler manifolds, Invent. Math. 74 (1983), 169-198. [9] Kato Ma.: Compact complex manifolds containing “global spherical shells”, Pro- ceedings of the Int. Symp. Alg. Geometry, Kyoto 1977, Kinokuniya Book Store, Tokyo 1978. [10] Lamari, A.: Courants kählériens et surfaces compactes, Ann. Inst. Fourier 49 (1999), 263-285. [11] Lamari, A.: Le cône Kählérien d’une surface, J. Math. Pures Appl. 78 (1999), 249-263. [12] Méo, M.: Image inverse d’un courant positif fermé par une application analytique surjective, C. R. Acad. Sci. Paris Sr. I Math. 322 (1996), 1141–1144. [13] Nakamura I.: On surfaces of class VII0 with curves II, Tohôku Math. Jour. 42, (1990), 475-516. Institut de Mathématiques Elie Cartan, Nancy-Université, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France and Institute of Mathematics of the Romanian Academy. E-mail address : [email protected] URL: http://www.iecn.u-nancy.fr/∼toma/ http://www-fourier.ujf-grenoble.fr/~demailly/books.html 1. Introduction 2. The Kähler rank 3. Kato surfaces 4. The main results References

0704.1196

Novel algorithm to calculate hypervolume indicator of Pareto approximation set

arXiv:0704.1196v1 [cs.CG] 10 Apr 2007 Novel algorithm to calculate hypervolume indicator of Pareto approximation set Qing Yang1 and Shengchao Ding2,3 1 School of Computer Science and Technology, South-Central University for Nationalities, Wuhan, China 2 Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China 3 Graduate University of the Chinese Academy of Sciences, Beijing, China [email protected] Abstract. Hypervolume indicator is a commonly accepted quality mea- sure for comparing Pareto approximation set generated by multi-objective optimizers. The best known algorithm to calculate it for n points in d- dimensional space has a run time of O(nd/2) with special data structures. This paper presents a recursive, vertex-splitting algorithm for calculating the hypervolume indicator of a set of n non-comparable points in d > 2 dimensions. It splits out multiple child hyper-cuboids which can not be dominated by a splitting reference point. In special, the splitting refer- ence point is carefully chosen to minimize the number of points in the child hyper-cuboids. The complexity analysis shows that the proposed algorithm achieves O(( d )n) time and O(dn2) space complexity in the worst case. 1 Introduction Optimization for multiple conflicting objectives results in more than one optimal solutions (known as Pareto-optimal solutions). Although one of these solutions is to be chosen at the end, the recent trend in evolutionary and classical multi- objective optimization studies have focused on approximating the set of Pareto- optimal solutions. However, to assess the quality of Pareto approximation set, special measures are needed [1]. Hypervolume indicator is a commonly accepted quality measure for compar- ing approximation set generated by multi-objective optimizers. The indicator measures the hypervolume of the dominated portion of the objective space by Pareto approximation set and has received more and more attention in recent years [2, 3, 1, 4]. There have been some studies that discuss the issue of fast hypervolume cal- culation [5–8]. These algorithms partition the covered space into many cuboid- shaped regions, within which the approach considering the dominated hyper- volume as a special case of Klee’s measure problem is regarded as the cur- rent best one. This approach [8] adopts orthogonal partition tree which requires O(nd/2) storage and streaming variant [9]. Conceptual simplification of the im- plementation are concerned and thus the algorithm achieves an upper bound of http://arxiv.org/abs/0704.1196v1 O(n log n+ nd/2) for the hypervolume calculation. Ignoring the running time of sorting the points according to the d-th dimension, O(n log n), the running time of this approach is exponential of the dimension of space d. This paper develops novel heuristics for the calculation of hypervolume in- dicator. Special technologies are applied and the novel approach yields upper bound of O((d )n) runtime and consumes O(dn2) storage. The paper is orga- nized as follows. In the next section, the hypervolume indicator is defined, and some background on its calculation is provided. Then, an algorithm is proposed which uses the so-called vertex-splitting technology to reduce the hypervolume. The complexities of the proposed algorithm are analyzed in Section 4. The last section concludes this paper with an open problem. 2 Background Without loss of generality, for multi-objective optimization problems, if the d objective functions f = (f1, . . . , fd) are considered with fi to be minimized, not one optimal solution but a set of good compromise solutions are obtained since that the objectives are commonly conflicting. The compromise solutions are commonly called Pareto approximation solutions and the set of them is called the Pareto approximation set. For a Pareto approximation set M = {y1, y2, . . . , yn} produced in a run of a multi-objective optimizer, where yi = (yi1, . . . , yid) ∈ M ⊂ Rd, all the solutions are non-comparable following the well-known concept of Pareto dominance. Specially, we say that yi dominates yk at the j-th dimension if yij < ykj . The unary hypervolume indicator of a set M consists of the measure of the region which is simultaneously dominated by M and bounded above by a ref- erence point r = (r1, . . . , rd) ∈ R d such that rj ≥ maxi=1,...,n {yij}. In the context of hypervolume indicator, we call the solutions in M as the domina- tive points. As illustrated in Fig. 1(a), the shading region consists of an or- thogonal polytope, and may be seen as the union of three axis-aligned hyper- rectangles with one common vertex, i.e., the reference point r. Another example in three dimensional space is shown in Fig. 1(b), where five dominative points, y1 = (1, 2, 3), y2 = (4, 3, 2), y3 = (5, 1, 4), y4 = (3, 5, 1), y5 = (2, 2, 2.5), and the reference point r = (6, 6, 6) are considered. The volume is the union of the vol- umes of all the cuboids each of which is bounded by a vertex, where the common regions are counted only once. If a point yk is dominated by another point yi, the cuboid bounded by yk is completely covered by the cuboid bounded by yi. And thus only the non-dominated points contribute to the hypervolume. 3 The proposed algorithm In other works, e.g. the work of Beume and Rudolph [8], the hyper-cuboid in d-dimensional space are partitioned into child hyper-cuboids along the d-th di- mension and then all these child hypervolumes are gathered together by the inclusion-exclusion principle [10]. (a) A hypervolume indicator in the two- objective case (b) A hypervolume indicator in the three-objective case. To lay out the cuboids well, the axes are rotated where the reference point is shaded In this paper, we step in another way. The hyper-cuboid is partitioned into child hyper-cuboids at some splitting reference points and then all the child hy- pervolumes are gathered directly. More detailed, given a point yi ∈ M , each of other points inM must dominated yi at some dimensions for the non-comparable relation. If the parts over yi are handled, the problem of calculating the hyper- volume bounded by M and the reference point is figured out. The additional part partitioned out at the j-th dimension is also a d-dimensional hyper-cuboid whose vertices are ones beyond yi at such dimension. Their projections on the hyperplane orthogonal to dimension j are all dominated by yi, and thus are free from consideration. It should be noted that the reference point of child hyper-cuboid is altered to r′ = (r1, . . . , yij , . . . , rd), namely the j-th coordinate is replaced by yij . The other child hyper-cuboids are handled in the similar way. In these processes, the given point is called the splitting reference point. Obviously, the hyper-cuboids with more dominative points require more run time to calculate the hypervolumes. To reduce the whole run time for calculating all these child hyper-cuboids, the splitting reference point should be carefully selected. The strategy adopted in this paper is described as follows. (1) Let k = n − 1 and choose a point with the least dimensions on which the point dominated by other k points. (2) If some points tie, update k as k − 1 and then within these points, choose a point with the least dimensions on which the point dominated by other k points. (3) Repeat the similar process until only single point is left or k = 1. And if k = 1 and several points are left, the first found point is selected. By the above principle, as an example, not y2 or other points but y5 is chosen as the first splitting reference point for the case shown in Fig. 1(b). Two child cuboids each bounded by one points and another child cuboid bounded by two points are generated by splitting along y5. This is the optimal strategy in such case. The algorithm to calculate the hypervolume is shown in Algorithm 1. Some major parameters are as follows. – int[n][d] order The orders of all the dominative points at each dimension are represented by a two-dimensional array of integer. – int split The index of the point at which the hyper-cuboid is cut to generate multiple child hyper-cuboids is called split. – int[n] splitCount The numbers of k present in the split-th row of the array order are saved in splitCount, where k = 0, . . . , n− 1. – int[n] coveredCount The numbers of k present in the current checked row of the array order are save in coveredCount, where k = 0, . . . , n− 1. Moreover, some conventions are explained as follows. – The subscript of yij begins with 1 while the index of array begins with 0. Thus yij is same as y[i− 1][j − 1]. – Assume a and b are two arrays and n is an element. a[] ⇐ n means setting each element of a as n, while a[] ⇐ b[] means copying all the elements of b to a pairwise. – Assume S is a set and x is an element. S ⇐ S + {x} means appending a copy of x to S. The inputs of the algorithm are a set of non-dominated (dominative) points and a reference point, thus the hyper-cuboids are represented implicitly. In fact, when the hyper-cuboid is cut into two child hyper-cuboids, there may be some points dominated by the splitting reference point in the bigger cuboid, and thus such points could be removed from the points set H . In the proposed algorithm, it does not matter whether those points are removed or not. 4 Complexity Analysis Before discussing the time-space complexity of the proposed algorithm, some properties are presented firstly. Lemma 1. Let δij be the number of points dominating yi at the j-th dimension. (1) For d ≥ 2 and each i ∈ {1, . . . , n}, j=1 δij ≥ n− 1. (2) For d ≥ 2 and each j ∈ {1, . . . , d}, i=1 δij ≤ (n− 1). (3) For d = 2 and each i ∈ {1, . . . , n}, j=1 δij = n− 1. j=1 δij ≤ (n− 1). (5) For d ≥ 2 and each i ∈ {1, . . . , n}, j=1 δij ≤ (n− 1). Algorithm 1 Calculate Hypervolume, CalcV olume(H) Input: The hyper-cuboid H defined by the dominative points {y1, y2, . . . , yn} where yi = (yi1, . . . , yid), and the reference point r = (r1, . . . , rd), namely H = {y1, . . . , yn, r}. The initial number n of dominative points can be obtained from the length of H and the dimension d is known too. Output: The hypervolume of H , volume. 1: /* initialization */ 2: if n = 1 then 3: return |rj − y1j |; 4: end if 5: volume ⇐ 0; 6: splitCount[] ⇐ n; 7: /* count the numbers of points dominating every point at each dimension */ 8: for j = 1 to d do 9: sort y1j , . . . , ynj ; 10: for all i such that 1 ≤ i ≤ n do 11: order[i− 1][j − 1] ⇐ number of points dominating yij strictly; 12: end for 13: end for 14: /* estimate split based on the statistical results of order */ 15: for i = 1 to n do 16: coveredCount[] ⇐ 0; 17: for j = 1 to d do 18: coveredCount[order[i− 1, j − 1]]++; 19: end for 20: for k = n− 1 downto 0 do 21: if coveredCount[k] < splitCount[k] then 22: split ⇐ i; 23: splitCount[] ⇐ coveredCount[]; 24: break; 25: end if 26: end for 27: end for 28: /* cut H at each dimension through the point indexed by split */ 29: for j = 1 to d do 30: if order[split− 1][j − 1] > 0 then 31: H2 ⇐ {}; 32: for all yi in H\{ysplit, r} do 33: if yij is dominated strictly by ysplit,j then 34: H2 ⇐ H2 + {yi}; 35: yij ⇐ ysplit,j ; 36: end if 37: /* Here yi can be removed from H if yi is dominated strictly by ysplit */ 38: end for 39: r2 ⇐ r; 40: r2[j − 1] ⇐ ysplit,j ; 41: H2 ⇐ H2 + {r2}; 42: volume ⇐ volume+ CalcV olume(H2); 43: end if 44: end for 45: volume ⇐ volume+ |rj − ysplit,j |; 46: return volume; Proof. It is clear that (2) ⇒ (4) ⇒ (5). The follows show (1), (2) and (3). (1) (By Contradiction.) Assume to the contrary there is some i ∈ {1, . . . , n},∑d j=1 δij < n − 1. If this is the case, there are at least one yk where k 6= i such that each yij dominates ykj for all j ∈ {1, . . . , d}. It follows that yi dominates yk, which contradicts our assumption that all the elements in {y1, . . . , yn} are non-comparable. (2) Given j, sort all yij where i = 1, . . . , n and label each yij a sequence number I(i) which ranges from 0 to n − 1. Thus i=1 I(i) = (n − 1). There are two cases to consider. Firstly, if all yij are different each other, then δij = I(i). It follows that i=1 δij = (n − 1). Secondly, if there are same elements within {y1j, . . . , ynj}, without loss of generality, suppose yij = ykj and I(k) = I(i) + 1. Then δij = δkj = I(i) < I(k), it follows that∑n i=1 δij < (n− 1). This completes the proof. (3) (By contradiction.) For any yi, j=1 δij < n − 1 is excluded by (1) of this lemma. Thus j=1 δij > n − 1 for some yi is considered. If this is the case, we obtain j=1 δij > n(n − 1), contradicting (2) of this lemma, which implies j=1 δij = i=1 δij ≤ n(n − 1), namely∑n j=1 δij ≤ n(n− 1). Lemma 2. Let ωi(k) be the amount of k in all δij where j = 1, . . . , d, namely ωi(k) = |{j : δij = k, j = 1, . . . , d}|. Then (1) 0 ≤ ωi(k) ≤ d for any i and k; i=1 ωi(k) ≤ d for any k; k=0 kωi(k) ≤ (n− 1) for any i. Proof. By the definition of ωi(k), it is clear that all statements follows Lemma 1. Lemma 3. Let f(n, d) be the runtime of Algorithm 1 to compute a hypervolume with n dominative points in a d-dimensional space. Then (1) f(n, d) + f(m, d) > f(n− 1, d) + f(m+ 1, d) where n−m > 1; (2) f(n, d) > f(m, d) + f(n−m, d) where n > m; (3) f(n, d) is minimal when j=1 δij = n− 1 and |δij − δik| ≤ 1 for any j and (4) f(n, d) is maximal when j=1 δij = (n − 1) for any i and ωi(k) = any i and each k = 0, . . . , n− 1. Proof. (1) and (2) are clear. (3) By the process of Algorithm 1, given some i, f(n, d) = dn logn+ f(δij , d) (1) By (1) of Lemma 1, j=1 δij ≥ n − 1. It is clear that for a given i, it is necessary that j=1 δij = n− 1 to minimize f(n, d). In addition, all the δij must share alike, i.e. |δij − δik| ≤ 1 for any j and k. If this is not the truth, suppose δij − δik > 1. Thus by (1) of this lemma, f(δij , d) + f(δik, d) > f(δij − 1, d) + f(δik + 1, d) (2) Let δij′ = δij − 1 and δik′ = δik + 1. δij′ and δik′ can be modified in the similar way until |δij − δik| ≤ 1. This completes the proof. (4) By (5) of Lemma 1, j=1 δij ≤ (n− 1). It is clear that for a given i, it is necessary that j=1 δij = (n− 1) to maximize f(n, d). Hence Eqn. (1) is written as follows, f(n, d) = dn logn+ ω(k)f(k, d) (3) Suppose yi is the splitting reference point chosen by Algorithm 1, ωi(n−1) ≤ , or else contradicting i=1 ωi(n− 1) ≤ d. To maximize f(n, d) in Eqn. (3), let ωi(n − 1) = . Similarly, we get ωi(n − 2) = , . . ., ωi(1) = , and so on. It is exactly k=0 kωi(k) = (n− 1). This completes the proof. 4.1 Bounds of runtime at special cases First of all, it is clear that f(1, d) = d. By (3) of Lemma 3, the algorithm performs best when each δij shares alike for the chosen i. If d ≥ n− 1, δij ≤ 1 for any j. f(n, d) = dn logn+ (n− 1)f(1, d) (4) which implies f(n, d) = Ω(dn log n). If d < n−1, δij > 1 for any j. In the rough, we get f(n, d) = dn logn+ d · f( , d) < dn logn+ d · f( , d) (5) It can be obtained from Eqn. (5) that f(n, d) = Θ(dn log n logd n) even when f(n−1 , d) is relaxed to f(n , d). Fredman and Weide [11] have shown that Klee’s measure problem has a lower bound of Ω(n logn) for arbitrary d ≥ 1. Just as Beume and Rudolph [8] have mentioned, although it is unknown what the lower bound for calculating the hypervolume is, it is definitely not harder than solving KMP because it is a special case of KMP. Therefore, there is a gap between the lower bound of the proposed algorithm and the actual lower bound of calculating the hypervolume. In the average cases, suppose that for the given splitting reference point yi,∑d j=1 δij = (n− 1). Meanwhile, each δij shares alike, i.e. δij = . Thus, f(n, d) = dn logn+ d · f( , d) < dn logn+ d · f( , d) (6) which implies the runtime of the proposed algorithm is Θ(dnlog d) at the given cases. 4.2 Upper bound of runtime By (2) of Lemma 3, f(n− 1) > f(n− 1− k) + f(k) for any k = 1, . . . , n−2 . And by (4) of Lemma 3, at the worst cases, we have f(n, d) = dn logn+ d (f(n− 1) + f(n− 2) + . . .+ f(2) + f(1)) < dn logn+ d (1 + n−2 )f(n− 1) < dn logn+ d f(n− 1) which implies that the proposed algorithm for computing the hypervolume bounded by n points and a reference point in d-dimensional space has a runtime ofO((d at the worst cases. 4.3 Space complexity Let g(n, d) be the used storage by Algorithm 1. In the proposed algorithm, every child hypervolume is calculated one by one. Since the storage can be reused after the former computation has been completed, g(n, d) is only related to the maximum usage of all the computations of child hypervolumes. Hence, g(n, d) = dn+ max i∈{1,...,n},j∈{1,...,d} {g(δij , d) : 0 ≤ δij ≤ n− 1} (8) Thus the upper bound of space is as follows. g(n, d) = dn+ g(n− 1, d) (9) where g(1) = d. It is easy to obtain an O(dn2) space upper bound for the proposed algorithm. Combining the above analyses together, we obtain the time-space complexity of the proposed algorithm. Theorem 1. The hypervolume of a hyper-cuboid bounded by n non-comparable points and a reference point in d-dimensional space can be computed in time )n) using O(dn2) storage. 5 Conclusions A fast algorithm to calculate the hypervolume indicator of Pareto approxima- tion set is proposed. In the novel algorithm, the hyper-cuboid bounded by non- comparable points and the reference point is partitioned into many child hyper- cuboids along the carefully chosen splitting reference point at each dimension. The proposed approach is very different to the technique used in other works where the whole d-dimensional volume is calculated by computing the (d − 1)- dimensional volume along the dimension d. Such difference results in very differ- ent time bounds, namely O((d )n) for our work and O(n 2 ) for the best previous result. Neither kind of technique can exceed the other completely and each has his strong point. Additionally, the amount of storage used by our algorithm is only O(dn2) even no special technique is developed to reduce the space complex- As the context has mentioned, it is very important to choose appropriate splitting reference point for our algorithm.Well selected point can reduce number of points in separated parts and thus cut down the whole runtime. We do not know whether the strategy adopted in this paper is optimal or near optimal. Further investigations should be worked on. References 1. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Perfor- mance Assessment of Multiobjective Optimizers: An Analysis and Review. IEEE Transactions on Evolutionary Computation 7(2) (2003) 117–132 2. Zitzler, E., Thiele, L.: Multiobjective Optimization Using Evolutionary Algorithms —A Comparative Study. In Eiben, A.E., ed.: Parallel Problem Solving from Nature V, Amsterdam, Springer-Verlag (1998) 292–301 3. Zitzler, E., Thiele, L.: Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach. IEEE Transactions on Evolution- ary Computation 3(4) (1999) 257–271 4. Zitzler, E., Brockhoff, D., Thiele, L.: The hypervolume indicator revisited: On the design of pareto-compliant indicators via weighted integration. In: Proceedings of the 4th International Conference on Evolutionary Multi-Criterion Optimization (EMO 2007). Volume 4403., Springer-Verlag (2007) 862–876 5. While, L., Bradstreet, L., Barone, L., Hingston, P.: Heuristics for optimising the calculation of hypervolume for multi-objective optimisation problems. In: The 2005 IEEE Congress on Evolutionary Computation. Volume 3. (2005) 2225–2232 6. While, L., Hingston, P., Barone, L., Huband, S.: A faster algorithm for calculating hypervolume. IEEE Transactions on Evolutionary Computation 10(1) (2006) 29– 7. Fonseca, C.M., Paquete, L., nez, M.L.I.: An improved dimension-sweep algorithm for the hypervolume indicator. In: IEEE Congress on Evolutionary Computation (CEC 2006). (2006) 1157–1163 8. Beume, N., Rudolph, G.: Faster s-metric calculation by considering dominated hypervolume as klee’s measure problem. In Kovalerchuk, B., ed.: Proceedings of the Second IASTED Conference on Computational Intelligence, Anaheim, ACTA Press (2006) 231–236 9. Edelsbrunner, H., Overmars, M.H.: Batched dynamic solutions to decomposable searching problems. Journal of Algorithms 6(4) (1985) 515–542 10. Overmars, M.H., Yap, C.K.: New upper bounds in klee’s measure problem. SIAM Journal on Computing 20(6) (1991) 1034–1045 11. Fredman, M.L., Weide, B.: The complexity of computing the measure of [ai, bi]. Communications of ACM 21 (1978) 540–544

0704.1197

Statistical properties of giant pulses from the Crab pulsar

Astronomy & Astrophysics manuscript no. Popov2006˙6589 c© ESO 2018 October 23, 2018 Statistical properties of giant pulses from the Crab pulsar M.V. Popov1 and B.Stappers2 1 Astro Space Center of the Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117997 Russia 2 Astronomical Institute ”Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam,The Netherlands; Stichting ASTRON, Postbus 2, 7990 AA, Dwingeloo, The Netherlands Received ; accepted ABSTRACT Aims. We have studied the statistics of giant pulses from the Crab pulsar for the first time with particular reference to their widths. Methods. We have analyzed data collected during 3.5 hours of observations conducted with the Westerbork Synthesis Radio Telescope operated in a tied-array mode at a frequency of 1200 MHz. The PuMa pulsar backend provided voltage recording of X and Y linear polarization states in two conjugate 10 MHz bands. We restricted the time resolution to 4 µs to match the scattering on the interstellar inhomogeneities. Results. In total about 18000 giant pulses (GP) were detected in full intensity with a threshold level of 6σ. We analyzed cumulative probability distribution (CPD) of giant pulse energies for groups of GPs with different effective widths in the range 4 to 65 µs. The CPDs were found to manifest notable differences for the different GP width groups. The slope of a power-law fit to the high-energy portion of the CPD evolves from −1.7 to −3.2 when going from the shortest to the longest GPs. There are breaks in the CPD power- law fits indicating flattening at low energies with indices varying from −1.0 to −1.9 for the short and long GPs, respectively. The GPs with a stronger peak flux density were found to be of shorter duration. We compare our results with previously published data and discuss the importance of these peculiarities in the statistical properties of GPs for the theoretical understanding of the emission mechanism responsible for GP generation. Key words. pulsars: general – pulsars: individual: B0531+21 – Methods: statistical – Radiation mechanisms: non-termal 1. Introduction Giant pulses are one of the most striking phenomena of pulsar radio emission. Their flux density can exceed thou- sands of times the average pulse-integrated flux. Although re- cently there are reports of detecting excessively strong pulses from a number of pulsars (Johnston et al. 2001; Kramer et al. 2002; Romani & Johnston 2001; Johnston & Romani 2002; Kuzmin et al. 2004; Ershov & Kuzmin 2003; Knight et al. 2006), only the Crab pulsar and the millisecond pulsar B1937+21 generate giant pulses numerous enough to study their statistical properties. The very large fluxes of giant pulses are coupled with an extremely short duration. Indeed, the over- whelming majority of giant pulses from the millisecond pulsar B1937+21 are shorter than 15 ns (Soglasnov et al. 2004), while giant pulses from the Crab pulsar have a mean width of about few microseconds (Hankins 2000) with occasional bursts shorter than 2 ns (Hankins et al. 2003). The longitude position of giant pulses is remarkable in that it coincides with the position of high-energy emission (Moffett & Hankins 1996; Cusumano et al. 2003; Johnston et al. 2004; Knight et al. 2006). Furthermore, giant pulses originate in a very narrow phase window that in general does not correspond to the phase window of regular radio emission. In the millisec- ond pulsar B1937+21, giant pulses are observed at the very trail- ing edge of the average profile in both the main pulse and the interpulse (Cognard et al. 1996; Kinkhabwala & Thorsett 2000; Soglasnov et al. 2004). Popov et al. (2006) has recently sug- gested that giant pulses from the Crab pulsar are also seen at the Send offprint requests to: M.V. Popov Correspondence to: [email protected] trailing edge of the regular radio emission window, which they consider to be the precursor. In other words, they consider that radio emission in the main pulse and the interpulse of the Crab pulsar consists completely of giant pulses. Lastly, an essential property of giant pulses, by which they may be distinguished from normal strong pulses, is the distribution of their flux den- sities, which appears to follow a power law in contrast with the Gaussian or exponential flux distribution typical of regular (ordinary) individual pulses (Backer 1971; Hesse & Wielebinski 1974; Ritchings 1976). In this paper we present an analysis of giant pulse-energy distribution based on the observations con- ducted with the Westerbork Synthesis Radio Telescope (WSRT) at 1200 MHz using the PuMa (Voute et al. 2002) pulsar backend. 2. Observations and data reduction The observations were carried out in November 2003 as part of a multi-frequency observing campaign that also included Jodrell Bank at 1420 MHz, Effelsberg at 8350 MHz, Kalyazin at 600 MHz, Pushchino at 111 MHz, and Kharkov (UTR-2) at 23 MHz. Simultaneous optical observations were also made with the 6-m telescope of the Special Astrophysical Observatory and 2.8-m telescope at La Palma. The MAGIC and HESS gamma-ray telescopes in La Palma and Namibiya also partici- pated. While some results obtained at the separate observatories have already been published (Jessner et al. 2005; Popov et al. 2006), the joint analysis of multi-frequency observations will be presented in future publications. We have analyzed the data obtained in observations with the WSRT during about 3.5 hours in two conjugate bands of 10 MHz each at a central frequency of 1197 MHz. Baseband http://arxiv.org/abs/0704.1197v2 2 M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses voltages of X and Y linear polarization states were recorded with two-bit sampling at Nyquist frequency. The coherent predetection dedispersion technique originally developed by Hankins (Hankins 1971; Hankins & Rickett 1975) was used to remove dispersion smearing in the received pulsar signal. We took the value of dispersion measure (56.757 pc cm−3) and the timing model from the Jodrell Bank Monthly Ephemeris (Lyne 1982). The technique provides a formal time resolution of 100 ns, but the expected pulse-broadening time due to interstellar scintillations is in the range 2 to 4 µs, based on the estimation made by Kuzmin et al. (2002). Therefore, we averaged the recorded signal after dispersion removal and square-low detec- tion synchronously with a topocentric pulsar period into 8192 bins per period and with a resulting sampling interval close to 4.1 µs. The total intensity time series was finally formed and used for detection of giant pulses with the threshold level of 6σ. To provide better sensitivity for wider giant pulses, we developed a searching procedure which progressively tried to increase the averaging time interval (τi) by 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, and 48 samples, i.e.from 4.1 µs to 196 µs. The averaging time with the best signal-to-noise ratio (SNR) was selected as We for the GP, detected simultaneously at several averaging times. The pulse energy E (or average flux density) was calculated for the GP as a product of the SNR and the pulse width, equal to the averaging time, which corresponds to the best SNR value. In this approach we have different threshold levels both in peak flux density Fp and in pulse energy E for each separate averaging time. The value of the root-mean-square deviation (RMS or σ) goes down with increasing time averaging as the square root of time. Therefore, the threshold level in peak flux density Fp decreases with increasing averaging time as Fp(τ) ∝ 1/ τ, while the threshold level in pulse energy E increases with τ as E(τ) ∝ For the Crab pulsar, the system temperature is notably influ- enced by the impact of the Crab Nebula, whose flux density can be approximated by the relation Fν = 955ν −0.27 Jy (ν in GHz) (Allen 1973; Bietenholz et al. 1997), which gives F1200 = 909 Jy at 1200 MHz. In our observations we used all 14 dishes of the WSRT in tied-array mode, where all telescopes are added coherently, with the width of synthesized beam being equal to 34 arcsec, thus reducing the contribution from the Crab Nebula to the system noise by a factor fν = ΩA/ΩCN = 0.14 to the value FCN = 127.6 Jy. Here ΩCN is a solid angle of the Crab Nebula, and ΩA a solid angle of the intersection of the synthesized beam with the Crab Nebula. However, there is still the contribution from the individual dishes; the system tem- perature of every 25-m dish was increased by the Crab Nebula emission by ∆TCN = F1200G = 86.3 K, where G is the gain of a single telescope, equal to 0.095 at 1200 MHz. The intrinsic system temperature in the absence of the Crab Nebula is 30 K at 1200 MHz, and the resulting total system temperature is thus 116.3 K, equivalent to Fsys = 87.4 Jy for the gain in the tied array (1.33 K/Jy), which when combined with the nebula con- tribution to the tied array becomes Ftot = Fsys + FCN = 215 Jy. One can see that WSRT reduces the contribution of the Crab Nebula to the system noise considerably when compared with the single dish observations at this frequency. This improvement will permit us to follow the statistics of giant pulses from the Crab pulsar to the lower energies; namely, the limiting RMS (σ) value in one polarization is Ftot/ Bτ, with B = 10 MHz, and τ = 4.1 µs the RMS is equal to 33.5 Jy. For total intensity, the RMS will be reduced by 2, and the threshold of 6σ in peak flux density will be equal to 142 Jy. The threshold will be lower 1000 10000 100000 1e+06 1 10 100 1000 Averaging time We in microseconds Fig. 1. Peak flux density versus the effective puse width We for all detected giant pulses. for averaged records, with a limiting value of 20 Jy for giant pulses, which have a maximum effective width of about 200 µs. In total 17869 giant pulses were detected over 370000 pulse periods with 14994 giant pulses located at the longitude of the main pulse and 2875 giant pulses associated with the interpulse. The background rate was determined by counting events at quiescent phases of the pulsar period and the corresponding values were taken into account in our statistical calculations. In fact, at the lowest range, between 6σ and 7σ, only about 10% of all pulses we potentially incorrectly identified as giant pulses. 3. Results Figure 1 represents the complete set of data showing peak flux density Fp versus the effective pulse width We. The striped na- ture of the diagram is caused by the procedure of giant pulse de- tection, which is based on the discrete values of averaging time We. The diagram demonstrates the peculiarity of the statistics on giant pulses: the strongest pulses clearly have a shorter duration; namely, there are no pulses with peak flux density greater than 1000 Jy that are wider than 16 µs. As a consequence, giant pulses with different durations have different distributions in pulse en- ergy, and the distribution in giant pulse width depends on the range of pulse energy used. Therefore, in our analysis, we sepa- rated the giant pulses belonging to the main pulse longitudes into five groups by their widths: (a) – GPs with the effective width We of 4.1 µs, (b) – GPs with the We of 8.2 µs, (c) – combined group of GPs with the We of 12.3 and 16.4 µs, (d) – combined group of GPs with the We of 24.5 and 32.8 µs, and (e) – combined group of GPs with the We of 49.2 and 65.6 µs. A separate group (f) was formed for the GPs detected at the longitude of the inter- pulse with the We 4.1, 8.2, and 12.3 µs. Then, pulse energy, or integrated flux density, was calculated as E = Fp ×We. In our analysis we use cumulative probability distribution (CPD), which gives the number of pulses N(E), above pulse en- M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses 3 Table 1. The list of parameters of the power-law fits to the CPDs displayed in Fig.2. Group We(µs) γhigh γlow Ebreak(Jy · µs) (a) 4.1 1.7 1.1 4000 (b) 8.2 1.8 1.0 3500 (c) 12.3, 16.4 2.3 1.2 4000 (d) 24.5, 32.8 2.5 1.4 4000 (e) 49.2, 65.6 3.2 1.9 5000 (f) 4.1, 8.2, 12.3 1.6 1.6 ergy E. Some authors prefer to calculate the probability distri- bution (PD) by giving the number of pulses n(E) per interval of energy dE. If a power-law fit is used for the PD function as n(E) ∝ E−β, then a power-law fit will be also valid for the CPD function N(E) ∝ E−γ, since N(E > Ey) = n(E)dE ∝ E−β+1y with γ = β − 1 in the absolute value. Figure 2 displays the CPD of pulse energies for all the afore- mentioned groups of GPs. The CPD for the GPs belonging to the interpulse can be fitted with a power-law function with γ = 1.6 ± 0.1. The CPDs for the GPs in the main pulse manifest a break in the power-law index at certain values of pulse energy, which is slightly different for the different groups. Table 1 contains the results of a least-square solution for the power-law fit. The first column indicates the group of GPs, We is the averaging time for the group in µs, γhigh and γlow are indices of the power law function for the high energy tail and low energy portion of the CPD, respectively, and Ebreak indicates the pulse energy where the slope of the power law changes. The break energy values were obtained as the crossing point between two straight lines defined by the least-square solutions. Relative inaccuracy is within 10% at the RMS level. The absence of any break in the power-law fit for the CPD of the GPs belonging to the interpulse longitudes is good proof that the observed breaks in the CPDs of the main pulse GPs were not caused by selection effects, since we used exactly the same procedure for GPs detection irrespective of pulse longitude. A possible explanation is that we observe the regions where the GPs are generated in the the main pulse and the interpulse at different impact angles relative to the GP beam, which has a half width of about 5 degrees. Indeed, the comparison of the CPDs (combined for the width groups (a), (b), and (c)) for the main pulse and the interpulse shows that the CPDs would be identical if one were to multiply the interpulse GP energies by a factor of about 4 and increase their rate of occurrence by a factor of about 2. Therefore, the weakest detected interpulse GPs with a pulse energy of about 1000 Jy · µs, when observed at the same impact angle as the main pulse GPs (multiplied by 4), will have pulse energies of 4000 Jy · µs, which is above the break point. 4. Discussion and conclusions The CPD function of the GPs originating at the longitudes of the main pulse manifests gradual changes in the power-law in- dex from −1.7 to −3.2 at the high energy part of the CPD, and from −1.0 to −1.9 at the low energy portion of the CPD. As a rough guide, all GPs can be separated into two main groups: short GPs belonging to groups (a) and (b), and long GPs with an We greater than 8.1 µs. It is interesting to note that the CPD power-law index of about −1.7 for short GPs is close to the ex- ponent determined for the CPD of the GPs from the millisec- ond pulsar B1937+21, which was found to be equal to −1.8 at 430 MHz (Cognard et al. 1996). The value was confirmed by Kinkhabwala & Thorsett (2000), and recently estimated by Soglasnov et al. (2004) as −1.4 at 1650 MHz. All GPs from the millisecond pulsar are very short, lasting only 1-2 µs as mea- sured by Kinkhabwala & Thorsett (2000), while Soglasnov et al. (2004) have found that the majority of GPs from this pulsar are shorter than 15 ns. Argyle & Gower (1972) were the first to present the CPD for GPs from the Crab pulsar. They combined the results of observa- tions made at 146 MHz with the 26-m Penticton radio telescope and with the 46-m dish at the Algonquin Radio Observatory and found that the CPD was consistent with the power-law exponent of −2.5 for the main pulse and with the exponent for the inter- pulse events equal to −2.8. Recently Cordes et al. (2004) in their multifrequency study of the Crab pulsar’s giant pulses presented histograms of GP peak amplitudes (S) at 0.43 and 8.8 GHz. The histograms represent the PD functions, i.e. number of events per interval of SNR in logarithmic binning d(lg S ). They found power-law segments in the distributions with the slopes −2.3 and −2.9 at 0.43 and 8.8 GHz, respectively. To compare this result with our study of the CPD functions one has to convert the number of events pre- sented by Cordes et al. (2004) from logarithmic binning to linear binning d(lg S ) = ln 10dS/S . Then, for a power-law fit with lin- ear binning the slope will change from −β to −(β + 1): N(S ) ∝ (S )−βd(lg S ) ∝ (S )−(β+1)dS for the power-law fit of the CPD γ = β − 1, as was explained in Sect. 3. Thus, the values of the power-law exponent β found by Cordes et al. (2004) in their histograms may by immediately compared with our values of the exponents γ for the power-law fit of the CPDs. In fact the inspection of the histogram of Cordes et al. (2004) for 0.43 GHz has enabled us to distinguish a break at an S/N value of about 30 where the slope changes from −2.3 to −0.7 in the peak amplitude distribution of the main pulse GPs, while the distribution for the interpulse GPs does not manifest such a break. With the given equivalent flux system temperature of 1262 Jy, the receiver band of 12.5 MHz, and a sample interval of 128 µs, the S/N = 30 break point at 0.43 GHz will correspond to a pulse energy of about 120000 Jy · µs. In his PhD thesis Moffett (1997) presentes the CPD functions at 1.4 GHz based on observations conducted with the VLA with a time resolution of 160 µs. In Fig. 4.3 (page 62) Moffett distinguishes a break in power-law fit of the CPD for the main pulse GPs at a level of 12 Jy corresponding to a pulse energy of about 2000 Jy · µs, where the slope changes from −3.0 to −1.8 when going from the high flux densities to the lower. Again, the CPD for the GPs originated at the longitudes of the interpulse (Figure 4.4) does not show a break going straight with the slope of −1.7 until it merges with noise at a level of about 6 Jy. Comparing the values of break-point pulse energy (BPPE) 120000 Jy · µs at a frequency of 0.43 GHz (Cordes et al.(2004)), 5000 Jy · µs at a frequency of 1200 MHz (this paper), and 2000 Jy · µs at a frequency of 1400 MHz (Moffett (1997)), we have found that the BPPE follows a simple power-law frequency dependence BPPE(ν) ≈ 7(ν)−3.4kJy · µs, with the exponent −3.4 close to the mean spectral index for the main pulse component of the average profile −3.0 (Moffett 1997). The conclusion can be considered as support for the suggestion 4 M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses 1000 10000 1000 10000 Pulse energy (Jy*µs) 1000 10000 1000 10000 Pulse energy (Jy*µs) 1000 10000 1000 10000 Pulse energy (Jy*µs) 1000 10000 1000 10000 Pulse energy (Jy*µs) 1000 10000 1000 10000 Pulse energy (Jy*µs) 1000 10000 1000 10000 Pulse energy (Jy*µs) Fig. 2. The CPD of giant pulses with pulse energy above the displayed value for different groups of giant pulses classified by their effective width We as described in the Table 1. Straight lines represent the fit by power-law functions with the parameters indicated in the Table 1. The last plot (f) represents giant pulses belonging to the interpulse. that all emission in the main component consists entirely of giant pulses (Popov et al. 2006). Lundgren et al. (1995) has collected about 30000 GPs from the Crab pulsar at 812 MHz with the Green Bank 43-m radio telescope in 10 days of observations simultaneous with the Compton Gamma Ray Observatory (CGRO) in May 1991. They did not distinguish GPs belonging to the main pulse and the interpulse. Their flux-density distribution was fitted with a power-law function for S > 200 Jy, and the exponent was found to be equal to −3.46, the exponent being equal to −2.46 for the CPD used in our analysis. The BPPE value is expected to be 14 kJy · µs at 812 MHz according to the frequency relation we derive above. The value corresponds to 50 Jy flux density with 307.5 µs averaging time used by (Lundgren et al. 1995), and it is well below their threshold of 120 Jy. The dramatic roll-off at the rather high pulse energy of about 200 Jy × 300µs = 60kJy · µs found by (Lundgren et al. 1995) was not observed in the CPDs at 430, 1200, and 1400 MHz discussed above. It was not observed in the CPD at 600 MHz either, which goes straight with the slope of −2.2 at least down to a pulse energy of about M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses 5 30 kJy · µs (Popov et al. 2006). Therefore, it is difficult to reconcile their results with those from the many independent data sets mentioned above. The power-law form of the observed CPDs can be compared with the field statistics of possible emission mechanisms responsible for the generation of GPs, as discussed by Cairns (2004) who used the values of the exponents of PD functions for comparison. The very short-time flux density variations are of particular interest, since they are closely tied to the physics of the emission process. The power-law indices γ of the CPDs at high energies for the shortest GPs are similar enough for both the Crab pulsar and the PSR B1937+21, and the values were found to be in the range 1.4 to 1.8. Converting to indices of the probability distribution function (PD) β = γ + 1 gives the range 2.4 – 2.8 for the values of β. According to the normalization conditions used by Cairns, the exponent α of the PD of the field (P(E) ∝ E−α) is connected with the exponent β of the observed PD power-law fit by the relation α = 2β − 1, giving us the range of α from 3.8 to 4.6 to be compared with the theoretical predictions of the field statistics determined by the emission mechanism and propagation effects. The observed breaks in the slope of the CPD functions at certain energies have to be included in theoretical explanations, giving us an extra parameter to constrain the source physics and emission mechanism. The break in the slope of the CPD functions has important con- sequences for the estimation of the total rate of GP generation. Popov et al. (2006) made such an estimation for the Crab pulsar under the suggestion that radio emission in the main pulse and in the interpulse consists entirely of giant pulses. They found that about 10 giant pulses are generated during one rotation period of the neutron star. In the estimation they considered that pulse energies of the GPs follow a power-law function with the exponent of −2.2 down to the threshold of about 100 Jy in the peak flux density at a frequency of 600 MHz, and with the threshold considered as a real lower limit equivalent to the minimum pulse energy of about 5000 Jy · µs. With their threshold of about 20000 Jy · µs in GP detection, they did not notice the break in the CPD slope. The break from −2.2 at high energies to −1.2 at low energies will notably change the estimate of the lower limit for GP energies to about 1000 Jy · µs, and the rate of GP generation will increase at least by a factor of 2. The lower energy limit for GP generation, if it exists, will serve as a crucial constraint on the physics of the emission mechanism. To solve the problem, it is necessary to test the low-intensity portion of the CPD function with better sensitivity. Such a study would be possible using VLA or GMRT observations in phased array mode, thereby, significantly reducing the impact from the Crab Nebula. A notable increase in the recording band from 10 to 160 MHz recently achieved for the PuMa II recording systems also makes new observations with the WSRT very promising. Finally, we summarize the main results of our analysis: 1) The CPDs were found to be notably different for the GPs detected at the longitudes of the main pulse and the interpulse. We suppose that the difference can be explained by the simple attenuation caused by a beaming factor. 2) For the main pulse longitudes, the CPD are different for the GPs of different effective widths with breaks in the CPD power-law indices indicating steepening at high energies (see Table 1). 3) The CPD power-law indices (γhigh ≈ 1.7) for the group of short GPs for the Crab pulsar (We < 10µs) are close to the value observed for the millisecond pulsar B1937+21, which seems to generate only the shortest GPs; 4) GPs with a stronger peak flux density were found to be of shorter duration. The last of these properties testifies in favor of nonlinear temporal models that suggest that the higher the intensity, the narrower the pulse width. Such an emission model was considered, for example, by Mikhailovskii et al. (1985), who treated the micropulses as solitons of the radio-wave envelope propagating through the magnetospheric plasma of the pulsar. Acknowledgements. This investigation was supported in part by the Russian Foundation for Fundamental Research (project number 04-02-16384). References Allen, C. W. 1973, Astrophysical quantities (University of London, Athlone Press) Argyle, E. & Gower, J. F. R. 1972, ApJ, 175, L89 Backer, D. C. 1971, PhD thesis, Cornell Univ., Ithaca, NY Bietenholz, M. F., Kassim, N., Frail, D. A., et al. 1997, ApJ, 490, 291 Cairns, I. H. 2004, ApJ, 610, 948 Cognard, I., Shrauner, J. A., Taylor, J. H., & Thorsett, S. E. 1996, ApJ, 457, L81 Cordes, J. M., Bhat, N. D. R., Hankins, T. H., McLaughlin, M. A., & Kern, J. 2004, ApJ, 612, 375 Cusumano, G., Hermsen, W., Kramer, M., et al. 2003, A&A, 410, L9 Ershov, A. A. & Kuzmin, A. D. 2003, AstL, 29, 91, transl. from: PAZh, 2003, 29, 111 Hankins, T. H. 1971, ApJ, 169, 487 Hankins, T. H. 2000, in ASP Conf. Ser. 202, Pulsar Astronomy – 2000 and be- yond, ed. M. Kramer, N. Wex, & R. Wielebinski (San Francisco: ASP), 165 Hankins, T. H., Kern, J. S., Weatherall, J. C., & Eilek, J. A. 2003, Nature, 422, Hankins, T. H. & Rickett, B. J. 1975, in Methods in Computational Physics: Advances in Research and Applications, Vol. 14: Radio Astronomy, ed. B. Alder, S. Fernbach, & M. Rotenberg (New York: Academic Press, Inc.), Hesse, K. H. & Wielebinski, R. 1974, A&A, 31, 409 Jessner, A., Slowikowska, A., Klein, B., et al. 2005, AdSpR, 35, 1166 Johnston, S. & Romani, R. W. 2002, MNRAS, 332, 109 Johnston, S., Romani, R. W., Roger, W., Marshall, F. E., & Zhang, W. 2004, MNRAS, 355, 31 Johnston, S., Van Straten, W., Kramer, M., & Bailes, M. 2001, ApJ, 549, L101 Kinkhabwala, A. & Thorsett, S. E. 2000, ApJ, 535, 365 Knight, H. S., Bailes, M., Manchester, R. N., Ord, S. M., & Jacoby, B. A. 2006, ApJ, 640, 941 Kramer, M., Johnston, S., & Van Straten, W. 2002, MNRAS, 334, 523 Kuzmin, A. D., Ershov, A. A., & Losovsky, B. Y. 2004, AstL, 30, 247, transl. from: PAZh, 2004, 30, 285 Kuzmin, A. D., Kondrat’ev, V. I., Kostyuk, S. V., et al. 2002, AstL, 28, 251, transl. from: PAZh, 2004, 28, 292 Lundgren, S. C., Cordes, J. M., Ulmer, M., et al. 1995, ApJ, 453, 433 Lyne, A. G. 1982, Jodrell Bank Crab Pulsar Monthly Ephemeris (http://www.jb.man.ac.uk/ pulsar/crab.html) Mikhailovskii, A. B., Onishchenko, O., & Smolyakov, A. I. 1985, SvAL, 11, 78, transl. from: PAZh, 1984, 11, 190 Moffett, D. A. 1997, PhD thesis, New Mexico Inst. Mining and Technology Moffett, D. A. & Hankins, T. H. 1996, ApJ, 468, 779 Popov, M. V., Soglasnov, V. A., Kondratiev, V. I., et al. 2006, ARep, 83, 660, transl. from: AZh, 2006, 83, 717 Ritchings, R. T. 1976, MNRAS, 176, 249 Romani, R. W. & Johnston, S. 2001, ApJ, 557, L93 Soglasnov, V. A., Popov, M. V., Bartel, N., et al. 2004, ApJ, 616, 439 Voute, J. L. L., Kouwenhoven, M. L. A., Langerak, J. J., et al. 2002, A&A, 385, Introduction Observations and data reduction Results Discussion and conclusions

0704.1198

A Doubly Distributed Genetic Algorithm for Network Coding

A Doubly Distributed Genetic Algorithm for Network Coding Minkyu Kim∗, Varun Aggarwal†, Una-May O’Reilly†, Muriel Médard∗ ∗Laboratory for Information and Decision Systems †Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, MA 02139, USA {minkyu@, varun ag@, unamay@csail., medard@}mit.edu ABSTRACT We present a genetic algorithm which is distributed in two novel ways: along genotype and temporal axes. Our algo- rithm first distributes, for every member of the population, a subset of the genotype to each network node, rather than a subset of the population to each. This genotype distri- bution is shown to offer a significant gain in running time. Then, for efficient use of the computational resources in the network, our algorithm divides the candidate solutions into pipelined sets and thus the distribution is in the temporal domain, rather that in the spatial domain. This temporal distribution may lead to temporal inconsistency in selection and replacement, however our experiments yield better effi- ciency in terms of the time to convergence without incurring significant penalties. Categories and Subject Descriptors: C.2.1 [Computer- Communication Networks]: Network Architecture and De- General Terms: Algorithms Keywords: Distributed genetic algorithm, network coding, optimization 1. INTRODUCTION We present a GA which is distributed in two novel ways: along genotype and temporal axes. In contrast to a con- ventional GA spatially distributed on the population axis, our doubly distributed algorithm first distributes, for every member of the population, a subset of the genotype to each network node rather than a subset of the population to each. The motivation for this genotype axis of distribution is to distribute the fitness evaluation steps of the Network Cod- ing GA (NCGA) [8] which relies on network codes generated randomly and in a decentralized manner. Self-referentially, the GA solving the network coding problem must be em- bedded in the same network for which it is searching for the Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. GECCO’07, July 7–11, 2007, London, England, United Kingdom. Copyright 2007 ACM 978-1-59593-697-4/07/0007 ...$5.00. optimal coding. With just this axis of distribution, the dis- tributed NCGA equals the performance of the (centralized) NCGA in terms of solution quality. However, as experiments herein suggest, it can lead to a significant gain in running time. The motivation for the second axis of distribution is to maximize the efficient use of the computational nodes in the network by minimizing their idle duration during the GA search. Along this second, temporal axis of distribution, successive sets of candidate solutions are pipelined through the network, from source to sinks and back. A time lag is incurred as the selected candidate travels through the net- work to undergo variation and fitness evaluation before it is inserted back into the population. This creates an age gap between the population from which a candidate solu- tion is selected and the population into which it is inserted and leads to the question of how selection and replacement in the doubly distributed GA should proceed. The ap- proach that is least efficient in terms of time, treats multiple pipelined sets of candidates as components of a single pop- ulation that proceeds in an age-synchronized, generational style for selection and replacement. It sends pipelined sets of selected candidates through the network but waits un- til every set has emerged back out before replacing any of them. We show that this approach, which we term “Gen- erational/Single Population,” incurs a cost of priming and flushing the pipeline but is faster than not pipelining at all. To avoid intermittently flushing the pipeline and then needing to prime it again, our first approach is to divide the population into a number of subpopulations and insert se- lected then genetically varied individuals back into the same subpopulation they were selected from. Migration between sub-populations occurs at some specified frequency regard- less of a slight age difference which maintains close tempo- ral consistency. We call this approach “Generational/Multi- population.” Alternatively, we can be intentionally “sloppy” and forgo any temporal consistency. Much like a steady state GA, a single population is steadily updated. However, unlike a steady state GA, regardless of the time gap (between when a candidate is selected, genetically varied, then evaluated for fitness and when an attempt is made to insert it into the population), insertion simply proceeds with the cur- rent population as new candidates emerge processed from the network. In addition to yielding a simple algorithm, the “temporally sloppy” approach crudely approximates the asynchronously timed selection, reproduction and replace- http://arxiv.org/abs/0704.1198v1 ment events of a naturally evolving population. We dub this “Non-generational/Single population.” Pipelining increases the number of evaluations per time unit. The Generational/Multi-population and Generational/ Single population approaches are constrained to respect age synchrony between selection and replacement. But the Non- generational/Single population approach does not and, there- fore, will have different and as yet unexplored dynamics. Will it converge with more or less fitness evaluations? Does the efficiency of pipelining produce a faster time to con- vergence? Will it find quality solutions? We explore these questions in the experiments. Though the proposed algorithm is discussed in the context of network coding, the contributions of this paper are not limited within that scope. 1) A genetic algorithm with the proposed two novel methods of distribution can be readily applied to a variety of other optimization scenarios arising in communication networks (e.g., routing, resource alloca- tion, etc.) or other connected systems where local decision variables are to be specified for the optimal performance of the whole system. 2) Furthermore, the proposed frame- work of temporal axis distribution can be combined with, not only the pipelining methods considered in this paper, a fairly general class of state-of-the-art strategies for paral- lel management of populations and communication between populations (e.g., [2, 14]), because it imposes essentially no constraint on the implementation of any such strategies ex- cept that there is slight temporal inconsistency between pop- ulations, which as shown in this paper may also have little effect to other strategies. The rest of the paper is organized as follows. Section 2 describes and formulates the network coding problem. Sec- tion 3 describes the NCGA which serves as a baseline. Sec- tion 4 motivates and describes distributing the NCGA along the genotype axis. Section 5 motivates the distribution along the temporal axis and describes three pipelined ap- proaches. Section 6 experimentally quantifies the advan- tage of distribution on the genotype axis and compares the pipelined approaches. Section 7 concludes. 2. NETWORK CODING Network coding is a novel technique that generalizes rout- ing. In traditional routing, each interior network node, which is not a source or sink node, simply forwards the received data or sends out multiple copies of it. In contrast, net- work coding allows interior network nodes to perform arbi- trary mathematical operations, e.g., summation or subtrac- tion, to combine the data received from different links. It is well known that network throughput can be significantly increased by network coding [1, 12]. While network coding is assumed to be done at all possible nodes in most of the network coding literature, it is often the case that network coding is required only at a subset of nodes to achieve the desired throughput. Consider Example 1: Example 1. In the canonical example of network B (Fig- ure 1(a)) [1], where each link has unit capacity, source s can send 2 units of data simultaneously to sinks t1 and t2, which is not possible with routing alone. But only node z needs to combine its two inputs while all other nodes perform routing only. If we suppose that link (z, w) in network B has capac- ity 2, which we represent by two parallel unit-capacity links in network B′ (Figure 1(b)), a multicast of rate 2 is possible without network coding. In network C (Figure 1(c)), where node s is to transmit data at rate 2 to the 3 leaf nodes, net- work coding is required either at node a or at node b, but not at both. � a a!b b a!b a!b (a) Network B a ba b (b) Network B′ a!b a!b (c) Network C Figure 1: Sample networks for Example 1 Example 1 leads us to the following question: To achieve the desired throughput, at which nodes does network coding need to occur? The problem of determining a minimal set of nodes where coding is required is NP-hard; its decision problem, which decides whether the given multicast rate is achievable without coding, reduces to a multiple Steiner subgraph problem, which is NP-hard [13]. For a GA, the problem can be posed as the minimization of coding cost (in links or nodes) subject to the constraint of feasibility (achieving the desired throughput). 2.1 Problem Formulation We assume that a network is given by a directed multi- graph G = (V,E) as in [10] where each link has a unit capacity whose unit can be arbitrarily chosen, e.g., P bits per second for a constant P , or a fixed size packet per unit time, etc. Links with larger capacities are represented by multiple links. Only integer flows are allowed, hence there is either no flow or a unit rate of flow on each link. We con- sider the single multicast scenario in which a single source s ∈ V wishes to transmit data at rate R to a set T ⊂ V of sink nodes. Rate R is said to be achievable if there exists a transmission scheme that enables all |T | sinks to receive all of the information sent. We only consider linear coding, where a node’s output on an outgoing link is a linear combi- nation of the inputs from its incoming links. Linear coding is sufficient for multicast [12]. Given an achievable rate R, we wish to determine a mini- mal set of nodes where coding is required in order to achieve this rate. However, whether coding is necessary at a node is determined by whether coding is necessary at at least one of the node’s outgoing links and thus, as pointed out also in [11], the number of coding links is in fact a more accurate estimator of the amount of computation incurred by coding. We assume hereafter that our objective is to minimize the number of coding links rather than nodes. It is clear that no coding is required at a node with only a single input since these nodes have nothing to combine with [8]. For a node with multiple incoming links, which we refer to as a merging node, if the linearly coded output to a particular outgoing link weights all but one incoming message by zero, effectively no coding occurs on that link; even if the only nonzero coefficient is not identity, there is another coding scheme that replaces the coefficient by iden- tity [11]. Thus, to determine whether coding is necessary at an outgoing link of a merging node, we need to verify whether we can constrain the output of the link to depend on a single input without destroying the achievability of the given rate. As in network C of Example 1, the necessity of coding at a link depends on which other links code and thus the problem of deciding where to perform network cod- ing in general involves a selection out of exponentially many possible choices. Employing a GA-based search method effi- ciently addresses the large and exponentially scaling size of the space. 3. NETWORK CODING GA ("A") In the network research community, [8] and [9] have doc- umented results that demonstrate the benefit of the NCGA over other existing approaches in terms of reducing the num- ber of coding links or nodes and its applicability to a variety of generalized scenarios. In the GA community, [7] has in- vestigated two different genotype encodings1 and associated operators. Reference [7]’s main finding is that the encoding and the genetic operators that respect the block structure of the problem, which will be detailed later, substantially outperforms those do not. It is also claimed that such supe- rior performance is mainly due to the modularity enforced by the block-wise genetic operators. We first describe the elements of the NCGA that uses a standard generation-based GA control loop with centralized operations. This centralized NCGA, which we refer to as “Algorithm A,” serves as a baseline approach in compar- ison with the distributed versions of the algorithm, which share the GA elements introduced in this section. 3.1 Genotype Encoding Suppose a merging node with k(≥ 2) incoming links. To consider the transmission to each of its outgoing links, we as- sign a binary variable to each of its k incoming links, whose being 1 indicates that the link state is active (the input from the associated incoming link is transmitted to the outgoing link) and 0 indicates it is inactive. Given that network cod- ing is required for the transmission only if two or more link states are active, we may need to consider those k variables together. We refer to the set of the k variables as a block of length k (see Figure 2 for an example). x x x y y (a) Merging node v x x x x x x       block for y block for y (b) Two blocks for outgoing links of v Figure 2: Node v with 3 incoming and 2 outgoing links results in 2 blocks, each with 3 variables in- dicating the states of incoming links (x1, x2, x3) onto the associated outgoing link. We notice that once a block has at least two 1’s, coding is already required on the outgoing link associated with that block, and thus replacing all the remaining 0’s with 1’s has 1To minimize confusion, throughout the paper, the term “encoding” refers to “genotype encoding” only, while the term “coding” means “network coding.” no effect on whether coding is done. Moreover, it can be shown that substituting 0 with 1, as opposed to substitut- ing 1 with 0, does not hurt the feasibility. Therefore, for a feasible genotype (which is defined below), any block with two or more 1’s can be treated the same as the block with all 1’s. Thus we could group all the states with two or more active links into a single state, coded transmission. This state is rounded out by k states for the uncoded transmis- sions of the input received from one of the k single incom- ing links and one state indicating no transmission. Thus, each block of length k can only take one of the following (k + 2) strings: “111...1”, “100...0”, “010...0”, “001...0”, ..., “000...1”, “000...0”. If we denote by dvin and d out the in- degree and the out-degree of node v, node v has dvout blocks of length dvin, and thus we have the search space of size (dvin + 2) out , where V is the set of all merging nodes. 3.2 Constraint and Fitness Function A genotype is called feasible if there exists a network cod- ing scheme that achieves the given rate R with the link states determined by the genotype. For the feasibility test of a genotype, we rely on the algebraic method described in [9], which later enables a distributed feasibility test. Given the feasibility of genotype y, its fitness value F is assigned as F (y) = number of coding links, if y is feasible, ∞, if y is infeasible, where the number of coding links can be easily calculated by counting the number of blocks in the genotype with at least two 1’s. 3.3 Genetic Operators To preserve the above encoding structure, we need to de- fine a new set of genetic operators, which we refer to as block- wise genetic operators. For block-wise uniform crossover, we let two genotypes subject to crossover exchange each block, rather than bit, independently with the given crossover prob- ability. For block-wise mutation, we let each block under mutation take another string chosen uniformly at random out of (k + 1) other strings for a length-k block. 3.4 Other Elements The NCGA evaluates fitness in a multi-step way: 1) each merging node consults the corresponding genotype blocks to compute random linear combinations of the inputs2, 2) alternately routed messages reach the sinks, 3) the feasibility of the genotype is assessed at the sinks, 4) if feasible, the coding links are counted. The NCGA uses tournament selection and terminates at some maximum number of generations. Afterward, the best solution of the run is optimized with greedy sweep: each of the remaining 1’s is switched to 0 if it can be done without violating feasibility. This procedure can only improve the solution, and sometimes the improvement can be substantial 4. GENOTYPE AXIS DISTRIBUTION ("B") Decentralizing the NCGA enables a network coding proto- col where the resources used for coding are optimized on the 2See [6] for an explanation of why this is sufficient. fly in a setup phase. Plus, distribution reduces the computa- tional efficiency of the algebraic feasibility test (see Section 4.4 for details). We refer to this genotype(only)-distributed NCGA as “Algorithm B.” 4.1 Overview Because of the way network coding depends on each merg- ing node contributing to the coding, and because each merg- ing node references its corresponding block on a genotype, the appropriate way to distribute the NCGA is to have each node handle only the blocks it needs from every member of the population. So, instead of dividing up the popula- tion and giving each island a subset of genotypes, we divide up the genotype of every population member and give each merging node a population wide set of that genotype sub- set. Thus, in contrast to a conventional distributed GA, the axis of distribution is genotype rather than population as illustrated in Figure 3. The previously centralized fitness evaluation steps are transformed into: 1) forward evaluation stage from merging nodes to each sink 2) backward evalua- tion stage from sinks to source and 3) fitness calculation at the source. With some amount of additional message infor- mation and coordination, all genetic operations can be done locally at each merging node. See Figure 4 for the overall structure.       ...       ...       ...       ... : : : : : : :: Genotypes Each block indicates transmission state of an outgoing link Each set of blocks determines local operations at a node, thus can be managed locally at that node Figure 3: Structure of Population [P1] preliminary processing; (all nodes) [P2] initialize population; (merging nodes) [P3] run forward evaluation phase; (all nodes) [P4] run backward evaluation phase; (all nodes) [P5] calculate fitness; (source) [P6] while termination criterion not reached (source) { [P7] calculate coordination vector; (source) [P8] run forward evaluation phase; (all nodes) [P9] perform selection, crossover, mutation; (merging nodes) [P10] run backward evaluation phase; (all nodes) [P11] calculate fitness; (source) } [P12] perform greedy sweep; (all nodes) Figure 4: Flow of Genotype-Distributed NCGA 4.2 Assumptions While we assume that each link can transmit one packet with the fixed size, say P bits, per time unit in the given direction, each link is also assumed to be able to send some amount of feedback data, typically much smaller than the packet size, in the reverse direction. Also, we assume that each interior node operates in a burst-oriented mode; i.e., for the forward (backward) evaluation phase, each node starts updating its output only after an updated input has been received from all incoming (outgoing) links. 4.3 Details of Genotype-Distributed Algorithm 4.3.1 Preliminary Processing [P1] The source initiates the algorithm by transmitting the “optimize” signal containing the following predetermined parameters: target multicast rate R, population size N , the size q of the finite field to be used, crossover probability, and mutation rate. Each participating node that has received the signal passes the signal to its downstream nodes. 4.3.2 Population Initialization [P2] Each merging node with din(≥ 2) incoming links will man- age a coding vector indicating the link states per population member. To initialize its subset of the population, each merging node generates N · din · dout binary numbers ran- domly. Then, for the coding vectors corresponding to the first of the N chromosomes, all the components are set to 1 [8]. 4.3.3 Forward Evaluation Phase [P3, P8] For the feasibility test of a chromosome, each node trans- mits a vector consisting of R components, which we refer to as a pilot vector. Each of its the components is from the finite field Fq and the i-th component represents the coeffi- cient used to encode the i-th source data. We assume that a set of N pilot vectors is transmitted together by a single packet. The source initiates the forward evaluation phase by send- ing out on each of its outgoing links a set of N random pilot vectors. Each non-merging node simply forwards all the pi- lot vectors received from its incoming link to all its outgoing links. Each merging node transmits on each of its outgoing links a random linear combination of the received pilot vectors, computed based on the node’s coding vectors as follows. Let us consider a particular outgoing link and denote the associated din coding vectors by v1, v2, ..., vdin . For the i-th (1 ≤ i ≤ N) output pilot vector ui, we denote the i-th input pilot vectors received form the incoming links by w1, w2, ..., wdin . Define the set J of indices as J = {1 ≤ j ≤ din| the i-th component of vj is 1}. Then, wj · rand(Fq), where rand(Fq) denotes a random element from Fq. If the set J is empty, ui is assumed to be zero. 4.3.4 Backward Evaluation Phase [P4, P10] To calculate a chromosome’s fitness value, two kinds of information need to be gathered: 1) whether each sink can decode data of rate R and 2) how many links are used for coding at each merging node. Each sink can determine whether data of rate R is de- codable for each of the N chromosomes by computing the rank of the collection of received pilot vectors. It is worth to point out that this is the same algebraic evaluation method described in [8], but the difference is that, rather than com- puting the system matrix with randomized elements cen- trally, now we actually construct random linear codes over the network in a decentralized fashion. Hence, this feasi- bility test also bears the same, but uncritical, possibility of errors as in the centralized case. Regarding the number of coding links, each merging node can simply count the num- ber links where coding is required by inspecting its coding vectors used in the forward evaluation phase. For the feedback of this information, each node transmits a vector consisting of N components, which is referred to as a fitness vector. The backward evaluation phase proceeds as follows: • After the feasibility tests of the N chromosomes are done, each sink generates a fitness vector whose i-th (1 ≤ i ≤ N) component is zero if the i-th chromosome is feasible at the sink, and infinity otherwise. Each sink then initiates the backward evaluation phase by transmitting its fitness vector to all of its parents. • Each interior node calculates its own fitness vector whose i-th (1 ≤ i ≤ N) component is the number of coding links at the node for the i-th chromosome plus the sum of all the i-th components of the received fitness vectors. Each node then transmits the calcu- lated fitness vector to only one of its parents, and an all-zero fitness vector (for just signaling) to the other parent nodes. Note that, since the network is assumed to be acyclic, each coding link of a chromosome contributes exactly once to the corresponding component of the source node’s fitness vector, and thus the above update procedure provides the source with the correct total number of coding links. 4.3.5 Fitness Calculation [P5, P11] The source calculates the fitness values of N chromosomes simply by component-wise summation of the received fitness vectors. Note that if an infinity were generated by any of the sinks, it should dominate the summations all the way up to the source, and thus the source can calculate the correct fitness value for the infeasible chromosome. 4.3.6 Termination Criterion [P6] The source can determine when to terminate the optimiza- tion by counting the number of generations iterated thus far. 4.3.7 Coordination Vector Calculation [P7] Since the population is divided into subsets that are man- aged at the merging nodes, genetic operations also need to be done locally at the merging nodes. However, some amount of coordination is required for consistent genetic operations throughout all the merging nodes, more specif- ically, for 1) selection of chromosomes, 2) paring of chro- mosomes for crossover, and 3) whether each pair is subject to crossover. This information is carried by a coordination vector, calculated at the source, consisting of the indices of selected chromosomes that are randomly paired and 1-bit data for each pair indicating whether the pair needs to be crossed over. The coordination vector is transmitted together with the pilot vectors in the next forward evaluation phase. 4.3.8 Genetic Operations [P8] Based on the received coordination vector, each merging node can locally perform genetic operations and renew its portion of the population as follows: • For selection, each node only retains the coding vec- tors that correspond to the indices of selected chromo- somes. • For block-wise crossover, each node independently de- termines whether each block is crossed over. Since no block is shared by multiple merging nodes, this can be done independently at each merging node. • For block-wise mutation, each node independently de- termines whether each block is mutated without any coordination with other nodes either. 4.3.9 Greedy Sweep [P12] Greedy sweep requires an additional protocol where, after the iteration terminates, the source is notified of the merg- ing nodes with at least one coding link, for each of which the source sends out a packet to test if uncoded transmission is possible on the link(s) where currently coding is required. Since this additional protocol requires more extensive coor- dination between nodes, we may leave this procedure op- tional, whose detailed description is omitted owing to space limitations. 4.4 Complexity The computational complexity required for evaluation of a single chromosome is O( dvind outR+ w∈V \V dwout + t∈T d R), which can be substantially less than that for the centralized version of the algorithm, i.e., O(|T |·(|E|2.376+ R3)) or O(|T | · (|E′|2 |V ′|)) [9]. 5. TEMPORAL AXIS DISTRIBUTION A unique characteristic of the genotype-distributed NCGA is that once each generation is initiated at the source (pro- cedure [P7] in Figure 4), the fitness values of N genotypes become only available after the forward and backward eval- uation phases are done, i.e., when the last fitness vector arrives at the source. Let us assume that the time required for each node to calculate its outgoing pilot vectors based on the received ones is negligible compared with the time required for packet transmissions. Then, if we denote by l the length of the longest path from the source to any of the sinks, the time lag between the initiation of the generation and the termination of the backward evaluation phase is 2l time units (see Figure 5(a)). Let us now define the evaluation efficiency, which we de- note by εv, as the number of fitness evaluations performed per unit time throughout the iteration of the GA. Then, for Algorithm B(genotype(only)-distributed NCGA), εv is only N/2l. For better efficiency, we may still utilize the network re- sources, while waiting for the fitness vectors to return to the source, to evaluate more genotypes. Suppose that, after initiating the forward evaluation phase of the n-th genera- tion at time t, we initiate additional k−1 forward evaluation phases at times t+1, ..., t+k−1. When k = 2l, the network resources become fully utilized by the time when the fitness values of the first set of N genotypes are available. Note that in fact k may even exceed 2l, but then the evaluation of the (n + 1)-th generation starts delayed at time t + k, rather than t + 2l. For simplicity, we assume k ≤ 2l in the following. Network ...0 1 2 2l 2l+1 2l+2 2l+k-1 4l 4l+1 4l+2k-1 ... ... ... Set # Pilot Vectors Fitness Vector Coordination Vector Pilot Vectors ... 4l+k-1 ... (a) Timing Diagram of Algorithm B: Genotype(only)-distributed NCGA. ...0 1 2 2l 2l+1 2l+2 2l+k-1 4l 4l+1 4l+2k-1 ... ... ... ... ... Set # Network ... 4l+k-1 ... Pilot Vectors Fitness Vector Coordination Vector Pilot Vectors (b) Timing Diagram of Algorithm D: This doubly distributed algorithm (Generational/ Multi-population) exploits pipelining, does not require intermittent flushing, respects age consistency between selected and replecement, and respects close age consistency in mi- gration. Figure 5: Comparison of Algorithms B and D via Timing Diagrams. 5.1 Generational / Single Population ("C") If we consider the k sets of N genotypes as a single popu- lation, we have to wait additional k−1 time units, after the first backward evaluation phase ends (at time t+2l), to pro- ceed to the next generation. In other words, we must flush the pipeline (and prime it again). Hence, the evaluation efficiency is given by 2l + k − 1 whose maximum is obtained when k = 2l such that εv = . For later comparison, we refer to this algorithm with k = 2l as “Algorithm C.” Avoiding the inefficiency of flushing the pipeline would generate a better εv and consequently faster convergence, provided that the algorithm requires a similar number of evaluations for the solutions of the same quality. Depending on how to manage those k sets of N genotypes, we may consider two different approaches as follows. 5.2 Generational / Multi-Population ("D") In this approach, referred to as “Algorithm D,” we re- gard each of those k sets of N genotypes as a subpopulation which occasionally exchanges individuals with other sub- populations. It is worth to point out that, unlike typical island parallel GAs [3] where subpopulations are spatially distributed over different locations of computation, we have subpopulations that are temporally distributed over differ- ent times of evaluation. We assume that migration is done at every f generations such that, before selection, each subpopulation replaces its worst k−1 individuals with the collection of k−1 individuals, one from each of the other k − 1 subpopulations. Since we have no constraint on the (spatial) connections between the subpopulations, we can freely choose to assume and exploit the complete connectivity between subpopulations. On the other hand, our algorithm imposes a different kind of constraint on migration, which is regarding the time syn- chronization between subpopulations. Let us assume that there is no delay in the network, so the backward evalua- tion phase of a particular subpopulation ends exactly after 2l time units its forward evaluation phase started. Suppose now that migration is about to happen at time t + 1 while constructing the first subpopulation for the (n+ 1)-th gen- eration. At that time, only the first subpopulation has the fitness values for the n-th generation, while all other k − 1 subpopulations still wait for their fitness values for the n- th generation to become available. Similarly, at time t + j (1 ≤ j ≤ k), only the first j subpopulations have their fit- ness values for the n-th generation, while the remaining k−j subpopulations do not. If we choose to perform migration in a age-synchronized, i.e., temporally consistent manner such that all the subpopulations exchange the best individuals of the same generation, we have to wait until time t + k without being able to renew any subpopulation. Hence, we alternatively perform the age-mixed, i.e., temporally closely consistent, migration, where we collect the best individu- als from the other k − 1 subpopulations of the most recent generation for which the fitness values are available. For instance, when we renew the j-th (2 ≤ j ≤ k − 1) subpopu- lation at time t + j, we take the best individual from each of the 1, ..., (j − 1)-th subpopulations at generation n, and from each of the (j+1), ..., k-th subpopulations at generation n− 1. Algorithm D proceeds in a completely pipelined manner (see Figure 5(b)), yielding the evaluation efficiency (g + 1)2l + k − 1 where g is the number of generations at the termination of the iteration. Note that, when k = 2l and g ≫ 1, εv ≈ N . Note that most changes in Algorithm D, compared with the genotype-distributed NCGA in Section 4, are regard- ing the computational aspects at the source. Hence, Algo- rithm D can be implemented within the same framework the genotype-distributed NCGA, with slight changes in the structure of the coordination vector and the increased num- ber of coding vectors that each merging node keeps. Owing to space limits, further implementational details are omit- 5.3 Non-Generational / Single Population ("E") Rather than managing k separate subpopulations, this ap- proach, referred to as “Algorithm E,” operates on a sin- gle population of size M = kN . The population is up- dated when the fitness values of each of the k sets of N genotypes, referred to as offspring, become available (i.e., “just-in-time”). This is a temporally “sloppy“ approach. From time 1 to k, the forward evaluation phases for the initial (random) k offspring are initiated. At time 2l + j (1 ≤ j ≤ k), the fitness values for the j-th offspring can be calculated at the source and all those N genotypes are just added to the population. We then calculate the coordina- tion vector for the j-th offspring, by performing tournament selection out of the current population, which is partially filled until time 2l + k, and initiate the forward evaluation phase for the second generation. At time 4l+ j (1 ≤ j ≤ k) and on, we update the population as follows: First combine the j-th offspring, whose fitness values are just calculated, with the existing population, and then pick the best kN individuals, out of those (k + 1)N individuals, to form the updated population. Considering each window of 2l time units from the begin- ning, we notice that except for the first and the last windows, kN genotypes are evaluated in each window (see Figure 6). Hence, if we assume that the total number of elapsed time units is large (≫ 1), we have εv ≈ , and when k = 2l, we obtain the maximum εv ≈ N . Algorithm E can also be implemented similarly to the genotype-distributed NCGA with some changes in the coor- dination and coding vectors, whose details are omitted. 6. EXPERIMENTS 6.1 Effect of Genotype Axis Distribution Since the genotype-distributed NCGA (Algorithm B) shares the same computational part of GA with the centralized one (Algorithm A), the two algorithms show the same perfor- mance in terms of solution quality. However, as described in Section 4.4, the computational complexity required by Algorithm B depends only on local topological parameters, which can often lead to a significant gain in terms of the running time. To compare the elapsed running time of the two algorithms, we run a test on a created set of topologies with high connectivity such that there exists a link between each pair of numbered nodes i and j (i < j), where the source is node 1 and the sinks are the last 10 nodes. The test is done by a simulation on a single machine while each node’s function is performed by a separate thread, thus it is pessimistic since it cannot benefit from the multi-processing gain whereas it only suffers from additional computational burdens for managing a number of threads. Table 1 shows that, nevertheless, Algorithm B exhibits an advantage in running time as the size of the network grows. Number of nodes 15 20 25 30 35 40 Algorithm A 0.3 1.5 4.3 13.5 29.5 65.6 Algorithm B 1.8 2.7 4.4 6.3 10.8 15.4 Table 1: Running Time Per Generation (seconds) 6.2 Effect of Temporal Axis Distribution To compare the doubly distributed approaches, we con- struct network G by cascading 15 copies of network B′ in Example 1(Figure 1(b)) in the form of a depth-4 binary tree such that the source of each subsequent copy of B′ is replaced by an earlier copy’s sink. The source is the tree’s root node and the sinks are the 16 leaf nodes. Setting P , the unit packet size, to 1500 bytes as a typical ethernet packet, we can calculate that N , the number of genotypes handled by a single packet, is around 200. Since l = 16 in network G, k = 2l = 32. Parameters on Population B Pop. size: 200 C Pop. size: 6400 D10 Subpops. (size, #): (200,32), Migration freq.: 10 D1 Subpops. (size, #): (200,32), Migration freq.: 1 E Pop. size: 6400, Offspring size: 200 Table 2: Population Parameters for Algorithms Table 2 summarizes the parameters for five algorithms we experiment with. Migration frequency (f) is changed from 10 to 1 from Algorithm D10 to D1. We set the tournament size to the half of the (sub)population size in each algo- rithm, i.e., 100, 3200, 100, 3200 for Algorithms B, C, D, E, respectively. The mixing ratio and the crossover prob- ability are both 0.8 and the mutation rate is 0.015 for all algorithms. We perform 30 runs for each algorithm until the algorithm converges to the optimal solution, which for network G is known to be zero. Table 3 shows the elapsed time units with the time efficiency εt, which we define as the algorithm’s speedup with respect to Algorithm B, and the total number of evaluations with the evaluation efficiency εv obtained from the experiments, which indeed matches the theoretical values almost exactly. For elapsed time and number of evaluations, p-value resulting from paired t-test with the next best (i,e., smallest) one is reported. Time p-value εt #Eval p-value εv B 13,907 - 1.00 86,920 1.38e-14 6.25 C 5,427 1.66e-08 2.56 542,720 2.10e-03 100.00 D10 2,497 1.58e-04 5.57 492,920 0.307 197.44 D1 4,157 7.55e-03 3.35 824,980 - 198.46 E 3,968 0.691 3.50 781,100 0.691 198.39 Table 3: Result of Experiments Pipelining is intended to be efficient by reducing the idle time of network nodes, hence Algorithm B, which does not pipeline, has the lowest εv. Though Algorithm C, which pipelines but stop to flush and re-prime, has much increased εv, Algorithms D10, D1, and E, which operate fully pipelined, offer the highest εv. Note, however, that the different dy- namics of these algorithms may impact the number of fitness evaluations required to reach the optimal solution, hence as can be observed in Table 3, the number of evaluations (and consequently, the realized εt) do not reveal εv in propor- tion. Figure 7 shows that evaluation efficiency comes at the cost of additional fitness evaluations. Algorithms D10 and B dominate all others yet not each other; Algorithm B is less efficient (it does not pipeline) but requires less fitness eval- uations, while D10 is more efficient but requires more eval- uations. Algorithm D10 gives a speedup (εt) of more than 5 times over algorithm B. Algorithms C, D1 and E, though ...0 1 2 2l 2l+1 2l+2 2l+k-1 4l 4l+1 4l+2k-1 ... ... ...Time Offspring # 2 3 k... Single Pop. 1 2 3 k Fill Population 1 2 3 k... 1 2 3 1 2 3 Network ... 4l+k-1 ... Update Population Pilot Vectors Fitness Vector Pilot Vectors Coord. Vector Figure 6: Timing Diagram of Algorithm E: This doubly distributed algorithm (Non-generational/Single pop- ulation) exploits pipelining and does not require intermittent flushing. It is “sloppy” with respect to temporal consistency between selection and replacement by using a single population with just-in-time updating. dominated by D10, still offer higher εt than B. These algo- rithms thus merit additional investigation because they may give better performance for different network topologies or other problems. Algorithms D1 and D10, though distributed temporally, resemble a spatially distributed GA (referred to as multiple- deme GA in [3]) in that they incur no communication over- head and can assume a fully-connected processor topology. The only difference in algorithm dynamics is that migration takes place between sub-populations that differ in age by one generation (see Section 5.2). Thus the performances of D1 and D10 as compared to B are in fact foreseeable from the observation that, in general, multiple-deme GAs require a greater number of evaluations than a standard GA while offering speedups due to parallelism, which is equivalent to higher εv. However, in our experiments, the size and the number of subpopulations are determined to maximize εv rather than the performance of GA. Determining the mi- gration strategy for multiple-GAs is an open question and probably problem dependent [4]. Algorithm E is a completely new algorithm, where the selection from the population and the replacement of off- springs are temporally inconsistent. A (slightly) similar property can be found in the second prototype for paral- lel GA in [5], where the algorithm sends out individuals to processors to be evaluated, and inserts and re-selects them opportunistically, i.e., when their fitness becomes available. Such, rather radical, changes in algorithm dynamics may raise a question whether Algorithm E would even work, which is verified by our experiments. The performance of E is similar to that of D1, hence surpassed by D10, which can be explained by the observation that the temporal mix- ing of E is similar to D1’s frequent mixing. Together, these two results suggest that the doubly distributed GA is robust to age mixing (i.e., temporal sloppiness), which deserves fur- ther in-depth analysis in the future. Evaluation efficiency 2001000 Dominated Non-dominated Figure 7: Tradeoff Plot 7. CONCLUSIONS We have presented a GA which is distributed in two novel ways: along genotype and temporal axes. In order to dis- tribute the fitness evaluation for the network coding prob- lem, our doubly distributed algorithm first distributes, for every member of the population, a subset of the genotype to each network node rather than a subset of the population to each. To maximize the efficient use of the computational nodes in the network, the second axis divides the candidate solutions into pipelined sets and thus the distribution is in the temporal domain, rather that in the spatial domain. We have found that this temporal distribution may lead to tem- poral inconsistency in selection and replacement, however our experiments have yielded better efficiency in terms of the time to convergence without incurring significant penal- ties. 8. REFERENCES [1] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung. Network information flow. IEEE Trans. Inform. Theory, 46(4):1204–1216, 2000. [2] E. Alba, F. Luna, and A. J. Nebro. Parallel heterogeneous genetic algorithms for continuous optimization. In Proc. IPDPS, 2003. [3] E. Cantú-Paz. A survey of parallel genetic algorithms. Calculateurs Parallèles, Réseaux et Systèms Répartis, 10(2):141–171, 1998. [4] E. Cantú-Paz and D. E. Goldberg. Efficient parallel genetic algorithms: Theory and practice. Comput. Methods Appl. Mech. Engrg., 186:211–238, 2000. [5] J. J. Grefenstette. Parallel adaptive algorithms for function optimization. Technical Report CS-81-19, Vanderbilt Univ. Computer Science Dept., 1981. [6] T. Ho, R. Koetter, M. Médard, D. R. Karger, and M. Effros. The benefits of coding over routing in a randomized setting. In Proc. IEEE ISIT, 2003. [7] M. Kim, V. Aggarwal, U.-M. O’Reilly, and M. Médard. Genetic representations for evolutionary minimization of network coding resources. In Proc. EvoComnet, 2007. [8] M. Kim, C. W. Ahn, M. Médard, and M. Effros. On minimizing network coding resources: An evolutionary approach. In Proc. NetCod, 2006. [9] M. Kim, M. Médard, V. Aggarwal, U.-M. O’Reilly, W. Kim, C. W. Ahn, and M. Effros. Evolutionary approaches to minimizing network coding resources. In Proc. IEEE Infocom, 2007. [10] R. Koetter and M. Médard. An algebraic approach to network coding. IEEE/ACM Trans. Networking, 11(5):782–795, 2003. [11] M. Langberg, A. Sprintson, and J. Bruck. The encoding complexity of network coding. IEEE Trans. Inform. Theory, 52(6):2386–2397, 2006. [12] S.-Y. R. Li, R. W. Yeung, and N. Cai. Linear network coding. IEEE Trans. Inform. Theory, 49(2):371–381, 2003. [13] M. B. Richey and R. G. Parker. On multiple Steiner subgraph problems. Networks, 16(4):423–438, 1986. [14] E.-G. Talbi and H. Meunier. Hierarchical parallel approach for GSM mobile network design. J. Parallel Distrib. Comput., 66:274–290, 2006. Introduction Network Coding Problem Formulation Network Coding GA ("A") Genotype Encoding Constraint and Fitness Function Genetic Operators Other Elements GENOTYPE Axis Distribution ("B") Overview Assumptions Details of Genotype-Distributed Algorithm Preliminary Processing [P1] Population Initialization [P2] Forward Evaluation Phase [P3, P8] Backward Evaluation Phase [P4, P10] Fitness Calculation [P5, P11] Termination Criterion [P6] Coordination Vector Calculation [P7] Genetic Operations [P8] Greedy Sweep [P12] Complexity TEMPORAL Axis Distribution Generational / Single Population ("C") Generational / Multi-Population ("D") Non-Generational / Single Population ("E") Experiments Effect of Genotype Axis Distribution Effect of Temporal Axis Distribution Conclusions References

0704.1199

Study of resonant processes for multi-pion production in $\bar p +p\to\ell ^++\ell^- +n_\pi \pi$ annihilation

Study of resonant processes for multi-pion production in p̄+ p→ ℓ+ + ℓ− + nππ annihilation E. A. Kuraev,∗ C. Adamušč́ın,† and E. Tomasi-Gustafsson‡ DAPNIA/SPhN, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France F. Maas CNRS/IN2P3, Institut de Physique Nucléaire, UMR 8608 and Univ. Paris-Sud, Orsay, F-91405 France (Dated: September 24, 2021) Abstract In frame of a phenomenological approach based on Compton-like Feynman amplitudes, we study multi-pion production in antiproton nucleon collisions. The main interest of this reaction is related to the possibility to study the properties of the presumable N̄N atom and of its resonances. For the case of formation of a scalar or pseudoscalar resonant state, with IG(JP ) = 1−(0±), 0+(0−) numerical results are obtained. The differential cross section in an experimental set-up where the pions invariant mass is measured, is given with explicit dependence on the lepton pair and pions invariant mass. PACS numbers: ∗JINR-BLTP, 141980 Dubna, Moscow region, Russian Federation †Department of Theoretical Physics, IOP, Slovak Academy of Sciences, Bratislava, Slovakia ‡Electronic address: [email protected] http://arxiv.org/abs/0704.1199v1 mailto:[email protected] In a previous paper [1] we calculated the differential cross section for single pion produc- tion in the reactions p̄+N → π+ ℓ++ ℓ−, and showed that such reactions are measurable at upcoming facilities, bringing unique information on electromagnetic and axial nucleon form factors in the unphysical region of time-like momentum transfer squared, q2. The purpose of the present work is to study the resonant production of a number nπ of pions accompanied by a lepton pair, in the antiproton nucleon annihilation process. We limit our discussion to quantum numbers of the excited resonances: IG(JP ) = 1+(0±) and 0+(0±). The existence a N̄N bound state and of its resonances was predicted in a series of papers, see Ref. [2, 3]. A partial wave analysis performed in [3] in the range 1960 < M∗ < 2410 combines data in different decay channels 3π0, π0η, π0ηη′ and reveals a series of resonant states in the pp̄ system. Some of the states were precisely identified, in particular two states with quantum numbers I(JPC) = 1(0−+), with masses M∗ = 2360 and 2070 MeV, respectively and widths about 300 MeV. Higher spin states were also identified. Such states show similar masses as dibaryon states, except that they exhibit large widths, which are explained by their decay through hadronic states, which is impossible for dibaryons [4]. NN̄ state has hydrogen-like atomic structure. Similarly to positronium, in addition to strong interaction, QED effects are present, due to the large masses (small distances) involved. Special attention is payed to resonances with masses ∼ 2M , where M is the nucleon mass. The resonances with zero orbital momentum are expected to have large width, whereas higher values of orbital momentum lead to a smaller width given by [2]: [(2ℓ+ 1)!!]2 MeV , (1) which gives, for example, Γ = 100 MeV for ℓ = 0, Γ = 10 MeV for ℓ = 1, and Γ = 0.5 MeV for ℓ = 2. Experimental evidence of such series of dibaryon resonances exist [4], although it is still controversial. An investigation of this problem will be possible at the upcoming FAIR facility [5], where the planned antiproton beams render possible the measurement of the reactions investigated in the present paper. The emission of a lepton pair permits to select the appropriate kinematics adapted to the excitation of such resonances, when the total p̄N center of mass (CMS) energy exceeds the mass of the resonances, due to a known mechanism, called ’return to resonances’ [6]: the lepton pair carries away the extra energy and momentum, providing the condition of π1(q1) π2(q2) πn(qn) p̄(−p1) ℓ−(p−) ℓ+(p+) p(p2) Jp = 0± γ∗(q) q − p1 π1(q1) π2(q2) πn(qn) p̄(−p1) p(p2) Jp = 0± ℓ+(p+) ℓ−(p−) γ∗(q) p2 − q (b)(a) FIG. 1: Feynman diagrams for the reaction p̄+ p → ℓ+ + ℓ− + nππ: lepton pair emission form the antiproton (a) and from the proton (b). exciting the resonances. Experimentally it manifests through a deformation of the Breit Wigner distribution, yielding a ’radiative’ tail. For the excitation of a narrow resonance, the emission of soft quanta alters the shape of the resonance curve: the height of the resonance curve decreases and a radiative tail arises. Indeed, for total energy W , higher than the resonance mass M ′, the most favourable mechanism is when the energy ”excess” W −M ′ is absorbed by the photon emission (with energy ω =W −M ′). Here the small extra factor α is compensated by the resonance denominator. Let us consider the annihilation of an antiproton and a proton in a lepton pair, and a resonant state which subsequently decays into a set of nπ pions: p̄(p1) + p(p2) → γ∗(q) + R(JP ,M ′) → ℓ+(p+) + ℓ−(p−) + π1(q1) + ... + πnπ(qnπ). Such state can have the quantum number of scalar, pseudoscalar, axial or vector particle. The Feynman diagrams for such reactions are plotted in Fig. 1. We neglect direct photon emission from the resonant state, assuming that it is (indirectly) included via the width of the resonance. To describe the kinematics of the process it is convenient to introduce two kinematical variables: the invariant mass squared of the lepton pair q2 = (p+ + p−) 2 (2) and the one of the pions: s1 = ( 2 = (p1 + p2 − q)2. (3) One can write the phase volume as: dΓx = (2π)−2δ4 p1 + p2 − p+ − p− − Πxi=1 2ǫi(2π)3 dΓQq (4) with ∫ dΓq = 2ǫ+(2π)3 2ǫ−(2π)3 δ4(q − p+ − p−), Πnπi=1 2ǫi(2π)3 (2π)4, dΓQq = d4qd4Qδ4(p1 + p2 − q −Q). It is convenient to calculate this last quantity in CMS, where ~q = −~Q, ~p1 = −~p2. In this reference frame we have: d4q = dqO|~q|dq2dOq = dΓQq, (5) where dOq = 2πd cos θq is the element of solid angle, integrated on the azymuthal angel φ. Then, using the relations s = q0 + Q0; (s1 − q2)/ s = Q0 − q0, and s = (p1 + p2)2, one finds: dΓQq = ∫ √s−√q2 dq2d cos θq Λ(s, s1, q2), Λ(a, b, c) = a 2+ b2+ c2−2(ab+ac+ bc), with θq = ~̂q, ~p1. The function Λ(s, s1, q 2) must be positive in the physical region. Let us restrict our considerations to scalar JP = 0+ and pseudoscalar JP = 0− states. The corresponding matrix elements are: Mires = 2)Jµv̄(p1)Oiµu(p2) gi1g2 s1 −M ′2 + iM ′Γ′ where M ′ and Γ′ are the mass and the width of the resonance, Vρ(q 2) = m2ρ/(m ρ − q2) corresponds to the coupling of the photon to the ρ meson (mρ is the ρ meson mass), when interacting with a nucleon, Jµ(q) = v̄(p+)γµu(p−) is the leptonic current and Oiµ Oiµ = Fpµ(q) −p̂1 + q̂ +M (p1 − q)2 −M2 γ(i) + p̂2 − q̂ +M (p2 − q)2 −M2 Fpµ(q) (8) (where γ(i) = γ5 for a pseudoscalar resonant state and γ (i) = 1 for a scalar state) contains the hadronic structure: Fpµ(q) = F 2)γµ + (q̂γµ − γµq̂), (9) which is parametrized in terms of the Pauli and Dirac proton form factors, F1 and F2 (in time-like region). Let us note that the hadron current J hµ = v̄(p1)Oiµu(p2) obeys the gauge invariance: J hµ qµ = 0. In Eq. (7), the coupling constant g2 is related to the width of the decay channel: Γ′nπ = Q , (10) where Γ′nπ is the width of the decay of the pp̄ state to nπ pions, g1 represents the coupling constant at the p̄p vertex to the resonant state. When the resonance is heavier than twice the nucleon mass its value can be related to the p̄p decay width: (gi1) 8πΓp̄p M ′Ai , (11) with Ai = β for J P = 0−, and Ai = β 3 for JP = 0+ and β = 1− (4M2/M ′2). The corresponding differential cross sections are: dq2ds1 Γ′nπΓ (s1 −M ′2)2 +M ′2Γ2i Λ(s, s1, q2) Di, for M ′ > 2M, (12) dq2ds1 α2M ′Γ′nπ 12π2q2 (s1 −M ′2)2 +M ′2Γ2i Λ(s, s1, q2) Di, for M ′ < 2M, (13) with r = 1− (4M2/s) and d cos θq gµν − S(i)µν , (14) S(i)µν = Tr(p̂1 −M)Oiµ(p̂2 +M)Oi∗ν . (15) Two dimensionless variables, X = q2/s and Y = s1/s can been defined, using the invariant mass of the lepton pair, q2, Eq. (2), and the invariant mass of the pions, s1, Eq. (3). The kinematical region scanned by the reaction, is defined by the conditions Λ(s, s1, q 2) > 0 and it corresponds to Y ≤ (1− X)2, as it is shown in Fig. 2. FIG. 2: Allowed kinematical region, as a function of X = q2/s and Y = s1/s (shaded area). The angular integration entering in Di, Eq. (14), can be calculated with the help of the following relations: d cos θq d cos θq q2ss1 +M2Λ(s, s1, q2) d cos θq d cos θq Λ(s, s1, q2) ∣∣∣∣∣ q2 + s1 − s+ r Λ(s, s1, q2) q2 + s1 − s− r Λ(s, s1, q2) ∣∣∣∣∣ , (16) with d1,2 = q 2 − 2p1,2q and d1 + d2 = s1 + q2 − s. The integration is performed in the center of mass of the initial particles. The differential cross sections for heavy scalar and pseudoscalar states have been cal- culated for the reaction p̄ + p → ℓ+ + ℓ− + nππ, as a function of the s and of the total momentum carried by the pions. Since the coupling constant g1 and the partial width Γp̄p are not known, the results for the differential cross section are rescaled by these quantities. For the numerical application, we focus on the scalar resonance f0(980) (Figs 3) and on the pseudoscalar resonance, with η(1405) (Fig. 4). The electromagnetic nucleon form factors have been parametrized according to [8]. The two dimensional plots show a structured landscape, limited by the kinematical cut due to the physical accessible region. Peaks appear in the projection to the q2 axis (inte- gration over s1), and are due to the vector meson resonances coming from enhanced photon Graph2D_y Entries 290 Mean 0.9608 RMS 0.2355 ]2 [GeV1s 0 1 2 3 4 5 Graph2D_y Entries 290 Mean 0.9608 RMS 0.2355 [GeV1 -1010 Graph2D_x Entries 290 Mean 0.6289 RMS 0.111 ]2 [GeV2q 0 1 2 3 4 5 6 Graph2D_x Entries 290 Mean 0.6289 RMS 0.111 FIG. 3: (Color online) Double differential cross section for the reaction p̄ + p → M∗ + nππ, M∗ ≡ f0(980), as a function of q2 and s1. vector meson coupling included in the FFs model. The projection to the s1 axis (integration over q2) shows a bump which is due to the excitation of the pp̄ resonant state. The absolute value of the cross section is in general about two orders of magnitude smaller than in the case of one pion production, as calculated in Ref. [1], but still measurable in the resonance region. To avoid double counting, we do not consider direct emission of a virtual photon from the resonance, as well as from charged component from its decay, as we imply that it is implicitly taken into account in the resonance width. Let us discuss now the possibility for the creation of an atomic state of p̄N type. In the case of a charged atom (N = n) the situation is similar to the deuteron case, with a weakly bound state. One may expect several narrow states with masses ≤ 2M and binding energy of ≃ 2− 3 MeV. The experimental observation of such states has been reported [4]. The coupling constant g1 related to the creation of the bound state (p̄n) can be related (following [7]) to the wave function of this state: g1 =M −3/2|Ψ(0)|, |ψ|2d3r = 1, Ψ(0) = −M V (r)rΨ(r)dr, (17) where Ψ(r) and r0 are the wave function and the radius of the p̄n bound state and V (r) is Graph2D_y Entries 403 Mean 1.947 RMS 0.2783 ]2 [GeV1s 0 1 2 3 4 5 Graph2D_y Entries 403 Mean 1.947 RMS 0.2783 Graph2D_x Entries 403 Mean 0.6166 RMS 0.09974 ]2 [GeV2q 0 1 2 3 4 5 6 -1010 Graph2D_x Entries 403 Mean 0.6166 RMS 0.09974 [GeV1 -1210 -1110 -1010 FIG. 4: (Color online) Double differential cross section for the reaction p̄ + p → M∗ + nππ, M∗ ≡ η(1405) as a function of q2 and s1. the potential of the p̄− n interaction. The reaction p̄ + n → π1 + ... + πnπ can be calculated using a similar formalism with evident modifications, replacing the proton FFs with the neutron FFs and adding a contact term proportional to (F n1 − F 1 )qµ/q 2. One may expect that the general behavior of the distributions will be similar to the case of p̄p collisions. The size of the relevant resonance should be be much smaller than for p̄p one (several fermi), as for a deuteron-like object. Neutral atoms, (N = p) are of electromagnetic nature, their typical size can be estimated to 25 fm, and their binding energy of the order of 20 keV. The width of these resonances in S state is rather large. Such resonances, with orbital momentum L = 0 have typically strong interaction width of order of 100 MeV. For L 6= 0, the width should be smaller, see Eq. (1). In conclusions, we have calculated the double differential cross section for the multipion production in proton antiproton collision, with emission of a leptonic pair. Numerical esti- mations show that the cross section is measurable in the kinematical region which will be accessible at FAIR and will allow to investigate the formation and the resonant structure of a possible NN̄ system. [1] C. Adamušč́ın, E. A. Kuraev, E. Tomasi-Gustafsson and F. Maas, arXiv:hep-ph/0610429, to appear in Phys. Rev. C. [2] L. N. Bogdanova, O. D. Dalkarov and I. S. Shapiro, Annals Phys. 84 (1974) 261. [3] A. V. Anisovich et al., Phys. Lett. B 517 (2001) 273; Phys. Lett. B 517 (2001) 261. [4] B. Tatischeff et al., Phys. Rev. C 59 (1999) 1878; B. Tatischeff and E. Tomasi-Gustafsson, arXiv:nucl-ex/0411044. [5] An International Accelerator Facility for Beams of Ions and Antiprotons, Conceptual Design Report, http://www.gsi.de. [6] V.Baier and V. S. Fadin, Phys. Lett. B 27 (1968) 223; G. Bonneau and F. Martin, Nucl. Phys. B 27 (1971) 381; F. Berends and R. Gastmans, ”Electromagnetic Interactions of hadrons, II, Plenum Press, New York-London, 1978. [7] A.I Akhiezer and I. Pomeranchuk, ’Some questions in hadron theory’, Chapter I, Moscow 1950 (in russian). [8] F. Iachello, A. D. Jackson and A. Lande, Phys. Lett. B 43, 191 (1973); F. Iachello, eConf C0309101, FRWP003 (2003) [arXiv:nucl-th/0312074]; F. Iachello and Q. Wan, Phys. Rev. C 69, 055204 (2004). http://arxiv.org/abs/hep-ph/0610429 http://arxiv.org/abs/nucl-ex/0411044 http://www.gsi.de http://arxiv.org/abs/nucl-th/0312074 References

0704.1200

Low frequency dispersive estimates for the Schrodinger group in higher dimensions

Low frequency dispersive estimates for the Schrödinger group in higher dimensions Simon Moulin and Georgi Vodev Abstract For a large class of real-valued potentials, V (x), x ∈ Rn, n ≥ 4 , we prove dispersive estimates for the low frequency part of eit(−∆+V )Pac, provided the zero is neither an eigen- value nor a resonance of −∆+ V , where Pac is the spectral projection onto the absolutely continuous spectrum of −∆ + V . This class includes potentials V ∈ L∞(Rn) satisfying V (x) = O 〈x〉−(n+2)/2−ǫ , ǫ > 0. As a consequence, we extend the results in [4] to a larger class of potentials. 1 Introduction and statement of results Let V ∈ L∞(Rn), n ≥ 4, be a real-valued function satisfying |V (x)| ≤ C〈x〉−δ , ∀x ∈ Rn, (1.1) with constants C > 0, δ > (n + 2)/2. Denote by G0 and G the self-adjoint realizations of the operators −∆ and −∆+V on L2(Rn), respectively. It is well known that the absolutely continuous spectrums of the operators G0 and G coincide with the interval [0,+∞), and that G has no embedded strictly positive eigenvalues nor strictly positive resonances. However, G may have in general a finite number of non-positive eigenvalues and that the zero may be a resonance. We will say that the zero is a regular point for G if it is neither an eigenvalue nor a resonance in the sense that the operator 1− V∆−1 is invertible on L1 with a bounded inverse. Let Pac denote the spectral projection onto the absolutely continuous spectrum of G. When n ≥ 3, Journé, Sofer and Sogge [4] proved the following dispersive estimate ∥∥eitGPac L1→L∞ ≤ C|t|−n/2, t 6= 0, (1.2) provided the zero is neither an eigenvalue nor a resonance, for potentials satisfying (1.1) with δ > n+ 4 as well as the condition V̂ ∈ L1. (1.3) This was later improved by Yajima [9] for potentials satisfying (1.1) with δ > n+ 2. When n = 3, the estimate (1.2) in fact holds without (1.3). In this case, it was proved in [2] for potentials satisfying (1.1) with δ > 3 and was later improved in [6] and [10] for potentials satisfying (1.1) with δ > 5/2. Goldberg [1] has recently showed that (1.2) holds for potentials V ∈ L3/2−ǫ∩L3/2+ǫ, 0 < ǫ≪ 1, which includes potentials satisfying (1.1) with δ > 2. When n = 2, (1.2) is proved by Schlag [5] for potentials satisfying (1.1) with δ > 3. Given any a > 0, set χa(σ) = χ1(σ/a), where χ1 ∈ C∞(R), χ1(σ) = 0 for σ ≤ 1, χ1(σ) = 1 for σ ≥ 2. Set ηa = χ(1 − χa), where χ denotes the characteristic function of the interval [0,+∞). Clearly, ηa(G) + χa(G) = Pac. When n ≥ 4, dispersive estimates with loss of (n − 3)/2 derivatives for the operator eitGχa(G), ∀a > 0, have been recently proved in http://arxiv.org/abs/0704.1200v2 [7] under the assumption (1.1), only. The loss of derivatives in this case is a high frequency phenomenon and cannot be avoided unless one imposes some regularity condition on the potential (see [3]). The condition (1.3) in [4] plays this role but it seems too strong. The natural conjecture would be that we have dispersive estimates for eitGχa(G) with loss of ν derivatives, 0 ≤ ν ≤ (n − 3)/2, provided V ∈ C(n−3)/2−ν(Rn) (with a suitable decay at infinity). It turns out that no regularity on the potential is needed in order to get dispersive estimates for the low frequency part eitGηa(G), a > 0 small. One just needs some decay at infinity. In fact, the low frequency analysis turns out to be easier in dimensions n ≥ 4 compared with the cases of n = 2 and n = 3, and can be carried out for a larger class of potentials satisfying (with some 0 < ǫ≪ 1) |x− y|−n+2 + |x− y|−(n−2)/2+ǫ |V (x)|dx ≤ C < +∞. (1.4) Clearly, (1.4) is fulfilled for potentials satisfying (1.1). Our main result is the following Theorem 1.1 Let n ≥ 4, let V satisfy (1.4) and assume that the zero is a regular point for G. Then, there exists a constant a0 > 0 so that for 0 < a ≤ a0 we have the estimate ∥∥eitGηa(G) L1→L∞ ≤ C|t|−n/2, t 6= 0. (1.5) Remark 1. We expect that (1.5) holds true for the larger class of potentials satisfying |x− y|−n+2 + |x− y|−(n−1)/2 |V (x)|dx ≤ C < +∞, (1.6) but the proof in this case would require a different approach. Combining (1.5) with the estimates of [7], we obtain the following Theorem 1.2 Let n ≥ 4, let V satisfy (1.1) and assume that the zero is a regular point for G. Then, we have the estimates, ∀t 6= 0, 0 < ǫ≪ 1, ∥∥eitGPacf ≤ C|t|−n/2 ∥∥∥〈G〉(n−3)/4f , (1.7) ∥∥eitGPacf ≤ Cǫ|t|−n/2 ∥∥∥〈x〉n/2+ǫf . (1.8) Remark 2. The proof in [7] is based on uniform estimates for the operator ψ(h2G), 0 < h ≤ 1, ψ ∈ C∞0 ((0,+∞)) (see Lemma 2.2 of [7] or Lemma 2.3 of [8]). In the proof of this lemma (which is given in [8]), however, there is a mistake. That is why, we will give a new proof in Appendix 1 of the present paper. Remark 3. We conjecture that the estimates (1.7) and (1.8) hold true for potentials satis- fying (1.1) with δ > (n+ 1)/2. Theorem 1.1 also allows to extend the results in [4] to a larger class of potentials. More precisely, we have the following Theorem 1.3 Let n ≥ 4, let V satisfy (1.1) with δ > n − 1 as well as (1.3), and assume that the zero is a regular point for G. Then, the estimate (1.2) holds true. Theorem 1.3 follows from (1.5) and the dispersive estimate for eitGχa(G) proved in Ap- pendix 2. To prove (1.5) we adapt the semi-classical approach of [7] based on the semi-classical version of Duhamel’s formula (which in our case is of the form (3.4) or (3.5)). While in [7] the estimates had to be uniform with respect to the semi-classical parameter 0 < h ≤ 1, in the case of low frequency we need to make them uniform for h ≫ 1 (see (3.1)). This, however, turns out to be easier (when n ≥ 4) as we can absorb the remaining terms taking h big enough (see Section 3). That is why, we do not need any more to work on weighted L2 spaces (as in [7]), which in turn allows to cover a much larger class of potentials. As mentioned in Remark 1, the natural class of potentials for which the low frequency analysis works out (for n ≥ 4) is given by (1.6), and the fact that the crucial Proposition 2.1 below holds true under (1.6) is a strong indication for that. In fact, (1.4) is used in the proof of Proposition 2.3, only. 2 Preliminary estimates Let ψ ∈ C∞0 ((0,+∞)). We will first prove the following Proposition 2.1 Let n ≥ 4, let V satisfy (1.6) and assume that the zero is a regular point for G. Then, there exist positive constants C, β and h0 so that the following estimates hold ∥∥ψ(h2G0) L1→L1 ≤ C, h > 0, (2.1) ∥∥ψ(h2G) L1→L1 ≤ C, h ≥ h0, (2.2) ∥∥ψ(h2G)− ψ(h2G0)T L1→L1 ≤ Ch−β, h ≥ h0, (2.3) where the operator 1− V∆−1 : L1 → L1 (2.4) is bounded by assumption. Proof. Set ϕ(λ) = ψ(λ2). We are going to take advantage of the formula ψ(h2G) = (z)(h2G− z2)−1zL(dz), (2.5) where L(dz) denotes the Lebesgue measure on C, ϕ̃ ∈ C∞0 (C) is an almost analytic contin- uation of ϕ supported in a small complex neighbourhood of suppϕ and satisfying ∣∣∣∣ ≤ CN |Im z| N , ∀N ≥ 1. For ±Im z > 0, denote R±0,h(z) = (h 2G0 − z2)−1, R±h (z) = (h 2G− z2)−1. The kernel of the operator R±0,h(z) is of the form R h (|x− y|, z), where R±h (σ, z) = ±h −2 iσ 4(2π)ν H±ν (σz/h) = h−nR±1 (σh−1, z), where ν = (n − 2)/2, H±ν (λ) = λνH±ν (λ), H±ν being the outgoing and incoming Henkel functions of order ν. It is well known that these functions satisfy the bound ∣∣H±ν (λ) ∣∣ ≤ C〈λ〉(n−3)/2e−|Imλ|, ∀λ, ±Imλ ≥ 0, (2.6) while near λ = 0 they are of the form H±ν (λ) = a±ν,1(λ) + λn−2 logλa ν,2(λ), (2.7) where a±ν,j are analytic functions, a ν,2 ≡ 0 if n is odd. By (2.6) and (2.7), we have ∣∣H±ν (λ)−H±ν (0) ∣∣ ≤ C|λ|1/2〈λ〉(n−4)/2, ∀λ, ±Imλ ≥ 0. (2.8) Hence, the functions R±h satisfy the bounds (for z ∈ C±ϕ := {z ∈ supp ϕ̃, ±Im z ≥ 0}, σ > 0, h ≥ 1) ∣∣R±h (σ, z) ∣∣ ≤ Ch−2 σ−n+2 + σ−(n−1)/2 , (2.9) ∣∣R±h (σ, z)−R h (σ, 0) ∣∣ ≤ Ch−5/2 σ−n+5/2 + σ−(n−1)/2 . (2.10) Using the above bounds we will prove the following Lemma 2.2 For z ∈ C±ϕ , we have ‖VR0,h(z)‖L1→L1 ≤ Ch −2, h ≥ 1, (2.11) ‖VR0,h(z)− VR0,h(0)‖L1→L1 ≤ Ch −5/2, h ≥ 1, (2.12) ‖VRh(z)‖L1→L1 ≤ Ch −2, h ≥ h0, (2.13) ∥∥∥R±0,h(z) L1→L1 ≤ C|Im z|−q, h > 0, Im z 6= 0, (2.14) ∥∥R±h (z) L1→L1 ≤ C|Im z|−q, h ≥ h0, Im z 6= 0, (2.15) with constants C, q, h0 > 0 independent of z and h. Proof. In view of (2.9), the norm in the LHS of (2.11) is upper bounded by |V (x)| ∣∣R±h (|x − y|, z) ≤ Ch−2 sup |V (x)| |x− y|−n+2 + |x− y|−(n−1)/2 dx ≤ Ch−2. The estimate (2.12) follows in the same way using (2.10). To prove (2.14), we use (2.6) to get (for z ∈ C±ϕ , Im z 6= 0) ∣∣R±1 (σ, z) ∣∣ ≤ Cσ−2ν〈σ〉(n−3)/2e−σ|Im z| ≤ Cσ−2ν〈σ〉−5/2|Im z|−(n+2)/2. (2.16) By (2.16), the norm in the LHS of (2.14) is upper bounded by ∣∣R±h (σ, z) ∣∣ dσ = C ∣∣R±1 (σ, z) ∣∣ dσ ≤ C|Im z|−(n+2)/2 〈σ〉−3/2dσ ≤ C|Im z|−(n+2)/2. To prove (2.13) and (2.15), we will use the identity R±h (z) 1 + h2VR±0,h(z) = R±0,h(z), ±Im z > 0. (2.17) Observe that 1 + h2VR±0,h(0) = 1 − V∆−1, which is supposed to be invertible on L1 with a bounded inverse denoted by T . Thus, it follows from (2.12) that there exists a constant h0 > 0 so that for h ≥ h0 the operator 1 + h2VR±0,h(z) is invertible on L1 with an inverse satisfying ∥∥∥∥ 1 + h2VR±0,h(z) )−1∥∥∥∥ L1→L1 ≤ C, z ∈ C±ϕ , (2.18) with a constant C > 0 independent of z and h. Hence, we can write R±h (z) = R 0,h(z) 1 + h2VR±0,h(z) . (2.19) Now (2.13) follows from (2.11), (2.18) and (2.19), while (2.15) follows from (2.14), (2.18) and (2.19). ✷ Clearly, (2.1) and (2.2) follow from (2.5) and (2.14), (2.15), respectively. To prove (2.3) we rewrite the identity (2.19) in the form R±h (z)−R 0,h(z)T = R±0,h(z)T h2VR±0,h(z)− h 2VR±0,h(0) h2VR±0,h(z)− h 2VR±0,h(0) (2.20) By Lemma 2.2, (2.18) and (2.20) we conclude ∥∥∥R±h (z)−R 0,h(z)T L1→L1 ≤ Ch−β |Im z|−q, h ≥ h0, z ∈ C±ϕ , Im z 6= 0, (2.21) with constants C, q, β > 0 independent of z and h. Now (2.3) follows from (2.5) and (2.21). Let ψ1 ∈ C∞0 ((0,+∞)), ψ1 = 1 on suppψ. Proposition 2.3 Under the assumptions of Theorem 1.1, there exist positive constants h0 and β so that we have the estimates ∥∥V eitG0ψ(h2G0) L1→L1 dt ≤ Ch−β, h ≥ 1, (2.22) ∥∥V ψ(h2G)eitG0ψ1(h2G0) L1→L1 dt ≤ Ch−β , h ≥ h0. (2.23) Proof. It is shown in [7] (Section 2) that the kernel of the operator eitG0ψ(h2G0) is of the form Kh(|x− y|, t) with a function Kh satisfying Kh(σ, t) = h −nK1(σh −1, th−2), |K1(σ, t)| ≤ C|t|−s−1/2σs−(n−1)/2, 0 ≤ s ≤ (n− 1)/2, σ > 0, t 6= 0. Hence, for all 0 ≤ s ≤ (n− 1)/2, σ > 0, t 6= 0, h > 0, we have |Kh(σ, t)| ≤ Chs−(n−1)/2|t|−s−1/2σs−(n−1)/2, which together with (1.4) imply ∥∥V eitG0ψ(h2G0) L1→L1 ≤ Chs−(n−1)/2|t|−s−1/2, 1/2− ǫ ≤ s ≤ 1/2 + ǫ, (2.24) where 0 < ǫ≪ 1. Clearly, (2.22) follows from (2.24). Furthermore, using (2.5), (2.13), (2.14) and (2.24), we get ψ(h2G)− ψ(h2G0) eitG0ψ1(h L1→L1 ≤ Ch2 ∥∥∥VR±h (z)V e itG0ψ1(h 2G0)R±0,h(z) L1→L1 L(dz) ≤ Ch2 ∥∥VR±h (z) L1→L1 ∥∥V eitG0ψ1(h2G0) L1→L1 ∥∥∥R±0,h(z) L1→L1 L(dz) ≤ Chs−(n−1)/2|t|−s−1/2 ∣∣∣∣ |Im z| −qL(dz) ≤ Chs−(n−1)/2|t|−s−1/2, 1/2− ǫ ≤ s ≤ 1/2 + ǫ, (2.25) which clearly implies (2.23). ✷ 3 Proof of Theorem 1.1 Denote Ψ(t, h) = eitGψ(h2G)− T ∗eitG0ψ(h2G0)T, T being given by (2.4). We will first show that (1.5) follows from the following Proposition 3.1 Under the assumptions of Theorem 1.1, there exist positive constants C, h0 and β so that for h ≥ h0 we have ‖Ψ(t, h)‖L1→L∞ ≤ Ch −β |t|−n/2, t 6= 0. (3.1) Recall that χa(σ) = χ1(σ/a), a > 0 small. Then we can write the function ηa as follows ηa(σ) = ψ(σθ) , σ > 0, where ψ(σ) = σχ′1(σ) ∈ C∞0 ((0,+∞)). Thus, we obtain from (3.1), ∥∥eitGηa(G) − T ∗eitG0ηa(G0)T L1→L∞ ∥∥∥Ψ(t, L1→L∞ ≤ C|t|−n/2 θ−1−β/2dθ ≤ C|t|−n/2, (3.2) provided a is taken small enough. Clearly, (1.5) follows from (3.2). Proof of Proposition 3.1. We will first prove the following Proposition 3.2 Under the assumptions of Theorem 1.1, there exist positive constants C, h0 and β so that for h ≥ h0 we have ∥∥V eitGψ(h2G) L1→L1 dt ≤ Ch−β. (3.3) Proof. Using Duhamel’s formula eitG = eitG0 + i ei(t−τ)GV eiτG0dτ, we get the identity eitGψ(h2G) = ψ(h2G)eitG0ψ1(h 2G0)T + e itGψ(h2G) 2G)− ψ1(h2G0)T ψ(h2G)ei(t−τ)GV eiτG0ψ1(h 2G0)Tdτ. (3.4) Using Propositions 2.1 and 2.3, (3.4) together with Young’s inequality we obtain ∥∥V eitGψ(h2G) L1→L1 dt ≤ Ch−β + Ch−β ∥∥V eitGψ(h2G) L1→L1 ∥∥∥V ψ(h2G)ei(t−τ)G L1→L1 ∥∥V eiτG0ψ1(h2G0) L1→L1 ≤ Ch−β + Ch−β ∥∥V eitGψ(h2G) L1→L1 which clearly implies (3.3) if we take h large enough. ✷ Using Duhamel’s formula eitG = eitG0 + i ei(t−τ)G0V eiτGdτ, we get the identity Ψ(t;h) = Ψj(t;h), (3.5) where Ψ1(t;h) = T ∗ψ1(h 2G0)e ψ(h2G)− ψ(h2G0)T 2G)− T ∗ψ1(h2G0) eitGψ(h2G), Ψ2(t;h) = i T ∗ψ1(h 2G0)e i(t−τ)G0V eiτGψ(h2G)dτ. By (2.1) and (2.3) together with the well known estimate ∥∥eitG0 L1→L∞ ≤ C|t|−n/2, we get ‖Ψ1(t;h)f‖L∞ ≤ Ch −β |t|−n/2‖f‖L1 + Ch−β ‖Ψ(t;h)f‖L∞ , t 6= 0. (3.6) By Propositions 2.3 and 3.2, ∀f ∈ L1, t > 0, we have tn/2 ‖Ψ2(t;h)f‖L∞ ∫ t/2 (t− τ)n/2 ∥∥∥ψ1(h2G0)ei(t−τ)G0 L1→L∞ ∥∥V eiτGψ(h2G)f ∥∥∥ψ1(h2G0)ei(t−τ)G0V L∞→L∞ ∥∥eiτGψ(h2G)f ∥∥V eiτGψ(h2G)f +C sup t/2≤τ≤t ∥∥eiτGψ(h2G)f ∥∥V eiτG0ψ1(h2G0) L1→L1 ≤ Ch−β‖f‖L1 + Ch−β sup t/2≤τ≤t ∥∥eiτGψ(h2G)f . (3.7) Combining (3.5), (3.6) and (3.7), we conclude, ∀f ∈ L1, t > 0, tn/2 ‖Ψ(t;h)f‖L∞ ≤ Ch −β‖f‖L1 + Ch−βtn/2 ‖Ψ(t;h)f‖L∞ +Ch−β sup t/2≤τ≤t τn/2 ‖Ψ(τ ;h)f‖L∞ . (3.8) Taking h big enough we can absorb the second and the third terms in the RHS of (3.8), thus obtaining (3.1). Clearly, the case of t < 0 can be treated in the same way. ✷ A Appendix 1 We will prove the following Lemma A.1 Let ψ ∈ C∞0 ((0,+∞)). Then, for all h > 0, s ≥ 0, we have the estimates ∥∥ψ(h2G0) L1→L1 ≤ C, (A.1) ∥∥〈x〉sψ(h2G0)〈x〉−s L1→L2 ≤ Ch−n/2〈h〉s, (A.2) where the constant C is of the form C = C′ sup 0≤k≤k0 |∂kλψ(λ)|, (A.3) with some integer k0 independent of ψ and a constant C ′ > 0 depending on the support of ψ, only. Furthermore, if V satisfies (1.1) with δ > n/2, we have the estimates (for 0 < h ≤ 1) ∥∥ψ(h2G)− ψ(h2G0) L1→L1 ≤ Ch2, (A.4) ∥∥〈x〉δ ψ(h2G)− ψ(h2G0) L1→L2 ≤ Ch2−n/2. (A.5) Proof. The estimate (A.1) is proved in Section 2 using the formula (2.5) and (2.14). It can be also seen by using the fact that the kernel of the operator ψ(h2G0) is of the form kh(|x− y|) with a function kh satisfying kh(σ) = h −nk1(σ/h), (A.6) |k1(σ)| ≤ Cm〈σ〉−m, ∀σ > 0, (A.7) for all integers m ≥ 0, with a constant Cm of the form Cm = C m sup 0≤j≤jm |∂jλψ(λ)|, (A.8) where jm is some integer independent of ψ, while C m > 0 depends on the support of ψ. By Young’s inequality, the norm in the LHS of (A.1) is upper bounded by |kh(|ξ|)|dξ = |k1(|ξ|)|dξ ≤ Cn+1. The norm in the LHS of (A.2) is upper bounded by 〈x〉2s〈y〉−2s|kh(|x− y|)|2dx 〈x− y〉2s|kh(|x − y|)|2dx ≤ C〈h〉s 〈ξ/h〉2s|kh(|ξ|)|2dξ = C〈h〉sh−n/2 〈ξ〉2s|k1(|ξ|)|2dξ ≤ Csn〈h〉sh−n/2, where sn is some integer depending on n and s. To prove (A.4) observe that by (2.5) we ψ(h2G)− ψ(h2G0) = (z)(h2G0 − z2)−1V (h2G− z2)−1zL(dz). (A.9) Clearly, (A.4) would follow from (A.9), (2.14) and the estimate (for z ∈ supp ϕ̃) ∥∥(h2G− z2)−1 L1→L1 ≤ C|Im z|−q, 0 < h ≤ 1, Im z 6= 0. (A.10) Let φ ∈ C∞0 ([1, 2]) be such that φ(θ2)θ−1dθ = 1. Given a parameter 0 < ε ≪ 1, we decompose the free resolvent as follows (h2G0 − z2)−1 = Aε(z;h) + Bε(z;h), (A.11) where Aε(z;h) = (εθh)2G0; (εθ) Bε(z;h) = (εθh)2G0; (εθ) where f(λ;µ; z) = λµ−1 − z2 . It is easy to see that there exist constants 0 < µ1 < µ2 so that the function f satisfies the following bounds ∣∣∣∂jλf(λ;µ; z) ∣∣∣ ≤ Cjµ, 0 < µ ≤ µ1, (A.12) ∣∣∣∂jλf(λ;µ; z) ∣∣∣ ≤ C′j |Im z|−j−1, µ1 ≤ µ ≤ µ2, (A.13) ∣∣∣∂jλf(λ;µ; z) ∣∣∣ ≤ C′′j , µ ≥ µ2, (A.14) for every integer j ≥ 0. By (A.1), (A.3) and (A.12), we have (εθh)2G0; (εθ) L1→L1 ≤ C(εθ)2, 0 < θ ≤ 1, (A.15) provided ε > 0 is taken small enough. We deduce from (A.15), ‖Aε(z;h)‖L1→L1 ≤ Cε 2, z ∈ supp ϕ̃, (A.16) with a constant C > 0 independent of z, h and ε. By (A.2), (A.3), (A.12)-(A.14), we have ∥∥〈x〉sf (εθh)2G0; (εθ) 〈x〉−s L1→L2 ≤ C(εθh)−n/2〈εθh〉s|Im z|−q, (A.17) with constants C, q > 0 independent of z, θ, h and ε. We deduce from (A.17), ‖Bε(z;h)‖L1→L2 ≤ C −n/2|Im z|−q θ−1−n/2dθ ≤ Cεh−n/2|Im z|−q, z ∈ supp ϕ̃, (A.18) with a constant Cε > 0 independent of z and h. It follows from (A.16) that the operator 1+ h2VAε(z;h) is invertible on L1, provided ε > 0 is taken small enough, independent of h. Therefore, we can write the identity h2G− z2 h2G0 − z2 h2G− z2 h2G0 − z2 , (A.19) in the form ( h2G− z2 = M(z;h) + h2 h2G− z2 )−1 N (z;h), (A.20) where the operators M(z;h) = h2G0 − z2 )−1 ( 1 + h2VAε(z;h) N (z;h) = V Bε(z;h) 1 + h2VAε(z;h) satisfy the estimates ‖M(z;h)‖L1→L1 + ‖N (z;h)‖L1→L1 ≤ C|Im z| −q, (A.21) ‖N (z;h)‖L1→L2 ≤ Ch −n/2|Im z|−q. (A.22) By (A.20) we have h2G− z2 M(z;h)N (z;h)j + h2J h2G− z2 )−1 N (z;h)J M(z;h)N (z;h)j + h2J h2G0 − z2 )−1 N (z;h)J +h2J+2 h2G0 − z2 h2G− z2 )−1 N (z;h)J , (A.23) for every integer J ≥ 1. By (A.22) and (2.14), we obtain h2G0 − z2 h2G− z2 )−1 N (z;h) L1→L1 ≤ ‖V ‖L2 h2G0 − z2 )−1∥∥∥ L1→L1 h2G− z2 )−1∥∥∥ L2→L2 ‖N (z;h)‖L1→L2 ≤ Ch−n/2|Im z|−q2 . (A.24) Now, (A.10) follows from (A.21), (A.23) and (A.24). To prove (A.5) we rewrite (A.20) in the form h2G− z2 )−1 − h2G0 − z2 Fj(z;h), (A.25) where F1(z;h) = h2Aε(z;h)VAε(z;h) 1 + h2VAε(z;h) F2(z;h) = h2Bε(z;h)VAε(z;h) 1 + h2VAε(z;h) F3(z;h) = h2 h2G− z2 V Bε(z;h) 1 + h2VAε(z;h) It is easy to see that we have the estimate ∥∥∥〈x〉s h2G− z2 )−1 〈x〉−s L2→L2 ≤ C|Im z|−q, z ∈ supp ϕ̃, 0 < h ≤ 1, (A.26) for every s ≥ 0 with constants C, q > 0 depending on s but independent of z and h. By (A.18) and (A.26), ∥∥〈x〉δF3(z;h) L1→L2 ≤ Ch2−n/2|Im z|−q, z ∈ supp ϕ̃, 0 < h ≤ 1. (A.27) Observe now that we can write the operator Aε(z;h) in the form Aε(z;h) = χ(3)ε (h2G0) h2G0 − z2 where χ(3)ε (σ) = ∫ εσ1/2 φ(θ2) Similarly, we can decompose the operator Bε(z;h) as B(1)ε + B(2)ε , where B(j)ε (z;h) = χ(j)ε (h2G0) h2G0 − z2 , j = 1, 2, χ(1)ε (σ) = ∫ ε−1σ1/2 εσ1/2 φ(θ2) , χ(2)ε (σ) = ε−1σ1/2 φ(θ2) Taking ε > 0 small enough we can arrange that suppχ ε ∩ suppϕ = ∅, j = 2, 3, so the operator-valued functions Aε(z;h) and B(2)ε (z;h) are analytic on supp ϕ̃. Therefore, we can write (A.9) in the form ψ(h2G)− ψ(h2G0) = (z)Fj(z;h)zL(dz), (A.28) where F4(z;h) = h2B(1)ε (z;h)VAε(z;h) 1 + h2VAε(z;h) By (A.17) (with s = δ), we have ∥∥〈x〉δF4(z;h) L1→L2 ≤ Ch2 ∥∥∥〈x〉δB(1)ε (z;h)〈x〉−δ L1→L2 ≤ Ch2−n/2|Im z|−q, z ∈ supp ϕ̃, 0 < h ≤ 1. (A.29) Now (A.5) follows from (A.27)-(A.29). ✷ B Appendix 2 Combining some ideas from [6],[7] and [4] we will prove the following Theorem B.1 Let n ≥ 4, let V satisfy (1.1) with δ > n − 1 as well as (1.3). Then, for every a > 0 we have the estimate ∥∥eitGχa(G) L1→L∞ ≤ C|t|−n/2, t 6= 0. (B.1) Remark. Note that (B.1) is proved in [4] for potentials satisfying (1.1) with δ > n, the condition (1.3) as well as an extra technical assumption. Here we eliminate this extra as- sumption. Proof. The key point in the proof in [4] is the bound ∥∥e−itG0V eitG0 L1→L1 ≤ ‖V̂ ‖L1 , ∀t. (B.2) Combining (B.2) with Duhamel’s formula one easily gets ∥∥e−itG0V eitG L1→L1 ≤ C, |t| ≤ 1, (B.3) with a constant C > 0 independent of t. In what follows we will derive (B.1) from (B.2) and (B.3). To this end, given a function ψ ∈ C∞0 ((0,+∞)) and a parameter 0 < h ≤ 1, as in [6], [7], denote Ψ(t, h) = eitGψ(h2G)− eitG0ψ(h2G0), F (t) = i ei(t−τ)G0V eiτG0dτ. As in these papers, it is easy to see that (B.1) follows from the following Theorem B.2 Under the assumptions of Theorem B.1, there exist constants C, β > 0 so that we have the estimates (for 0 < h ≤ 1, t 6= 0) ‖F (t)‖L1→L∞ ≤ C|t| −n/2, (B.4) ∥∥Ψ(t, h)− F (t)ψ(h2G0) L1→L∞ ≤ Chβ |t|−n/2. (B.5) Proof. Clearly, (B.4) follows from (B.2) for |t| ≤ 2. Let |t| ≥ 2. Without loss of generality we may suppose t ≥ 2. Write F = F1 + F2, where F1(t) = i ∫ t−1 ei(t−τ)G0V eiτG0dτ, F2(t) = i ei(t−τ)G0V eiτG0dτ. It follows from (B.2) that F2(t) satisfies (B.4). To deal with the operator F1(t), observe that its kernel is of the form U(|x− ξ|2/4, |y − ξ|2/4, t)V (ξ)dξ, where cn is a constant and U(σ1, σ2, t) = ∫ t−1 eiσ1/(t−τ)+iσ2/τ (t− τ)−n/2τ−n/2dτ. To prove that F1(t) satisfies (B.4), it suffices to show that |U(σ1, σ2, t)| ≤ Ct−n/2 1 + σ , ∀σ1, σ2 > 0, t ≥ 2. (B.6) To do so, observe that U(σ1, σ2, t) = t u(σ1t −1, σ2t −1, t−1) + u(σ2t −1, σ1t −1, t−1) , (B.7) where u(σ′1, σ 2, κ) = ∫ 1/2 /(1−τ ′)+iσ′ /τ ′(1− τ ′)−n/2(τ ′)−n/2dτ ′. It is easy to see that (B.6) follows from (B.7) and the bound |u(σ′1, σ′2, κ)| ≤ Cκ−(n−3)/2(σ′2)−1/2, ∀σ′1, σ′2 > 0, 0 < κ ≤ 1/2. (B.8) To prove (B.8), we make a change of variables µ = 1/τ ′ and write the function u in the form u(σ′1, σ 2, κ) = ∫ κ−1 eiϕ(µ,σ µn/2−2dµ, where ϕ(µ, σ′1, σ 2) = µσ µ− 1σ We have |u(σ′1, σ′2, κ)| ≤ C ∫ κ−1 µn/2−2dµ ≤ Cκ−(n−2)/2. (B.9) Furthermore, observe that ϕ′(µ) = = σ′2 − (µ− 1)2 , so ϕ′ vanishes at µ0 = 1 + (σ 1/2. We will consider now two cases. Case 1. µ0 6∈ [3/2, 3κ−1/2]. Then, we have |ϕ′(µ)| ≥ σ′2 |µ− µ0| µ− 1 ≥ , µ ∈ [2, κ−1]. Therefore, integrating by parts, we obtain u(σ′1, σ 2, κ) = ∫ κ−1 (iϕ′)−1 µn/2−2deiϕ = eiϕ(iϕ′)−1 µn/2−2 ∣∣∣∣∣ ∫ κ−1 eiϕf(µ)dµ, (B.10) where f(µ) = (iϕ′)−1 µn/2−2 = (iϕ′)−1 µn/2−2 µn/2−2. Since ∣∣∣∣ ∣∣∣∣ ≤ 2σ′1(µ− 1)−2 (µ− 1) |σ′2 − σ′1(µ− 1)−2| µ− 1 , we have (for µ ≥ 2) |f(µ)| ≤ C(σ′2)−1µn/2−3. (B.11) By (B.10) and (B.11), |u(σ′1, σ′2, κ)| ≤ C(σ′2)−1κ−(n−4)/2. (B.12) Clearly, in this case (B.8) follows from (B.9) and (B.12). Case 2. µ0 ∈ [3/2, 3κ−1/2]. Denote I(µ0) = [9µ0/10, 11µ0/10] ∩ [2, κ−1]. We write the function u as u1 + u2, where I(µ0) µn/2−2dµ = µ n/2−1 Ĩ(µ0) eiλφ(z)g(z)dz, (B.13) where we have made a change of variables µ = µ0(1 + z), Ĩ(µ0) ⊂ [−1/10, 1/10], λ = µ0σ′2, g(z) = 1 + z 1 + z − µ−10 (1 + z)n/2−2, φ(z) = (1 + z) (µ0 − 1)2 µ0(1 + z)− 1 = µ0 + µ0 − 1 z2 +O(z3), |z| ≪ 1, uniformly in µ0. It is easy to see that we have the estimate eiλφ(z)g(z)dz ∣∣∣∣ ≤ Cλ −1/2, |a| ≤ 1/10. (B.14) Indeed, the functions g(z) and φ(z) are analytic in |z| ≤ 1/10 with |g(z)| bounded there uniformly in µ0. Therefore, we can change the contour of integration to obtain (with some 0 < γ ≪ 1) eiλφ(z)g(z)dz ∣∣∣∣ ≤ eiλφ(e iγy)g(eiγy)dy ∣∣∣∣+ ∣∣∣∣a eiλφ(e iθa)g(eiθa)dθ e−Cλy dy + C′1 ′λθdθ = O(λ−1/2), with some constants C,C′, C1, C 1 > 0. By (B.13) and (B.14) we conclude |u1| ≤ C(σ′2)−1/2µ (n−3)/2 0 ≤ C̃(σ′2)−1/2κ−(n−3)/2. (B.15) On the other hand, if µ ∈ [2, κ−1] \ I(µ0), then |µ− µ0| µ− 1 ≥ C > 0, so we can bound from below |ϕ′(µ)|. Therefore, the function u2 can be treated in the same way as does u in Case 1. Thus, u2 satisfies (B.8) and hence, in view of (B.15), so does u. This completes the proof of (B.4). It suffices to prove (B.5) for 0 < h ≤ h0 with some constant 0 < h0 ≤ 1, since for h0 ≤ h ≤ 1 it follows from (B.4) and the estimate of the L1 → L∞ norm of Ψ(t, h) proved in [7] for the larger class of potentials satisfying (1.1) with δ > (n+2)/2 (without using (1.3)). Without loss of generality we may suppose t > 0. Now, using Duhamel’s formula as in [6], [7] we get the identity Ψ(t;h)− F (t)ψ(h2G0) = Ψj(t;h), (B.16) where Ψ1(t;h) = ψ1(h 2G0)e ψ(h2G)− ψ(h2G0) 2G)− ψ1(h2G0) eitG0ψ(h2G0) + 2G)− ψ1(h2G0) Ψ(t;h), Ψ2(t;h) = i 2G0)e i(t−τ)G0V eiτGψ(h2G)dτ, Ψ3(t;h) = −i ei(t−τ)G0V eiτG0ψ(h2G0)dτ, Ψ4(t;h) = i ∫ t−γ 2G0)e i(t−τ)G0VΨ(τ ;h)dτ, Ψ5(t;h) = −i ∫ t−γ (1− ψ1)(h2G0)ei(t−τ)G0V eiτG0ψ(h2G0)dτ, where ψ1 ∈ C∞0 ((0,+∞)), ψ1 = 1 on suppψ, and 0 < γ ≪ 1 is a parameter to be fixed later on, depending on h. In view of (A.4), we have ‖Ψ1(t;h)f‖L∞ ≤ Ch 2t−n/2‖f‖L1 + Ch2 ‖Ψ(t;h)f‖L∞ , ∀f ∈ L 1. (B.17) By (B.2) and (B.3), ‖Ψj(t;h)f‖L∞ ≤ Cγt −n/2‖f‖L1 + Cγ ‖Ψ(t;h)f‖L∞ , ∀f ∈ L 1, j = 2, 3, (B.18) with a constant C > 0 independent of t, h and γ. Proposition B.3 Let V satisfy (1.1) with δ > n− 1. Then, there exist constants C, β1 > 0 so that for 0 < h ≤ 1, t ≥ 2γ, we have the estimate ‖Ψ4(t, h)‖L1→L∞ ≤ Ch β1γ−(n−3)/2t−n/2. (B.19) Proof. We will make use of the following estimates proved in [7]. Proposition B.4 Let V satisfy (1.1) with δ > (n + 2)/2. Then, for every 0 < ǫ ≪ 1, 1/2− ǫ/4 ≤ s ≤ (n− 1)/2, 0 < h ≤ 1, t 6= 0, we have the estimates ∥∥∥ψ(h2G0)eitG0〈x〉−s−1/2−ǫ L2→L∞ ≤ Chs−(n−1)/2|t|−s−1/2, (B.20) ∥∥∥Ψ(t, h)〈x〉−s−1/2−ǫ L2→L∞ ≤ Chs−(n−3)/2−ǫ/4|t|−s−1/2. (B.21) By (B.20) and (B.21), we get (with some 0 < ε0 ≪ 1) ‖Ψ4(t, h)‖L1→L∞ ∫ t/2 ∥∥∥ψ1(h2G0)ei(t−τ)G0〈x〉−n/2−ε0 L2→L∞ ∥∥∥〈x〉−(n−2)/2−ε0Ψ(τ, h) L1→L2 ∫ t−γ ∥∥∥ψ1(h2G0)ei(t−τ)G0〈x〉−(n−2)/2−ε0 L2→L∞ ∥∥∥〈x〉−n/2−ε0Ψ(τ, h) L1→L2 ≤ Chε0/4t−n/2 ∫ t/2 τ−(n−2)/2dτ + Chε0/4t−n/2 ∫ t−γ (t− τ)−(n−2)/2dτ ≤ Chε0/4γ−(n−3)/2t−n/2. Proposition B.5 Let V satisfy (1.1) with δ > n−1. Then, for every 0 < ǫ≪ 1, 0 < h ≤ 1, t ≥ 2γ, we have the estimate ‖Ψ5(t, h)‖L1→L∞ ≤ Cǫh ǫγ−(n−3)/2−ǫt−n/2. (B.22) Proof. We will make use of the fact that the kernel of the operator eitG0ψ(h2G0) is of the form Kh(|x− y|, t), where Kh(σ, t) = (2π)ν+1 Jν(σλ)ψ(h2λ2)λdλ = h−nK1(σh−1, th−2), (B.23) where Jν(z) = zνJν(z), Jν(z) = (H+ν (z) +H−ν (z)) /2 is the Bessel function of order ν = (n− 2)/2. So, the kernel of the operator Ψ5 is of the form Wh(|x− ξ|, |y − ξ|, t, γ)V (ξ)dξ, where Wh(σ1, σ2, t, γ) = −i ∫ t−γ K̃h(σ1, t− τ)Kh(σ2, τ)dτ = h−2n+2W1(σ1h −1, σ2h −1, th−2, γh−2), (B.24) where K̃h is defined by replacing in the definition of Kh the function ψ by 1−ψ1. It is easy to see that (B.22) follows from the bound (for all σ1, σ2, γ > 0, 0 < ǫ≪ 1, t ≥ 2γ) |Wh(σ1, σ2, t, γ)| ≤ Cǫhǫγ−(n−3)/2−ǫt−n/2 σ−n+21 + σ 1 + σ 2 + σ . (B.25) In view of (B.24), it suffices to prove (B.25) with h = 1. Now, observe thatW1 =W where 1 (σ1, σ2, t, γ) = (σ1σ2) 4(2π)2ν+2 ei(t−γ)λ +iγλ2 2ρ(λ21, λ 2)Jν(σ1λ1)Jν(σ2λ2)dλ21dλ22, 1 (σ1, σ2, t, γ) = (σ1σ2) 4(2π)2ν+2 ei(t−γ)λ +iγλ2 1ρ(λ21, λ 2)Jν(σ1λ1)Jν(σ2λ2)dλ21dλ22, where the function ρ(λ21, λ (1− ψ1)(λ21)ψ(λ22) λ22 − λ21 = (1 − ψ1)(λ21)ψ1(λ22) ψ(λ22)− ψ(λ21) λ22 − λ21 satisfies the bound ∣∣∣∂α1 ρ(λ21, λ ∣∣∣ ≤ Cα1,α2〈λ21〉−1−α1 , ∀(λ1, λ2). (B.26) Given any integers 0 ≤ k,m < n/2, since Jν(z) = O(zn−2) as z → 0, we can integrate by parts to get 1 (σ1, σ2, t, γ) = i −m−k(t− γ)−kγ−m (σ1σ2) 4(2π)2ν+2 ei(t−γ)λ +iγλ2 ×∂kλ2 ρ(λ21, λ 2)Jν(σ1λ1)Jν(σ2λ2) dλ21dλ 1 (σ1, σ2, t, γ) = i −m−k(t− γ)−kγ−m (σ1σ2) 4(2π)2ν+2 ei(t−γ)λ +iγλ2 ×∂mλ2 ρ(λ21, λ 2)Jν(σ1λ1)Jν(σ2λ2) dλ21dλ Using the inequality ∣∣∣∣ ϕ(λ)dλ ∣∣∣∣ ≤ C|t| −1/2 ‖ϕ̂‖L1 , (B.27) we obtain (for t ≥ 2γ) ∣∣∣W (1)1 (σ1, σ2, t, γ) ∣∣∣ ≤ Ct−k−1/2γ−m(σ1σ2)−2ν eiτλ1+iγλ ×λ1∂kλ2 ρ(λ21, λ 2)Jν(σ1λ1)Jν(σ2λ2) dλ1dλ ∣∣∣ dτ, (B.28) ∣∣∣W (2)1 (σ1, σ2, t, γ) ∣∣∣ ≤ Ct−k−1/2γ−m(σ1σ2)−2ν eiτλ2+iγλ ×λ2∂mλ2 ρ(λ21, λ 2)Jν(σ1λ1)Jν(σ2λ2) dλ2dλ ∣∣∣ dτ. (B.29) Recall now that the function Jν is of the form Jν(z) = eizb+ν (z) + e−izb−ν (z), where b±ν (z) are symbols of order (n− 3)/2 for z ≥ 1, while near z = 0 the function Jν(z) is equal to z2ν times an analytic function. Therefore, it satisfies the bounds ∣∣∂jzJν(z) ∣∣ ≤ Czn−2−j〈z〉j−(n−1)/2, ∀z > 0, 0 ≤ j ≤ n− 2, (B.30) ∣∣∂jzJν(z) ∣∣ ≤ Cj〈z〉(n−3)/2, ∀z > 0, j ≥ 0. (B.31) Moreover, the functions b±ν (z) are of the form (near z = 0) b±ν (z) = b ν,1(z) + z n−2 log z b±ν,2(z), where b±ν,j(z) are analytic functions, b ν,2(z) ≡ 0 if n is odd. Therefore, we have ∣∣∂jzb±ν (z) ∣∣ ≤ C, 0 < z ≤ 1, 0 ≤ j ≤ n− 3, ∣∣∂jzb±ν (z) ∣∣ ≤ Cǫz−ǫ, 0 < z ≤ 1, j = n− 2, ∣∣∂jzb±ν (z) ∣∣ ≤ Cjzn−2−j, 0 < z ≤ 1, j ≥ n− 1, which imply ∣∣∂jzb±ν (z) ∣∣ ≤ C〈z〉(n−3)/2−j , ∀z > 0, 0 ≤ j ≤ n− 3, (B.32) ∣∣∂jzb±ν (z) ∣∣ ≤ Cǫz−ǫ〈z〉−(n−1)/2+ǫ, ∀z > 0, j = n− 2, (B.33) ∣∣∂jzb±ν (z) ∣∣ ≤ Cjzn−2−j〈z〉−(n−1)/2, ∀z > 0, j ≥ n− 1. (B.34) ± (λ1, λ2, σ1, σ2) = λ1e ∓iσ1λ1∂kλ2 ρ(λ21, λ ±iσ1λ1b±ν (σ1λ1)Jν(σ2λ2) ± (λ1, λ2, σ1, σ2) = λ2e ∓iσ2λ2∂mλ2 ρ(λ21, λ 2)Jν(σ1λ1)e±iσ2λ2b±ν (σ2λ2) By (B.26), (B.30)-(B.34), we have (with ℓ = 0, 1) ∣∣∣∂ℓλ1A ± (λ1, λ2, σ1, σ2) ≤ C〈σ1〉k+(n−3)/2σn−22 〈σ2〉m−(n−1)/2〈λ1〉(n−3)/2−k−1, ∀(λ1, λ2), (B.35) ∣∣∣∂ℓλ2A ± (λ1, λ2, σ1, σ2) ≤ Cσn−21 〈σ1〉m−(n−1)/2〈σ2〉k+(n−3)/2〈λ1〉(n−3)/2−m−2, ∀(λ1, λ2). (B.36) Using the inequality ‖ϕ̂(τ)‖L1 ≤ C‖〈τ〉ϕ̂(τ)‖L2 ≤ C ∥∥∂ℓλϕ(λ) ∣∣∂ℓλϕ(λ) we obtain from (B.28) and (B.35) (if k > (n− 3)/2) ∣∣∣W (1)1 (σ1, σ2, t, γ) ∣∣∣ ≤ Ct−k−1/2γ−m(σ1σ2) eiτλ1+iγλ ± (λ1, λ2, σ1, σ2)dλ1dλ ∣∣∣∣ dτ Ct−k−1/2γ−m(σ1σ2) −2ν sup λ1,λ2 ∣∣∣∂ℓλ1A ± (λ1, λ2, σ1, σ2) ≤ Ct−k−1/2γ−mσ−(n−2)/21 〈σ1〉k+(n−3)/2〈σ2〉m−(n−1)/2, (B.37) where we have made a change of variables τ → τ ± σ1. Similarly, by (B.29) and (B.36), we get (if m > (n− 3)/2) ∣∣∣W (2)1 (σ1, σ2, t, γ) ∣∣∣ ≤ Ct−k−1/2γ−m(σ1σ2) eiτλ2+iγλ ± (λ1, λ2, σ1, σ2)dλ2dλ ∣∣∣∣ dτ Ct−k−1/2γ−m(σ1σ2) −2ν sup ∣∣∣∂ℓλ2A ± (λ1, λ2, σ1, σ2) ∣∣∣ dλ21 ≤ Ct−k−1/2γ−mσ−(n−2)/22 〈σ2〉k+(n−3)/2〈σ1〉m−(n−1)/2. (B.38) We would like to apply (B.37) and (B.38) with k = (n−1)/2,m = (n−3)/2+ǫ, 0 < ǫ≪ 1. To this end, we need to show that these estimates are valid for all real (n−3)/2 < m ≤ (n−2)/2, (n−2)/2 ≤ k < n/2 if n is even, and for k = (n−1)/2 and all real (n−3)/2 < m ≤ (n−1)/2 if n is odd. This can be done by interpolation as follows. Let φ ∈ C∞0 (R), φ(λ) = 1 for |λ| ≤ 1, φ(λ) = 0 for |λ| ≥ 2. Decompose W (j)1 as X(j) + Y (j), j = 1, 2, where X(j) and Y (j) are defined by replacing in the definition of W 1 the function ρ by φ(λj)ρ and (1−φ)(λj)ρ, respectively. Clearly, the functions X(j) satisfy (B.37) and (B.38), respectively, for all integers 0 ≤ k,m < n/2, while the functions Y (j) satisfy (B.37) and (B.38) for all integers k > (n − 3)/2, (n − 3)/2 < m < n/2, respectively. When n is odd, this is fulfilled with k = (n− 1)/2. To show this in the case of even n, we write the function φ as φ(λ) = with some function φ1 ∈ C∞0 (R), φ1(λ) = 0 in a neighbourhood of λ = 0. Thus, X(j) = X(j)p , whereX p is defined by replacing in the definition ofX (j) the function φ(λj) by φ1(2 pλj). As above, one can see that the functions X p , j = 1, 2, satisfy (B.37) and (B.38), respectively, with an extra factor in the RHS of the form 2p(k−n/2) for all integers k ≥ (n − 2)/2, and hence, by interpolation, for all real k ≥ (n−2)/2. Therefore, summing up these estimates we conclude that X(j), j = 1, 2, satisfy (B.37) and (B.38), respectively, for all real (n− 2)/2 ≤ k < n/2, and in particular for k = (n− 1)/2. Hence, so do the functions W (j)1 . Furthermore, 1 satisfies (B.37) for all integers 0 ≤ m < n/2, and hence, by interpolation, for all real 0 ≤ m ≤ (n− 1)/2 if n is odd, and for all real 0 ≤ m ≤ (n− 2)/2 if n is even. In particular, this is valid with m = (n − 3)/2 + ǫ. To show that the function W (2)1 satisfies (B.38) with m = (n − 3)/2 + ǫ, we decompose it as Z +N , where Z and N are defined by replacing in the definition of W 1 the function ρ by φ(λ1)ρ and (1 − φ)(λ1)ρ, respectively. Clearly, the function Z satisfies (B.37) for all integers 0 ≤ m < n/2, and hence, by interpolation, for all real 0 ≤ m ≤ (n − 1)/2 if n is odd, and for all real 0 ≤ m ≤ (n− 2)/2 if n is even. To deal with the function N , we write the function 1− φ as (1 − φ)(λ) = −pλ), with some function φ2 ∈ C∞0 (R), φ2(λ) = 0 in a neighbourhood of λ = 0. Thus, where Np is defined by replacing in the definition of N the function (1−φ)(λ1) by φ2(2−pλ1). Now, the functionsNp satisfy (B.38) with an extra factor in the RHS of the form 2 −p(m−(n−3)/2) for all integers 0 ≤ m < n/2, and hence, by interpolation, for all real 0 ≤ m ≤ (n − 1)/2 if n is odd, and for all real 0 ≤ m ≤ (n − 2)/2 if n is even. Therefore, summing up these estimates we conclude that N satisfies (B.38) for all real (n− 3)/2 < m ≤ (n − 1)/2 if n is odd, and for all real (n− 3)/2 < m ≤ (n− 2)/2 if n is even. In particular, this is valid with m = (n− 3)/2 + ǫ. By (B.37) and (B.38) with k = (n− 1)/2, m = (n− 3)/2 + ǫ, we obtain |W1(σ1, σ2, t, γ)| ≤ Ct−n/2γ−(n−3)/2−ǫ σ−n+21 〈σ1〉n−2〈σ2〉−1+ǫ + σ 2 〈σ2〉n−2〈σ1〉−1+ǫ ≤ Ct−n/2γ−(n−3)/2−ǫ σ−n+21 + σ 2 + 〈σ1〉−1+ǫ + 〈σ2〉−1+ǫ ≤ Ct−n/2γ−(n−3)/2−ǫ σ−n+21 + σ 2 + σ 1 + σ which is the desired bound. ✷ Taking γ = hβ with a suitably chosen constant β′ > 0, we deduce from (B.4), (B.16)- (B.19) and (B.22), ∥∥Ψ(t;h)f − F (t)ψ(h2G0)f ≤ Chβt−n/2‖f‖L1 + Chβ ∥∥Ψ(t;h)f − F (t)ψ(h2G0)f , ∀f ∈ L1, (B.39) with some constant β > 0. Taking h small enough, we can absorb the second term in the RHS of (B.39), thus obtaining (B.5). ✷ References [1] M. Goldberg, Dispersive bounds for the three dimensional Schrödinger equation with almost critical potentials, GAFA 16 (2006), 517-536. [2] M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Commun. Math. Phys. 251 (2004), 157-178. [3] M. Goldberg and M. Visan, A conterexample to dispersive estimates for Schrödinger operators in higher dimensions, Commun. Math. Phys. 266 (2006), 211-238. [4] J.-L. Journé, A. Sofer and C. Sogge, Decay estimates for Schrödinger operators, Commun. Pure Appl. Math. 44 (1991), 573-604. [5] W. Schlag, Dispersive estimates for Schrödinger operators in two dimensions, Com- mun. Math. Phys. 257 (2005), 87-117. [6] G. Vodev, Dispersive estimates of solutions to the Schrödinger equation, Ann. H. Poincaré 6 (2005), 1179-1196. [7] G. Vodev, Dispersive estimates of solutions to the Schrödinger equation in dimensions n ≥ 4, Asymptot. Anal. 49 (2006), 61-86. [8] G. Vodev, Dispersive estimates of solutions to the wave equation with a potential in dimensions n ≥ 4, Commun. Partial Diff. Equations 31 (2006), 1709-1733. [9] K. Yajima, The W k,p continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan 47 (1995), 551-581. [10] K. Yajima, Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue, Commun. Math. Phys. 259 (2005), 475-509. Université de Nantes, Département de Mathématiques, UMR 6629 du CNRS, 2, rue de la Houssinière, BP 92208, 44332 Nantes Cedex 03, France e-mail: [email protected] e-mail: [email protected] Introduction and statement of results Preliminary estimates Proof of Theorem 1.1 Appendix 1 Appendix 2

0704.1201

Evaluation of the Axial Vector Commutator Sum Rule for Pion-Pion Scattering

arXiv:0704.1201v3 [hep-ph] 1 Jun 2007 FTUAM 06-07 April 9, 2007 arXiv: 0704.1201 [hep-ph] Evaluation of the Axial Vector Commutator Sum Rule for Pion-Pion Scattering Stephen L. Adler Institute for Advanced Study Einstein Drive Princeton, NJ 08540, USA F. J. Ynduráin Departamento de F́ısica Teórica, C-XI Universidad Autónoma de Madrid, Canto Blanco, E-28049, Madrid, Spain Abstract We consider the sum rule proposed by one of us (SLA), obtained by taking the expectation value of an axial vector commutator in a state with one pion. The sum rule relates the pion decay constant to integrals of pion-pion cross sections, with one pion off the mass shell. We remark that recent data on pion- pion scattering allow a precise evaluation of the sum rule. We also discuss the related Adler–Weisberger sum rule (obtained by taking the expectation value of the same commutator in a state with one nucleon), especially in connection with the problem of extrapolation of the pion momentum off its mass shell. We find, with current data, that both the pion-pion and pion-nucleon sum rules are satisfied to better than six percent, and we give detailed estimates of the experimental and extrapolation errors in the closure discrepancies. http://arxiv.org/abs/0704.1201v3 Typeset with PHys MaTEX -evaluation of the axial vector commutator sum rule for pion-pion scattering- 1. Introduction. In refs. 1, 2 a sum rule was obtained by taking matrix elements of the commutator, [χ+(t), χ−(t)] = 2I3, (1.1) between one-nucleon states. Here χ± are the chiral charges, χ±(x0) = d3xA0 I3 is the third component of isospin and the A are axial currents. In terms of quarks, +(x) = ū(x)γ µγ5d(x), A (x) = d̄(x)γµγ5u(x). One may note that, in QCD, the commutation relation (1.1) is an exact theorem, that follows from the global symmetries of the QCD Lagrangian; in fact, (1.1) may be obtained by integrating the local axial vector commutation relation δ(x0 − y0)[A0+(x), Aν−(y)] = 2δ4(x− y)V ν3 (x) with V ν3 (x) = (ūγνu − d̄γνd) the third component of the vector (isospin) current. This commutation relation is exact in QCD, irrespective of the value of the u, d quark masses. The sum rule obtained in refs. 1, 2 relates the axial weak charge, gA, to integrals over pion-nucleon cross sections, with the pion off-shell (and of zero mass), g2rKNNπ(0) (MN+Mπ)2 s−M2N σ+0p(s)− σ 0p(s) . (1.2) Here, σ±0p is the total cross section for scattering of a zero mass π ± on a proton, this latter on its mass shell (with averages over the spin of the nucleon implicit). Later, in ref. 3, it was noted that one can also take matrix elements of (1.1) between one pion states. In this way one obtains the relation g2rKNNπ(0) s−M2π σ−0π(s)− σ 0π(s) , (1.3) where σ±0π are the respective total cross sections for scattering of a zero mass π ± on an on-shell π−. Both in (1.2) and (1.3) the exact Goldberger–Treiman relation[4] (see Sect. 2 below for the meaning of this) grKNNπ(0)Fπ was used to relate the χ± to the pion field. In these formulas, Mπ = 139.57 MeV is the (charged) pion mass, MN the nucleon mass, gA is the weak axial coupling, gr the on-shell strong NNπ coupling, and KNNπ(0) the nucleon-nucleon-pion vertex, for a zero mass pion, normalized to KNNπ(M π) = 1. Fπ is the pion decay constant. One can rewrite (1.2), (1.3) in a form that is more convenient by expressing the constant prefactor in terms of Fπ, corresponding to writing the PCAC definition of the pion field in terms of Fπ, ∂ ·A± = πφ± (1.4) (φ± are the fields for π ±) and also expressing the sum rule in terms of scattering amplitudes, rather than cross sections. In this case we find that (1.2), (1.3) are replaced by the relations g2A = 1 + 8πF (MN+Mπ)2 (s−M2N)2 Im Fπ+,p(k 2 = 0; s, t = 0)− Im Fπ−,p(k2 = 0; s, t = 0) (1.5) 1 = 4πF 2π (s−M2π)2 Im Fπ+,π−(k 2 = 0; s, t = 0)− Im Fπ−,π−(k2 = 0; s, t = 0) . (1.6) – 1 – -s. l. adler and f. j. ynduráin- Here Fπ±,p(k 2, s, t = 0) stands for the forward pion-proton scattering amplitude, π±(k) +N(p) → π±(k) +N(p) with the nucleon N(p) on its mass shell, p2 = M2N , while the momentum of the pion is off-shell, k 2 = 0 in (1.5). Likewise, Fπ±,π−(k 2 = 0; s, t = 0) is the forward scattering amplitude for a π±(k) with momentum k2 = 0 on a π−(p) on its mass shell, p2 = M2π : π±(k) + π−(p) → π±(k) + π−(p). In the form (1.6), the pionic sum rule is seen as a remarkable relation that allows one to calculate the pion decay constant in terms of (off-shell) pion-pion scattering amplitudes. In ref. 3, this sum rule for the pion case could not be fully evaluated. In 1965 data for ππ scattering in the S0 wave were so scanty that it was only possible to note that fulfillment of the sum rule requires a strong ππ interaction in this S0 wave. The situation has improved dramatically at present; precise and reliable experimental data on the P and S0 waves for ππ scattering, and also on high energy scattering, have appeared in the last years, which will allow us to present an evaluation of (1.3), (1.6) to a few percent accuracy. The nucleon sum rule, known as the Adler–Weisberger sum rule, (1.2), (1.5) was already evaluated in refs. 1, 2, 3; however, we will present here a new calculation: the precision of the pionic sum rule is such that the question of the extrapolation to the off-shell pion in (1.3), (1.6) becomes important, and the corresponding discussion is illuminated by considering also what happens in the nucleon case. In the present paper we will only consider the sum rules (1.5), (1.6); but, by taking expectation values of (1.1) between kaon states one can get a sum rule identical to (1.5), replacing the g2A in the left hand side by zero, the nucleon mass by the kaon mass, and pion-nucleon by pion-kaon amplitudes: e.g., for a K+ state, 1 = 8πF 2π (MK+Mπ)2 (s−M2K)2 Im Fπ−,K+(k π = 0; s, t = 0)− Im Fπ+,K+(k2π = 0; s, t = 0) . (1.7) It is also possible to find a complementary relation to (1.7), but extrapolating in the mass of the kaon. Consider the currents A = ūγµγ5s, A = s̄γµγ5u; A = d̄γµγ5s, A = s̄γµγ5d, and the corresponding chiral charges, χK+ , χK− ; χK0 , χK̄0 . We have [χK+ , χK− ]− [χK0 , χK̄0 ] = 2I3. Taking expectation values on a π− state we now find the sum rule 1 = 4πF 2K (MK+Mπ)2 (s−M2π)2 Im FK+,π−(k K = 0; s, t = 0) + FK̄0,π−(k K = 0; s, t = 0) − Im FK−,π−(k2K = 0; s, t = 0)− Im FK0,π−(k2K = 0; s, t = 0) =8πF 2K (MK+Mπ)2 (s−M2π)2 Im FK+,π−(k K = 0; s, t = 0)− Im FK+,π+(k2K = 0; s, t = 0) (1.8) the last relation using isospin and charge conjugation invariance. We here leave the pion on its mass shell, and extrapolate in the kaon mass. Although there are some studies on pion-kaon scattering, variously using chiral perturbation theory, dispersive techniques and phenomenological information (see ref. 5 at low energies, and at high energies, the Regge calculations of ref. 10), the experimental information on πK scattering amplitudes is much less precise than that for ππ scattering. However, having two different masses to extrapolate (Mπ and MK) would perhaps give some insight on the matter of extrapolation: the only differences between (1.7) and (1.8) are FK = 113.00 ± 1.06 MeV instead of Fπ, the different masses that are sent to zero, and the masses that appear in the denominators in the integrals. Similarly, as discussed by Weisberger, ref. 5, by taking appropriate chiral charges in SU(3), and taking expectation values on nucleons one obtains relations involving kaon-nucleon scattering. The pion- pion, pion-kaon or pion-nucleon sum rules require only an extrapolation in the pion mass to yield an on-shell relation, while the kaon-nucleon [or kaon-pion, Eq. (1.8)] sum rule requires extrapolation in the much larger kaon mass to give a physical relation, which is therefore expected to be less precise than the ones studied here, but could also give information on the question of extrapolation on the meson masses. Nevertheless, and as already stated we will, in the present paper, concern ourselves only with the pion-pion and pion-nucleon – 2 – -evaluation of the axial vector commutator sum rule for pion-pion scattering- sum rules, for which good experimental data exist, and which require extrapolation only to M2π ; leaving the rest for future work. 2. The Goldberger–Treiman relation Although the Goldberger–Treiman relation[4] is not the object of this paper, we say a few words about it as we will use some information connected with it. We first have what may be called the exact or off-shell Goldberger–Treiman relation, KNNπ(0)grFπ = 1. (2.1) This involves the pion-nucleon coupling, with the pion off its mass shell (with momentum k2 = 0). One can define the quantity (the G.T. discrepancy) ∆G.T. ≡ 1− , (2.2) which measures the validity of the approximationKNNπ(0) ≃ KNNπ(M2π) = 1 that is used to get the on-shell Goldberger–Treiman relation Fπ = MNgA/gr. With the present values gA = 1.2695 ± 0.0029, Fπ = 92.42 ± 0.26 (both from the Particle Data Tables[6]) and with MN = 938.9 (average n-p mass) and taking the on-shell pion nucleon coupling constant gr = 13.2± 0.2, the relation (2.2) equals ∆G.T. = 0.023± 0.016, (2.3) so the effect of approximating KNNπ(0) by unity appears to be small. This corrects most of the mismatch studied by Pagels and Zepeda,[8] which turns out to be largely due to an underestimated gA (a possibility that they actually considered); the remainder in (2.3) can easily be attributed, as was done in ref. 8, to the contribution of the π(1300) resonance to KNNπ(0), expected to be of O[M π(1300) ] ∼ 1%. One can look at the off-shell Goldberger–Treiman relation in a different way, as providing the value of the quantity KNNπ(0): (2.1) tells us that KNNπ(0) = = 0.977± 0.015. (2.4) In fact, from the careful analysis of Pagels and Zepeda, it follows that most of the deviation of KNNπ(0) from unity is due to the fact that, since the pion is off-shell, and the corresponding Green’s function is not amputated, the NNπ vertex must contain a factor M2πΠ(0), where Π(k 2) is the pion propagator normalized to 1 = (M2π − k2)Π(k2)|k2=M2π . Thus, one expects KNNπ(0) ≃ M2πΠ(0), (2.5) which will play a role in the extrapolation discussion for the sum rules below. In fact, the same factor M2πΠ(0) will appear in the sum rules because, both in the pion-pion and pion-nucleon cases, the propagator corresponding to the off-shell pion line is not amputated. Use of ∆G.T. to estimate off-shell extrapolation corrections has also been discussed by Dominguez.[8] 3. Calculation of the sum rule on pions 3.1. The sum rules We present here a sketch of the derivation of the sum rules, for ease of reference; more details may be found in refs. 1, 2, 3. We will treat in detail the pionic case derived in ref. 3, and indicate the modifications necessary for the nucleon case. – 3 – -s. l. adler and f. j. ynduráin- We first take the expectation value of the commutator (1.1) between physical π− states, and intro- duce a sum over a complete set of states: we find 〈π−(p′)|2I3|π−(p)〉 = − 2× 2p0δ(p− p′) d(q2) 〈π−(p′)|χ+(t)|q; INT〉〈q; INT|χ−(t)|π−(p)〉 − (+ ↔ −), (3.1) where by |q; INT〉 we denote a physical intermediate state with total momentum q and internal degrees of freedom “INT”. Next, using the PCAC definition (1.4), one can relate 〈q; INT|χ±(x0)|π−(p)〉 = p0 − q0 π〈q; INT| d3xφ±(x)|π−(p)〉 with φ±(x) the field operator for π ±. Working in the infinite momentum frame (p → ∞) gives the sum rule 1 = 2πF 2π (s−M2π)2 |F (π+(k2 = 0), π−(p) → q; INT)|2 − |F (π−(k2 = 0), π−(p) → q; INT)|2 (3.2a) here s = q2, and F (π±(k2 = 0), π−(p) → q; INT) = (2π)5/2〈q; INT|M2πφ±(0)|π−(p)〉 (3.2b) is the amplitude for a pseudoscalar current, with virtual four momentum k, k2 = 0, to scatter off a physical π−, p2 = M2π , into the physical intermediate state |q; INT〉. Using extended unitarity, this may be written as the sum rule (1.6), which we repeat here in the form of a discrepancy, ∆π = 0, with ∆π ≡ 4πF 2π (s−M2π)2 Im Fπ+,π−(k 2 = 0; s, t = 0)− Im Fπ−,π−(k2 = 0; s, t = 0) − 1. (3.3) In these formulas, the Fπ±,π−(k 2 = 0; s, t = 0) are the forward scattering amplitudes for an off-shell pion with zero mass. For an on-shell pion, the corresponding scattering amplitude is obtained by replacing, in (3.2b), F (π±(k2 = 0), π−(p) → q; INT) = (2π)5/2〈q; INT|M2πφ±(0)|π−(p)〉 = (2π)5/2〈q; INT|(M2π + )φ±(0)|π−(p)〉 → (2π)5/2〈q; INT|(M2π + )φ±(0)|π−(p)〉 k2=M2 (3.4) and using unitarity to perform the sum over intermediate states. In QCD one expects the mass scale for the internal dynamics to be given by a parameter µ0 of the order of the parameter Λ ∼ 0.4 GeV or the rho resonance mass, Mρ; thus, to an error M2π/µ20 ∼ 10% or smaller, we can relate (3.3) to physical quantities by approximating the off-shell scattering amplitudes by the physical scattering amplitudes as in (3.4). The sum rule (1.2), (1.5) on nucleons is derived in a similar manner. The only differences are the replacement of the pion mass by the nucleon mass in the denominator corresponding to (3.3), the different isospin of the proton (that results in a factor −1/2 with respect to that for the π−) and that, due to the existence of the neutron intermediate state, we find the extra term proportional to g2A in (1.5) because the proton-neutron matrix element of the divergence 〈p|(∂ · A|n〉 is proportional to gA. The pion-kaon and kaon-pion sum rules are also similar to the pion-pion one. 3.2. The pion sum rule in the on-shell approximation In the form (3.3), the sum rule is an exact theorem, following from the commutation relation (1.1) and the definition (1.4); but, of course, direct comparison with experiment is precluded by the fact that the amplitudes that appear in (3.3) involve a pion off its mass shell. As stated in the previous subsection, a first approximation is obtained by neglecting the fact that (3.3) is defined for off-shell pseudoscalar currents, i.e., working in the approximation (3.4). The sum rule is then, – 4 – -evaluation of the axial vector commutator sum rule for pion-pion scattering- 1 ≃ 4πF 2π (s−M2π)2 {ImFπ−(s)− Im Fπ+(s)} (3.5) with F±(s) physical, forward π ±π− scattering amplitudes. One can write these scattering amplitudes in terms of amplitudes with well-defined isospin in the t or s channels, Fπ−(s)− Fπ+(s) = F (It=1) = 13F (Is=0) + 1 F (Is=1) − 5 F (Is=0); (3.6) the F are normalized so that, for pions on their mass shell, and in the elastic region, one has F (Is) = 2 2s1/2 (2l + 1) sin δ l . (3.7) The factor 2 in front of the right hand side is due to identity of the particles and k = 1 s− 4M2π is the center of mass momentum, for physical pions. In this case, we can use the precise determinations of the pion-pion scattering amplitudes, obtained fitting experimental data, that have been found recently[9,10] thanks to the availability of very precise data: on the low energy S0 wave from kaon decays and, for the P wave, from determinations of the pion form factor.[11] One finds, for the contributions of the various waves to the right hand side of (3.5) for energy below 1420 MeV,[9] S0; s1/2 ≤ 932 MeV : 0.408± 0.013 S0; 932 ≤ s1/2 ≤ 1420 MeV : 0.043 D0 : 0.097± 0.003 P : 0.403± 0.003 F : 0.0016 S2 : −0.090± 0.005 D2 : −0.0023; (3.8a) errors are only given for the more significant pieces. The results are similar to those already obtained in ref. 3 (although, of course, now much more precise) for the contributions of P, D0, S2 waves. What is new is the contribution of the S0 wave, which turns out to be the most important of all, thus confirming the prediction in ref. 3 that an important S0 wave contribution is needed to saturate the sum rule. For the (Regge) contribution above 1420 MeV we use the Regge formula ImF (1)(s) = (1.22± 0.14)(s/1 GeV2)0.42 and then find[10] s1/2 > 1420 MeV : 0.167± 0.017. (3.8b) Altogether, the right hand side of (3.5) now reads 1.027± 0.022, i.e., ∆π = 0.027± 0.022. (3.9) The error is due to the experimental errors in the pion-pion scattering amplitudes in (3.8). Therefore, we only have a discrepancy of (2.7± 2.2)% in the fulfillment of the sum rule in this approximation. A few words may be said on the smallness of the error in (3.9). This is due to the fact that, as stated, recent experimental results have allowed us to get very precise fits to data at low energy. Moreover, the larger contributions to the final result come from independent sources, so one can add their errors quadratically. And, finally, all the large contributions (S0, P, D0 waves and Regge region) are positive: only the relatively small S2 and D2 contributions produce cancellations. As we will see, the situation is less favourable for the sum rule on nucleons, where large cancellations take place. – 5 – -s. l. adler and f. j. ynduráin- 3.3. Extrapolation To improve the evaluation using (3.5) one can calculate by taking the recipe of ref. 3 for extrapolation, which takes into account threshold kinematic effects by replacing (3.7), in the elastic region,1 by 4s1/2 (2l + 1) sin δ l , (3.10) where k(0) = (s −M2π)/2s1/2 is the center of mass momentum for a pion of zero mass incident on a target pion that is on mass shell. This does not take into account the effects of the extrapolation at high energies, s1/2 > 1.42 GeV, likely below the 1% level, which we neglect. We have verified that this recipe works in a model calculation in which the interactions are generated by effective Lagrangians gρ(~φ× ∂ µ~φ)ρµ, gf ~φ( gµν − ∂µ∂ν)~φfµν2 , . . . coupling pions to various resonances [ρ, f2(1275), . . . ]. In these models, the off shell correction is valid not only at threshold but throughout the resonance region, which justifies our using it here in the elastic region. The recipe in (3.10) then gives a discrepancy ∆π = 0.069± 0.023. (3.11) This deteriorates the sum rule, which seems to imply that the dynamical effects of the extrapolation are not negligible. In fact, this could be expected; the replacement (3.10) does not affect the S waves, in particular the S0 wave, which is the one that contributes most to the sum rule. A possible way to estimate at least part of the dynamical correction is to assume it to be universal, and take it from the Goldberger–Treiman relation: thus multiplying the right hand side of (3.11) by the factor K2NNπ(0) = 0.955±0.03, as was done in ref. 3; see also ref. 8. A motivation for this was already given when we discussed the Goldberger–Treiman relation (end of Sect. 2); the motivation for this in ref. 3 was the observation that in a field theory of pions and nucleons, a zero mass pion must couple to a physical pion through a virtual nucleon loop, and so the factor KNNπ(0) should be present (of course, both motivations are not exclusive, but complementary). In terms of QCD, an analogous observation is that an off-shell pion couples to both nucleons and pions through a coupling to a single quark line, suggesting that a universal off-shell factor may be present. It would be of interest to pursue this idea further within a QCD framework; see also the discussion in Sect. 5 below. Including this factor K2NNπ(0) improves the fulfillment of the sum rule to ∆π = 0.021± 0.023, (3.12) an agreement as good as one can wish. However, as we will see in the case of the nucleon sum rule, following the same procedure of including a K2NNπ(0) factor depreciates, rather than improves, agreement with experiment. Hence a more conservative procedure is to take the difference between (3.11) and (3.12) as a measure of the uncertainty in the extrapolation procedure and thus write for the discrepancy, Eq. (3.3b), ∆π = 0.021± 0.023 (Exp.)± 0.048 (Extr.), (3.13) showing explicitly the error arising from experimental errors in the pion-pion amplitudes and the estimated error of the extrapolation procedure. This will be our final result for the sum rule on pions. 3.4. Connection with chiral perturbation theory Consider the forward dispersion relation for the (physical) amplitude F (It=1)(s), F (It=1)(s) = 2s− 4M2π ImF (It=1)(s′) (s′ − s)(s′ + s− 4M2π) 1 Actually, we make the corresponding replacement up to the Regge region, s1/2 ≃ 1.42 GeV. – 6 – -evaluation of the axial vector commutator sum rule for pion-pion scattering- Evaluating it at threshold, and writing F (It=1)(4M2π) in terms of scattering lengths, we find the so-called (first) Olsson sum rule[12] [which is thus an exact consequence of the dispersion relation, independent of the current algebra commutator (1.1)]: 0 − 5a 0 = 3Mπ ImF (It=1)(s) s(s− 4M2π) . (3.14) One can evaluate the scattering lengths to lowest order (l.o.) in chiral perturbation theory, i.e., to lowest order in M2π (actually, this is strictly equivalent to the old Weinberg [13] evaluation) finding 0 − 5a 4πF 2π and, substituting in (3.14), the l.o. relation 1 = 4πF 2π ImF (It=1)(s) s(s− 4M2π) . (3.15) To l.o. (which implies M2π ≈ 0) Eq. (3.15) is equivalent to (1.6) since ImF (It=1)(s) s(s− 4M2π) ImF (It=1)(s) (s−M2π)2 ImF (It=1)(k2 = 0; s, t = 0) (s−M2π)2 The same conclusion is reached if we use the Froissart–Gribov representation of the P wave scattering length. Of course, (3.15) differs in status from (1.6) as the latter is valid exactly, whereas the l.o. expression 0 − 5a = 3Mπ/4πF π is known to have large loop corrections. Another connection with results based on chiral perturbation theory is obtained by remarking that, using chiral methods to two loops, plus analyticity (in the form of Roy equations) and extra experimental information, has led Colangelo, Gasser and Leutwyler[15] to propose parametrizations of the S0, S2 and P waves at low energy consistent with these requirements. Substituting them into Eq. (3.5) gives the result ∆π = 0.046, to be compared with what we found using experimental data in Eq. (3.9), 0.027±0.022. Likewise, implementing off-shell corrections as in (3.10) gives ∆π = 0.086 [to be now compared with 0.069 ± 0.023 from Eq. (3.11)] and, finally, applying the correction deduced from the off-shell Goldberger–Treiman relation under the universality assumption gives ∆π = 0.033, a number which is also similar to what was found using experimental ππ data in Eq. (3.12), viz., 0.021± 0.023.2 We emphasize that it is not our intention in this paper to give a detailed review of either chiral perturbation theory or the Olsson sum rule, which are substantial topics in their own right. We are only interested in elucidating the connection between the Olsson relation and the pion-pion sum rule (1.6), which, as far as we know, can only be established to leading order in chiral perturbation theory. Tests of the Olsson relation by itself have been made extensively in the literature; see, for example, refs. 14, 15 for evaluations using two-loop chiral perturbation theory, and the papers of Kamiński, Peláez and Ynduráin and of Peláez and Ynduráin in ref. 9 for calculations using experimental data prior to 2006. The Olsson relation has also been verified including more recent data (those of the NA48/2 collaboration, ref. 11) as in our evaluation of the pion-pion sum rule here, by Kamiński, Peláez, and Ynduráin (unpublished work in progress). 4. The Adler–Weisberger sum rule 4.1. An approximate calculation The Adler–Weisberger sum rule (1.2), (1.5), has been evaluated in a number of papers (besides the original ones); a recent article is ref. 17. If we approximate the amplitudes in (1.5) by the scattering amplitudes for 2 The numbers 0.046, 0.086 and 0.033 were kindly communicated to us by H. Leutwyler[16] as an independent check on the evaluation of Sect. 3.2, using different methods. – 7 – -s. l. adler and f. j. ynduráin- a physical pion, k2 = M2π , then we get the sum rule g2A ≃ 1 + 8πf2π (MN+Mπ)2 (s−M2N)2 ImFπ+(s)− Im Fπ−(s) (4.1) with Fπ±(s) the physical, forward π ±p scattering amplitudes, normalized so that the pion-proton cross sections are σπ±p(s) = λ1/2(s,M2π ,M ImFπ±(s), with the function λ(a, b, c) as defined below. In terms of s-channel isospin amplitudes, we have Fπ+(s)− Fπ−(s) = 23 F (Is=3/2) − F (Is=1/2) where the partial wave expansion in the elastic region is F (I)(s) = 2s1/2 (l + 1) sin δ l+ + l sin δ ; (4.2) the phase shifts δ l± correspond to isospin I, orbital angular momentum l, and total angular momentum j = l± 1 . The center of mass momentum is k = 1 λ(s,M2π ,M , λ(a, b, c) = [a2 − (b + c)2][a2 − (b− c)2]. The high energy part, s1/2 > 2374 MeV of the integral in (4.1) is easily evaluated with the fit to πN cross sections in ref. 10: Regge formulas are good approximations for kinetic energies above 1 GeV. We will use the Regge formula ImFπ+(s)− Im Fπ−(s) = (0.42± 0.04)(s/1 GeV2)0.42, and integrate this from s1/2 = 2374 MeV to infinity. This would give a result of −0.091. Alternatively, we can use numerical values of the cross sections given in ref. 18 from s1/2 = 2374 MeV to s1/2 = 3004 MeV, and use the Regge formula above this latter energy. This would give −0.087. We consider this last number to be the more reliable one and then write, taking errors into account, 8πf2π (2374 MeV)2 (s−M2N )2 ImFπ+(s)− Im Fπ−(s) = −0.087± 0.006. (4.3a) On the other hand, numerical evaluation of the low energy piece, using the numerical cross sections collected in ref. 18 gives 8πf2π ∫ (2374 MeV)2 (MN+Mπ)2 (s−M2N )2 ImFπ+(s)− Im Fπ−(s) = 0.460± 0.024. (4.3b) This large error is due to the fact that the number in the right hand side is the difference between two large numbers: specifically, 0.46 = 1.71 [from π+]− 1.25 [from π−]. (4.4) Substituting the value of gA the sum rule (4.1) reads 1.612± 0.006 = 1 + 0.373± 0.025. (4.5) We may define a discrepancy ∆A.W. as the difference between g A and the right hand side of Eq. (1.5), and express the result in (4.5) as a largish mismatch, ∆A.W. = 0.239± 0.025. In the present case, there are various substantial cancellations: as already stated, there are cancel- lations between the π+p and π−p cross sections at low energy, but there is also a cancellation between the – 8 – -evaluation of the axial vector commutator sum rule for pion-pion scattering- low energy region and higher energy (s1/2 > 1.390 GeV) contributions. For example, if we only integrated up to and including the ∆(3, 3) resonance region, we would have obtained 8πf2π ∫ (1390 MeV)2 (MN+Mπ)2 (s−M2N )2 ImFπ+(s)− Im Fπ−(s) = 0.70. These cancellations amplify the errors in the sum rule, and indicate that the effects of the extrapolation to a zero mass pion, which affect mostly low energies, are now very important, as already remarked in refs. 1 and 3. 4.2. Extrapolation To perform the extrapolation, we repeat the method used for the case of the sum rule on pions, and replace the expression (4.2) for the scattering amplitude by 0 (s) = 2s1/2 (l + 1) sin δ l+ + l sin δ (4.7) with k(0) the momentum for an incident pion of zero mass, k(0) = 1 λ(s, 0,M2N) We integrate this in the elastic region, s1/2 <∼ 1.5 GeV, using the parametrizations of ref. 19. These parametrizations have been obtained by fitting up to energies of, respectively, 1.3 GeV and 1.38 GeV; however, we have verified that they continue to fit the experimental cross sections up to ∼ 150 MeV above their nominal maximum. Between these energies and 1.9 GeV, one can use a resonance saturation model, with the resonance parameters taken from the PDT;[6] plus a background, estimated as the tail of the lower energy resonances. Above 1.9 GeV, there are not well determined values for the resonance parameters and, moreover, a resonance model will cease to be valid as one is entering the Regge regime. Fortunately, one can likely neglect the effects of the extrapolation above such energy, and so this will be done here. The results one gets for the extrapolation correction are 0.225 : up to 1460 MeV; −0.065 : resonances, from 1460 MeV; 0.005 : background, from 1460 MeV. (4.8) Adding this and considering as an error estimate the variation of the result if we vary the matching point of the parametrizations and the resonance model from 1460 to 1420 MeV or 1520 MeV, we get a correction of 0.165± 0.009, and hence the sum rule becomes 1.612± 0.006 = 1 + (0.373± 0.025) + (0.165± 0.009) = 1.538± 0.034. (4.9) We have added the errors linearly, as they are clearly correlated. The results show reasonable fulfillment of the Adler–Weisberger sum rule, ∆A.W. = 0.074± 0.034. If we include a global correction, as we did in the pionic case, multiplying the r.h.s. of (4.9) by the factor K2NNπ(0) = 0.955± 0.03, the sum rule now deteriorates to 1.612± 0.006 = 1 + [(0.373± 0.025) + (0.165± 0.009)]× (0.955± 0.030) = 1.514± 0.038. (4.10) We can now write our final result, as we did for the pionic case, as (4.10), adding as an extra error the difference between (4.10) and (4.9): ∆A.W. = 0.098± 0.006 [g2A]± 0.031 [Exp.]± 0.035 [Extr.] (4.11) To compare with the results obtained in refs. 1 and 3, we note that these papers did not multiply through by a factor of g2A = 1.612, and included the K NNπ(0) extrapolation factor. Thus, the relevant number to use is [cf. (4.10)] (0.514± 0.038)/1.612 = 0.319± 0.024. The comparable number in refs. 1 and 3 is (4M2N/g r)(R1 + R2 + R3) = 0.254 + 0.155− 0.061 = 0.348, with a roughly estimated error of ±0.025. Hence our current evaluation, and the 1965 evaluation of refs. 1 and 3, are in satisfactory agreement. This should come as no surprise, since good pion-nucleon cross sections were already available in 1965. What – 9 – -s. l. adler and f. j. ynduráin- has changed dramatically since then, and is the motivation for the present paper, is the status of data on pion-pion scattering. 4.3. Connection with lowest order chiral perturbation theory As we did for the pion case, the Adler–Weisberger relation can be connected with (lowest order) chiral perturbation theory, by comparing e.g., (4.1) with the forward dispersion relation for the πp scattering amplitude for exchange of isospin unity at threshold,3 evaluating the amplitude at threshold in terms of the scattering lengths combination a3 − a1, and calculating the latter in terms of Fπ, as in the Tomozawa– Weinberg articles.[21] Details of this may be found in ref. 17. 5. Comments We have shown that with current data, the pion-pion sum rule, as well as the pion-nucleon one, is satisfied to better than six percent. To improve on this precision, it will be necessary to have an improved understanding of dynamical extrapolation corrections that account for the appearance of a zero mass, off-shell pion in the sum rules. We make two remarks in this regard. The first is that extrapolation of the incident pion to zero mass is not the same as taking the chiral limit of QCD, since the target pion or nucleon, and all intermediate state particles, remain on mass shell. The second is that while for a generic pion interpolating field the results of this extrapolation are not well-defined, the sum rules involve a very specific choice of pion interpolating field: the divergence of the axial vector current, which is a well defined entity in QCD, as are the on shell pion-pion and pion-nucleon scattering amplitudes. Thus, the question of estimating extrapolation corrections, that are needed for a very accurate comparison of the sum rules with experiment, is a well-posed one in QCD. Modern lattice methods may permit improvement on the estimates that have been used here and in refs. 1 and 3. In this respect, it is amusing to remark that, unlike the situation in 1965, the precision of the pionic sum rule is now greater than that of the pion-nucleon one. This is very likely due to the fact that the latter involves small differences of large numbers, so any small alteration is amplified here. 3 This is generally known as the Goldberger–Miyazawa–Oehme sum rule.[20] – 10 – -evaluation of the axial vector commutator sum rule for pion-pion scattering- Acknowledgements We wish to thank H. Leutwyler for supplying numbers that give an independent check of the pion-pion sum rule evaluation, and also for correspondence that assisted one of us (SLA) in reconstructing the reasoning behind the extrapolation method used for the pion-pion case in ref. 3. The work of SLA was supported in part by the U. S. Department of Energy under Grant No. DE-FG02-90ER40542. FJY is grateful to J. R. Peláez for interesting discussions in the preliminary stages of this paper, and to the Spanish DGI of the MEC under contract FPA2003-04597 for financial support. References 1 Adler, S. L., Phys. Rev. Letters 14, 1051 (1965). 2 Weisberger, W. I., Phys. Rev. Letters 14, 1047 (1965). 3 Adler, S. L., Phys. Rev. 140, B736 (1965) and (E) Phys. Rev. 149, 1294 (1966) and Phys. Rev. 175, 2224 (1968). 4 Goldberger, M. L., and Treiman, S. B., Phys. Rev. 109, 193 (1958). 5 References on pion-kaon scattering: M. Jamin, J., Oller, J. A., and Pich, A., Nucl. Phys. B587, 331 (2000); Gómez-Nicola, A., and Peláez, J. R., Phys. Rev. D65, 054009 (2002); Descotes-Genon, S., and Moussallam, B., Eur. Phys. J. C33, 409 (2004); Zhou, Z. Y., and Zheng, H. Q., Nucl. Phys. A775, 212 (2006). Theoretical proposal of kaon-nucleon sum rules: Weisberger, W. I., Phys. Rev. 143, 1302 (1966). 6 Particle Data Tables: Yao, W.-M., et al., Journal of Physics G33, 1 (2006). 7 Saino, M. E., hep-ph/9912337 and, especially, Schindler, M. R., et al., nucl-th/0611083. The second article reviews previous evaluations. 8 Pagels, H., and Zepeda, A., Phys. Rev. D5, 3262 (1972). For a more recent discussion of correc- tions to the Goldberger–Treiman relation, and their relation to off-shell extrapolation corrections, see Dominguez, C. A., Phys. Rev. D15, 1350 (1977) and Phys. Rev. D16, 2320 (1977). 9 S0 wave, low energy (s1/2 ≤ 932 MeV): Ynduráin, F. J., Garćıa Mart́ın, R., and Peláez, J. R., hep- ph/0701025. P wave, low energy (s1/2 ≤ 932 MeV): de Trocóniz, J. F., and Ynduráin, F. J., Phys. Rev., D65, 093001, (2002), and Phys. Rev. D71, 073008 (2005). S2 wave, D2 and F wave, s1/2 ≤ 1420 MeV: Peláez, J. R., and Ynduráin, F. J., Phys. Rev. D71, 074016 (2005). S0 and P waves, medium energy (up to s1/2 ≤ 1420 MeV); D0 wave: Kamiński, R., Peláez, J. R., and Ynduráin, F. J., Phys.Rev. D74, 014001 (2006) and (E) D74, 079903 (2006). 10 High energy, s1/2 > 1420 MeV: Peláez, J. R., and Ynduráin, F. J., Phys. Rev. D69, 114001 (2004). Actually, in the present calculation one uses the slightly improved Regge parametrization in the paper of Kamiński, Peláez and Ynduráin in ref. 9; see also Cudell, J. R., et al., Phys. Letters B587, 78 (2004) and Peláez, J. R., in Proc. Blois. Conf. on Elastic and Diffractive Scattering (hep-ph/0510005) for the pion-nucleon case. 11 Kl4 decays: Rosselet, L., et al., Phys. Rev. D15, 574 (1977); Pislak, S., et al., Phys. Rev. Lett., 87, 221801 (2001); NA48/2 ( CERN/SPS experiment); Bloch-Devaux, B., presented at QCD06 in Montpellier (France), 3-7 July 2006 and Masetti, L., presented at ICHEP06 in Moscow (Russia), 26 July to 2 August 2006. K → 2π decays: Aloisio, A., et al., Phys. Letters, B538, 21 (2002). Pion form factor data: Novosibirsk, ρ region: L. M. Barkov et al., Nucl. Phys. B256, 365 (1985); R. R. Akhmetshin et al., Phys. Letters B527, 161 (2002). τ decays: ALEPH: R. Barate et al., Z. Phys. C76, 15 (1997); OPAL: K. Ackerstaff et al., Eur. Phys. J. C7, 571 (1999); CLEO: S. Anderson et al., Phys. Rev., D61, 112002 (2000). 12 Olsson, M. G., Phys. Rev.162, 1338 (1967). 13 Weinberg, S., Phys. Rev. Letters 17, 616 (1966). 14 Gasser, J., and Leutwyler, H., Ann. Phys. (N.Y.) 158, 14 (1984). 15 Colangelo, G., Gasser, J., and Leutwyler, H., Nucl. Phys. B603, 125, (2001). See also Ananthanarayan, B., et al., Phys. Rep., 353, 207, (2001). The Roy equations were derived in Roy, S. M., Phys. Letters, 36B, 353, (1971). – 11 – http://arxiv.org/abs/hep-ph/9912337 http://arxiv.org/abs/nucl-th/0611083 http://arxiv.org/abs/hep-ph/0701025 http://arxiv.org/abs/hep-ph/0701025 http://arxiv.org/abs/hep-ph/0510005 -s. l. adler and f. j. ynduráin- 16 Leutwyler, H. (private communication). 17 Kondratyuk, S., et al., Nucl. Phys. A736, 339 (2004). 18 The COMPAS Group compilations can be traced from Hagiwara, K., et al., Phys. Rev. D66, 010001 (2002). 19 Rowe, G., Salomon, M., and Landau, R. H., Phys. Rev. C18, 584 (1978); Ebrahim, A. A., and Peterson, R. J., Phys. Rev. C54, 2499 (1996). 20 Goldberger, M. L, Miyazawa, H., and Oehme, R., Phys. Rev. 99, 986 (1955). For a very recent evaluation and further references, see Abaev, V. V., Metsä, P., and Sainio, M. E., arXiv:0704.3167 [hep-ph]. 21 Tomozawa, Y., Nuovo Cimento 46A, 707 (1966); Weinberg, S., Phys. Rev. Lett. 19, 616 (1966). – 12 – http://arxiv.org/abs/0704.3167

0704.1202

Colour pairs for constraining the age and metallicity of stellar populations

Mon. Not. R. Astron. Soc. 000, 1–10 (2002) Printed 24 October 2018 (MN LATEX style file v2.2) Colour Pairs for Constraining the Age and Metallicity of Stellar Populations ⋆ Zhongmu Li1,2 † and Zhanwen Han1 1National Astronomical Observatories/Yunnan Observatory, the Chinese Academy of Sciences, Kunming, 650011, China 2Graduate School of the Chinese Academy of Sciences Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11 ABSTRACT Using a widely used stellar population synthesis model, we study the ability of using pairs of AB system colours to break the well-known stellar age–metallicity degeneracy and give constraints on two luminosity-weighted stellar-population parameters (age and metallicity). The relative age and metallicity sensitivities of AB system colours that relate to u, B, g, V , r, R, i, I, z, J ,H , andK bands are presented, and the abilities of various colour pairs for breaking the age–metallicity degeneracy are quantified. Our results suggest that a few pairs of colours can be used to constrain the two above stellar-population parameters. This will be very useful for exploring the stellar populations of distant galaxies. In detail, colour pairs [(r−K), (u−R)] and [(r−K), (u− r)] are shown to be the best pairs for estimating the luminosity-weighted stellar ages and metallicities of galaxies. They can constrain two stellar-population parameters on average with age uncertainties less than 3.89 Gyr and metallicity uncertainties less than 0.34 dex for typical colour uncertainties. The typical age uncertainties for young (Age < 4.6 Gyr) populations and metal-rich (Z > 0.001) populations are small (about 2.26 Gyr) while those for old populations (Age > 4.6 Gyr) and metal-poor (Z < 0.001) populations are much larger (about 6.88 Gyr). However, the metallicity uncertainties for metal-poor populations (about 0.0024) are much smaller than for other populations (about 0.015). Some other colour pairs can possibly be used for constraining the two parameters, too. As a whole, the estimation of stellar-population parameters is likely to reliable only for early-type galaxies with small colour errors and globular clusters, because such objects contain less dust. In fact, no galaxy is totally dust-free and early-type galaxies are also likely have some dust, e.g., E(B-V)∼ 0.05, which can change the stellar ages about 2.5 Gyr and metallicities (Z) about 0.015. When we compare the photometric estimates with previous spectroscopic estimates, we are shown some dif- ferences, especially when comparing the stellar ages determined by two methods. The differences mainly result from the young populations of galaxies. Therefore, it is dif- ficult to get the absolute values of stellar ages and metallicities, but the results are useful for getting some relative values. In addition, our results suggest that colours relating to both UBV RIJHK and ugriz magnitudes are much better than either UBV RIJHK colours or ugriz colours for breaking the well-known degeneracy. The results also show that the stellar ages and metallicities of galaxies observed by the Sloan Digital Sky Survey (SDSS) and the Two-Micron All-Sky Survey (2MASS) can be estimated via photometry data. Key words: galaxies: stellar content — galaxies: photometry — galaxies: elliptical and lenticular, cD. ⋆ The data are available at the CDS or on request to the authors. † E-mail: [email protected] 1 INTRODUCTION The formation and evolution of galaxies is one of the hottest topics in astronomy and astrophysics. Great progress in the field has been had (see, e.g., Thomsen & Baum c© 2002 RAS http://arxiv.org/abs/0704.1202v4 2 Zhongmu Li and Zhanwen Han 1989, Kormendy & Djorgovski 1989, Kodama et al. 1998, van Dokkum & Stanford 2003, Kauffmann & Charlot 1998, Baugh et al. 1996, Baugh et al. 1998, Cole et al. 2000, De Lucia et al. 2006, and Thomas 1999), and two luminosity-weighted stellar-population parameters (age and metallicity) of galaxies are crucial. A lot of stellar population synthesis models such as Bruzual & Charlot (2003)(hereafter BC03), Worthey (1994), Vazdekis (1999) , Fioc & Rocca-Volmerange (1997), and Zhang et al. (2005) have been brought forward for stellar population studies and a great deal of observational data have been supplied by big surveys such as the Sloan Digital Sky Survey (SDSS) and the Two-Micron All-Sky Survey (2MASS). However, it is difficult to measure the stellar ages and metallicities of some distant galaxies (e.g., galaxies with redshifts greater than 0.3) via the spectra-like methods. The reason is that reliable spectra or line-strength indices (see, e.g., Worthey 1994, Gallazzi et al. 2005, Li et al. 2006) are usually avail- able for some nearby galaxies. If we can use one of important observational results, i.e., photometry, we will be able to ex- plore the stellar populations of some distant galaxies. Be- cause colours are easier to be obtained than spectra and are independent of the distances of objects, they are good can- didates for studying the stellar populations of such galaxies. Many works have been tried in this way, e.g., Dorman et al. (2003), Yi (2003), Wu et al. (2005), James et al. (2006), Li et al. (2007); Li & Han (2007), and Kaviraj et al. (2006). It seems possible to break the stellar age–metallicity de- generacy (Worthey 1994) by colours. Many pairs of colours (hereafter colour pairs) are used in previous works, e.g., (U −R) and (R−K) by Peletier & Balcells (1996), (B−R) and (R−K) by Bell & de Jong (2000), (V −I) and (V −K) by Puzia et al. (2002), (B−K) and (J−K) by James et al. (2006), (B−V ) and (B−K) by Li et al. (2007), (B−V ) and (V −K) by Lee et al. (2007), and most colours used are only in UBV RIJHK bands. As advocated by, e.g., Kong et al. (2006), colours relating to both UBV RIJHK and ugriz bands, i.e., (B − z) and (z − Ks), can be used to select galaxies with various redshifts. Similar colours can possibly be used to break the stellar age–metallicity degeneracy. We intend to give some investigations in this work, using the BC03 stellar population synthesis model. The paper is organized as follows. In Sect. 2, we briefly introduce the BC03 model and the calculation of colours. In Sect. 3, we present the age and metallicity sensitivities of colours. In Sect. 4, we try to search for colour pairs that can be used to constrain stellar age and metallicity. In Sect. 5, we give our discussions and conclusions. 2 THE BC03 MODEL AND CALCULATION OF COLOURS The BC03 model is a widely used model in stellar popula- tion synthesis study. Its standard model takes the Chabrier (2003) initial mass function (IMF) and uses Padova 1994 library tracks to calculate integrated colours. A few alter- native stellar spectral libraries are considered to give the spectral predictions of simple stellar populations (SSPs). The model provides us magnitudes and colours on both Johnson-Cousins-Glass (UBV RIJHKLM bands) and AB (ugriz bands) systems. More detailed information about Table 1. Relative metallicity sensitivities of 66 AB system colours. colour rms colour rms colour rms (B −K) 2.0294 (I − z) 0.8218 (B −H) 0.5758 (R−K) 1.8610 (i−K) 0.8094 (u− g) 0.5654 (u−K) 1.6904 (g − i) 0.7863 (g − V ) 0.5400 (I − J) 1.6125 (B − r) 0.7554 (i−H) 0.5178 (V −K) 1.5774 (B −R) 0.7335 (H −K) 0.5105 (r −K) 1.4114 (V − I) 0.7237 (R− I) 0.5074 (i− J) 1.4031 (g − r) 0.7027 (r −R) 0.5012 (I −K) 1.2654 (g − R) 0.6952 (r −H) 0.4834 (R− J) 1.2160 (V −R) 0.6896 (J −K) 0.4707 (g − J) 1.1094 (V − r) 0.6877 (R− i) 0.4675 (B − z) 1.0914 (u− I) 0.6725 (u− V ) 0.4440 (u− J) 1.0474 (r − i) 0.6722 (z −H) 0.4180 (B − J) 1.0455 (R − z) 0.6675 (g −H) 0.4116 (V − J) 1.0145 (u−B) 0.6632 (u− r) 0.4000 (V − z) 1.0069 (r − z) 0.6534 (B − g) 0.3984 (u− z) 0.9985 (g − z) 0.6404 (B − V ) 0.3908 (z − J) 0.9805 (r − I) 0.6382 (u−R) 0.3887 (g −K) 0.9719 (u−H) 0.6320 (V −H) 0.3835 (r − J) 0.9247 (i− z) 0.6151 (z −K) 0.3637 (B − i) 0.8788 (u− i) 0.5909 (R−H) 0.3344 (V − i) 0.8719 (I −H) 0.5867 (i− I) 0.2262 (B − I) 0.8414 (g − I) 0.5856 (J −H) 0.1670 the model please refer to Bruzual & Charlot (2003). In this work, BV RIJHK magnitudes on AB system are recalcu- lated from those given on Johnson-Cousins-Glass system, by taking -0.1, 0.0, 0.2, 0.45, 0.9, 1.4, and 1.9 as the differences between the zero points of AB system and Johnson-Cousins- Glass system magnitudes, in B, V , R, I , J , H and K bands, respectively 1. 3 AGE AND METALLICITY SENSITIVITIES OF COLOURS We study the age and metallicity sensitivities by a rela- tive metallicity sensitivity (rms) technique, which was used by Worthey (1994) and Li et al. (2007), and find age- and metallicity-sensitive colours based on the rms of colours. The rms method estimates the rms of each colour by the ratio of percentage change of age to that of metallicity when they lead to the same change in a colour respectively. Colours with large rms (>1.0) are more sensitive to metallic- ity and those with small rms (<1.0) to stellar age. Following Li et al. (2007), we calculated the rms of each colour in this work. The rms of 66 AB system colours are calculated in the work. The detailed data are listed in Table 1. As we see, (B − K), (R − K), (u − K), (I − J), (V − K), (r − K), (i− J), (I −K), (R− J), and (g − J) are more sensitive to stellar metallicity while (J −H), (i− I), (R−H), (z −K), (V −H), (u−R), (B−V ), (B− g), (u− r), and (g−H) to stellar age. We select the former ten colours as age-sensitive colours while the latter ten as metallicity-sensitive colours. 1 http://www.astro.livjm.ac.uk/∼ikb/convert-units/node1.html c© 2002 RAS, MNRAS 000, 1–10 http://www.astro.livjm.ac.uk/$^\sim $ikb/convert-units/node1.html Colour Pairs for Constraining Stellar-Population Parameters 3 4 COLOUR PAIRS FOR BREAKING THE AGE–METALLICITY DEGENERACY 4.1 Colour pairs for general studies Using the 10 metallicity-sensitive colours and the 10 age- sensitive colours presented in the last paragraph of Sect. 3, we buildup 100 colour pairs. Each colour pair includes a metallicity-sensitive colour and an age-sensitive colour. Us- ing the colour pairs one by one, we fit the stellar ages and metallicities of 500 testing stellar populations. To make the results useful for estimating the parameters of all kinds of stellar populations, the ages and metallicities of the test- ing populations are generated randomly within the ranges of 0.1–15 Gyr and 0.0001–0.05, respectively. The averages of uncertainties in age and metallicity, ∆t and ∆Z, are then calculated by taking typical uncertainties for input colours. The typical uncertainties for U , B, V , R, I , J , H , K, u, g, r, i, and z magnitudes are taken as 0.109, 0.116, 0.059, 0.03, 0.07, 0.08, 0.09, 0.126, 0.11, 0.01, 0.007, 0.007 and 0.012 mag, respectively. These values are estimated using the data sup- plied by the NASA/IPAC Extragalactic Database (NED), the 2MASS and SDSS surveys. Then we investigate the abil- ity of each colour pair for breaking the well-known stellar age–metallicity degeneracy by comparing the ∆t and ∆Z. A least-square method is used to fit the ages and metal- licities of stellar populations in the work. The uncertain- ties in stellar-population parameters are estimated by tak- ing the maximum uncertainties in the two parameters when considering the uncertainties of colours. One can refer to Denicoló et al. (2005) or Li et al. (2006) for more details. The main results are listed in Table 2. Because colour pairs that can well break the stellar age– metallicity degeneracy lead to small uncertainties in stel- lar age and metallicity, pairs with small ∆t and ∆Z are better for constraining two stellar-population parameters. However, as we see, some colour pairs have only small ∆t or small ∆Z. In this case, it is difficult to compare the abilities of various colour pairs. We defined a parameter, uncertainty parameter (UP ), to solve this problem. The UP is calcu- lated by taking the average of the relative uncertainties of stellar ages and metallicities of the 500 testing stellar pop- ulations. According to the calculation of UP , colour pairs with small UP s are more suitable for breaking the stellar age–metallicity degeneracy. The UP s of colours are shown in Table 2, together with ∆ts and ∆Zs. Considering actual applications, we only list the results of colour pairs that have UP s smaller than 2.0 when taking typical uncertain- ties for colours. In the table, colours are sorted by an in- creasing order of UP . Note that we also list the average metallicity uncertainty in dex. As we see, [(r−K), (u−R)] and [(r −K), (u− r)] are the best colour pairs for breaking the well-known degeneracy. Given typical uncertainties for colours, the two colour pairs can constrain stellar-population parameters with relative uncertainties smaller than about 96%, which corresponds to average uncertainties in stel- lar age and metallicity smaller than 3.89 Gyr and 0.34 dex, respectively. Some other pairs, i.e., [(R−K), (u−R)], [(I −K), (u−R)], [(R −K)], (u− r)] and [(i− J), (u−R)] can possibly be used to give constraints on the stellar ages and metallicities of galaxies as they have small UP s (6 1.13) for typical uncertainties of colours. In addition, colour pairs relating to both UBV RIIHJK bands and ugriz bands are shown to be much better than those only relating to one of the two kinds of bands. To see the abilities of colour pairs for breaking the age–metallicity degeneracy, we show the colour–colour grids of four pairs in Fig. 1. However, as we note from Table 2, the uncertainties in two stellar-population parameters are very big. This results from the large observational uncertainties. Because the ob- servational uncertainties depend on surveys and they will possibly be decreased in future surveys, we tried to find the best pairs for various uncertainties (0.02, 0.05 and 0.10) in colours. We assumed that all colours have the same uncer- tainty in each test. The results are shown in Table 2. As we see, if the uncertainties of colours are less than 0.05 mag, the stellar ages and metallicities of galaxies can be constrained with uncertainty less than 1 Gyr and 0.0076, respectively, via (I −K) and (u−R). 4.2 Colour pairs for special studies Because we often need to estimate the ages and metallic- ities of some special stellar populations, e.g., the old pop- ulations of globular clusters, we tried to find some colour pairs that are suitable for estimating the two luminosity- weighted stellar-population parameters of such populations. In detail, the best colour pairs for estimating the stellar ages and metallicities of young (t < 4.6 Gyr), old (t > 4.6 Gyr), metal poor (Z < 0.001), and metal rich (Z > 0.001) stellar populations are found by taking the above typical uncer- tainties of colours. The best ten colour pairs for studying each kind of special population are listed in Table 3. As we see, some colour pairs for studying different kinds of stel- lar populations are various. However, [(r −K), (u− r)] can be used to constrain the age and metallicity of all kinds of stellar populations. We can also find that when colour pair [(r−K), (u− r)] is used to study the stellar-population pa- rameters of various populations, the uncertainties of results are different. The age uncertainties of old or metal-poor pop- ulations are usually larger than those of young or metal-rich populations. 4.3 Composite colour pairs In practice, we can also use colour pairs including magni- tudes on different systems, as colours relating to the same bands but on different systems usually have similar proper- ties for breaking the well-known stellar age–metallicity de- generacy (Worthey 1994). For example, we can use colour pair [(r−Ks), (u− r)], in which ur magnitudes are on AB- system and Ks magnitude on 2MASS system instead of [(r−K), (u−r)], in which all magnitudes are on AB-system. Of course, we can use Johnson-Cousins-Glass system colours together with AB system colours. In the work, we analyzed colour pairs relating to five AB-system bands (ugriz) and three 2MASS bands (JHKs). According to the results, colour pairs [(u − r), (r − Ks)], [(u−r), (i−J)] and [(u−Ks), (z−Ks)] are more suitable for constraining stellar-population parameters than others. The UP s of the three colour pairs can refer to Table 2. Because these pairs have colour-colour grids similar to those shown in Fig. 1, we do not show them here. c© 2002 RAS, MNRAS 000, 1–10 4 Zhongmu Li and Zhanwen Han Table 2. The abilities of various colour pairs for breaking the age–metallicity degeneracy. Here ∆t and ∆Z (or ∆[Z/H]) are the average uncertainties of stellar ages and metallicities, while UP is the uncertainty parameter, i.e., the average of relative uncertainties of ages and metallicities of 500 testing stellar populations. Uncertainty (mag) typical 0.02 0.05 0.10 Colour pair ∆t ∆Z ∆[Z/H] UP ∆t ∆Z UP ∆t ∆Z UP ∆t ∆Z UP (Gyr) (dex) (Gyr) (Gyr) (Gyr) [(r −K), (u− R)] 3.89 0.0176 0.33 0.95 0.37 0.0047 0.17 1.01 0.0085 0.35 2.13 0.0145 0.66 [(r −K), (u− r)] 3.57 0.0176 0.34 0.96 0.40 0.0045 0.17 1.14 0.0093 0.40 2.58 0.0146 0.78 [(R−K), (u−R)] 4.15 0.0191 0.37 1.06 0.33 0.0044 0.16 0.96 0.0084 0.35 1.98 0.0142 0.63 [(I −K), (u− R)] 3.89 0.0217 0.44 1.09 0.27 0.0032 0.12 0.89 0.0076 0.31 2.33 0.0131 0.58 [(R−K), (u− r)] 4.33 0.0192 0.38 1.11 0.32 0.0040 0.15 1.04 0.0087 0.37 2.19 0.0142 0.67 [(i− J), (u−R)] 3.16 0.0222 0.40 1.13 0.45 0.0078 0.25 1.28 0.0148 0.55 3.54 0.0228 1.19 [(i− J), (u− r)] 3.18 0.0224 0.41 1.16 0.50 0.0089 0.29 1.44 0.0148 0.63 3.71 0.0206 1.19 [(I −K), (u− r)] 4.12 0.0218 0.45 1.19 0.28 0.0034 0.12 0.92 0.0078 0.32 2.34 0.0133 0.61 [(u−K), (z −K)] 5.21 0.0188 0.37 1.34 0.34 0.0041 0.16 1.48 0.0081 0.39 4.01 0.0144 0.90 [(g − J), (z −K)] 6.21 0.0176 0.35 1.37 0.48 0.0073 0.31 2.27 0.0082 0.44 4.56 0.0132 0.98 [(r −K), (R −H)] 4.20 0.0255 0.39 1.54 0.48 0.0073 0.31 3.54 0.0181 1.08 3.79 0.0211 1.26 [(B −K), (z −K)] 6.81 0.0187 0.37 1.55 0.32 0.0072 0.28 2.13 0.0084 0.43 4.61 0.0146 0.96 [(i− J), (g −H)] 5.42 0.0243 0.47 1.56 0.42 0.0096 0.34 2.61 0.0168 0.77 7.28 0.0257 2.26 [(i− J), (z −K)] 5.09 0.0175 0.30 1.59 0.42 0.0096 0.34 2.47 0.0158 0.83 5.65 0.0146 1.71 [(V −K), (i− I)] 5.61 0.0249 0.49 1.59 0.42 0.0096 0.34 2.80 0.0201 1.05 3.08 0.0203 1.10 [(V −K), (u− R)] 5.38 0.0230 0.51 1.62 0.38 0.0050 0.18 1.19 0.0101 0.44 2.82 0.0152 0.80 [(V −K), (u− r)] 5.56 0.0227 0.50 1.72 0.57 0.0052 0.21 1.32 0.0107 0.48 3.13 0.0151 0.98 [(I − J), (z −K)] 4.72 0.0160 0.31 1.76 2.43 0.0110 0.93 2.72 0.0110 0.74 5.54 0.0153 2.05 [(r −K), (g −H)] 7.77 0.0226 0.43 1.86 0.66 0.0064 0.29 2.99 0.0096 0.55 6.85 0.0176 1.44 [(g − J), (u− r)] 5.58 0.0262 0.53 1.94 0.91 0.0140 0.53 2.36 0.0182 1.02 6.45 0.0264 2.74 [(I − J), (J −H)] 5.23 0.0280 0.59 1.95 0.91 0.0140 0.53 2.36 0.0182 1.02 6.06 0.0215 2.06 [(R− J), (u− r)] 5.53 0.0224 0.55 1.97 0.40 0.0074 0.23 1.95 0.0178 0.94 5.09 0.0233 1.88 [(R−K), (g −H)] 8.02 0.0227 0.47 1.98 0.60 0.0077 0.35 2.79 0.0094 0.54 6.13 0.0167 1.27 Table 3. Best colour pairs for estimating the ages and metallicities of old (Age > 4.6 Gyr), young (Age < 4.6 Gyr), metal-rich (Z > 0.001), and metal-poor (Z < 0.001) stellar populations. Symbols have the same meanings as in Table 2. The values are calculated using the typical colour uncertainties. Note that the age uncertainties of metal-poor populations are not always right because some testing populations are out of the colour-colour grid when taking their colour uncertainties into account. Old Population Young or Metal-rich Population Metal-poor Population Colour pair ∆t ∆Z UP Colour pair ∆t ∆Z UP Colour pair ∆t ∆Z UP [(r −K), (u− r)] 6.88 0.0141 0.73 [(r −K), (u− r)] 2.26 0.0174 0.76 [(r −K), (u− r)] 3.64 0.0024 4.96 [(i− J), (u− r)] 7.17 0.0146 0.73 [(i− J), (u− r)] 1.34 0.0222 0.78 [(r −K), (u− R)] 3.98 0.0024 4.96 [(i− J), (u−R)] 7.58 0.0149 0.75 [(i− J), (u−R)] 1.43 0.0220 0.81 [(R −K), (u− R)] 4.10 0.0026 5.52 [(r −K), (u− R)] 7.51 0.0144 0.76 [(r −K), (u− R)] 2.54 0.0173 0.81 [(g − J), (z −K)] 1.85 0.0032 5.69 [(R−K), (u− r)] 8.17 0.0170 0.89 [(R−K), (u− r)] 2.70 0.0195 0.85 [(R −K), (u− r)] 3.89 0.0030 5.79 [(R−K), (u−R)] 8.47 0.0174 0.91 [(R−K), (u−R)] 2.64 0.0195 0.86 [(B −K), (z −K)] 3.67 0.0032 6.18 [(V −K), (u− R)] 9.57 0.0188 1.07 [(R− J), (u− r)] 2.23 0.0216 0.93 [(i− J), (B − V )] 3.42 0.0031 6.23 [(i− J), (i− I)] 11.29 0.0177 1.10 [(I −K), (u− R)] 2.46 0.0235 0.95 [(u−K), (z −K)] 6.33 0.0029 6.26 [(V −K), (u− r)] 9.19 0.0187 1.13 [(I −K), (u− r)] 2.50 0.0235 0.96 [(I −K), (B − V )] 3.87 0.0030 6.47 [(I −K), (z −K)] 9.79 0.0170 1.16 [(R− J), (u− R)] 2.29 0.0250 1.06 [(r −K), (B − V )] 3.44 0.0031 6.67 5 APPLICATION OF COLOURS AND COLOUR PAIRS 5.1 Using colour pairs to constrain stellar-population parameters To test the application of colour pairs to estimate the ages and metallicities of stellar populations, we select 1 646 lu- minous (absolute magnitude Mr < −22 and r-band Pet- rosian magnitude < 17.77) early-type (concentration index C > 2.8) galaxies observed by both 2MASS and the sec- ond release of SDSS (SDSS-DR2). All the sample galaxies have small magnitude uncertainties (< 0.15 mag). Note that only the galaxies with colour-fitted stellar ages smaller than 15 Gyr and stellar metallicities richer than 0.008 are se- lected for our sample galaxies, because the age of the uni- verse was shown to be smaller than about 15 Gyr (e.g., Shafieloo et al. 2006) and the results for populations with c© 2002 RAS, MNRAS 000, 1–10 Colour Pairs for Constraining Stellar-Population Parameters 5 Figure 1. Colour–colour grids of four colour pairs that are suitable for constraining stellar age and metallicity. Colours are on AB system. Solid and dashed lines represent constant age and metallicity, respectively. Note that we did not mark for constant ages of 6, 8, 10, 12 Gyr, as the limitation of the space. The four panels are for [(r−K), (u−R)], [(r−K), (u− r)], [(R−K), (u− r)], and [(I−K), (u−R)], respectively. metallicities poorer than 0.008 seems unreliable as large un- certainties. Then we use (r −K) and (u− r) colours to es- timate two stellar-population parameters of these galaxies, according to the results shown in Tables 2 and 3. The K- band magnitudes of galaxies are calculated from Ks-band magnitudes supplied by 2MASS, by using the same method as Bessell (2005). The k-corrections of Ks-band magnitudes are estimated as −6log(1 + z), as used by Girardi et al. (2003), where z is the redshift. The galactic extinctions of K-band magnitudes are calculated using the model of Burstein & Heiles (1982). The u and z bands magnitudes, and their k-corrections and galactic extinctions are taken from SDSS. The differences between SDSS magnitudes and AB magnitudes are taken into account according to the val- ues supplied by SDSS. In Fig. 2, the sample galaxies are shown on the (u − r) versus (r − K) grid. For clearly, we only plotted the error bars for the first 10 galaxies of our sample, which are marked in black. We see that a rough estimation for the two parameters of galaxies can be ob- tained, although the uncertainties are somewhat big. As ex- amples, we list the stellar-population parameters of a few galaxies in Table 4, in which the data of three subsets of galaxies are listed. In fact, even if we take Lick indices for such work, the uncertainties in stellar ages and metallicities of these galaxies are very big because of the large obser- vational uncertainties. For our sample galaxies, the uncer- tainties in Hβ and [MgFe] indices are typically about 0.3 Å, which will lead to large uncertainties in stellar-population parameters. The Lick indices and uncertainties of our sam- ple galaxies are taken from the Garching SDSS catalogs (http://www.mpa-garching.mpg.de/SDSS/DR4/). Further- more, we found that 24 galaxies are out of the theoreti- cal grid of BC03 models. This possibly results from the ef- fects of young stellar populations of galaxies, the limitation of theoretical models, and large observational uncertainties of colours. In special, we found that most galaxies can fall inside the theoretical colour-colour grid when taking their colour uncertainties into account. This means that most of the 24 galaxies could have physical ages, but as pointed out by Li et al. (2007), the presence of young populations in such early-type galaxies can also make metal-rich popula- tions outside the theoretical colour-colour grids. c© 2002 RAS, MNRAS 000, 1–10 http://www.mpa-garching.mpg.de/SDSS/DR4/ 6 Zhongmu Li and Zhanwen Han Table 4. Stellar ages and metallicities of old (Age > 4.6 Gyr), young metal-rich (Age < 4.6 Gyr and Z > 0.02), and young metal- poor (Age < 4.6 Gyr and Z < 0.02) galaxies. The results are measured by two kinds of methods (photometric and spectroscopic). The photometric results are determined via (u − K) and (z − K) colours in the work, and the spectroscopic results are obtained by Gallazzi et al. (2005). The symbol “objID” is the unique SDSS identifier of each galaxy. Note that the definition of metal-poor galaxy here is different from that in other places of the paper, as the limitation of our sample galaxies. Photometric results Spectroscopic results Population type objID Age Error Z Error Age Error Z Error (Gyr) (Gyr) (Gyr) (Gyr) 587727177921921120 4.6 0.0293 +0.0110 −0.0125 0.0411 +0.0077 −0.0171 Old 587727225693143113 5.7 +10.7 0.0300 +0.0143 −0.0176 0.0291 +0.0207 −0.0088 587727225692160083 9.1 0.0416 +0.0082 −0.0108 0.0350 +0.0141 −0.0127 587727177921986670 2.7 0.0478 +0.0022 −0.0121 0.0241 +0.0220 −0.0133 Young and metal-rich 587727225160466548 2.2 0.0385 +0.0089 −0.0081 0.0301 +0.0162 −0.0114 587727225689538771 2.7 0.0432 +0.0068 −0.0124 0.0305 +0.0146 −0.0109 587727230524063879 3.7 +8.0 0.0189 +0.0128 −0.0121 7.4 +2.9 0.0226 +0.0184 −0.0097 Young and metal-poor 587727227302707220 3.8 0.0166 +0.0038 −0.0085 0.0251 +0.0179 −0.0073 588848898845114541 2.9 +2.7 0.0192 +0.0097 −0.0044 3.5 +2.5 0.0135 +0.0304 −0.0054 In the work, we fitted the stellar ages and metallicities of our sample galaxies via BC03 SSPs with ages within 0.1– 19.96 Gyr and metallicities within 0.0001–0.05. In Fig. 3, we compare the parameters determined by colours with those by a few Lick indices (Gallazzi et al. 2005). The results of Gallazzi et al. (2005) were obtained by comparing D4000, Hβ, HδA+HγA, [Mg2Fe], and [MgFe] ′ indices of galaxies to the values of theoretical stellar populations and have taken the effects of young populations (YSPs) into account. The detailed data about the sample galaxies can be obtained on request to the authors or via the CDS in the future. From panels (a) and (c) of Fig. 3, we find that the stellar ages and metallicities determined by colours are respectively smaller and somewhat richer than those determined by Lick indices. The reason is that there are composite stellar populations (CSPs) in galaxies and the YSPs make the results derived from colours bias younger and richer in metal compared to those of the dominant populations (DSPs) of galaxies (Li & Han 2007). In detail, because the age of a YSP affects the colours of a star system more stronger than the mass fraction of the YSP, YSPs with only a few percent stellar mass can make the colour-fitted stellar populations younger and metal-richer than DSPs of galaxies if the YSPs are not too old. Then in panels (b) and (d) Fig. 3, we compare the stellar-population parameters measured by colours and cor- rected for the effects of YSPs to those determined by Lick in- dices. The correction is accomplished using the possible dis- tributions of the differences between the stellar-population parameters of the DSPs of CSPs and the parameters derived from two colours of CSPs (Li & Han 2007). The above distri- butions were obtained by a statistical method, in which the fractions of YSPs were assumed to depend on the ages of the DSPs and YSPs, and the fraction of a YSP was assumed to decline exponentially with decreasing age of the YSP (see, e.g., Thomas et al. 2005). In special, the distributions can be used to give a rough correction for the effects of YSPs. Here, we get the corrected stellar age of a galaxy by submit- ting from the colour-fitted result a random value that fits to the distribution of the difference between the stellar ages of DSPs and those derived from two colours. A similar method is used to get the corrected stellar metallicities. For more clearly, we show the distributions of the uncorrected and corrected parameters in Fig. 4. We are shown that after the correction, the distributions of stellar ages obtained by two methods, especially the peaks, become more similar after the correction (Fig. 4). However, it is also shown that the distri- butions of stellar metallicities obtained by various methods are similar. Therefore, it seems that without any correction, we can get correct distributions of stellar metallicities of lu- minous early-type galaxies via colours. Note that because we used a least-square fitting method similar to Denicoló et al. (2005) and Li et al. (2006), the effects of the uncertainties of colours were not taken into account here. The uncertain- ties may effect the above distributions slightly. Furthermore, the dust may contribute to the difference between the two kinds of results. In fact, a small amount of dust can make (u−r) colours of galaxies redder, and then lead to additional uncertainties in stellar ages. In detail, a dust of E(B − V ) = 0.05 will change the (u − r) and (r − K) colours about 0.12 mag, and then lead to about 2.5 Gyr age uncertainty and 0.015 metallicity uncertainty. In addition, the average stellar-population parameters obtained by photometry and spectroscopy are found to be similar (about 7 Gyr for age and 0.02 for metallicity). Therefore, although it is difficult to get the accurate stellar-population parameters of each galaxy via colours, we can get reliable values for the aver- age age and metallicity of a sample of galaxies, as can the distributions of the two parameters. c© 2002 RAS, MNRAS 000, 1–10 Colour Pairs for Constraining Stellar-Population Parameters 7 Figure 2. Our sample galaxies (1 646 galaxies) in the (u− r) versus (r−K) grid. Dark points with error bars show the first 10 galaxies of our sample. The colour uncertainties of other galaxies can refer to those shown. Lines have the same meanings as in Fig. 1. Here we did not mark for the constant age of 8 and 16 Gyr. 5.2 Using colours in conjunction with spectroscopic indices Because some colours are shown to be sensitive to stellar age or metallicity, we can possibly use colours in conjunc- tion with Lick indices to study stellar-population parame- ters. For example, we can use a metallicity-sensitive colour together with an age-sensitive Lick index to estimate the stellar ages and metallicities of galaxies. We have a try in this work. The above galaxy sample are used here. The stellar- population parameters are fitted via Hβ and (r − K) in- dices using the same method as Sect 5.1. The results are compared to those determined by Gallazzi et al. (2005) in panels (a) and (b) of Fig. 5. Note that the results did NOT correct for the effects of young populations. We see that using Hβ instead of (u − r), the stellar ages obtained are closer to those of Gallazzi et al. (2005). This means that line indices are affected by young populations of galaxies more slightly. However, the fitted stellar metallicities of some galaxies are poorer than those obtained by Gallazzi et al. (2005). It should mainly result from the effects of YSPs in galaxies. In fact, YSPs make the (r −K) colour of galaxies obviously bluer than those of the DSPs, but they affect the Hβ index more slightly. When (r − K) is used in conjunc- tion with Hβ to estimate the stellar-population parameters of galaxies, the two indices will lead to more poor metallic- ities compared those determined by a colour pair. One can see a Hβ versus [MgFe] grid for comparison, as the shape of Hβ versus (u − r) grid is similar to that of Hβ versus [MgFe]. Therefore, it is difficult to use photometry in con- junction with spectroscopy to estimate the metallicities of galaxies. Furthermore, we tried to give some final results for the stellar-population parameters of our sample galaxies us- ing Hβ, [MgFe], (u−r) and (r−K) indices together. Because the uncertainties of colours and Lick indices are different, we use a χ2 fit (see, e.g., Press et al. 1992) to estimate the stellar-population parameters. The stellar metallicities are shown to be different significantly from those determined by previous work. The comparison can be seen in panels (c) and (d) of Fig. 5. The results did not correct for the effects of young populations, either. Because the results of (Gallazzi et al. 2005) have taken the effects of YSPs into ac- count, the above results are certainly affected by the YSPs in galaxies. The results may also effected by the large ob- servational uncertainties. However, it seems that the YSPs effect the results stronger, because when we tried to esti- mate the stellar-population parameters via Hβ and [MgFe], we got poorer metallicities than those of the previous work. As a whole, our results suggest that colour pairs can be used c© 2002 RAS, MNRAS 000, 1–10 8 Zhongmu Li and Zhanwen Han Figure 3. Comparison of stellar ages and metallicities derived from colours with those from Lick indices. The results derived from colours are obtained in this work and those derived from Lick indices were obtained by Gallazzi et al. (2005). The typical errors of stellar ages obtained by colours and Lick indices are about 3.80 Gyr and 4.17 Gyr, respectively. The symbol “lick” denotes the results of Gallazzi et al. (2005) while “colour” means the results derived from (r − K) and (u − r). The suffix “corrected” means the results corrected for the effects of young stellar populations. Dashed lines show a ± 3.5 Gyr spread about the unity (solid) line for stellar ages in panels a) and b) while show a ± 0.015 spread in stellar metallicities in panels c) and d). more conveniently for estimating the stellar-population pa- rameters of distant galaxies, as colours can be obtained more easily than spectra and can constrain the stellar-population parameters with uncertainties similar to Lick indices (typi- cally 4 Gyr for age and 0.015 for metallicity). 6 DISCUSSIONS AND CONCLUSIONS We investigated the relative metallicity sensitivities of AB system colours relating to u, B, g, V , r, R, i, I , z, J , H , and K bands in the first place. Then we studied the abilities of colour pairs for constraining luminosity-weighed stellar ages and metallicities. The results showed that [(r−K), (u−R)] and [(r − K), (u − r)] are the best colour pairs for break- ing the stellar age–metallicity degeneracy while colour pairs such as (R−K), (u−R)], [(I−K), (u−R)], [(R−K)], (u−r)] and [(i − J), (u − R)] can also be used. Colour pairs [(r − K), (u−R)] and [(r −K), (u− r)] can measure two stellar- population parameters with small uncertainties (∆t 6 3.89 Gyr, ∆[Z/H] 6 0.34 dex for typical uncertainties in colours). However, the age uncertainties for old populations (Age > 4.6 Gyrs) and metal-poor populations (Z < 0.001) are al- ways significantly larger than for young populations (Age < 4.6 Gyrs) and metal-rich populations (Z > 0.001). The rea- son is that the colours of some old populations or metal-poor populations are largely indistinguishable within present typ- ical errors. One can see Fig. 3 for comparison. However, the metallicity uncertainty of metal-poor populations (about 0.0024) is much less than that of other populations (about 0.015). In the work, we did not take the uncertainties in the- oretical stellar population models into account. A detailed study about it please refer to the work of Yi (2003). Further- c© 2002 RAS, MNRAS 000, 1–10 Colour Pairs for Constraining Stellar-Population Parameters 9 Figure 5. Stellar-population parameters measured using colours in conjunction with line indices are plotted versus those measured by a few line indices. The suffix “comp” represents the results derived from Hβ and (r−K) while “all” for the results measured by two colours in conjunction with two line indices. Lines have meanings similar to Fig. 4. The typical uncertainties for stellar ages and metallicities are 4 Gyr and 0.015, respectively. more, colours are usually affected by the dust of galaxies, even if early-type ones. A typical effect of dust in early-type galaxies, E(B-V)∼ 0.05, will change the (r−K) and (u− r) colours of galaxies about 0.12 mag, and then lead to addi- tional uncertainties in stellar-population parameters (about 2.5 Gyr for age and 0.015 for metallicity). Thus it is more suitable to use colours to measure the stellar-population pa- rameters of galaxies with poor dust and gas, e.g., luminous early-type ones (with small colour uncertainties). Actually, to quantify the age and metallicity errors induced by a typ- ical dust is in study. When we compare the results deter- mined by photometry with those determined by Lick in- dices, we found that the stellar ages determined by colours are less than those determined by Lick indices. The differ- ence mainly results from the effects of young populations in these galaxies. We also found that the average uncertainties of stellar-population parameters determined by colours and Lick indices are similar (typically 4 Gyr for age and 0.015 for metallicity). Therefore, it is actually difficult to get the ab- solute values of stellar ages and metallicities via colours, but we can get some relative values. This will be useful for some statistical studies of the stellar populations of galaxies. In addition, our results suggest that it is better to use colours relating to both UBV RIJHK and ugriz bands than to use those only relating to one of the two kinds of bands for esti- mating the two stellar-population parameters. According to the results, we can estimate the stellar-population parame- ters of some distant galaxies, via the photometry data sup- plied by, e.g., SDSS and 2MASS. The possible uncertainties of using various colour pairs can be estimated. The results presented in the paper can also help us to choose suitable bands for the observation of stellar population studies. We also tried to find some colour pairs that are suitable for estimating the luminosity-weighted ages and metallicities of some special stellar populations. The results show that some colour pairs for estimating the two stellar-population c© 2002 RAS, MNRAS 000, 1–10 10 Zhongmu Li and Zhanwen Han Figure 4. Comparison of the distributions of stellar-population parameters obtained by two kinds of methods. Solid lines are for the results obtained by Lick indices (Gallazzi et al. 2005). Dotted lines in panels a) and c) show the results obtained by [(r − K), (u−r)] and without any correction, while those in in panels b) and d) show the results corrected for the effects of young populations using the results of Li & Han (2007). parameters of young (Age < 4.6 Gyr), old (Age > 4.6 Gyr), metal-poor (Z < 0.001), and metal-rich (Z > 0.001) popula- tions are different. However, [(r−K), (u−r)] can be used to estimate the ages and metallicities of all stellar populations. Although we took the BC03 model for our work, the re- sults can be used for other models as most stellar population synthesis models predict similar colours for the same popu- lation. Furthermore, we suggest to choose a colour pair for estimating the two stellar-population parameters of galax- ies, although a few colours can give smaller range for the two parameters. The reason is that galaxies usually contain more than two populations, and via comparing observational colours with predictions of simple stellar populations, we can only measure less stellar ages and richer stellar metallicities for galaxies compared to the dominant stellar populations (DSPs) of galaxies. For a sample of galaxies, the effects of young populations on the ages and metallicities of dominant stellar populations can be roughly corrected and some reli- able estimations for the averages and distributions of two stellar-population parameters can be obtained (Li & Han 2007). Note that the BC03 model is a single stellar popula- tion model, which does not take the binary interactions into account. This is actually different from the real stellar popu- lations of galaxies. If stars of a population evolve as binaries rather than single stars, the colours of the population will be different with those of a single stellar population. Typically, the (u− r) and (r −K) colours predicted by binary stellar populations will be bluer about 0.05 mag than those pre- dicted by single populations. Using binary populations in- stead of single populations, the stellar ages and metallicities will be about 1.14 Gyr older and 0.0093 richer, respectively. A detailed study about this will be given in the future. ACKNOWLEDGMENTS We greatly acknowledge the anonymous referee for some use- ful comments. We also thank Profs. Gang Zhao, Xu Kong, and Dr. Fenghui Zhang for useful discussions, Dr. Anna Gal- lazzi for some useful discussions and her group for the line indices, stellar ages and metallicities of our sample galaxies, the Sloan Digital Sky Survey (SDSS), the Two-Micron All- Sky Survey (2MASS) and the NASA/IPAC Extragalactic Database (NED) for the photometry data of galaxies. This work is supported by the Chinese National Science Foun- dation (Grant Nos 10433030, 10521001), and the Chinese Academy of Science (No. KJX2-SW-T06). REFERENCES Baugh C. M., Cole S., Frenk C. S., 1996, MNRAS, 283, Baugh C. M., Cole S., Frenk C. S., Lacey C. G., 1998, ApJ, 498, 504 Bell E. F., de Jong R. S., 2000, MNRAS, 312, 497 Bessell M. S., 2005, ARA&A, 43, 293 Bruzual G., Charlot S., 2003, MNRAS, 344, 1000 Burstein D., Heiles C., 1982, AJ, 87, 1165 Chabrier G., 2003, ApJ, 586, L133 Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 2000, MNRAS, 319, 168 De Lucia G., Springel V., White S. D. M., Croton D., Kauff- mann G., 2006, MNRAS, 366, 499 Denicoló G., Terlevich R., Terlevich E., Forbes D. A., Ter- levich A., 2005, MNRAS, 358, 813 Dorman B., O’Connell R. W., Rood R. T., 2003, ApJ, 591, Fioc M., Rocca-Volmerange B., 1997, A&A, 326, 950 Gallazzi A., Charlot S., Brinchmann J., White S. D. M., Tremonti C. A., 2005, MNRAS, 362, 41 Girardi M., Mardirossian F., Marinoni C., Mezzetti M., Rigoni E., 2003, A&A, 410, 461 James P. A., Salaris M., Davies J. I., Phillipps S., Cassisi S., 2006, MNRAS, 367, 339 Kauffmann G., Charlot S., 1998, MNRAS, 294, 705 Kaviraj S., Rey S. ., Rich R. M., Lee Y. ., Yoon S. ., Yi S. K., 2006, ArXiv Astrophysics e-prints Kodama T., Arimoto N., Barger A. J., Arag’on-Salamanca A., 1998, A&A, 334, 99 Kong X., Daddi E., Arimoto N., Renzini A., Broadhurst T., Cimatti A., Ikuta C., Ohta K., da Costa L., Olsen L. F., Onodera M., Tamura N., 2006, ApJ, 638, 72 Kormendy J., Djorgovski S., 1989, ARA&A, 27, 235 Lee H.-c., Worthey G., Trager S. C., Faber S. M., 2007, ApJ, 664, 215 Li Z., Han Z., 2007, A&A, 471, 795 Li Z., Han Z., Zhang F., 2007, A&A, 464, 853 Li Z.-M., Zhang F.-H., Han Z.-W., 2006, ChJAA, 6, 669 Peletier R. F., Balcells M., 1996, AJ, 111, 2238 Press W., Teukolsky S., Vetterling W., Flannery, B., Nu- merical Recipes in FORTRAN: Second Edition, Cam- bridge university press, 1992 Puzia T. H., Zepf S. E., Kissler-Patig M., Hilker M., Minniti D., Goudfrooij P., 2002, A&A, 391, 453 Shafieloo A., Alam U., Sahni V., Starobinsky A. A., 2006, MNRAS, 366, 1081 c© 2002 RAS, MNRAS 000, 1–10 Colour Pairs for Constraining Stellar-Population Parameters 11 Thomas D., 1999, MNRAS, 306, 655 Thomas D., Maraston, C., Bender, R., 2005, ApJ, 621, 673 Thomsen B., Baum W. A., 1989, ApJ, 347, 214 van Dokkum P. G., Stanford S. A., 2003, ApJ, 585, 78 Vazdekis A., 1999, ApJ, 513, 224 Worthey G., 1994, ApJS, 95, 107 Wu H., Shao Z., Mo H. J., Xia X., Deng Z., 2005, ApJ, 622, 244 Yi S. K., 2003, ApJ, 582, 202 Zhang F., Li L., Han Z., 2005, MNRAS, 364, 503 This paper has been typeset from a TEX/ LATEX file prepared by the author. c© 2002 RAS, MNRAS 000, 1–10 Introduction The BC03 model and calculation of colours age and metallicity sensitivities of colours colour pairs for breaking the age–metallicity degeneracy Colour pairs for general studies Colour pairs for special studies Composite colour pairs Application of colours and colour pairs Using colour pairs to constrain stellar-population parameters Using colours in conjunction with spectroscopic indices discussions and conclusions

0704.1204

The effect of supernova asymmetry on coalescence rates of binary neutron stars

THE EFFECT OF SUPERNOVA ASYMMETRY ON COALESCENCE RATES OF BINARY NEUTRON STARS K.A. POSTNOV, A.G. KURANOV Sternberg Astronomical Institute, Universitetskij pr. 13, 119992 Moscow, Russia We study the effect of the kick velocity – neutron star spin alignment observed in young radio pulsars on the coalescence rate of binary neutron stars. The effect is shown to be especially strong for large kick amplitudes and tight alignments, reducing the galactic rate of binary neutron star coalescences up to an order of magnitude with respect to the rates calculated for random kicks. The spin-kick correlation also leads to much narrower NS spin- orbit misalignments compared to random kicks. 1 Introduction Kick velocity imparted to a newborn neutron star is known to be an important phenomenological parameter of core collapse supernovae. The origin of the kicks remains unclear (see, for example, 1 and references therein). Recently, new important observational results appeared suggesting possible NS spin-kick alignment. Tight spin-kick alignment follows from measurements of radio pulsar polarization 2, as well as from X-ray observations of pulsar wind nebulae around young pulsars3,4. Implications of these observations to the formation of double pulsars were discussed by Wang et al. 5. Here we explore the effect of NS spin – kick correlation on the formation and galactic coalescence rate of double neutron stars (DNS) which are primary targets for modern gravitational wave detectors. We show that the tighter alignement, the smaller is the DNS merging rate with respect to models with random kick orientation. The effect is especially important for large kick amplitudes (∼ 400 km/s). 2 Effect on the binary neutron star coalescence rates The effect of NS kick velocity on merging rates of compact binaries was studied earlier (e.g. 6). The observed tight NS spin – kick alignment may have important implications to the formation and evolution of binary compact stars (see especially earlier paper by Kalogera 7). Let us consider the standard evolutionary scenario leading to formation of binary NS from a massive binary system (see8 for discussion and references) focusing on the effect of the NS kick velocity. We shall assume that the kick velocity vector is confined within a cone which is coaxial with the progenitor’s rotation axis and characterized by angle θ < π/2. We shall consider only central kicks thus ignoring theoretically feasible off-center kicks affecting the NS spin 9,10. The value of the kick velocity is assumed to obey the Maxellian distribution f(v) ∼ v2 exp(−(v/v0) 2), as suggested by pulsar proper motion measurements11. The velocity v0 varied from 0 to 400 km/s. http://arxiv.org/abs/0704.1204v2 0 50 100 150 200 250 300 350 400 450 Kick velocity (km/s) Double NS coalescence rate 0,0 0,2 0,4 0,6 0,8 1,0 (in units /2) 50 km/s 75 km/s 100 km/s 125 km/s 150 km/s 200 km/s 400 km/s Figure 1: Galactic coalescence rate of DNS vs. kick parameter v0 (random kicks). An almost exponential decay with v0 is seen for v0 > 100 km/s Figure 2: Relative change of DNS merging rate for NS spin-kick correlation The rotational axes of both components are assumed to be aligned with the orbital angular momentum before the primary collapses to form first NS. The SN explosion is treated in a standard way as instantaneous loss of mass of the exploding star. The effect of the kick on the post-explosion binary orbital parameters is treated from the point of view of energy-momentum conservation in two point-mass body problem (see e.g. in 7,12). The first SN explosion most likely occurs when the binary orbit is circular (unless the initial binary is very wide so that tidal circularization is ineffective), while the second explosion can happen before the orbit has been tidally circularized; in the latter case we choose the position of the star in the orbit distributed according to Kepler’s 2d law. We use the population synthesis method to calculate the expected coalescence rate of DNS (see 6,8 and references therein). The standard assumptions about binary evolution have been made: Salpeter’s mass function for the primary’s mass, f(M1) ∼ M −2.35, flat initial mass ratio (q = M2/M1 < 1) distribution f(q) = const, initial semi-major axes distribution in the form d log a = const. The common envelope phase is treated in the standard way8 with the efficiency αCE = 0.5. The calculations were normalized to the galactic star formation rate 3M⊙ per year, with binary fraction 50%. We also have carefully taken into account rotational evolution of magnetized compact stars, as described in13,14, assuming no magnetic field decay. The galactic DNS merging rate is shown in Fig. 1 as a function of the kick parameter v0 and assuming random central kicks. Note an almost exponential decay of the rate with v0 for v0 > 100 km/s. Fig. 2 shows the relative change in the DNS merging rate when we allow for NS spin-kick alignment with different values of the confinement angle θ. It is seen that tight alignment generally reduces the DNS merging rate, with the effect being especially strong for large kick velocity amplitudes. Such a reducing relative to calculations with random kicks is in fact expected, because the NS spin – kick correlation excludes kicks in the orbital plane which, if directed opposite to the orbital velocity, can additionally bind the post-explosion binary system. 3 Neutron star spin – orbit misalignment There is another observational consequence of the kick in DNS systems: NS spin – orbit mis- alignment, which can be tested by geodetic precession measurements in binary pulsars 15. Such a misalignment is potentially very interesting for GW studies 16. After SN explosion in a bi- nary system, additional kick imparted to newborn NS results, in general, in a misalignment between the new orbital angular momentum and the NS (and the secondary component’s) spin -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 Cos ( v=100 km/s 0,0 0,2 0,4 0,6 0,8 1,0 v=100 km/s Cos ( -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 Cos ( v=100 km/s 0,0 0,2 0,4 0,6 0,8 1,0 v=100 km/s Cos ( -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 Cos ( v=400 km/s 0,0 0,2 0,4 0,6 0,8 1,0 v=400 km/s Cos ( -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 Cos ( v=400 km/s 0,0 0,2 0,4 0,6 0,8 1,0 v=400 km/s Cos ( Figure 3: NS spin-orbit misalignment cosΨ in coalescing DNS for v0 = 100 km/s (upper row) and 400 km/s (bottom row) and different NS spin-kick alignment angles θ0. NS1: spin aligned with orbit before SN explosion; NS2: spin aligned with original binary’s orbital angular momentum vector characterized by angle Ψ1. After the second SN explosion in the system, there are several possibilities for the NS spin-orbit misalignment. 1) In sufficiently wide binaries, when tidal interactions between the components are inef- ficient, the orientation of the NS1 spin and the secondary’s spin vector may remain unchaged until the second SN explosion, after which the orbital angular momentum vector changes again due to NS2 kick. So in this case we would expect two coaxial NS with spins misaligned by angle Ψ2 with orbital angular momentum. However, such binaries, unless very eccentric, may be too wide to coalesce over the Hubble time. 2) In close binaries, tidal interactions tend to rapidly align angular momentum vector of the normal star with the orbital angular momentum. To spin-up the NS rotation up to observed ms periods (in binary ms pulsars), a modest amount of matter (∼ 0.1M⊙) should be accreted by NS. This amount is sufficient to align the NS rotation with the orbital angular momentum. So if NS1 accretes matter before the second SN explosion, both NS1 and the secondary component’s spins are most likely aligned with orbital angular momentum (see discussion in5). Note that the NS1 spin tends to align with the orbital angular momentum even if NS1 does not accrete matter but spins-down by the propeller mechanism before the second SN explosion, since in that case very strong currents must flow through its polar cap and the alignment torques is as strong as during accretion. So the NS1 remains misaligned prior to the second SN explosion only in rare cases where the secondary collapses shortly after the first SN in the binary. If both NS1 and secondary were aligned with orbital angular momentum prior to the second SN explosion, both neutron stars will be equally misaligned with orbital angular momentum, Ψ1 = Ψ2. In our population synthesis simulations we take into account the discussed spin alignment effects. In Fig. 3 we show the calculated distribution between the NS1 and NS2 spins and orbital angular momentum in coalescing DNS systems (angles Ψ1 and Ψ2, respectively) assuming spin- orbit alignment (angle Ψ1) and conservation of the secondary’s angular momentum (angle Ψ2). Clearly, the real distribution must be intermediate between the two, depending on the degree of misalignment of the secondary’s angular momentum prior to the collapse. It is seen that the Table 1: Mean NS spin-orbit misalignment Ψ (in units π/2) Kick v0 NS1 (Ψ1) NS2 (Ψ2) (km/s) Kick confinement angle θ (in units π/2) 0.01 0.1 0.2 0.5 1.0 0.01 0.1 0.2 0.5 1.0 50 0.061 0.060 0.059 0.073 0.212 0.307 0.310 0.316 0.308 0.345 100 0.110 0.110 0.109 0.186 0.444 0.378 0.381 0.381 0.423 0.633 200 0.192 0.195 0.207 0.337 0.535 0.417 0.419 0.442 0.575 0.813 400 0.257 0.262 0.291 0.451 0.670 0.442 0.447 0.481 0.670 0.909 misalignment angles can be very different (and even with negative cosines) for random or loosely constrained (θ ∼ π/2) kicks, while tight spin-kick alignment (θ ≪ π/2) results in much narrow distributions (see also 7). The mean misalignement angles Ψ are presented in Table 1. 4 Conclusions We have shown that the spin-velocity correlation observed in radio pulsars, suggesting NS spin- kick velocity alignment, may have very important implications to GW studies. First, the tight alignment reduces the galactic rate of double neutron star coalescences (especially for large kicks 300-400 km/s – up to ten times) relative to models with random kicks. Second, the spin-kick correlation results in specific distribution of NS spin – orbit misalignments, which can be tested by analysing GW signals from DNS mergings. Acknowledgments KAP acknowledges the financial support from the Meeting Organizers and RFBR grant 07-02- 08065z. References 1. D. Lai in Cosmic Explosions in Three Dimensions: Asymmetries in Supernovae and Gamma-ray Bursts, ed. P. Hoflich, P. Kumar, and J. C. Wheeler (Cambridge Univ. Press, Campridge, 2004), p. 276. 2. S. Johnston et al., MNRAS 364, 1397 (2005). 3. D.J. Helfand et al, Astrophys. J. 556, 380 (2001). 4. O. Kargaltsev et al, arXiv:astro-ph/0611599. 5. C. Wang et al, Astrophys. J. 639, 1007 (2006). 6. V.M. Lipunov et al, MNRAS 288, 245 (1997). 7. V. Kalogera, Astrophys. J. 541, 319 (2000). 8. K.A. Postnov and L.R. Yungelson, LRR 9, 6 (2006). 9. H.C. Spruit and E.S. Phinney, Nature 393, 139 (1998). 10. C. Wang et al, Astrophys. J. 656, 399 (2007). 11. G. Hobbs et al, MNRAS 360, 974 (2005). 12. L.P. Grishchuk et al, Physics-Uspekhi 44, 1 (2001); arXiv:astro-ph/0008481. 13. V.M. Lipunov, Astrophysics of Neutron Stars (Springer Verlag, Berlin, 1992) 14. V.M. Lipunov, K.A. Postnov and M.E. Prokhorov, in The scenario machine: Binary star population synthesis, ed. R.A. Sunyaev (Harwood Academic Publ., Amsterdam, 1996). 15. M. Bailes, Astron. Astrophys. 202, 109 (1988). 16. T. Apostolatos et al, Phys. Rev. D 49, 6274 (1994). http://arxiv.org/abs/astro-ph/0611599 http://arxiv.org/abs/astro-ph/0008481 Introduction Effect on the binary neutron star coalescence rates Neutron star spin – orbit misalignment Conclusions

0704.1205

Electronic structure of the zigzag spin-chain compound In$_2$VO$_5$

Electronic structure of the zigzag spin-chain compound In2VO5 U. Schwingenschlögl Institut für Physik, Universität Augsburg, 86135 Augsburg, Germany (Dated: November 28, 2018) Band structure calculations within the local spin-density approximation are presented in order to investigate the electronic and magnetic properties of the zigzag spin-chain compound In2VO5. The essential structural feature of the system is a double chain of VO6-octahedra, which leads to compet- ing intrachain and interchain magnetic couplings. Frustration of the spin-chains is expected for the proposed antiferromagnetic ordering at low temperatures. However, the band calculations indicate that the experimental room temperature crystal structure is incompatible with antiferromagnetism. Both the intrachain and interchain coupling is found to be ferromagnetic. PACS numbers: 71.20.-b, 71.20.Be Keywords: density functional theory, band structure, magnetism, spin-chain Magnetism in low-dimensional quantum spin systems results in fascinating physical properties when the spin ordering is frustrated due to geometrical restrictions. In this context, the S = 1/2 antiferromagnetic zigzag spin-chain gives rise to one of the most fundamental model systems for analyzing the interplay of frustration and magnetism. Recently, In2VO5 has been put for- ward for consideration as a promising candidate for a frustrated zigzag spin-chain. Vanadium oxides in gen- eral are very susceptible to electronic ordering phenom- ena, see [1, 2, 3], for instance, and the references given there. To be more specific, the essential structural fea- ture of In2VO5 is a double chain of corner sharing VO6- octahedra along the crystallographical b-axis. As indium realizes the oxidation state In3+, the vanadium ions are left with a single electron in the 3d shell. This formal V4+ valence with a 3d1 electronic configuration comes along with S = 1/2 spins at the vanadium sites, sepa- rated by non-magnetic oxygen sites. Because of nearest and next-nearest neighbour V-V interactions of the same order of magnitude, competence between the intrachain and interchain exchange coupling is typical for In2VO5. The compound crystallizes in the simple orthorhombic space group Pnma, where a unit cell comprises four for- mula units. Senegas et al. [4] have obtained the lattice parameters a = 7.232 Å, b = 3.468 Å, and c = 14.82 Å FIG. 1: (Color online) Schematic structure of the zigzag VO- chains in In2VO5. The magnetic V sites are surrounded by distorted O-octahedra, sharing corners along the chains. by means of single crystal x-ray analysis at room tem- perature. Figure 1 illustrates the spacial arrangement of the zigzag VO-chains. The coordination polyhedron of the magnetic V sites is a distorted O-octahedron with one strongly elongated VO-bond of length 2.23 Å. This bond connects two adjacent VO-chains, therefore giving rise to the characteristical double chain geometry. The VO-bond in trans-position accordingly is shortened to 1.76 Å. Moreover, the intrachain bond length amounts to 1.81 Å in both directions and the remaining VO-bonds in the equatorial plane of the coordination polyhedron have lengths of 2.01 Å and 2.03 Å. All V sites are crys- tallographically equivalent, as are the intrachain O sites. Within the zigzag VO-chains we have V-O-V bond an- gles of 146◦, thus considerable deviations from a straight line configuration. In contrast, the V-O-V bond angles between V sites in adjacent chains amount to only 107◦. Nearest neighbour magnetic sites therefore are located in different VO-chains, whereas intrachain V sites are next- nearest neighbours. Sign and strength of the magnetic coupling constants in In2VO5 very recently have been analyzed by Volkova [5] based on a phenomenological theoretical method for quantitatively estimating the magnetic coupling in low- dimensional crystalline compounds. The only input into the calculation is the crystal structure [6]. This crystal chemical approach results in an antiferromagnetic near- est neighbour interaction given by the coupling constant J1 = −0.9mRyd. Competing magnetic interaction along the zigzag VO-chains is likewise antiferromagnetic, with coupling constant J2 = −1.6mRyd. Despite two paths for the nearest neighbour exchange and a single path for the intrachain exchange, see figure 1, the next-nearest neighbour magnetic coupling therefore is found to domi- nate. Furthermore, since the ratio J2/J1 = 1.68 exceeds the Majumdar-Ghosh point 0.5, the zigzag spin-chains in In2VO5 are expected to be frustrated. The following electronic structure results for In2VO5 rely on the scalar-relativistic augmented spherical wave (ASW) method [7]. The implementation in use particu- larly accounts for the non-spherical contributions to the charge density inside the atomic spheres. The structural http://arxiv.org/abs/0704.1205v2 input for the calculation is taken from Senegas et al. [4]. For a correct representation of the crystal potential in voids of the In2VO5 structure the physical spheres have to be complemented by additional augmentation spheres at carefully selected interstitial sites. It turns out that it is sufficient to dispose 84 additional spheres from 15 crys- tallographically inequivalent classes in order to keep the linear overlap of the physical spheres below 15% and the overlap of any pair of spheres below 20%. Since we have 32 physical spheres, the unit cell entering the calculation thus comprises 116 augmentation spheres in total. The basis set taken into account in the secular matrix con- sists of In 5s, 5p, 4d, V 4s, 4p, 3d, and O 2s, 2p states, as well as states of the additional augmentation spheres. During the course of the band structure calculation the Brillouin zone is sampled with an increasing number of up to 56 k-points in the irreducible wedge, which ensures convergence of the results with respect to the fineness of the k-space grid. For the exchange-correlation functional the Vosko-Wilk-Nusair parametrization is used. As long as the augmentation spheres are selected carefully and the crystal structure is not altered, the ASW method is highly reliable for the comparison of magnetic energies. We start the discussion of the electronic structure of In2VO5 by addressing results of a spin-degenerate band structure calculation. They allow us to study general is- sues concerning the anisotropy of the electronic states, the chemical bonding, and hybridization effects. After- wards we will investigate the magnetic coupling by spin- polarized calculations for various spin patterns. Figure 2 shows the band structure for the spin-degenerate case along selected high symmetry lines in the first Brillouin zone of the simple orthorhombic lattice, where the high symmetry points are defined by the standard reciprocal lattice vectors Γ = (0, 0, 0), X = (1 , 0, 0), S = (1 , 0), Y = (0, 1 , 0), Z = (0, 0, 1 ), U = (1 , 0, 1 ), R = (1 and T = (0, 1 ). In the vicinity of the Fermi energy, we have several electronic bands revealing little dispersion throughout the first Brillouin zone. Because these bands originate from the V 3d states, they are responsible for a remarkable peak in the V 3d density of states (DOS), compare the DOS curves in figure 2. In the energy range shown, we have almost only contributions from the V 3d and O 2p states. Fully occupied In 4d bands give rise to a pronounced structure around −15 eV, with respect to the Fermi level. The gross features of the partial V 3d and O 2p DOS are typical for compounds based on VO6-octahedra. As to be expected from a molecular orbital picture, we can identify two groups of bands in the energy ranges from −8.5 eV to −2.8 eV and from −0.2 eV to 4.2 eV. Inter- action between V 3d and O 2p atomic orbitals leads to bonding and antibonding molecular states. The bonding bands are fully occupied, whereas the antibonding bands cross the Fermi level and cause In2VO5 to be a metal at room temperature. Even though the bonding and anti- bonding states are dominated by oxygen and vanadium contributions, respectively, non-vanishing admixtures of ZTRUZΓYSXΓ -8 -6 -4 -2 0 2 4 (E - EF) (eV) FIG. 2: (Color online) Band structure and partial V 3d and O 2p DOS (per atom), as resulting from a spin-degenerate calculation. intrachain coupling interchain coupling energy gain fe fe 12mRyd fe af 10mRyd af fe 4mRyd TABLE I: Comparison of the energy gain (per V site) due to the exchange coupling for various spin patterns. the other states are present in figure 2. They trace back to significant VO-hybridization, particularly between or- bitals mediating σ-type overlap. For investigating the magnetic coupling in In2VO5, we have to consider the following three spin patterns, since nearest and next-nearest neighbour exchange interaction is relevant. First, we assume the magnetic coupling to be ferromagnetic both along the VO-chains and between neighbouring chains. Afterwards, we assume either the intrachain or the interchain coupling to be antiferromag- netic, while keeping the other coupling ferromagnetic. In -1 0 1 2 3 4 5 (E - E ) (eV) V 3d spin-majority, fe -1 0 1 2 3 4 5 (E - EF) (eV) V 3d spin-minority, fe FIG. 3: (Color online) Partial V 3d spin-majority (top) and spin-minority (bottom) DOS (per atom) for ferromagnetic in- trachain coupling. each case, spin-polarized band structure calculations re- sult in a lowering of the total energy as compared to the spin-degenerate solution. Values for the energy gain per magnetic site are summarized in table I. The largest en- ergy gain of 12mRyd is obtained when both the nearest and next-nearest neighbour exchange interaction is ferro- magnetic. With respect to this value, intrachain and in- terchain antiferromagnetic coupling raises the energy by 2mRyd and 8mRyd, respectively. The magnetic ground state of In2VO5 hence is found to be ferromagnetic, i.e. the room temperature crystal structure of Senegas et al. [4] is incompatible with antiferromagnetism within the VO-chains as well as between the chains. We next study the effects of the intrachain magnetic coupling on the electronic structure of ferromagnetically coupled VO-chains. Partial V 3d spin-majority and spin- minority densities of states for ferromagnetic and anti- ferromagnetic exchange along the chains are shown in figures 3 and 4. The width of the spin-majority bands in -1 0 1 2 3 4 5 (E - E ) (eV) V 3d spin-majority, af -1 0 1 2 3 4 5 (E - EF) (eV) V 3d spin-minority, af FIG. 4: (Color online) Partial V 3d spin-majority (top) and spin-minority (bottom) DOS (per atom) for antiferromagnetic intrachain coupling. figure 3 amounts to 4.8 eV. It therefore is about 0.4 eV larger than for the spin-degenerate bands, see figure 2, whereas the width of the spin-minority bands hardly al- ters. In contrast, the spin-majority and spin-minority band widths are rather similar in figure 4. While for fer- romagnetic intrachain coupling the spin-minority group of states is observed at higher energies, leaving only V 3d spin-majority states occupied, both spin components contribute at the Fermi level for antiferromagnetic cou- pling. As a consequence, the local V magnetic moment accumulates to only 0.71µB in the latter case. Oxygen magnetic moments are neglectible. On the contrary, the ferromagnetic coupling results in magnetic moments of 0.92µB for the V and 0.05µB for the intrachain O sites, which sum up to 4µB per unit cell. Due to a strong spin splitting of nearly 0.8 eV, see figure 3, a large number of occupied states is shifted to lower energies, paving the way for the ferromagnetic ground state. In conclusion, electronic structure calculations using density functional theory indicate that the experimental room-temperature crystal structure of the zigzag spin- chain compound In2VO5 is incompatible with both an- tiferromagnetic intrachain and interchain coupling. Fer- romagnetism is stabilized instead, which contradicts the crystal chemical estimates by Volkova [5]. This discrep- ancy probably traces back to hybridization between the V 3d and O 2p states, as reflected by remarkable oxygen magnetic moments. Nevertheless, the antiferromagnetic coupling likewise comes along with energy gain as com- pared to the non-magnetic solution. Because of narrow V 3d bands at the Fermi energy, see figure 2, electronic correlations beyond the local density approximation can play a role and further stabilize antiferromagnetic inter- action. However, this seems not to be the case here, as recent experiments point at a transition from ferromag- netic to antiferromagnetic exchange at low temperature, accompanied by structural alterations [8]. Since strong coupling of the electronic system to the crystal lattice is typical for transition metal oxides [9], slight changes in the crystal structure may induce relevant modifications of the magnetic exchange. A large variety of phase tran- sitions is known for compounds with octahedrally coor- dinated transition metal atoms. The vanadium and tita- nium Magnéli phases, for example, are subject to metal- insulator transitions accompanied by distinct structural alterations [10, 11]. A low temperature structural phase transition in In2VO5 therefore still could cause an anti- ferromagnetic ground state with frustrated zigzag spin- chains. In order to solve this question, a detailed inves- tigation of the In2VO5 crystal structure is required. Acknowledgement Valuable discussions with L.M. Volkova are gratefully acknowledged. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 484). [1] J.B. Goodenough, Prog. Solid State Chem. 5, 145 (1971). [2] W. Brückner, H. Oppermann, W. Reichelt, J.I. Terukow, F.A. Tschudnowski, and E. Wolf, Vanadiumoxide (Aka- demie, Berlin, 1983). [3] U. Schwingenschlögl and V. Eyert, Ann. Phys. (Leipzig) 13, 475 (2004). [4] J. Senegas, J.-P. Manaud, and J. Galy, Acta Cryst. B31, 1614 (1975). [5] L.M. Volkova, J. Phys.: Condens. Matter 19, 176208 (2007). [6] L.M. Volkova and S.A. Polyshchuk, J. Supercond. 18, 583 (2005). [7] V. Eyert, Int. J. Quantum Chem. 77, 1007 (2000); The Augmented Spherical Wave Method – A Comprehensive Treatment, Lecture Notes in Physics (Springer, Heidel- berg, 2007). [8] A. Möller and V. Kataev, private communication. [9] V. Eyert, U. Schwingenschlögl, and U. Eckern, Europhys. Lett. 70, 782 (2005); Chem. Phys. Lett. 390, 151 (2004). [10] U. Schwingenschlögl, V. Eyert, and U. Eckern, Europhys. Lett. 64, 682 (2003); Europhys. Lett. 61, 361 (2003). [11] I. Leonov, A.N. Yaresko, V.N. Antonov, U. Schwingen- schlögl, V. Eyert, and V.I. Anisimiov, J. Phys.: Condens. Matter 18, 10955 (2006).

0704.1207

Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations

Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations Säıd Benachour Institut Elie Cartan-Nancy, Université Henri Poincaré BP 239, F-54506 Vandœuvre lès Nancy cedex, France E-mail: [email protected] Grzegorz Karch Instytut Matematyczny, Uniwersytet Wroc lawski pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland Institute of Mathematics, Polish Academy of Sciences, Warsaw (2002-2003) E-mail: [email protected] Philippe Laurençot Mathématiques pour l’Industrie et la Physique CNRS UMR 5640, Université Paul Sabatier-Toulouse 3 118 route de Narbonne, F-31062 Toulouse cedex 4, France E-mail: [email protected] November 10, 2021 Abstract The large time behavior of solutions to the Cauchy problem for the viscous Hamilton-Jacobi equation ut − ∆u + |∇u|q = 0 is classified. If q > qc := (N + 2)/(N + 1), it is shown that non-negative solutions corresponding to integrable initial data converge in W 1,p(RN ) as t → ∞ toward a multiple of the fundamental solution for the heat equation for every p ∈ [1,∞] (diffusion-dominated case). On the other hand, if 1 < q < qc, the large time asymptotics is given by the very singular self-similar solutions of the viscous Hamilton-Jacobi equation. 2000 Mathematics Subject Classification: 35K15, 35B40. http://arxiv.org/abs/0704.1207v1 2 S. Benachour, G. Karch, & Ph. Laurençot For non-positive and integrable solutions, the large time behavior of solutions is more complex. The case q ≥ 2 corresponds to the diffusion-dominated case. The diffusion profiles in the large time asymptotics appear also for qc < q < 2 provided suitable smallness assumptions are imposed on the initial data. Here, however, the most important result asserts that under some conditions on initial conditions and for 1 < q < 2, the large time behavior of solutions is given by the self-similar viscosity solutions to the non-viscous Hamilton-Jacobi equation zt + |∇z|q = 0 supplemented with the initial datum z(x, 0) = 0 if x 6= 0 and z(0, 0) < 0. Résumé Nous classifions le comportement asymptotique des solutions du problème de Cauchy pour l’équation de Hamilton-Jacobi avec diffusion ut−∆u+|∇u|q = 0. Si q > qc := (N+2)/(N+1), nous montrons que, lorsque t → ∞, les solutions intégrables et positives convergent dans W 1,p(RN ) vers un multiple de la solution fondamentale de l’équation de la chaleur pour tout p ∈ [1,∞] (diffusion dominante). Ensuite, si 1 < q < qc, le comportement asymptotique est décrit par la solution très singulière auto-similaire de l’équation de Hamilton-Jacobi avec diffusion. En ce qui concerne les solutions intégrables et négatives, la situation est plus complexe. Le terme de diffusion est de nouveau dominant si q ≥ 2, ainsi que lorsque qc < q < 2 pourvu que la donnée initiale soit suffisamment petite. Ensuite, pour 1 < q < 2, nous identifions une classe de données initiales pour laquelle le comportement asymptotique des solutions est donné par une solution de viscosité auto-similaire de l’équation de Hamilton- Jacobi zt + |∇z|q = 0 avec la condition initiale (non continue) z(x, 0) = 0 si x 6= 0 et z(0, 0) < 0. Keywords: Diffusive Hamilton-Jacobi equation, self-similar large time behavior, Lapla- cian unilateral estimates. Mots-clés : Equation de Hamilton-Jacobi diffusive, comportement asymptotique auto- similaire, estimations unilatérales du Laplacien. 1 Introduction We investigate the large time behavior of integrable solutions to the Cauchy problem for the viscous Hamilton-Jacobi equation ut −∆u+ |∇u|q = 0 , x ∈ RN , t > 0 ,(1.1) u(x, 0) = u0(x) , x ∈ RN ,(1.2) where q > 1. The dynamics of the solutions to (1.1)-(1.2) is governed by two competing effects, namely those resulting from the diffusive term −∆u and those corresponding to the “hyperbolic” nonlinearity |∇u|q. Our aim here is to figure out whether one of Viscous Hamilton-Jacobi equations 3 these two effects rules the large time behavior, according to the values of q and the initial data u0. Since the nonlinear term |∇u|q is non-negative, it acts as an absorption term for non-negative solutions and as a source term for non-positive solutions. We thus consider separately non-negative and non-positive solutions. Let us outline our main results now. For non-negative initial data, it is already known that diffusion dominates the large time behavior for q > qc := (N +2)/(N +1) and that the nonlinear term only becomes effective for q < qc [1, 4, 6, 8]. We obtain more precise information in Theorems 2.1 and 2.2 below. In particular, if q ∈ (1, qc) and the initial datum decays sufficiently rapidly at infinity, there is a balance between the diffusive and hyperbolic effects: the solution u(t) behaves for large t like the very singular solution to (1.1), the existence and uniqueness of which have been established in [5, 3, 23]. For non-positive initial data, there are two critical exponents q = qc and q = 2, as already noticed in [21], and the picture is more complicated. More precisely, the diffusion governs the large time dynamics for any initial data if q ≥ 2 and for sufficiently small initial data if q ∈ (qc, 2), and we extend the result from [21, Proposition 2.2] in that case (cf. Theorem 2.3, below). On the other hand, when q ∈ (1, 2), we prove that, for sufficiently large initial data, the large time behavior is governed by the nonlinear reaction term. This fact is also true for any initial datum u0 6≡ 0 if N ≤ 3 and q is sufficiently close to 1. We actually conjecture that the nonlinear reaction term always dominates in the large time for any non-zero initial datum as soon as q ∈ (1, qc). Let us finally mention that, when q ∈ (qc, 2), there is at least one (self-similar) solution for which there is a balance between the diffusive and hyperbolic effects for large times [7]. Before stating more precisely our results, let us recall that for every initial datum u0 ∈ W 1,∞(RN) the Cauchy problem (1.1)-(1.2) has a unique global-in-time solution which is classical for positive times, that is u ∈ C(RN × [0,∞)) ∩ C2,1(RN × (0,∞)) . In addition, this solution satisfies the estimates ‖u(t)‖∞ ≤ ‖u0‖∞ and ‖∇u(t)‖∞ ≤ ‖∇u0‖∞ for all t > 0.(1.3) Moreover, by the maximum principle, u0 ≥ 0 implies that u ≥ 0 and u0 ≤ 0 ensures that u ≤ 0. We refer the reader to [1, 4, 17] for the proofs of all these preliminary results. In addition, a detailed analysis of the well-posedness of (1.1)-(1.2) in the Lebesgue spaces Lp(RN) may be found in the recent paper [7]. Notations. The notation to be used is mostly standard. For 1 ≤ p ≤ ∞, the Lp-norm of a Lebesgue measurable real-valued function v defined on RN is denoted by ‖v‖p. We will always denote by ‖·‖X the norm of any other Banach space X used in this paper. Also, W 1,∞(RN) denotes the Sobolev space consisting of functions in L∞(RN) 4 S. Benachour, G. Karch, & Ph. Laurençot whose first order generalized derivatives belong to L∞(RN). The space of compactly supported and C∞-smooth functions in RN is denoted by C∞c (RN), and C0(RN) is the set of continuous functions u such that |x|≥R {|u(x)|} = 0 . For a real number r, we denote by r+ := max {r, 0} its positive part and by r− := max {−r, 0} its negative part. The letter C will denote generic positive constants, which do not depend on t and may vary from line to line during computations. Throughout the paper, we use the critical exponent qc := N + 2 N + 1 2 Results and comments As already outlined, the large time behavior of solutions to (1.1)-(1.2) is determined not only by the exponent q of the nonlinear term |∇u|q but also by the sign, size, and shape of the initial conditions. In the present paper, we attempt to describe this variety of different asymptotics of solutions, imposing particular assumptions on initial data. In order to present our results in the most transparent form, we divide this section into subsections. 2.1 Non-negative initial conditions In Theorems 2.1 and 2.2 below, we always assume that u0 is a non-negative function in L 1(RN ) ∩W 1,∞(RN ) , u0 6≡ 0 ,(2.1) and we denote by u = u(x, t) the corresponding non-negative solution of the Cauchy problem (1.1)-(1.2). In that case, we recall that t 7−→ ‖u(t)‖1 is a non-increasing function and that |∇u| belongs to Lq(RN × (0,∞)). In addition, I∞ := lim u(x, t) dx = u0(x) dx− |∇u(x, s)|q dx ds(2.2) satisfies I∞ > 0 if q > qc and I∞ = 0 if q ≤ qc (cf. [1, 4, 6], for details). Since we would have I∞ = ‖u0‖1 > 0 for the linear heat equation, we thus say that diffusion dominates the large time behavior when I∞ > 0, that is, when q > qc. We first consider the diffusion-dominated case. Theorem 2.1 Suppose (2.1) and that q > qc. For every p ∈ [1,∞], t(N/2)(1−1/p)‖u(t)− I∞G(t)‖p = 0(2.3) Viscous Hamilton-Jacobi equations 5 t(N/2)(1−1/p)+1/2‖∇u(t)− I∞∇G(t)‖p = 0.(2.4) Here, G(x, t) = (4πt)−N/2 exp(−|x|2/(4t)) is the fundamental solution of the heat equa- tion. When p = 1, the relation (2.3) is proved in [8] and Theorem 2.1 extends the convergence of u(t) towards a multiple of G(t) to W 1,p(RN), p ∈ [1,∞]. Remark 2.1 Theorem 2.1 holds true when I∞ = 0 (i.e. for q ≤ qc) as well, but in that case, the relation (2.3) says only that ‖u(t)‖p tends to 0 as t → ∞ faster than t−(N/2)(1−1/p). Our next theorem is devoted to the balance case 1 < q < qc when a particular self-similar solution of (1.1) appears in the large time asymptotics. Theorem 2.2 Suppose (2.1). Assume that q ∈ (1, qc) and, moreover, that ess lim |x|→∞ |x|au0(x) = 0 with a = q − 1 .(2.5) For every p ∈ [1,∞], t(N/2)(1−1/p)+(a−N)/2‖u(t)−W (t)‖p = 0(2.6) t(N/2)(1−1/p)+(a−N)/2+1/2‖∇u(t)−∇W (t)‖p = 0,(2.7) where W (x, t) = t−a/2W (xt−1/2, 1) is the very singular self-similar solution to (1.1). For the existence and uniqueness of the very singular solution to (1.1), we refer the reader to [5, 3, 23]. Notice also that the initial datum u0 is integrable by assumption (2.5) since a > N for 1 < q < qc. Remark 2.2 In the critical case q = qc, it is also expected that u(t) converges towards a multiple of G(t) with a correction in the form of an extra logarithmic factor resulting from the absorption term. This conjecture is supported by what is already known for non-negative solutions to the Cauchy problem wt −∆w + w(N+2)/N = 0 (see, e.g., [25] and the references therein). 2.2 Non-positive initial conditions We now turn to non-positive solutions and assume that u0 is a non-positive function in L 1(RN) ∩W 1,∞(RN) , u0 6≡ 0 .(2.8) 6 S. Benachour, G. Karch, & Ph. Laurençot We denote by u = u(x, t) the corresponding non-positive solution of the Cauchy prob- lem (1.1)-(1.2). In that case, we recall that t 7−→ ‖u(t)‖1 is a non-decreasing function and put I∞ := inf u(x, t) dx = − sup ‖u(t)‖1 ∈ [−∞,−‖u0‖1] .(2.9) Substituting u = −v in (1.1)-(1.2) we obtain that v = v(x, t) is a non-negative solution to vt −∆v − |∇v|q = 0, v(x, 0) = −u0(x),(2.10) which has been studied in [7, 16, 17, 21]. We start again with the diffusion-dominated case. Theorem 2.3 Suppose (2.8). a) Assume that q ≥ 2. Then I∞ > −∞ and |∇u| belongs to Lq(RN × (0,∞)). In addition, I∞ is given by (2.2) and the relations (2.3) and (2.4) hold true for every p ∈ [1,∞]. b) Assume that q ∈ (qc, 2). There exists ε = ε(N, q) such that, if ‖u0‖1‖∇u0‖(N+1)q−(N+2)∞ < ε ,(2.11) then the conclusions of part a) are still valid. The fact that I∞ > −∞ under the assumptions of Theorem 2.3 is established in [21], together with the relation (2.3) for p = 1. We extend here this convergence to W 1,p(RN), p ∈ [1,∞]. The smallness assumption imposed in (2.11) is necessary to obtain the heat ker- nel as the first term of the asymptotic expansion of solutions. This is an immediate consequence of the following theorem and the subsequent discussion. Theorem 2.4 Suppose (2.8) and that q ∈ (qc, 2). a) There exists a non-positive self-similar solution V = V (x, t) = t−(2−q)/(2(q−1))V (xt−1/2, 1) to (1.1) such that t(N/2)(1−1/p)‖V (t)‖p = ∞ and lim t(N/2)(1−1/p)+1/2‖∇V (t)‖p = ∞ for all p ∈ [1,∞]. b) There is a constant K = K(q) ≥ 0 such that, if u0 ∈ W 2,∞(RN) satisfies ‖u0‖∞ ∥(∆u0) 1−2/q > K(2.12) ‖u(t)‖∞ > 0.(2.13) Viscous Hamilton-Jacobi equations 7 The first part of Theorem 2.4 is proved in [7] while the second assertion is new. Let us point out here that, for the Hamilton-Jacobi equation wt+ |∇w|q = 0, the L∞-norm of solutions remains constant throughout time evolution, while it decays to zero for the linear heat equation. We thus realize that, under the assumptions of Theorem 2.4 b), the diffusive term is not strong enough to drive the solution to zero in L∞ as t → ∞ and the large time dynamics is therefore ruled by the Hamilton-Jacobi term |∇u|q. Unfortunately, the conditions (2.11) and (2.12) do not involve the same quantities. Still, we can prove that if u0 fulfils ‖u0‖∞ ‖D2u0‖1−2/q∞ > K (which clearly implies (2.12) since q < 2), the quantity ‖u0‖1‖∇u0‖(N+1)q−(N+2)∞ cannot be small. Indeed, there is a constant C depending only on q and N such that ‖u0‖∞ ‖D2u0‖1−2/q∞ )q(N+1)/2 ≤ C‖u0‖1‖∇u0‖q(N+1)−(N+2)∞ .(2.14) For the proof of (2.14), put B = ‖u0‖∞‖D2u0‖1−2/q∞ and note that the Gagliardo- Nirenberg inequalities ‖u0‖∞ ≤ C ‖∇u0‖N/(N+1)∞ ‖u0‖ 1/(N+1) ‖∇u0‖∞ ≤ C ‖D2u0‖(N+1)/(N+2)∞ ‖u0‖ 1/(N+2) imply that ‖∇u0‖(2−q)(N+2)∞ ≤ C ‖D2u0‖(2−q)(N+1)∞ ‖u0‖ = C B−q(N+1) ‖u0‖q(N+1)∞ ‖u0‖ ≤ C B−q(N+1) ‖∇u0‖qN∞ ‖u0‖21 , whence the above claim. We next show that the second assertion of Theorem 2.4 is also true when q ∈ (1, qc). Theorem 2.5 Suppose (2.8) and that q ∈ (1, qc]. There is a constant K = K(q) ≥ 0 such that, if u0 ∈ W 2,∞(RN ) fulfils (2.12), then (2.13) holds true. Furthermore, if N ≤ 3 and 1 < q < 4/ 1 + 2N , then K(q) = 0. We actually conjecture that K(q) = 0 for any q ∈ (1, qc), but we have yet been unable to prove it. The last result confirms the domination of the Hamilton-Jacobi term for large times when (2.13) holds true and provides precise information on the large time behavior. Theorem 2.6 Let q ∈ (1, 2). Assume that u0 ∈ C0(RN ) fulfils (2.8) and is such that M∞ := lim ‖u(t)‖∞ > 0 .(2.15) 8 S. Benachour, G. Karch, & Ph. Laurençot ‖u(t)− ZM∞(t)‖∞ = 0 ,(2.16) where ZM∞ is given by ZM∞(x, t) := − M∞ − (q − 1) q−q/(q−1) )q/(q−1) (2.17) for (x, t) ∈ RN × (0,∞). In fact, ZM∞ is the unique viscosity solution in BUC(RN × (0,∞)) to zt + |∇z|q = 0 in RN × (0,∞)(2.18) with the bounded and lower semicontinuous initial datum z(x, 0) = 0 if x 6= 0 and z(0, 0) = −M∞. The last assertion of Theorem 2.6 follows from [24]. Moreover, ZM∞ is actually given by the Hopf-Lax formula ZM∞(x, t) = inf −M∞ 1{0}(y) + (q − 1) q−q/(q−1)|x− y|q/(q−1) t−1/(q−1) for (x, t) ∈ RN × (0,∞), where 1{0} denotes the characteristic function of the set {0}. Observe that ZM∞ is a self-similar solution to (2.18) since ZM∞(x, t) = ZM∞(xt −1/q, 1). If N = 1, the convergence stated in Theorem 2.6 extends to the gradient of u. Proposition 2.1 Assume that N = 1 and consider a non-positive function u0 in W 1,1(R) ∩ W 1,∞(R). Under the assumptions and notations of Theorem 2.6, we have t(1−1/p)/q ‖ux(t)− ZM∞,x(t)‖p = 0 for p ∈ [1,∞). In fact, if N = 1 and u0 ∈ W 1,1(R) ∩W 1,∞(R), the function U := ux is a solution to the convection-diffusion equation Ut − Uxx + (|U |q)x = 0 , x ∈ R , t > 0 ,(2.19) with initial datum U(0) = u0x and satisfies U(x, t) dx = u0x(x) dx = 0 , t ≥ 0 .(2.20) The large time behavior of non-negative or non-positive integrable solutions to (2.19) is now well-identified [12, 13] but this is far from being the case for solutions satisfying (2.20). In this situation, some sufficient conditions on U(0) are given in [19] for the Viscous Hamilton-Jacobi equations 9 solution to (2.19) to exhibit a diffusion-dominated large time behavior. Also, conver- gence to N -waves is studied in [20] but, for solutions satisfying (2.20), no condition is given in that paper which guarantees that U(t) really behaves as an N -wave for large times. As a consequence of our analysis, we specify such a condition and also provide several new information on the large time behavior of solutions to (2.19) satisfying (2.20). Results on the large time behavior of solutions to equation (2.19) satisfying the condition (2.20) are reviewed in the companion paper [2]. We finally outline the contents of the paper: the next section is devoted to some preliminary estimates. Theorems 2.1 and 2.3 (diffusion-dominated case) are proved in Section 4 and Theorem 2.2 in Section 5. The remaining sections are devoted to the “hyperbolic”-dominated case: Theorems 2.4 and 2.5 are proved in Section 5 and Theorem 2.6 and Proposition 2.1 in Section 6. 3 Preliminary estimates Let us first state a gradient estimate for solutions to (1.1) which is a consequence of [4, Theorem 1] (see also [17, Theorem 2]). Note that, in this section, we do not impose a sign condition on the solution u to (1.1). Proposition 3.1 Assume that u = u(x, t) is the solution to (1.1)-(1.2) corresponding to the initial datum u0 ∈ W 1,∞(RN). For every q > 1, there is a constant C1 > 0 depending only on q such that ‖∇u(t)‖∞ ≤ C1 ‖u0‖1/q∞ t−1/q , for all t > 0 .(3.1) Proof. Setting v = u+‖u0‖∞, it readily follows from (1.1) and the maximum principle that v is a non-negative solution to (1.1). By [4, Theorem 1], there is a constant C depending only on q such that ∥∇v(q−1)/q(t) ≤ C t−1/q , t > 0 . Since ∇v = (q/(q − 1)) v1/q ∇v(q−1)/q and |u(x, t)| ≤ ‖u0‖∞, we further deduce that ‖∇u(t)‖∞ = ‖∇v(t)‖∞ ≤ C ‖v(t)‖1/q∞ ∥∇v(q−1)/q(t) ≤ C ‖u0‖1/q∞ t−1/q , whence (3.1). � Next, we derive estimates for the second derivatives of solutions to (1.1)-(1.2) when q ∈ (1, 2]. 10 S. Benachour, G. Karch, & Ph. Laurençot Proposition 3.2 Under the assumptions of Proposition 3.1, if q ∈ (1, 2], the Hessian matrix D2u = uxixj 1≤i,j≤N of u satisfies D2u(x, t) ≤ ‖∇u0‖ q (q − 1) t Id ,(3.2) D2u(x, t) ≤ C2 ‖u0‖(2−q)/q∞ Id ,(3.3) for (x, t) ∈ RN × (0,∞), where C2 is a positive constant depending only on q. Furthermore, if u0 ∈ W 2,∞(RN), D2u(x, t) ≤ ‖D2u0‖∞ Id .(3.4) In Proposition 3.2, Id denotes the identity matrix of MN(R). Given two matrices A and B in MN(R), we write A ≤ B if Aξ · ξ ≤ Bξ · ξ for every vector ξ ∈ RN . For q = 2, the estimates (3.2) and (3.3) follow from the analysis of Hamilton [18] (since, if f is a non-negative solution to the linear heat equation ft = ∆f , the function − ln f solves (1.1) with q = 2). In Proposition 3.2 above, we extend that result to any q ∈ (1, 2]. Remark 3.1 The estimates (3.2) and (3.3) may also be seen as an extension to a multidimensional setting of a weak form of the Oleinik type gradient estimate for scalar conservation laws. Indeed, if N = 1 and U = ux, then U is a solution to Ut − Uxx + (|U |q)x = 0 in R× (0,∞). The estimates (3.2) and (3.3) then read Ux ≤ C ‖U(0)‖2−q∞ t−1 and Ux ≤ C ‖u0‖(2−q)/q∞ t−2/q for t > 0, respectively, and we thus recover the results of [15, 20] in that case. Proof of Proposition 3.2. For 1 ≤ i, j ≤ N , we put wij = uxixj . It follows from equation (1.1) that wij,t −∆wij = −q |∇u|q−2 uxk wjk = −q |∇u|q−2 wik wjk − q |∇u|q−2 uxk wjk,xi(3.5) − q (q − 2) |∇u|q−4 uxk wik uxk wjk Consider now ξ ∈ RN \ {0} and set wij ξi ξj . Viscous Hamilton-Jacobi equations 11 Multiplying (3.5) by ξi ξj and summing up the resulting identities yield ht −∆h = −q |∇u|q−2 wik ξi − q |∇u|q−2 ∇u · ∇h − q (q − 2) |∇u|q−4 uxj wij ξi .(3.6) Thanks to the following inequalities |∇u|q−4 uxj wij ξi ≤ |∇u|q−4 |uxj |2 wij ξi ≤ |∇u|q−2 wik ξi h2 ≤ |ξ|2 wik ξi and since q ≤ 2, the right-hand side of identity (3.6) can be bounded from above. We thus obtain ht −∆h ≤ −q (q − 1) |∇u|q−2 wik ξi − q |∇u|q−2 ∇u · ∇h ≤ −q |∇u|q−2 ∇u · ∇h− q (q − 1) |∇u|q−2 Consequently, Lh ≤ 0 in RN × (0,∞) ,(3.7) where the parabolic differential operator L is given by Lz := zt −∆z + q |∇u|q−2 ∇u · ∇z + q (q − 1) |∇u|q−2 On the one hand, since q ∈ (1, 2] and |∇u(x, t)| ≤ ‖∇u0‖∞, it is straightforward to check that H1(t) := ‖h(0)‖∞ q (q − 1) t |ξ|2 ‖∇u0‖2−q∞ , t > 0 , satisfies LH1 ≥ 0 with H1(0) ≥ h(x, 0) for all x ∈ RN . The comparison principle then entails that h(x, t) ≤ H1(t) for (x, t) ∈ RN × (0,∞), from which we conclude that h(x, t) ≤ ‖h(0)‖∞ ≤ ‖D2u0‖∞ |ξ|2 , 12 S. Benachour, G. Karch, & Ph. Laurençot h(x, t) ≤ |ξ| 2 ‖∇u0‖2−q∞ q (q − 1) t In other words, (3.2) and (3.4) hold true. On the other hand, we infer from (3.1) that H2(t) := 1 |ξ|2 q2 (q − 1) ‖u0‖(2−q)/q∞ , t > 0 , satisfies LH2 ≥ 0 with H2(0) = +∞ ≥ h(x, 0) for all x ∈ RN . We then use again the comparison principle as above and obtain (3.3). � Remark 3.2 Since q ∈ (1, 2] and ∇u may vanish, the proof of Proposition 3.2 is somehow formal because of the negative powers of |∇u| in (3.6). It can be made rigorous by first considering the regularised equation uεt −∆uε + |∇uε|2 + ε2 for ε ∈ (0, 1), and then letting ε → 0 as in [4]. In fact, we need a particular case of Proposition 3.2. Corollary 3.1 Under the assumptions of Proposition 3.2 ∆u(x, t) ≤ C3 ‖∇u0‖ ,(3.8) ∆u(x, t) ≤ C4 ‖u0‖(2−q)/q∞ ,(3.9) for (x, t) ∈ RN × (0,+∞), where C3 and C4 are positive constants depending only on q and N . Furthermore, if u0 ∈ W 2,∞(RN), ∆u(x, t) ≤ sup ∆u0(x) , t ≥ 0 .(3.10) Proof. Consider i ∈ {1, . . . , N} and define ξi = (ξij) ∈ RN by ξii = 1 and ξij = 0 if j 6= i. We take ξ = ξi in (3.7) and obtain that Luxixi ≤ 0, that is, (uxixi)t −∆uxixi + q |∇u| q−2 ∇u.∇uxixi + q (q − 1) |∇u|q−2 u2xixi ≤ 0 in RN × (0,∞). Summing the above inequality over i ∈ {1, . . . , N} and recalling that |∆u|2 ≤ N u2xixi , we end up with −∆(∆u) + q |∇u|q−2 ∇u.∇ (∆u) + q (q − 1) |∇u| |∆u|2 ≤ 0 in RN × (0,∞). We next proceed as in the proof of Proposition 3.2 to complete the proof of Corollary 3.1. � Viscous Hamilton-Jacobi equations 13 4 Diffusion-dominated case The proofs of Theorems 2.1 and 2.3 rely on some properties of the non-homogeneous heat equation which we state now. Similar results have already been used in [8, 21]. Theorem 4.1 Assume that u = u(x, t) is the solution of the Cauchy problem to the linear non-homogeneous heat equation ut = ∆u+ f(x, t), x ∈ RN , t > 0,(4.1) u(x, 0) = u0(x), x ∈ RN ,(4.2) with u0 ∈ L1(RN) and f ∈ L1(RN × (0,∞)). Then ‖u(t)− I∞G(t)‖1 = 0 ,(4.3) where I∞ := lim u(x, t) dx = u0(x) dx+ f(x, t) dx dt . Assume further that there is p ∈ [1,∞] such that f(t) ∈ Lp(RN) for every t > 0 t1+(N/2)(1−1/p) ‖f(t)‖p = 0 .(4.4) t(N/2)(1−1/p)‖u(t)− I∞G(t)‖p = 0 ,(4.5) t(N/2)(1−1/p)+1/2‖∇u(t)− I∞∇G(t)‖p = 0 .(4.6) Proof. We first observe that the assumptions on u0 and f warrant that I∞ is finite, and we refer to [8] for the proof of (4.3). We next assume (4.4) and prove (4.6). Let T > 0 and t ∈ (T,∞). By the Duhamel formula, ∇u(t) = ∇G(t− T ) ∗ u(T ) + ∇G(t− τ) ∗ f(τ) dτ. It follows from the Young inequality that t(N/2)(1−1/p)+1/2 ‖∇u(t)−∇G(t− T ) ∗ u(T )‖p ≤ C t(N/2)(1−1/p)+1/2 ∫ (T+t)/2 (t− τ)−(N/2)(1−1/p)−1/2 ‖f(τ)‖1 dτ + C t(N/2)(1−1/p)+1/2 (T+t)/2 (t− τ)−1/2 ‖f(τ)‖p dτ )(N/2)(1−1/p)+1/2 ∫ ∞ ‖f(τ)‖1 dτ 14 S. Benachour, G. Karch, & Ph. Laurençot + C sup τ (N/2)(1−1/p)+1 ‖f(τ)‖p (T+t)/2 (t− τ)−1/2 τ−1/2 dτ )(N/2)(1−1/p)+1/2 ∫ ∞ ‖f(τ)‖1 dτ + C sup τ (N/2)(1−1/p)+1 ‖f(τ)‖p Also, classical properties of the heat semigroup (see, e.g., [11]) ensure that t(N/2)(1−1/p)+1/2 ∇G(t− T ) ∗ u(T )− u(x, T ) dx ∇G(t− T ) = 0 , t(N/2)(1−1/p)+1/2 ‖∇G(t− T )−∇G(t)‖p = 0 for every p ∈ [1,∞]. Since, by elementary calculations, we have ‖∇u(t)− I∞ ∇G(t)‖p ≤ ‖∇u(t)−∇G(t− T ) ∗ u(T )‖p ∇G(t− T ) ∗ u(T )− u(x, T ) dx ∇G(t− T ) u(x, T ) dx− I∞ ‖∇G(t− T )‖p + |I∞| ‖∇G(t− T )−∇G(t)‖p , the previous relations imply that lim sup t(N/2)(1−1/p)+1/2 ‖∇u(t)− I∞ ∇G(t)‖p ‖f(τ)‖1 dτ + sup τ (N/2)(1−1/p)+1 ‖f(τ)‖p u(x, T ) dx− I∞ The above inequality being valid for any T > 0, we may let T → ∞ and conclude that (4.6) holds true. The assertion (4.5) then follows from (4.3) and (4.6) by the Gagliardo-Nirenberg inequality. � Proof of Theorem 2.1. Since u is non-negative, we infer from [4, Eq. (17)] that there is a constant C = C(q) such that ‖∇u(q−1)/q(t)‖∞ ≤ C ‖u(t/2)‖(q−1)/q∞ t−1/2 , t > 0 . Also, u is a subsolution to the linear heat equation and therefore satisfies ‖u(t)‖p ≤ ‖G(t) ∗ u0‖p ≤ C t−(N/2)(1−1/p) ‖u0‖1 , t > 0 , Viscous Hamilton-Jacobi equations 15 for every p ∈ [1,∞] by the comparison principle. Since ∇u = (q/(q−1)) u1/q ∇u(q−1)/q, we obtain that t(N/2)(1−1/p)+1 ‖|∇u(t)|q‖p ≤ C t(N+2−q(N+1))/2 −→ for p ∈ [1,∞], because q > (N + 2)/(N + 1). Theorem 2.1 then readily follows by Theorem 4.1 with f(x, t) = −|∇u(x, t)|q. � Proof of Theorem 2.3, part a). Since q ≥ 2, we infer from [21] that I∞ is finite and negative and that ∇u ∈ Lq(RN × (0,∞)) .(4.7) Setting b := ‖∇u0‖q−2∞ , it follows from (1.3) that ut −∆u ≥ −b |∇u|2 in RN × (0,∞). The comparison principle then entails that u ≥ w, where w is the solution to wt −∆w = −b |∇w|2 , w(0) = u0 . The Hopf-Cole transformation h := e−bw − 1 then implies that h solves ht −∆h = 0 , h(0) = e−bu0 − 1 . Therefore, for t > 0, 0 ≤ −bw(x, t) ≤ h(x, t) ≤ ‖h(t)‖∞ ≤ C t−N/2 ‖h(0)‖1 ≤ C t−N/2 , since u0 ∈ L1(RN) ∩ L∞(RN). Recalling that 0 ≥ u ≥ w, we end up with ‖u(t)‖∞ ≤ C t−N/2 , t > 0 .(4.8) It next follows from [17, Theorem 2] that ‖∇u(t)‖∞ ≤ C ‖u(t/2)‖∞ t−1/2 , t > 0 , which, together with (4.8), yields ‖∇u(t)‖∞ ≤ C t−(N+1)/2 , t > 0 .(4.9) Recalling (1.3), we also have ‖∇u(t)‖∞ ≤ C (1 + t)−(N+1)/2 , t ≥ 0 .(4.10) We next put A1(t) := sup τ∈(0,t) τ 1/2 ‖∇u(τ)‖1 16 S. Benachour, G. Karch, & Ph. Laurençot which is finite by [7]. Since q ≥ 2 and N ≥ 1, we infer from the Duhamel formula and (4.10) that, for α ∈ (0, 1/2), t1/2 ‖∇u(t)‖1 ≤ C ‖u0‖1 + C t1/2 (t− τ)−1/2 ‖∇u(τ)‖qq dτ ≤ C + C t1/2 (t− τ)−1/2 (1 + τ)−(q−1)(N+1)/2 ‖∇u(τ)‖1 dτ ≤ C + C t1/2 (t− τ)−1/2 (1 + τ)−1 τ−1/2 A1(τ) dτ ≤ C + C α−1/2 ∫ (1−α)t (1 + τ)−1 τ−1/2A1(τ) dτ + C t1/2 A1(t) (1−α)t (t− τ)−1/2 2 + t τ−1/2 dτ ≤ C + C α−1/2 (1 + τ)−1 τ−1/2A1(τ) dτ + C A1(t) (1− τ)−1/2 τ−1/2 dτ , whence 1− C α1/2 A1(t) ≤ C(α) (1 + τ)−1 τ−1/2A1(τ) dτ Consequently, there is α0 ∈ (0, 1/2) sufficiently small such that A1(t) ≤ B1(t) := C(α0) (1 + τ)−1 τ−1/2A1(τ) dτ for t ≥ 0. Now, for t ≥ 0, (t) = C(α0) (1 + t) −1 t−1/2A1(t) ≤ C(α0) (1 + t)−1 t−1/2B1(t) , from which we deduce that A1(t) ≤ B1(t) ≤ B1(0) exp C(α0) (1 + τ)−1 τ−1/2 dτ ≤ C(α0) . We have thus proved that ‖∇u(t)‖1 ≤ C t−1/2 , t > 0 .(4.11) We finally infer from (4.9), (4.11) and the Hölder inequality that t(N/2)(1−1/p)+1 ‖|∇u(t)|q‖p ≤ C t(N+2−q(N+1))/2 −→ Viscous Hamilton-Jacobi equations 17 for p ∈ [1,∞], and we conclude as in the proof of Theorem 2.1. � Proof of Theorem 2.3, part b). Since q ∈ (qc, 2), we obtain from [21] that there is ε > 0 such that, if u0 fulfils (2.11), then I∞ is finite and negative and there are C > 0 and δ > 0 such that ‖∇u(t)‖qq ≤ C t−1 (1 + t)−δ , t > 0 .(4.12) In particular, |∇u|q ∈ L1(RN × (0,∞)) and lim t ‖|∇u(t)|q‖1 = 0 .(4.13) We next claim that ‖∇u(t)‖∞ ≤ C t−(N+1)/2 , t > 0 .(4.14) Indeed, we fix r ∈ (qc, q) such that r < N/(N − 1) and define s = r/(r − 1) and a sequence (ri)i≥0 by and ri+1 = (N + 1) r − (N + 2) ri , i ≥ 0 . We now proceed by induction to show that, for each i ≥ 0, there is Ki ≥ 0 such that ‖∇u(t)‖∞ ≤ Ki t−(N+1)/2 + t−ri , t > 0 .(4.15) Thanks to (3.1), the assertion (4.15) is true for i = 0. Assume next that (4.15) holds true for some i ≥ 0. We infer from (4.12), (4.15) and the Duhamel formula that ‖∇u(t)‖∞ ≤ C ‖u0‖1 t−(N+1)/2 + C ∫ t/2 (t− τ)−(N+1)/2 ‖∇u(τ)‖qq dτ (t− τ)−(N/2)(1−1/r)−1/2 ‖∇u(τ)‖qsq dτ ≤ C t−(N+1)/2 ‖u0‖1 + ∫ t/2 ‖∇u(τ)‖qq dτ (t− τ)−(N/2)(1−1/r)−1/2 ‖∇u(τ)‖q/r∞ ‖∇u(τ)‖q/sq dτ ≤ C t−(N+1)/2 + C I(t) , where I(t) := (t− τ)−(N/2)(1−1/r)−1/2 τ−(N+1)/2 + τ−ri τ−1/s dτ . 18 S. Benachour, G. Karch, & Ph. Laurençot Since r < N/(N − 1) and q > qc, we have I(t) ≤ C (t− τ)−(N/2)(1−1/r)−1/2 τ−(q(N+1))/2r + τ−(qri)/r τ−1/s dτ ≤ C t−((N+1)r−(N+2))/2r t−(q(N+1))/2r + t−(qri)/r t−(N+1)/2 t−((N+1)q−(N+2))/2r + t−ri+1 t−(N+1)/2 + t−ri+1 for t ≥ 1. Consequently, for t ≥ 1, ‖∇u(t)‖∞ ≤ Ki+1 t−(N+1)/2 + t−ri+1 while (1.3) implies that the same inequality is valid for t ∈ [0, 1] for a possibly larger constant Ki+1. Thus (4.15) is true for i + 1, which completes the proof of (4.15). To obtain (4.14), it suffices to note that ri → ∞ since q > r. Now, owing to (4.13) and (4.14), we are in a position to apply Theorem 4.1 and conclude that (2.3) and (2.4) holds true for p = 1 and p = ∞. The general case p ∈ (1,∞) then follows by the Hölder inequality. � 5 Convergence towards very singular solutions The goal of this section is to prove Theorem 2.2. Recall that we assume that 1 < q < qc and that u0 is a non-negative and integrable function satisfying in addition ess lim |x|→∞ |x|a u0(x) = 0 ,(5.1) with a = (2− q)/(q − 1) ∈ (N,∞). We define R(u0) := inf {R > 0 , |x|a u0(x) ≤ γq a.e. in {|x| ≥ R}} , where γq := (q − 1)(q−2)/(q−1) (2− q)−1 and observe that R(u0) is finite by (5.1). Denoting by u the corresponding solution to (1.1) and introducing τ(u0) := (N + 2)− q(N + 1) (N + 1)q −N R(u0) we infer from [5, Lemma 2.2 & Proposition 2.4] that there is a constant C1 depending only on N and q such that t(a−N)/2 ‖u(t)‖1 + ta/2 ‖u(t)‖∞ + t(a+1)/2 ‖∇u(t)‖∞ ≤ C1(5.2) for each t > τ(u0) and u(x, t) ≤ Γq(|x| − R(u0)) , t > 0 , |x| > R(u0) .(5.3) Viscous Hamilton-Jacobi equations 19 Here, Γq is given by Γq(r) = γq r −a, r ∈ (0,∞). Let us observe at this point that decay estimates for ∇u(t) in Lp can be deduced from (5.2) and the Duhamel formula. Lemma 5.1 For p ∈ [1,∞], there is a constant C(p) depending only on N , q and p such that t((a+1)p−N)/2p ‖∇u(t)‖Lp ≤ C(p) for t > τ(u0) .(5.4) Proof. Indeed, since u is non-negative, it follows from [4, Theorem 1] that ‖∇u(q−1)/q(t)‖∞ ≤ C(q) t−1/q for t > 0, which, together with (5.2) and the Duhamel formula entails that, for t > τ(u0), ‖∇u(t)‖1 ≤ ‖∇G(t/2) ∗ u(t/2)‖1 + ‖∇G(t− s) ∗ |∇u|q‖ ≤ C t−1/2 ‖u(t/2)‖1 + C (t− s)−1/2 ∥∇u(q−1)/q(s) ‖u(s)‖1 ds ≤ C t−(a+1−N)/2 + C (t− s)−1/2 s−(a+2−N)/2 ds ≤ C t−(a+1−N)/2 . Interpolating between (5.2) and the above estimate yields (5.4). � In order to investigate the large time behavior of u, we use a rescaling method and introduce the sequence of rescaled solutions (uk)k≥1 defined by uk(x, t) = k a u(kx, k2t) , (x, t) ∈ RN × [0,∞) , k ≥ 1 . Lemma 5.2 For k ≥ 1, we have t(a−N)/2 ‖uk(t)‖1 + ta/2 ‖uk(t)‖∞ + t(a+1)/2 ‖∇uk(t)‖∞ ≤ C1(5.5) for t > τk := τ(u0) k −2 and uk(x, t) ≤ Γq |x| − R(u0) for |x| > R(u0) and t > 0 .(5.6) Proof. It is straightforward to check that, for each k ≥ 1, uk is the solution to (1.1) with initial datum uk(0) and satisfies estimates (5.5) and (5.6) as a consequence of (5.2) and (5.3). � We next use (1.1) and the non-negativity of uk to control the behavior of uk(x, t) for large x uniformly with respect to k. For k ≥ 1, t > 0 and R ≥ 0, we put Ik(R, t) := {|x|≥R} uk(x, t) dx+ {|x|≥R} |∇uk(x, t)|q dxdt .(5.7) 20 S. Benachour, G. Karch, & Ph. Laurençot Lemma 5.3 For every T > 0, we have t∈[0,T ] Ik(R, t) = 0 .(5.8) Proof. Let ̺ be a non-negative function in C∞(RN) such that 0 ≤ ̺ ≤ 1 and ̺(x) = 0 if |x| ≤ and ̺(x) = 1 if |x| ≥ 1 . For R > 0 and x ∈ RN , we set ̺R(x) = ̺(x/R). As uk is a non-negative solution to (1.1), we have Ik(R, t) ≤ uk(x, t) ̺R(x) dx+ |∇uk(x, s)|q ̺R(x) dxds uk(x, 0) ̺R(x) dx+ uk(x, s) |∆̺R(x)| dxds ≤ ka−N {|x|≥kR/2} u0(x) dx+ |∆̺|∞ {R/2≤|x|≤R} uk(x, s) dxds .(5.9) Owing to (5.1) and (5.6), we further obtain that, for R ≥ 1 + 4 R(u0), Ik(R, t) ≤ ka−N {|x|≥kR/2} |∆̺|∞ {R/2≤|x|≤R} |x| − R(u0) ≤ C R−(a−N) + T |∆̺|∞ {R/2≤|x|≤R} ≤ C(T, ̺) R−(a−N) . Lemma 5.3 then readily follows since a > N . � We finally study the behavior of uk for small times. Lemma 5.4 Let r > 0. There is a positive constant C(r) depending only on q, N and r such that {|x|≥r} uk(x, t) dx ≤ C(r) |x|≥kr/2 {|x|a u0(x)}+ t (5.10) for t > τk and k ≥ 4 R(u0)/r. Proof. We fix r > 0 and use the same notations as in the proof of Lemma 5.3. Thanks to the properties of ̺, we infer from (5.9) with R = r that, for t > τk and Viscous Hamilton-Jacobi equations 21 k ≥ 4 R(u0)/r, {|x|≥r} uk(x, t) dx ≤ uk(x, t) ̺r(x) dx ≤ ka−N {|x|≥kr/2} u0(x) dx |∆̺|∞ {r/2≤|x|≤r} uk(x, s) dxds ≤ C(̺, r) |x|≥kr/2 {|x|a u0(x)}+ t where we have used (5.6) to obtain the last inequality. � Proof of Theorem 2.2. Owing to Lemma 5.2 and Lemma 5.3 we may proceed as in [4, Theorem 3] to prove that there are a subsequence of (uk) (not relabeled) and a non-negative function u∞ ∈ C((0,∞);L1(RN)) ∩ Lq((s,∞)× RN)) ∩ L∞(s,∞;W 1,∞(RN)) satisfying u∞(t) = G(t− s) ∗ u∞(s)− G(t− τ) ∗ |∇u∞(τ)|q dτ τ∈[s,t] ‖uk(τ)− u∞(τ)‖1 = 0(5.11) for every s > 0 and t > s. It remains to identify the behavior of u∞ as t → 0. On the one hand, consider r > 0 and t > 0. Since τk → 0 as k → ∞, we have t > τk for k large enough and it follows from Lemma 5.4, (5.1) and (5.11) that {|x|≥r} u∞(x, t) dx ≤ C(r) t . Consequently, {|x|≥r} u∞(x, t) dx = 0 .(5.12) On the other hand, considerM > 0 and set kM := M 1/(a−N). For k ≥ kM , we denote by vk the solution to (1.1) with initial datum vk(0) given by vk(x, 0) := M k N u0(kx), x ∈ RN . Since a > N , we have vk(0) ≤ uk(0) for k ≥ kM and the comparison principle warrants that vk(x, t) ≤ uk(x, t) , (x, t) ∈ RN × [0,∞) , k ≥ kM .(5.13) 22 S. Benachour, G. Karch, & Ph. Laurençot We next observe that (vk(0)) converges narrowly towards (M ‖u0‖1) δ as k → ∞ (δ denoting the Dirac mass at x = 0). We then proceed as in [4] to conclude that τ∈[s,t] ‖vk(τ)− SM(τ)‖1 = 0 for every s > 0 and t > s, where SM denotes the unique non-negative solution to (1.1) with initial datum (M ‖u0‖1) δ [4]. Recalling (5.11) and (5.13), we realize that SM(x, t) ≤ u∞(x, t) , (x, t) ∈ RN × (0,∞) . The above inequality being valid for any M > 0, it is then straightforward to deduce {|x|≤r} u∞(x, t) dx = ∞ .(5.14) In other words, u∞ is a very singular solution to (1.1) and the uniqueness of the very singular solution to (1.1) (cf. [3, 23]) implies that u∞ = W , where W is the very singular solution to (1.1), see Theorem 2.2. The uniqueness of the limit actually entails that the whole sequence (uk)k≥1 converges towards W in C([s, t];L1(RN)) for s > 0 and t > s. Expressed in terms of u, we have thus shown that t(a−N)/2 ‖u(t)−W (t)‖1 = 0 .(5.15) Finally, it follows from (5.2), (5.15) and the Gagliardo-Nirenberg inequality that (2.6) holds true. The last step of the proof is to obtain the convergence (2.7) for the gradients. Consider p ∈ [1,∞], t > 0 and α ∈ (0, 1). By the Duhamel formula, we have Ap(t) := t ((a+1)p−N)/2p ‖∇(u−W )(t)‖Lp ≤ t((a+1)p−N)/2p ‖∇G((1− α)t) ∗ (u−W )(αt)‖Lp + t((a+1)p−N)/2p ‖∇G(t− s) ∗ (|∇u(s)|q − |∇W (s)|q)‖Lp ds ≤ C(α) t(a−N)/2 ‖(u−W )(αt)‖1 + C t((a+1)p−N)/2p (t− s)−1/2 s−1/2 ‖∇(u−W )(s)‖Lp ds , where we have used the fact that max {‖∇u(s)‖∞, ‖∇W (s)‖∞} ≤ C s−(a+1)/2 by (5.2) and the properties of W in order to obtain the last inequality. Consequently, by the definition of Ap(t) and the change of variables s 7→ ts, we obtain Ap(t) ≤ C(α) t(a−N)/2 ‖(u−W )(αt)‖1 Viscous Hamilton-Jacobi equations 23 + C t((a+1)p−N)/2p (t− s)−1/2 s−1/2 s−((a+1)p−N)/2p Ap(s) ds ≤ C(α) t(a−N)/2 ‖(u−W )(αt)‖1 (1− s)−1/2 s−1/2 s−((a+1)p−N)/2p Ap(st) ds . Now, introducing Ap(∞) := lim sup Ap(t) ≥ 0 , which is finite by (5.4), we may let t → +∞ in the above inequality and use (5.15) to conclude that Ap(∞) ≤ C (1− s)−1/2 s−1/2 s−((a+1)p−N)/2p ds Ap(∞) . Finally, the choice of α < 1 sufficiently close to 1 readily yields that Ap(∞) = 0, from which (2.7) follows. � 6 Proofs of Theorems 2.4 and 2.5 Proof of Theorem 2.4, part a). The required non-positive self-similar solution V = V (x, t) = t−(2−q)/(2(q−1))V x t−1/2, 1 is constructed and studied in [7, Theorem 3.5]. In particular, it is shown that the self- similar profile V(x) := V (x, 1) is a radially symmetric bounded C2 function. Moreover, the profile V and its first derivative V ′ both decay exponentially as |x| → ∞ (see [7, Proposition 3.14]) � Proof of Theorem 2.4, part b). Recall that by assumption (2.8), u = u(x, t) is a non-positive solution to (1.1). For t ≥ 0, we put m(t) = inf {u(x, t) , x ∈ RN} ≤ 0. The comparison principle ensures that t 7→ m(t) is a non-decreasing function of time m∞ := sup m(t) ∈ (−∞, 0] . Since u is a classical solution to (1.1), it follows from (1.1) that u(x, t) ≤ u0(x) + ∆u(x, τ) dτ ≤ u0(x) + ∆u(y, τ) dτ for every x ∈ RN and t ≥ 0. Therefore, m(t) ≤ −‖u0‖∞ + ∆u(y, τ) dτ , 24 S. Benachour, G. Karch, & Ph. Laurençot and we infer from (3.9) and (3.10) that m(t) ≤ −‖u0‖∞ + T ∥(∆u0) + C ‖u0‖(2−q)/q∞ τ−2/q dτ for T > 0 and t > T . Since q < 2, we may let t → ∞ in the above inequality and obtain with the choice T = ‖u0‖(2−q)/2∞ ‖(∆u0)+‖−q/2∞ that there is a constant K depending only on q such that m∞ ≤ −‖u0‖∞ +Kq/2 ∥(∆u0) (2−q)/2 ‖u0‖(2−q)/2∞ .(6.1) Therefore, if ‖u0‖∞ > K ‖(∆u0)+‖(2−q)/q∞ , we readily conclude from (6.1) that m∞ < 0, whence (2.13). � Proof of Theorem 2.5. The proof of the first assertion of Theorem 2.5 is the same as that of Theorem 2.4, part b), hence we skip it. We next assume that N ≤ 3 and that 1 < q < 4/(1 + 1 + 2N). For t > 0, we put ℓ(t) := ‖u(t)‖∞ ∥(∆u(t))+ 1−2/q Since u is a non-positive subsolution to the linear heat equation, we infer from classical properties of the heat semigroup that ‖u(t)‖∞ ≥ ‖G(t) ∗ u0‖∞ ≥ C t−N/2 for t large enough. As q < 2, this estimate and (3.9) entail that, for t large enough, ℓ(t) ≥ C t(4(2−q)−Nq2)/2q2 −→ since q < 4/(1 + 1 + 2N). Consequently, there exists t0 large enough such that ℓ(t0) > K(q) and we may apply the first assertion of Theorem 2.5 to t 7−→ u(t0 + t) to complete the proof. � Under the assumptions of Theorem 2.4, part b) or Theorem 2.5, we may actually bound the L1-norm of u(t) from below and improve significantly [21, Proposition 2.1]. Proposition 6.1 Assume that u0 satisfies (2.8) and that M∞ := lim ‖u(t)‖∞ > 0 . Then there is a constant C = C(N, q, u0) such that ‖u(t)‖1 ≥ C tN/q , t ≥ 0 .(6.2) Viscous Hamilton-Jacobi equations 25 Proof. We fix t > 0. For k ≥ 1, let xk ∈ RN be such that ‖u(t)‖∞−1/k ≤ −u(xk, t). For R > 0, it follows from (3.1) and the time monotonicity of ‖u(t)‖∞ that ‖u(t)‖1 ≥ − {|x−xk|≤R} u(x, t) dx {|x−xk|≤R} (−u(xk, t)− |x− xk| ‖∇u(t)‖∞) dx ‖u(t)‖∞ − N + 1 ‖u0‖1/q∞ t−1/q ≥ C RN − C ′ R t−1/q Letting k → ∞ and choosing R = /(2 C ′) yields the claim (6.2). � 7 Proof of Theorem 2.6 and Proposition 2.1 Proof of Theorem 2.6. STEP 1. Recall that, by (2.8), u0 is a non-positive function. We assume further that u0 is compactly supported in a ball B(0, R0) of R N for some R0 > 0. For λ ≥ 1, we introduce uλ(x, t) := u(λx, λ qt) , (x, t) ∈ RN × (0,∞) , which solves uλ,t + |∇uλ|q = λq−2 ∆uλ in RN × (0,∞)(7.1) with initial datum uλ(0). Lemma 7.1 There is a constant C = C(N, q, ‖u0‖∞) such that, for t ≥ 0 and λ ≥ 1, ‖uλ(t)‖∞ + t1/q ‖∇uλ(t)‖∞ + t ‖uλ,t(t)‖∞ ≤ C .(7.2) Proof. It first follows from (1.3) that ‖uλ(t)‖∞ = ‖u(λqt)‖∞ ≤ ‖u0‖∞ , while Proposition 3.1 yields ‖∇uλ(t)‖∞ = λ ‖∇u(λqt)‖∞ ≤ C1 ‖u0‖1/q∞ t−1/q . We next infer from [16, Theorem 5] that ‖uλ,t(t)‖∞ = λq ‖ut(λqt)‖∞ ≤ λq C(N, q) ‖u0‖∞ (λqt)−1 = C(N, q) ‖u0‖∞ t−1 , 26 S. Benachour, G. Karch, & Ph. Laurençot which completes the proof. � Owing to Lemma 7.1, we may apply the Arzelà-Ascoli theorem and deduce that there are a subsequence of (uλ) (not relabeled) and a non-positive function u∞ ∈ C(RN × (0,∞)) such that uλ −→ u∞ in C(B(0, R)× (t1, t2))(7.3) for any R > 0 and 0 < t1 < t2. It also follows from (7.3) and Lemma 7.1 that u∞(t) ∈ BUC(RN) and satisfies ‖u∞(t)‖∞ + t1/q ‖∇u∞(t)‖∞ + t ‖u∞,t(t)‖∞ ≤ C(7.4) for each t > 0. We next introduce the function Hλ : R× RN × SN(R) → R defined by Hλ(ξ0, ξ, S) := ξ0 + |ξ|q − λq−2 tr(S) , where SN(R) denotes the subset of symmetric matrices of MN(R) and tr(S) denotes the trace of the matrix S. On the one hand, we notice that (7.1) reads Hλ(uλ,t,∇uλ, D2uλ) = 0 in RN × (0,∞) and that Hλ is elliptic. On the other hand, Hλ converges uniformly on every compact subset of R×RN ×SN (R) towards H∞ : R×RN → R given by H∞(ξ0, ξ) := ξ0 + |ξ|q. Therefore, for every τ > 0, u∞(.+τ) is the unique viscosity solution to (2.18) with initial datum u∞(τ) ( see, e.g., [10, Proposition IV.1] and [9, Theorem 4.1]). In addition, since u∞(τ) is bounded and Lipschitz continuous by (7.4), we infer from [14, Section 10.3, Theorem 3] that u∞(.+ τ) is given by the Hopf-Lax formula u∞(x, t+ τ) = inf u∞(y, τ) + (q − 1) q−q/(q−1) |x− y|q/(q−1) t−1/(q−1) (7.5) for (x, t) ∈ RN × [0,∞). It remains to identify the behavior of u∞(t) as t → 0. Consider first x ∈ RN , t ∈ (0,∞) and s ∈ (0, t). We infer from (3.9) and (7.1) that uλ(x, t) ≤ uλ(x, s) + λq−2 ∆uλ(x, σ) dσ ≤ uλ(x, s) + λq−2 λ2 C (λqσ) ≤ uλ(x, s)− C λq−2 t(q−2)/q − s(q−2)/q Since q ∈ (1, 2), we may pass to the limit as λ → ∞ in the previous inequality and use (7.3) to deduce that t 7→ u∞(x, t) is non-increasing for every x ∈ RN . Since u∞ is bounded by (7.4), we may thus define u∞(0) by u∞(x, 0) := sup {u∞(x, t)} ∈ (−∞, 0] for x ∈ RN .(7.6) Viscous Hamilton-Jacobi equations 27 In particular, u∞(0) is a lower semicontinuous function as the supremum of continuous functions. More information on u∞(0) are consequences of the next result. Lemma 7.2 For each t > 0, there is ̺(t) > 0 such that u∞(x, t) = 0 if |x| > ̺(t) and ‖uλ(t)− u∞(t)‖∞ = 0 .(7.7) Moreover, ̺(t) → 0 as t → 0. Taking Lemma 7.2 for granted, we see that (7.6) and Lemma 7.2 imply that u∞(x, 0) = 0 for x 6= 0 since ̺(t) → 0 as t → 0. We set ℓ := −u∞(0, 0), so that u∞(x, 0) = −ℓ 1{0}(x) , x ∈ RN , and fix (x, t) ∈ RN × (0,∞). We will now proceed along the lines of [24] to show that u∞(x, t) = Zℓ(x, t) (recall that Zℓ is defined in (2.17)). Introducing the notation µ := (q−1) q−q/(q−1) t−1/(q−1), it follows from (7.6) and Lemma 7.2 that, for 0 < σ < τ and |y| ≤ ̺(σ), u∞(y, σ) + µ |x− y|q/(q−1) ≥ u∞(y, τ) + µ |x− y|q/(q−1) ≥ u∞(0, τ) + µ |x|q/(q−1) − ω(σ) , ω(σ) := sup |y|≤̺(σ) |u∞(y, τ)− u∞(0, τ)|+ µ sup |y|≤̺(σ) ∣|x− y|q/(q−1) − |x|q/(q−1) while, for 0 < σ < τ and |y| ≥ ̺(σ), u∞(y, σ) + µ |x− y|q/(q−1) ≥ 0 . The previous bounds from below and (7.5) entail that u∞(x, t+ σ) ≥ min 0, u∞(0, τ) + µ |x|q/(q−1) − ω(σ) for 0 < σ < τ . Since ̺(σ) → 0 as σ → 0 and u∞ ∈ C(RN × (0,∞)), we may pass to the limit as σ → 0 in the above inequality and obtain u∞(x, t) ≥ min 0, u∞(0, τ) + µ |x|q/(q−1) for τ > 0. Letting τ → 0 yields u∞(x, t) ≥ min 0,−ℓ+ µ |x|q/(q−1) = Zℓ(x, t) . 28 S. Benachour, G. Karch, & Ph. Laurençot On the other hand, (7.5) and (7.6) ensure that u∞(x, t + τ) ≤ inf u∞(y, 0) + µ |x− y|q/(q−1) = Zℓ(x, t) , whence u∞(x, t) ≤ Zℓ(x, t) by the continuity of u∞ in RN × (0,∞). We have thus shown that u∞ = Zℓ. In particular, ‖u∞(t)‖∞ = ℓ for t ≥ 0. But (2.15) and (7.7) imply ‖u∞(t)‖∞ = lim ‖uλ(t)‖∞ = lim ‖u(λqt)‖∞ = M∞ , whence ℓ = M∞ and u∞ = ZM∞ . For t > 0, the sequence (uλ(t)) has thus only one possible cluster point in L∞(RN ) as λ → ∞, from which we conclude that the whole family (uλ(t)) converges to ZM∞(t) in L ∞(RN) as λ → ∞. In particular, for t = 1, ‖uλ(1)− ZM∞(1)‖∞ = 0 . Setting λ = t1/q and using the self-similarity of ZM∞ , we are finally led to (2.16). STEP 2. We now consider an arbitrary function u0 ∈ C0(RN ) fulfilling (2.8) and such that (2.15) holds true. There is a sequence (un0 ) of non-positive functions in C∞c (RN) such that un0 −→ u0 in L∞(RN) . For n ≥ 1, we denote by un the solution to (1.1) with initial datum un0 and put Mn∞ := lim ‖un(t)‖∞ . By [17, Corollary 4.3], we have ‖un(t)− u(t)‖∞ ≤ ‖un0 − u0‖∞ for t ≥ 0 , from which we readily deduce that |Mn∞ −M∞| ≤ ‖un0 − u0‖∞ . Consequently, Mn∞ −→ M∞ as n → ∞ and (2.15) guarantees that Mn∞ > 0 for n large enough. The analysis performed in the previous step then implies that ‖un(t)− ZMn (t)‖∞ = 0 for n large enough. Therefore, ‖u(t)− ZM∞(t)‖∞ ≤ ‖u(t)− un(t)‖∞ + ‖un(t)− ZMn∞(t)‖∞ + ‖ZMn (t)− ZM∞(t)‖∞ ≤ ‖un0 − u0‖∞ + ‖un(t)− ZMn∞(t)‖∞ + |M ∞ −M∞| ≤ 2 ‖un0 − u0‖∞ + ‖un(t)− ZMn∞(t)‖∞ , Viscous Hamilton-Jacobi equations 29 whence lim sup ‖u(t)− ZM∞(t)‖∞ ≤ 2 ‖un0 − u0‖∞ for n large enough. Letting n → ∞ then completes the proof of Theorem 2.6. � Proof of Lemma 7.2. Let h0 be a non-positive function in C∞c (R) such that h0(y) = −‖u0‖∞ if y ∈ (−R0, R0) (recall that u0 is compactly supported in B(0, R0)). We denote by h the solution to the one-dimensional viscous Hamilton-Jacobi equation ht − hyy + |hy|q = 0 in R× (0,∞) , h(0) = h0 in R . For i ∈ {1, . . . , N} and (x, t) ∈ RN × (0,∞), we put hi(x, t) := h(xi, t) and notice that hi is the solution to (1.1) with initial datum hi(0) ≤ u0. The comparison principle then entails that h(xi, t) = h i(x, t) ≤ u(x, t) ≤ 0 , (x, t) ∈ RN × (0,∞) .(7.8) We next introduce w := hy and notice that w is the solution to the one-dimensional convection-diffusion equation wt − wyy + (|w|q)y = 0 in R× (0,∞) ,(7.9) w(0) = w0 := h0,y in R . The comparison principle then entails that b(y, t) ≤ w(y, t) ≤ a(y, t) , (y, t) ∈ R× (0,∞) ,(7.10) where b ≤ 0 and a ≥ 0 denote the solutions to (7.9) with initial data b(0) = −w−0 ≤ 0 and a(0) = w+0 ≥ 0. Since w0 ∈ L1(R), it follows from [13] that ‖b(t)− Σ−B(t)‖1 = lim ‖a(t)− ΣA(t)‖1 = 0 ,(7.11) where B := ‖b(0)‖1, A := ‖a(0)‖1, and, for M ∈ R, ΣM is the source solution to the one-dimensional conservation law ΣM,t + (|ΣM |q)y = 0 in R× (0,∞) , Σ(0) = M δ0 in R . Here, δ0 denotes the Dirac mass in R centered at y = 0. The source solution ΣM is actually given by ΣM(y, t) := y 1/(q−1) (qt)−1/(q−1) 1[0,ξM (t)](y) , ξM(t) := q q − 1 )(q−1)/q t1/q , 30 S. Benachour, G. Karch, & Ph. Laurençot if M ≥ 0, and ΣM(y, t) := −|y|1/(q−1) (qt)−1/(q−1) 1[−ηM (t),0](y) , ηM(t) := q q − 1 )(q−1)/q t1/q , if M ≤ 0 (see, e.g., [22]). In particular, ΣM satisfies λ ΣM(λy, λ qt) = ΣM (y, t) for (λ, y, t) ∈ (0,∞)× R× (0,∞) .(7.12) Now, let t > 0 and set ̺(t) := N1/2 max {ξA(t), η−B(t)} ≤ C t1/q . If x ∈ RN is such that |x| > ̺(t), there is i ∈ {1, . . . , N} such that |xi| > max {ξA(t), η−B(t)}, whence either xi > ξA(t) or xi < −η−B(t). In the latter case, we infer from (7.8), (7.10) and (7.12) that 0 ≥ uλ(x, t) ≥ h(λxi, λqt) = ∫ λxi w(y′, λqt) dy′ b(λy′, λqt) dy′ (b(λy′, λqt)− Σ−B(λy′, λqt)) dy′ ≥ −‖(b− Σ−B)(λqt)‖1 . Similarly, if xi > ξA(t), (7.8), (7.10) and (7.12) yield 0 ≥ uλ(x, t) ≥ −‖(a− ΣA)(λqt)‖1 . Therefore, if x ∈ RN is such that |x| > ̺(t), then |uλ(x, t)| ≤ max {‖(a− ΣA)(λqt)‖1, ‖(b− Σ−B)(λqt)‖1} .(7.13) Passing to the limit as λ → ∞ in (7.13) and using (7.3) and (7.11) provide the first assertion of Lemma 7.2. We next use once more (7.3) and (7.13) to conclude that (7.7) holds true. � Proof of Proposition 2.1. We keep the notations of the proof of Theorem 2.6 and introduce Uλ(x, t) := uλ,x(x, t) = λ ux(λx, λ qt) , (x, t) ∈ R× (0,∞) . It follows from (7.1) and Lemma 7.1 that Uλ,t + (|Uλ|q)x = λ q−2 Uλ,xx , (x, t) ∈ R× (0,∞) , Viscous Hamilton-Jacobi equations 31 ‖Uλ(t)‖1 ≤ ‖u0,x‖1 and t1/q ‖Uλ(t)‖∞ ≤ C(7.14) for t > 0. We recall that, by Theorem 2.6, the family (uλ) converges towards ZM∞ in C(RN × [t1, t2]) for any t2 > t1 > 0. Owing to (7.14), we readily conclude that (Uλ) converges weakly-⋆ towards ZM∞,x in L ∞(RN × (t1, t2)) for any t2 > t1 > 0. We may then proceed along the lines of [13, Section 3] to show that (Uλ) converges towards ZM∞,x in L 1(R) as λ → ∞. Expressing this convergence result in terms of U = ux and using (3.1) yield Proposition 2.1 by interpolation. � Acknowledgements. We thank Professor Herbert Koch for pointing out to us Ref. [18] and Professor Brian Gilding for useful comments on Proposition 3.2. The preparation of this paper was partially supported by the KBN grant 2 P03A 002 24, the POLONIUM project ÉGIDE–KBN No. 05643SE, and the EU contract HYKE No. HPRN-CT-2002-00282. References [1] L. Amour and M. Ben-Artzi, Global existence and decay for viscous Hamilton– Jacobi equations, Nonlinear Anal. 31 (1998), 621–628. [2] S. Benachour, G. Karch and Ph. Laurençot, Asymptotic profiles of solutions to convection-diffusion equations, C. R. Acad. Sci. Paris, Ser. I, to appear. [3] S. Benachour, H. Koch and Ph. Laurençot, Very singular solutions to a nonlinear parabolic equation with absorption. II – Uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, to appear. [4] S. Benachour and Ph. Laurençot, Global solutions to viscous Hamilton–Jacobi equations with irregular initial data, Comm. Partial Differential Equations 24 (1999), 1999–2021. [5] S. Benachour and Ph. Laurençot, Very singular solutions to a nonlinear parabolic equation with absorption. I. Existence, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 27–44. [6] M. Ben-Artzi and H. Koch, Decay of mass for a semilinear parabolic equation, Comm. Partial Differential Equations 24 (1999), 869–881. [7] M. Ben-Artzi, Ph. Souplet and F.B. Weissler, The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures Appl. 81 (2002), 343–378. [8] P. Biler, M. Guedda and G. Karch, Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation, J. Evolution Equations, in press. 32 S. Benachour, G. Karch, & Ph. Laurençot [9] M.G. Crandall, L.C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487–502. [10] M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42. [11] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 693–698. [12] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equa- tions in RN , J. Funct. Anal. 100 (1991), 119–161. [13] M. Escobedo, J.L. Vázquez and E. Zuazua, Asymptotic behavior and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal. 124 (1993), 43–65. [14] L.C. Evans, Partial Differential Equations, Graduate Stud. Math. 19, Amer. Math. Soc., Providence, 1998. [15] E. Feireisl and Ph. Laurençot, The L1-stability of constant states of degenerate convection-diffusion equations, Asymptot. Anal. 19 (1999), 267–288. [16] B. Gilding, M. Guedda and R. Kersner, The Cauchy problem for the KPZ equa- tion, prépublication LAMFA 28, Amiens, Décembre 1998. [17] B. Gilding, M. Guedda and R. Kersner, The Cauchy problem for ut = ∆u+|∇u|q, J. Math. Anal. Appl. 284 (2003), 733–755. [18] R.S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), 113–126. [19] G. Karch and M.E. Schonbek, On zero mass solutions of viscous conservation laws, Comm. Partial Differential Equations 27 (2002), 2071–2100. [20] Y.J. Kim, An Oleinik type estimate for a convection-diffusion equation and con- vergence to N-waves, J. Differential Equations, to appear. [21] Ph. Laurençot and Ph. Souplet, On the growth of mass for a viscous Hamilton- Jacobi equation, J. Anal. Math. 89 (2003), 367–383. [22] T.-P. Liu and M. Pierre, Source-solutions and asymptotic behavior in conserva- tion laws, J. Differential Equations 51 (1984), 419–441. [23] Y. Qi and M. Wang, The self-similar profiles of generalized KPZ equation, Pacific J. Math. 201 (2001), 223–240. Viscous Hamilton-Jacobi equations 33 [24] T. Strömberg, The Hopf-Lax formula gives the unique viscosity solution, Differ- ential Integral Equations 15 (2002), 47–52. [25] J.L. Vázquez, Asymptotic behaviour of nonlinear parabolic equations. Anomalous exponents, in “Degenerate Diffusions”, W.M. Ni, L.A. Peletier & J.L. Vázquez (eds.), IMA Vol. Math. Appl. 47, Springer, New York, 1993, 215–228. Introduction Results and comments Non-negative initial conditions Non-positive initial conditions Preliminary estimates Diffusion-dominated case Convergence towards very singular solutions Proofs of Theorems ?? and ?? Proof of Theorem ?? and Proposition ??

0704.1209

Characterization of fragment emission in ^{20}Ne (7 - 10 MeV/nucleon) + ^{12}C reactions

Characterization of fragment emission in 20Ne (7 - 10 MeV/nucleon) + 12C reactions Aparajita Dey1, C. Bhattacharya1, S. Bhattacharya1, S. Kundu1, K. Banerjee1, S. Mukhopadhyay1, D. Gupta1, T. Bhattacharjee1, S. R. Banerjee1, S. Bhattacharyya1, T. K. Rana1, S. K. Basu1, R. Saha1, K. Krishan1∗, A. Mukherjee1†, D. Bandopadhyay1‡, C. Beck2 Variable Energy Cyclotron Centre, Sector - 1, Block - AF, Bidhan Nagar, Kolkata - 700 064, India. Institut Pluridisciplinaire Hubert Curien, UMR7500, CNRS-IN2P3 et Universite Louis Pasteur, 23, Rue du Loess, B.P. 28, F-67037, Strasbourg Cedex 2, France. The inclusive energy distributions of the complex fragments (3 ≤ Z ≤ 7) emitted from the bom- bardment of 12C by 20Ne beams with incident energies between 145 and 200 MeV have been mea- sured in the angular range 10o ≤ θlab ≤ 50 o. Damped fragment yields in all the cases have been found to have the characteristic of emission from fully energy equilibrated composites. The binary fragment yields are compared with the standard statistical model predictions. Whereas Li and Be fragments yields are in agreement with statistical-model calculations, enhanced yields of entrance channel fragments (5 ≤ Z ≤ 7) indicate the survival of orbiting-like process in 20Ne + 12C system at these energies. PACS numbers: 25.70.Jj, 24.60.Dr, 25.70.Lm I. INTRODUCTION Several experiments have been done in recent years to understand the mechanism of complex fragment emission in low-energy (Elab <∼ 10 MeV/nucleon) light heavy-ion (Aprojectile + Atarget ∼ 60) reactions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. The origin of these fragments extends from quasi-elastic (QE)/ projectile breakup [2, 3], deep-inelastic (DI) transfer and orbiting [4, 6, 12, 13, 14, 15, 16], to fusion-fission (FF) [18, 19, 20, 21, 22, 23] processes; and in some cases the structure of the nuclei has been found to play an important role. In most of the reactions studied, the observed fully energy-damped yields of the fragments have been successfully explained in terms of fusion-fission (FF) mechanism [18, 19, 20, 21, 22, 23]. However, the reactions involving α - cluster nuclei (e.g., 20Ne + 12C [12, 13], 24Mg + 12C [16], 28Si + 12C [14, 24] etc.) deserved special attention, where the observations of large enhancement in yield and/or resonance-like excitation function in a few outgoing channels have been indicative of a competitive role played by the deep-inelastic orbiting mechanism [12, 13, 14, 15]. In the FF mechanism, a completely equilibrated compound nucleus (CN) is formed, which decays into various exit channels. The decay probability is governed by the available phase space and barrier penetration probabilities for the respective decay channels. The process occurs in a similar time scale which is required for the complete relaxation of the entrance channel energy and angular momentum. On the other hand, deep inelastic orbiting may be described in terms of the formation of a long-lived, dinuclear molecular complex [15], which acts as a “door way to fusion”, with a strong memory of the entrance channel. In this picture, the interacting ions are trapped in a more deformed configuration than that of the compound nucleus (trapped in the pocket of the ion-ion interaction potential due to combined effects of Coulomb and centrifugal barriers). Both orbiting and fusion-fission processes occur on similar time scale. In addition to that, for the light heavy-ion systems, the shapes of the orbiting dinuclear complexes are quite similar to the saddle and scission shapes obtained in course of evolution of the FF process. Therefore it is difficult to differentiate the signatures of the two processes. The enhancement of fully energy damped reaction yields in light systems was first observed in the study of 20Ne + 12C inelastic scattering at backward angles [12], where large cross sections have been observed in inelastic scattering yields near 180o. Subsequently, orbiting was observed in 28Si + 12C [14, 24] and 24Mg + 12C [16] reactions. Detailed study of 28Si + 12C system revealed that, at lower bombarding energies, the excitation spectra for the 12C frag- ments were dominated by single excitation and mutual excitations of the 12C and 28Si fragments, whereas at higher bombarding energies, the dominant strength for all these channels shifted to higher excitation energies [24]. For the higher bombarding energies, the most probable Q-values were found to be independent of detection angles and the resulting angular distributions were found to have dσ/dΩ ∝ 1/ sin θc.m. like angular dependence — characteristic of a long-lived, orbiting, dinuclear complex. Similar results have been obtained for 20Ne + 12C system [12, 13], where ∗ Present address : 306, VIP Enclave, VIP Road, Kolkata - 700 059, India † Present address : Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata - 700 064, India. ‡ Present address : TRIUMF, 4004 Westbrook Mall, Vancouver, Canada V6T2A3. http://arxiv.org/abs/0704.1209v2 resonance-like behaviour was also found in the excitation functions for several outgoing channels, which was similar to the observation made for symmetric 16O + 16O system [25, 26, 27, 28]. Enhancements of large angle, binary reaction yields have also been observed in somewhat heavier 28Si + 28Si, 24Mg + 24Mg systems [1], where significant non resonant background yield was observed at higher excitation energies. The general pattern which unfolds from these studies clearly suggests that the enhancements are manifestations of dynamics of damped nuclear reactions involving a large number of channels, rather than due to specific structure effect appearing only in a few select channels. The enhancement in elastic and inelastic channels may be explained in terms of a long lived dinuclear configuration that decays back to entrance channel due to weak absorption which inhibits the orbiting configuration from spreading into compound nuclear states. However, the enhancement in the elastic channel can be explained with the assumption of weak absorption of grazing partial waves only; on the contrary, deep-inelastic orbiting phenomenon in general suggests weak absorption in the angular momentum window between the critical angular momentum of fusion, lcr, and the reaction grazing angular momentum, lgr. Besides, substantial mass and charge transfer, due to the rapid mass equilibration in light systems, would also occur during the evolution of the orbiting dinuclear complex. So, the rearrangement channels are also of interest in probing the dynamics of the orbiting process involving light nuclear systems. It is, therefore evident that, though some qualitative understanding about the phenomenon of deep-inelastic orbiting reaction, in general (i.e., correlation with number of open reaction channels [29], or, alternatively, to weak absorption) has been arrived at, precise mechanism of the process is still unknown. The deep-inelastic orbiting process has been observed in several light α-like systems, for example, 20Ne + 12C [12, 13], 24Mg + 12C [16], 28Si + 12C [14, 24], 28Si + 16O [30] systems, where the number of open channels are small (∼ 10) [29]. However, 16O (116 MeV) + 28Si reaction [6] showed different behaviour so far as the shape of the energy distributions, variation of < Q > values with angle, and yields of the fragments are concerned. For a better understanding of the orbiting process, it is interesting to study how the orbiting process evolves with energy. Intuitively, survival of long lived dinuclear configuration other than fused composite is less probable at higher excitations and there are also indications that entrance channel effect becomes smaller at higher energies [1]. Shapira et al. [13] made detailed study of 20Ne + 12C system in the energy range Elab = 54 – 81 MeV and showed that there was large enhancement of strongly damped yields, the characteristic of a long-lived orbiting 20Ne + 12C dinuclear system. The aim of the present paper was to extend the investigation on fragment yield from 20Ne + 12C reaction at higher excitation energies, which might allow us to have a better understanding of orbiting vis-à-vis fusion-fission processes for 20Ne + 12C system. If a long-lived rotating dinuclear complex is formed in these reactions, mass and charge transfer should also occur, which leads to typical deep-inelastic reaction yields. Therefore, backangle measurements for rearrangement channels became of interest. In addition to that the excitation function measurements, for the dependence of the average total kinetic energy (EtotK ) loss on bombarding energy for each binary exit channel, would provide an important probe of the dynamical properties of the long-lived dinuclear complex. With this motivation, we have studied the fragment emission spectra from the reaction 20Ne + 12C at Elab = 145, 158, 170, 180 and 200 MeV, respectively. A part of the present data has already been reported [4], which showed enhancement in yield of Carbon and Boron fragments well above the standard statistical model predictions. The paper has been arranged as follows. Section II describes the experimental procedures. In Sec. III we present the analysis of 20Ne + 12C data. The results have been discussed in Sec IV. Finally, the summary and conclusions are presented in Sec. V. II. EXPERIMENTS The experiment was performed using accelerated 20Ne ion beams of energies 145, 158, 170, 180 and 200 MeV, respectively, from the Variable Energy Cyclotron at Kolkata. The target used was ∼ 550 µg/cm2 self-supporting 12C. Different fragments (5 ≤ Z ≤ 13) have been detected using two solid state [Si(SB)] telescopes (∼ 10 µm ∆E, 300 µm E) mounted on one arm of the 91.5 cm scattering chamber. Two solid state telescopes (∼ 50 µm, 100 µm ∆E [Si(SB)] and 5 mm E [Si(Li)]) were mounted on the other arm of the scattering chamber for the detection of light charged particles and light fragments (1 ≤ Z ≤ 4); the same detectors were also used as monitor detectors for normalization purposes. Typical solid angle subtended by each detector was ∼0.3 msr. The telescopes were calibrated using elastically scattered 20Ne ion from Au, Al targets and a Th-α source. The systematic errors in the data, arising from the uncertainties in the measurements of solid angle, target thickness and the calibration of current digitizer have been estimated to be ≈ 15%. Part of these uncertainties are due to the extrapolation procedures employed to extract the fully damped yields from the sequential decay components by the use of Monte Carlo simulations described in Sec. IV A. III. RESULTS A. Energy distribution Inclusive energy distributions for various fragments (3 ≤ Z ≤ 13) have been measured in the angular range 10o - 50o for all the bombarding energies. This covered backward angles in the center-of-mass (c.m.) frame, because of the inverse kinematics of the reactions. Typical energy spectra of the emitted fragments (3 ≤ Z ≤ 13) obtained at an angle 10o at Elab = 170 MeV are shown in Fig. 1. It is evident that the energy spectra for lighter fragents (3 ≤ Z ≤ 6) exhibit strong peaking in energy. The peaks are nearly Gaussian in shape having its centroid at the expected kinetic energies for the fission fragments obtained from the Viola systematics corrected by the corresponding asymmetry factors [31, 32] and are shown by arrows in Fig. 1. The shapes of the energy spectra for the other heavier fragments (7 ≤ Z ≤ 13) are quite different from those obtained for the lighter ones. The additional contributions from DI and QE processes have been seen in the higher energy part of the spectra. Moreover, there are contributions from the recoiling nuclei (energy corresponding to vCN cos θlab, shown by dashed lines in Fig. 1). All these contibutions, other than fission fragments, fall off rapidly as one moves away from the grazing angle. In this paper, we report the results from the lighter fragments (Z = 3–7). The inclusive energy distributions for the fragments Lithium (Z = 3), Beryllium (Z = 4), Boron (Z = 5), Carbon (Z = 6) and Nitrogen (Z = 7) obtained at an angle 10o at various bombarding energies are shown in Fig. 2. It is observed that at all bombarding energies the energy spectra of the ejectiles (Li, Be, B, C, N) are nearly Gaussian in shape and they have been fitted with a single Gaussian. The non-Gaussian shapes at the low-energy side of the spectra correspond to sequential decay processes – which can be simulated by Monte-Carlo statistical model calculations (described in Sec. IV A). The Gaussian fits so obtained are shown by solid lines in Fig. 2. The centroids (shown by arrows) are found to correspond to the scission of deformed dinuclear configuration [15, 31, 32]. This suggests that, in all cases, the fragments are emitted from fully energy relaxed composite – as expected for both FF and orbiting processes. The increasing yields at lower energies may also be due to the second kinematical solution, which is a signature of binary nature of emission process. B. Average velocity The average velocities of the fragments have been computed from the measured energies and from the Z values using the empirical relation proposed by Charity et al. [33]: A = Z × (2.08 + 0.0029× Z). (1) The average velocities of the fragments obtained at different bombarding energies have been plotted in the v‖ vs. v⊥ plane in Fig. 3. It is seen that at all energies, the average velocities of different fragments fall on a circle centered around the respective vCN , the compound nuclear velocity. This suggests that at all bombarding energies the average velocities (as well as kinetic energies) of the fragments are independent of the c.m. emission angles and indicates that at all these energies the fragments are emitted from a fully equilibrated CN-like source with full momentum transfer. The magnitude of the average fragment velocities (i.e., the radii of the circles in Fig. 3) decreases with the increase of fragment mass, which is indicative of the binary nature of the emission. C. Angular distribution The center-of-mass (c.m.) angular distributions of the fragments (Li, Be, B, C, and N) emitted in the 20Ne (145, 158, 170, 180 and 200 MeV) + 12C reactions are shown in Fig. 4. The transformations from the laboratory to center-of-mass system have been done with the assumption of a two body kinematics averaged over total kinetic energy distributions. The c.m. angular distributions of these fragments obtained at all bombarding energies follow the 1/sinθc.m. -like variation (shown by solid lines in Fig. 4) — which further corroborate the conjecture of emission from fully equilibrated composite. D. Average kinetic energy The average total kinetic energies in the center-of-mass, EtotK , for the fragments (3 ≤ Z ≤ 7) obtained at all bombarding energies, have been displayed as a function of scattering angle in Fig. 5. The average fragment kinetic energies in the center-of-mass have been obtained from the respective laboratory values assuming two body kinematics. It is observed from Fig. 5 that EtotK values are almost constant for each of the exit channel. The near constancy of EtotK indicates that at all energies the lifetime of the dinuclear complex is longer than the time needed to completely damp the energy in the relative motion [34, 35, 36, 37]. The predictions of Viola systematics [31] for fission fragment kinetic energies, corrected by an asymmetric factor [32], have been shown by solid lines in Fig. 5. The EtotK values predicted from Viola systematics are found to be in good agreement with the experimental data at all bombarding energies. E. Average Q-value distribution The variations of average Q-value, < Q >, with center-of-mass emission angle for the fragments Li, Be, B, C, and N obtained at different bombarding energies are shown in Fig. 6. It is observed that the < Q > of different fragments are independent of the center-of-mass emission angles at all bombarding energies. This is in contrast to the observation made earlier for other light systems (16O (116 MeV) + 27Al, 28Si, 20Ne (145 MeV) + 27Al, 59Co systems), where sharp fall off of < Q > with angle have been seen [5, 6]. The < Q > values remain nearly constant which further suggest that at all angles, the fragments are emitted from completely equilibrated source at all incident energies considered here. F. Equilibrium cross-section The energy distributions, velocity diagrams, angular distributions and < Q > distributions indicate that the yield of these fragments (Z = 3 – 7) originates from fully energy relaxed events associated with the decay of either compound nucleus or long-lived, orbiting dinuclear system. A detailed investigation have been made to decipher the role played by aforementioned processes in the fragment yield by comparing the experimental yields with the theoretical predictions of the standard statistical model [38], extended Hauser-Feshbach model (EHFM) [20]. The experimental angle integrated yields of the fragments emitted in the 20Ne + 12C reaction at different bombarding energies are shown in Fig. 7 by solid circles (taken from Ref. [13]) and triangles. The theoretical predictions of the statistical model code CASCADE [38] are shown by solid lines in Fig. 7. The calculations are done considering l values up to the critical angular momentum of fusion, lcr, at each energy (given in Table I). It is observed that at all energies the experimental yields of the fragments Li and Be are in fair agreement with the theoretical CASCADE prediction. However, the yields of the fragments B, C and N (near entrance channel) obtained at different energies are much higher than those predicted by the statistical model code CASCADE. A similar observation has been reported by Shapira et al. [12, 13] for the same system at lower energies for Carbon fragment. The theoretical predictions using EHFM have also been shown by dotted lines in Fig. 7 for the fragments B, C and N. The EHFM predictions also similar to those obtained from CASCADE calculations and the experimental yields are in fair excess of the theoretical estimates of both CASCADE and EHFM. G. Excitation energy dependence of < Q > The average Q values (< Q >) for the fragments Li, Be, B, C, and N have been plotted in Fig. 8 as a function of the incident energy. The linear dependence of < Q > with energy provides strong evidence that the long life time may be associated with an orbiting phenomenon. This linear dependence of < Q > can be expressed by simple equation of the form < Q >= c −m ∗ Ec.m., where m is the slope and c is the intercept (for example, < Q > = (14.9 ± 1.0) - (0.97 ± 0.02) Ec.m. for the fragment Carbon). The experimentally determined intercepts are found to be in fair agreement with kinetic energies calculated using Viola systematics. It is interesting to note that the < Q > values obtained in the present experiment between 145 MeV and 200 MeV fall on the same straight line extrapolated from the lower energy (∼54 – 81 MeV) data [13]. This means that the energy relaxation is complete for the fragment emission studied here up to the incident energy of 200 MeV. Moreover, it also means that the final kinetic energy = < Q > + Ec.m.) is nearly independent of bombarding energy — which may be due to the limitation on the maximum value of angular momentum beyond which the formation of di-nucleus is not allowed due to centrifugal repulsion [24]. IV. DISCUSSIONS In general, the energy distributions, the angular distributions and the total fragment yields measured for 20Ne + 12C reaction at incident energies between 145 MeV and 200 MeV are similar to those obtained at lower incident energies (∼ 50 - 80 MeV) for the same system (see Refs. [12, 13]). Large energy damping, 1/sinθc.m. dependence of angular distribution and near constancy of < Q > over a wide angular range signify that the fragment decay originates from a long lived, fully energy equilibrated system. However, the large enhancement of fragment emission cross section (5 ≤ Z ≤ 7) over the statistical-model predictions leads to the conjecture that the orbiting mechanism may still play a major role at these energies. The possibility of these enhancements, either due to feeding from the secondary deexcitation of the heavier fragments or due to orbiting mechanism, are investigated in great details and described in the following subsections. A. Contribution from secondary decay of heavier fragments There is a possibility that the primary heavier fragments (formed due to the binary decay of composite system) may have sufficient excitation energy to deexcite through the emission of light particles and γ-rays and contribute to the yield of lighter fragments. This additional contribution from the secondary decay increases the total elemental yield of the lighter fragments. To check whether the enhancement in B, C, and N yield could be due to feeding from the secondary decay of heavier fragments of various possible binary breakup combinations, we have performed detailed simulations of secondary decay using the Monte-carlo binary decay version of the statistical decay code LILITA [39] and the statistical model code CASCADE [38] and PACE4 [40]. Secondary decay of Si∗ (binary channel 28Si + 4He), Al∗ (binary channel 26,25Al + 6,7Li), Mg∗ (binary channel 24,25,23Mg + 8,7,9Be), Na∗ (binary channel 22,21Na + 10,11B), Ne∗ (binary channel 20Ne + 12C), F∗ (binary channel 18F + 14N), O∗ (binary channel 16O + 16O), N∗ (binary channel 14N + 18F), C∗ (binary channel 12C + 20Ne), B∗ (binary channel 10,11B + 22,21Na) and Be∗ (binary channel 8,7,9Be + 24,25,23Mg) have been studied. Indeed the LILITA calculations (using the parameter set proposed in the Appendix of Ref. [13] and assuming the excitation energy division follows the mass ratio [1]) are in qualitative agreement with the experimental results obtained at 9 MeV/nucleon by Rae et al. [41] for the sequential decay of 20Ne + 12C. It was found that even at the highest excitation energy, secondary decay of Si∗ and Al∗ do not reach up to N; the contribution of primary Mg∗, Na∗ decay to Z ≤ 7 were estimated to be <∼ 1% of the primary yield and that of Ne∗ decay to N, C, B yield were estimated to be ∼ 10–20%, ∼ 15–20% and ∼ 30–50% of the primary yield, respectively. Nearly ∼ 40–45% of the primary O∗ produced through binary exit channel 16O + 16O decays to C. The secondary decay yields from the primary excited fragments are shown in Fig. 9 for different bombarding energies. As the binary yield of O, F are small (∼ 10% of the binary Ne yield, as estimated from CASCADE [38]), overall secondary decay contribution from O, F are smaller than that from Ne. Moreover, the simulations of energy distributions of the secondary decay yield of C from Ne as well as F, O using the code LILITA show that they peak at much lower energies (typically, at ∼ 45–50 MeV for Ne, ∼ 48–55 MeV for F and ∼ 55–60 MeV for O, compared to the peak of the experimental energy distribution at ∼ 75–95 MeV). Now, the Gaussian fitting procedure for the extraction of primary fragment yield is fairly efficient in rejecting most of the low energy tail (typical rejection ratio ∼ 25–40% of the total yield). The energy distributions of the secondary decay have been shown in Fig. 10 for Carbon, in Fig. 11 for Boron and in Fig. 12 for Nitrogen. In the inset of these figures the spectra shown are the difference spectra; the difference between the experimental spectra and the Gaussian fitted to the spectra. It has been found that the secondary decay distributions reproduce the difference spectra very well for all the cases. It is thus evident that the secondary decay component does not interfere with the estimated primary yield for two reasons: firstly, total secondary decay yield is not quite large, and secondly, the Gaussian fitting procedure for the extraction of primary yield does take care, to a large extent, of the rejection of the contributions of the secondary decay components as their energy distributions are different from those of the primary components. B. Fragment cross-section It has been observed that the statistical model calculations do not reproduce most of the observed experimental yields, therefore, an additional reaction component corresponding to the orbiting mechanism has to be considered. The large measured cross sections for B, C and N fragments led to the suggestion that an orbiting, di-nuclear configuration is formed that decays back to the entrance channel. After the discovery of orbiting in the 28Si + 12C system, similar enhancements of large-angle, binary-reaction yields are also observed in the present data. It is expected that the orbiting mechanism will retain a greater memory of the entrance channel than the fusion-fission process. The trapped, dinuclear complex can either evolve with complete amalgamation into a fully equilibrated compound nucleus or, alternatively, escape into a binary exit channel by way of orbiting trajectories. Orbiting can therefore be described in terms of the formation of a long-lived di-nuclear molecular complex which acts as a “doorway” state to fusion with a strong memory of the entrance channel. The equilibrium orbiting model has been used to successfully explain both the observed cross sections and total kinetic energy (TKE) values of the fully damped fragments for several lighter nuclear systems at lower energies. The theoretical prediction of the equilibrium model for orbiting and fusion [15] is denoted by dash-dotted line in Fig. 7 for the fragment B, C and N, and it also fails to explain the large enhancement in the fragment yield. The curve displayed in Fig. 7 represents the “best fit” that can be obtained by the orbiting model with a reasonable choice of the Bass potential parameters (strengths, short range, and long range of the proximity potential). It is, therefore, evident that both the equilibrium orbiting and statistical decay (CASCADE, EHFM) models result in comparable disagreement with the data. It may be interesting to note here that Shapira et al. studied the same reaction at lower energies [12, 13] and came to the conclusion that the large enhancements in the energy damped fragment yield observed at those energies might be due to nuclear orbiting phenomenon. The shortcomings of the equilibrium model for orbiting does not imply that the presence of an orbiting mechanism, as distinct from fission, can be ruled out. On the contrary, there may be a large orbiting-like contribution from non fusion window (in the angular momentum window lcr ≤ l ≤ lgr). This is consistent, at least qualitatively, from the fact that, CASCADE calculation [38] performed with l values up to lgr (shown by dashed lines in Fig. 7) is found to reproduce the data fairly well. The values of lgr at different bombarding energies are given in Table I. Yields in the transfer channels (B, N, for example) are also found to be strongly affected by the orbiting process (yield enhancement), which may be due to stochastic nucleon exchanges during long lifetime of the dinuclear system. In Fig. 13, we show the ratio of Beryllium to Lithium (square), Boron to Lithium (circle), Carbon to Lithium (triangle) and Nitrogen to Lithium (inverted triangle) yield as a function of bombarding energies and the corresponding statistical model (CASCADE) calculations (solid lines). It is found that the observed Beryllium to Lithium ratio is well explained with the statistical model calculations. However, the other observed ratios are higher than the theoretically calculated ratios. This implies the dominance of orbiting yield over the compound nucleus yield. C. Comparison of the C and C reactions The large orbiting yields that account for the largest part of the fully damped yields of 20Ne + 12C can be qualitatively understood in the framework of the Number of Open Channels (NOC) model [29, 42]. The calculated NOC are shown in Table I for both the 20Ne + 12C and the 19F + 12C reactions [29] at several excitation energies along with the measured and calculated (fusion-fission cross sections as predicted by CASCADE) fully damped yields. The NOC for 20Ne + 12C exhibits the characteristic minimum for a grazing angular momentum of approximately lgr = 30h̄ [29]. This very deep minimum (NOC = 4.5) explains: (i) why resonant structures have been observed to be significant in 20Ne + 12C [27, 28] and, (ii) why the orbiting yields observed for C fragments are much larger than the CASCADE predictions. The comparison with 19F + 12C is instructive at E∗ = 60 MeV [7] (the corresponding value of the yield of fragment C for 20Ne + 12C, as given in Table I, has been extracted from Fig. 7 to permit a direct comparison with 19F + 12C). The large NOC value for 19F + 12C [29] (almost order of magnitude bigger) is consistent with the fact that essentially no resonances have been observed in this system [43, 44]. This was confirmed by the times scale measurements of Suaide et al. [45] who found that fusion-fission (with high NOC values), a very slow mechanism, is more competitive than a faster process such as orbiting (with small NOC values) in 19F + 12C. It is worth noting from Table I that CASCADE predicts almost identical fusion-fission cross section for both reactions at E∗ = 60 MeV. On the other hand, due to the survival of orbiting at energies larger than 7 MeV/nucleon, the fully damped yields are much more than a factor of two bigger for the 20Ne + 12C system at E∗ = 60 MeV. V. SUMMARY AND CONCLUSIONS The inclusive double differential cross-section for fragments having Z = 3 – 7 emitted in the reaction 20Ne (∼ 7 – 10 MeV/nucleon) + 12C have been measured. Total emission cross-section for the fragments Li to N have been estimated from the experimental distributions. The c.m. angular distributions for the fragments at all the bombarding energies are found to have a 1/sin θc.m.-type of dependence which signifies the emission of these fragments from a long-lived equilibrated composite. The average velocity plots in v‖ vs. v⊥ plane indicate that the fragments are emitted from fully equilibrated source moving with compound nucleus velocity. The average kinetic energy and the average Q-value of the fragments are independent of the emission angles. This also suggests the emission from a long-lived, equilibrated composite. The angle-integrated cross-section for Li and Be fragments agree well with the theoretical predictions of statistical model but the yield of B, C and N fragments (near to entrance channel) are in excess with the theoretically predicted values. This indicates the presence of other type of reaction process, namely orbiting. In contrast, the study of the nearby system 19F (96 MeV) + 12C clearly showed that the fragments are emitted in the fusion-fission process [7]. Low values of NOC [29] obtained for 20Ne + 12C system as compared to the same obtained for 19F + 12C system also confirms the conjecture of survival of orbiting in 20Ne + 12C system at higher excitations. It is interesting to mention at this point that, 16O, 20Ne + 28Si systems [6, 10, 11], even though α-like, do not show the characteristics of orbiting at these energies, but orbiting-like behaviour has been observed for 28Si + 16O reaction at lower energies [22]. The present analysis also indicates that the enhancement in fragment yield for 20Ne + 12C reactions can not be explained by the equilibrium orbiting model [15]. This may be due to the fact that, the equilibrium orbiting model in its present form seems to be inadequate to explain the phenomena at higher excitations, and a more complete understanding of orbiting and vis-a-vis the angular momentum dissipation (which plays a crucial role in defining orbiting trajectories and yield) will be required. Acknowledgements The authors like to thank the cyclotron operating crew for smooth running of the machine, and H. P. Sil for the fabrication of thin silicon detectors for the experiment. One of the authors ( A. D. ) acknowledges with thanks the financial support provided by the Council of Scientific and Industrial Research, Government of India. [1] S. J. Sanders, A. Szanto de Toledo, and C. Beck, Phys. Rep. 311, 487 (1999) and references therein. [2] N. Carlin-Filho, et al., Phys. Rev. C 40, 91 (1989). [3] S. J. Padalino, et al., Phys. Rev. C 41, 594 (1990). [4] C. Bhattacharya, et al., Phys. Rev. C 72, 021601R (2005). [5] C. Bhattacharya, et al., Phys. Rev. C 66, 047601 (2002). [6] C. Bhattacharya, et al., Phys. Rev. C 69, 024607 (2004). [7] C. Bhattacharya, et al., Phys. Rev. C 54, 3099 (1996). [8] C. Beck, et al. Eur. Phys. J. A2, 281 (1998). [9] C. Beck, et al., Phys. Rev. C 54, 227 (1996). [10] S. P. Barrow, et al., Phys. Rev. C 52, 3088 (1995). [11] K. Farrar, et al., Phys. Rev. C 54, 1249 (1996). [12] D. Shapira, J. L. C. Ford, Jr., J. Gomez del Campo, R. G. Stokstad and R. M. Devries, Phys. Rev. Lett., 43, 1781 (1979). [13] D. Shapira, J. L. C. Ford, Jr. and J. Gomez del Campo, Phys. Rev. C 26, 2470 (1982). [14] D. Shapira, R. Novotny, Y. C. Chan, K. A. Erb, J. L. C. Ford jr., J. C. Peng, and J. D. Moses, Phys. Lett. 114B, 111 (1982). [15] B. Shivakumar, S. Ayik, B. A. Harmon, and D. Shapira, Phys. Rev. C 35, 1730 (1987). [16] W. Dünnweber, et al., Phys. Rev. Lett. 61, 927 (1988). [17] Aparajita Dey, C. Bhattacharya, S. Bhattacharya, T. K. Rana, S. Kundu, K. Banerjee, S. Mukhopadhyay, S. R. Banerjee, D. Gupta, and R. Saha, Phys. Rev. C 75, 064606 (2007). [18] L. G. Moretto, Nucl. Phys. A247, 211 (1975). [19] S. J. Sanders, Phys. Rev. C 44, 2676 (1991). [20] T. Matsuse, C. Beck, R. Nouicer, and D. Mahboub, Phys. Rev. C 55, 1380 (1997). [21] A. K. Dhara, C. Bhattacharya, S. Bhattacharya, and K. Krishan, Phys. Rev. C 48, 1910 (1993). [22] A. Szanto de Toledo, S. J. Sanders, and C. Beck, Phys. Rev. C 56, 558 (1997). [23] A. Szanto de Toledo, B. V. Carlson, C. Beck, and M. Thoennessen, Phys. Rev. C 54, 3290 (1996). [24] D. Shapira, D. Schull, J. L. C. Ford, Jr., B. Shivakumar, R. L. Parks, R. A. Cecil and S. T. Thornton, Phys. Rev. Lett., 53, 1634 (1984). [25] A. Gobbi, et al., Phys. Rev. C 7, 30 (1973). [26] D.G. Kovar, et al., Phys. Rev. C 20, 1305 (1979). [27] J.J. Kolata, R.C. Fuller, R.M. Freeman, F. Haas, B. Heusch, and A. Gallmann, Phys. Rev. C 16, 891 (1977). [28] I. Tserruya, Y. Eisen, D. Pelte, A. Gavron, H. Oeschler, D. Berndt, and H.L. Harney, Phys. Rev. C 18, 1688 (1978). [29] C. Beck, Y. Abe, N. Aissaoui, B. Djerroud, and F. Haas, Phys. Rev. C 49, 2618 (1994). [30] J. M. Oliveira, et al., Phys. Rev. C 53, 2926 (1996). [31] V. E. Viola and K. Kwiatkowski and M. Walker, Phys. Rev. C 31, 1550 (1985). [32] C. Beck, et al., Z. Phys. A343, 309 (1992). [33] R. J. Charity et al., Nucl. Phys. A476, 516 (1988). [34] R. Eggers, M.N. Namboodri, P. Gonthier, K. Geoffroy, and J.B. Natowitz, Phys. Rev. Lett. 37, 324 (1976). [35] P. Braun-Munzinger, T.M. Cormier, and C.K. Gelbke, Phys. Rev. Lett. 37, 1582 (1976). [36] T.M. Cormier, P. Braun-Munzinger, P.M. Cormier, J.W. Harris, and L.L. Lee, Jr., Phys. Rev. C 16, 215 (1977). [37] J.B. Natowitz, M.N. Namboodri, R. Eggers, P. Gonthier, K. Geoffroy, R. Hanus, C. Towsley, and K. Das, Nucl. Phys. A277, 477 (1977). [38] F. Pühlhofer, Nucl. Phys. A280, 267 (1977). [39] J. Gomez del Campo and R. G. Stokstad, LILITA, a MonteCarlo Hauser-Feshbach code, ORNL-TM 7295. [40] A. Gavron, Phys. Rev. C 21, 230 (1980). [41] W. D. M. Rae, A. J. Cole, B. G. Harvey, and R. G. Stokstad, Phys. Rev. C 30, 158 (1984). [42] C. Beck, Y. Abe, N. Aissaoui, B. Djerroud, and F. Haas, Nucl. Phys. A583, 269 (1995). [43] N. Aissaoui, F. Haas, R.M. Freeman, C. Beck, A. Morsad, B. Djerroud, R. Caplar, G.A. Monnehan, A. Hachem, and M.Youlal, Z. Phys. A348, 237 (1994). [44] N. Aissaoui, F. Haas, R.M. Freeman, C. Beck, A. Morsad, B. Djerroud, R. Caplar, and A. Hachem, Phys. Rev. C 55, 516 (1997). [45] A.A.P Suaide et al., Phys. Rev. C 66, 014607 (2002). TABLE I: The angular momentum values, NOC and C fragment yield for different energies. System Elab E ∗ lcr lgr NOC a C yield (mb) (MeV) (MeV) (h̄) (h̄) expt CASCADEb Ne + C 109.5 60 20 26 5.2 122.6c 35.7 Ne + C 145 73 24 31 4.5 151.9±26.8 62.2 Ne + C 158 78 24 33 8.2 149.7±26.5 58.7 Ne + C 170 82 24 34 11.0 131.8±23.8 56.9 Ne + C 180 86 25 36 33.0 142.9±25.4 66.0 Ne + C 200 94 25 38 73.2 178.3±30.7 51.4 F + C 96 60 21 24 97.2 47.92±4.37 [7] 34.5 a Values taken from Ref. [29] b Calculation done using lcr values given in Table I c Extracted from Fig. 7 Energy (MeV) Ne + = 10 = 170 MeV 0 30 60 90 120 150 10 Mg 0 30 60 90 120 150 FIG. 1: Inclusive energy distributions of different fragments emitted in the reaction 20Ne (170 MeV) +12C at θlab = 10 o. The arrow corresponds to the expected fission fragment kinetic energy. The dashed line indicates the average energy of the recoiling nucleus. (a,2)(a,1) (a,3) (a,4) (a,5) 0.2 (b,1) (b,2) (b,3) (b,4) 0.8(b,5) (c,1) (c,2) (c,3) (c,4) (c,5) (d,1) (d,2) (d,3) (d,4) (d,5) 0 40 80 120 (e,1) 0 40 80 120 160 40 80 120 160 40 80 120 160 40 80 120 160 200 Energy (MeV) (e,5)(e,4)(e,3)(e,2) FIG. 2: Inclusive energy distributions for the fragments Lithium (a), Beryllium (b), Boron (c), Carbon (d) and Nitrogen (e) emitted in the reaction 20Ne +12C at an angle 10o for bombarding energies 145 MeV (1), 158 MeV (2), 170 MeV (3), 180 MeV (4) and 200 MeV (5), respectively. The arrow corresponds to the centroid of the fitted Gaussian distribution (solid curve). 170 MeV 0.04 158 MeV 0.04 145 MeV 0.04 180 MeV 0.00 0.04 0.08 0.12 0.16 0.20 200 MeV FIG. 3: The average velocities of the fragments plotted in v‖ vs v⊥ plane at different bombarding energies. The average velocities are denoted by filled triangles (Li), filled circles (Be), open triangles (B), open circles (C) and filled squares (N). The arrows correspond to the compound nucleus velocities. Ne,Li)Al Ne,Be)Mg Ne,B)Na Ne,C)Ne Ne,N)F 30 60 90 30 60 90 30 60 90 (degree) 30 60 90 30 60 90 FIG. 4: The c.m. angular distributions of fragments (Z = 3 – 7) obtained at different bombarding energies. Solid circles (145 MeV), triangles (158 MeV), inverted triangles (170 MeV), squares (180 MeV) and diamonds (200 MeV) correspond to the experimental data and the solid lines are f(θc.m.) ∼ 1/ sin θc.m. fit to the data. Lithium Berillium Boron Carbon Nitrogen 10 15 20 25 30 10 15 20 25 30 10 15 20 25 30 (degree) 10 15 20 25 30 10 15 20 25 30 FIG. 5: Average kinetic energy of different fragments obtained at Elab = 145, 158, 170, 180 and 200 MeV (denoted by inverted triangle, square, diamond, triangle and circle, respectively) plotted as a function of laboratory angle. 20 40 60 80 100 Carbon Lithium Berillium 145 MeV 158 MeV 170 MeV 180 MeV 200 MeV Boron 40 60 80 100 (degree) Nitrogen FIG. 6: Average Q-values of different fragments obtained at Elab = 145, 158, 170, 180 and 200 MeV (denoted by inverted triangle, square, diamond, triangle and circle, respectively) plotted as a function of c.m. emission angle. 2 Lithium Berillium Boron 2 Carbon 40 80 120 160 200 (MeV) Nitrogen FIG. 7: The excitation functions for the angle-integrated yield of the fragments. Triangles are the present data; lower energy data (filled circles) for Carbon fragments are taken from [13]. The solid curves are the predictions of the statistical model. The dash-dotted curves for B, C and N are prediction of equilibrium orbiting model and the dotted curves are the same from EHFM [20]. The dashed curves show CASCADE calculations using grazing angular momentum. 0 Fragment C Ec.m.(MeV) Fragment B 0 Fragment Li Fragment Be 20 40 60 80 0 Fragment N FIG. 8: Bombarding energy dependence of the average Q-values. The solid line shows the linear dependence of < Q > with bombarding energy. The < Q > values at lower energies for Carbon are taken from [13]. 200 MeV 4 5 6 7 8 145 MeV 170 MeV FIG. 9: Percentage secondary decay contribution from primary Ne∗ (circle), F∗ (square) and O∗ (triangle) to Z = 5 – 7. 145 MeV 0 40 80 120 4 180 MeV 0 40 80 120 0 40 80 120 160 200 Energy (MeV) 200 MeV 0 40 80 120 FIG. 10: Secondary decay contribution for Carbon fragments at different energies. The energy distribution at θlab = 10 o along with the fitted Gaussian are shown. Inset: Distribution shows the difference spectra (total spectra - Gaussian) and the solid line represents the total secondary decay contribution estimated using LILITA [39]. 1.0 145 MeV 0 40 80 120 180 MeV 0 40 80 120 0.2 0 40 80 120 160 200 1.5 200 MeV 0 40 80 120 Energy (MeV) FIG. 11: Same as Fig. 10 for Boron fragments. 3 145 MeV 0 40 80 120 4 180 MeV 0 40 80 120 0 40 80 120 160 200 200 MeV 0 40 80 120 160 Energy (MeV) FIG. 12: Same as Fig. 10 for Nitrogen fragments. Be/Li 0 C/Li 140 160 180 200 (MeV) FIG. 13: Bombarding energy dependence of the ratio of angle-integrated fragment yield. The Beryllium to Lithium (square), Boron to Lithium (circle), Carbon to Lithium (triangle), and Nitrogen to Lithium (inverted triangle) ratios are shown. The solid line shows the theoretical prediction using CASCADE. Introduction Experiments Results Energy distribution Average velocity Angular distribution Average kinetic energy Average Q-value distribution Equilibrium cross-section Excitation energy dependence of <Q> Discussions Contribution from secondary decay of heavier fragments Fragment cross-section Comparison of the 20Ne + 12C and 19F + 12C reactions Summary and Conclusions References

0704.1210

The dynamics of Jupiter and Saturn in the gaseous proto-planetary disk

The dynamics of Jupiter and Saturn in the gaseous proto-planetary disk Alessandro Morbidelli Aurélien Crida Observatoire de la Côte d’Azur Corresponding author: Alessandro Morbidelli B.P. 4229 06304 Nice Cedex 4, France email: [email protected] Received ; accepted http://arxiv.org/abs/0704.1210v1 – 2 – ABSTRACT We study the possibility that the mutual interactions between Jupiter and Saturn prevented Type II migration from driving these planets much closer to the Sun. Our work extends previous results by Masset and Snellgrove (2001), by exploring a wider set of initial conditions and disk parameters, and by using a new hydrodynamical code that properly describes for the global viscous evolution of the disk. Initially both planets migrate towards the Sun, and Saturn’s migration tends to be faster. As a consequence, they eventually end up locked in a mean motion resonance. If this happens in the 2:3 resonance, the resonant motion is particularly stable, and the gaps opened by the planets in the disk may overlap. This causes a drastic change in the torque balance for the two planets, which substantially slows down the planets’ inward migration. If the gap overlap is substantial, planet migration may even be stopped or reversed. As the widths of the gaps depend on disk viscosity and scale height, this mechanism is particularly efficient in low viscosity, cool disks. The initial locking of the planets in the 2:3 resonance is a likely outcome if Saturn formed at the edge of Jupiter’s gap, but also if Saturn initially migrated rapidly from further away. We also explore the possibility of trapping in other resonances, and the subsequent evolutions. We discuss the compatibility of our results with the initial conditions adopted in Tsiganis et al. (2005) and Gomes et al. (2005) to explain the current orbital architecture of the giant planets and the origin of the Late Heavy Bombardment of the Moon. – 3 – 1. Introduction The general theory of planet-gas disk interactions (see for instance Lin and Papaloizou, 1979; Goldreich and Tremaine, 1980; Papaloizou and Lin, 1984; Lin and Papaloizou, 1986; Ward, 1986, 1997) predicts a systematic migration of the planets towards the central star. This prediction received a spectacular confirmation with the discovery of the first extra-solar planets, on orbits with semi-major axes comparable to or smaller than those of the terrestrial planets in our Solar System (the so-called hot and warm Jupiters). Despite planet migration is undoubtedly a fact, it is not a general rule, or -at least- not a rule without exceptions. In our Solar System, Jupiter and Saturn should not have migrated substantially, despite they evidently formed in a massive gaseous disk in order to accrete their hydrogen rich atmospheres. In fact, the existence of Uranus and Neptune outside of Saturn’s orbit, and of the Kuiper belt beyond Neptune, constrains the inward migration of Jupiter and Saturn within a few AUs at most (probably much less; see below). In extra-solar systems, with the extension of the timescale of observations, new planets have been found at distances from the parent star comparable to that of Jupiter. Thus, it is important to study non-generic mechanisms that, in some cases, might stop planet migration, or slow it down significantly. A new tight constrain on the orbits of Jupiter and Saturn at the time of disappearance of the gas disk comes from a recent model developed to explain the origin of the Late Heavy Bombardment (LHB) of the terrestrial planets (Gomes et al., 2005). In addition to the LHB, this model explains the current orbital architecture of the giant planets (orbital separations, eccentricities and mutual inclinations; Tsiganis et al., 2005) the origin and orbital distribution of the Trojans of Jupiter (Morbidelli et al., 2005) and the structure of the Kuiper belt (Levison et al., 2007). If true, this model implies that, at the disappearance of the gas, the giant planets and the primordial trans-Neptunian planetesimal disk were – 4 – originally in a compact configuration. This limits severely the radial migration of the planets during the preceeding phase, dominated by the interactions with the gas. Moreover, the orbits of the planets were quasi-circular, and the ratio of the orbital periods of Saturn and Jupiter was smaller than 2. Therefore, it is important to investigate whether these constraints are consistent with the dynamics of the planets in the gas disk, as far as we understand it. A pioneer work in this direction has been done by Masset and Snellgrove (2001; MS01 hereafter). In that letter the authors presented the first numerical simulation of the evolution of Jupiter and Saturn in a gas disk. Saturn initially started at twice the heliocentric distance of Jupiter. After a phase of inward runaway migration, Saturn was captured into the 2:3 mean motion resonance with Jupiter. At that point, the planets reversed their migration, moving outward in parallel, while preserving their resonant relationship. This result is interesting for our purposes in two respects. First, it shows that a two-planet system in some configurations can avoid migration toward the central star. Second, the stable relative configuration achieved by Saturn and Jupiter is characterized by a ratio of orbital periods smaller than 2, as required by the LHB model discussed above. Therefore, we think that it is timely to investigate more in detail the mechanism unveiled by MS01, in particular exploring a wider range of parameters. In fact, there are a few open issues on the validity of the mechanism and its consistency with the solar system structure that we need to address: i) it is unclear how the mechanism depends on the adopted initial conditions. If the initial, inward runaway migration of Saturn is a crucial aspect, then the overall result might depend on the numerical resolution of the disk and on its initial state (namely, whether the planets are dropped into a virgin disk, or they are allowed to sculpt the disk for some time before they are let free to migrate). In fact, some researchers – 5 – could not successfully reproduce the MS01 simulation (Kley, private communication), possibly because of these issues. Moreover, a wide range inward migration of Saturn might be inconsistent with the compact orbital architecture of the giant planets invoked by the LHB model. ii) Jupiter –and to some lesser extent Saturn–, being a giant planet, undergoes type II migration. It is well known that this kind of migration follows the global evolution of the disk. The numerical algorithm used in MS01 could not model the evolution of the disk correctly, as it considered only an annulus of it, with wise –but nevertheless arbitrarily chosen– boundary conditions. In Crida et al. (2007) we have presented a new hybrid scheme that allows the computation of the global evolution of the disk, using a system of nested 1D and 2D grids. Consequently, this algorithm allows a correct simulation of Type II migration. It is true that MS01 argues that, once the planets are in the 2:3 resonance, they are no longer locked in the evolution of the disk (the reason for which they can move outward, despite the disk has a global motion towards the Sun). Nevertheless, the torque unbalance that MS01 measured depends critically on the mass of the disk inside the orbit of Jupiter, and the latter depends on the global evolution of the disk. In particular, it is known that giant planets can open cavities in the inner part of the disk (Rice et al., 2003; Varniere et al., 2006; Crida and Morbidelli, 2007). If this happened in this case, Jupiter and Saturn could not migrate outward. Thus, we think that it is important to re-simulate the dynamics of Jupiter and Saturn using our new, trustable algorithm. iii) the evolution of Jupiter presented in MS01 is probably inconsistent with the solar system architecture. In fact, after Jupiter and Saturn lock in their mutual 2:3 resonance, their outward migration is rather fast. Jupiter increases its orbital radius by ∼ 40% in 1,000 orbits. If this really occurred in the Solar System, Jupiter would – 6 – have been at some time in the middle of the asteroid belt. The properties of the asteroid belt (in particular the quite tight zoning of the taxonomic types) exclude this possibility. Thus, we need to find orbital evolutions that are much more stationary than the one presented in MS01. iv) MS01 claim that the trapping of Jupiter and Saturn into the 2:3 resonance is the most likely outcome of their evolution in the gas disk. This might be problematic in the context of the LHB model (Gomes et al., 2005). That model argues that the ratio of Saturn’s and Jupiter’s orbital periods was smaller than 2, but requires that it was not too much smaller than this value. Otherwise, the mass of the planetesimal disk remaining after the disappearance of the gas would not have been sufficient to drive Saturn across the 1:2 resonance with Jupiter, which is required to trigger the planetary instability responsible for the origin of the LHB. Thus, it is important to investigate if other resonances between Jupiter and Saturn are possible in the context of the mechanism of MS01, or if there are possibilities to leave the resonance towards the end of the gas disk lifetime. With these goals in mind, this paper is structured as follows. In Sect. 2 we briefly review MS01 result using a very similar simulation (and the same numerical code), and then we discuss the dependence of the results on the initial location of Saturn. In Sect. 3 we use the algorithm of Crida et al. (2007) to investigate how the dynamics of the Jupiter-Saturn couple depends on the disk’s aspect ratio and viscosity, and also on numerical parameters such as disk’s resolution and smoothing length. In Sect. 4 we briefly address the effect of the accretion of mass onto the planets, already partially discussed in MS01. In Sect. 5 we explore the dynamics for different values of the planet masses, to understand how generic the MS01 mechanism can be for extra-solar planet cases. In Sect. 6 we discuss possible ways to reconcile the MS01 mechanism with the LHB model of Gomes et al. Our conclusions are – 7 – then recollected in the summary section. 2. The Masset–Snellgrove mechanism Fig. 1 is a quite close reproduction of Fig. 1 in MS01. It has been obtained using the code presented in Masset (2000a,b) (the same as used in MS01), with similar parameters. Specifically, Jupiter is initially at r = 1, while Saturn is at r = 2. The disk extends from r = 0.3 to 5. It is modeled in two dimensions, using a grid with resolution 282 in radius and 325 in azimuth. Its initial surface density is uniform in radius and is equal to 6 × 10−4 in our units (the mass of the Sun is 1), which corresponds to the surface density of the minimal mass nebula (Hayashi, 1981) if the unit of length corresponds to 5 AU. The boundary conditions allow outflow, but not inflow. We adopt an α prescription for the viscosity (Shakura and Sunyaev, 1973), with α = 6 × 10−3, and assume a constant aspect ratio H/r = 4% in the disk’s equation of state. Both Jupiter and Saturn initially migrate inward, as expected. The migration of Saturn accelerates exponentially, in a runaway –also called Type III– regime (Masset and Papaloizou, 2003) that is well fitted by an exponential curve up to t ∼ 100. After a time of about 100 Jovian initial orbital periods, Saturn crosses the 1:2 mean motion resonance (MMR) with Jupiter, because its migration is faster than the threshold below which the capture into resonance is certain in the ‘adiabatic’ approximation (Malhotra, 1993): |ȧS|/(aSΩS) << 0.5j(j + 1)µJeS, for the j : j + 1 resonance, where µJ is the mass ratio of Jupiter to the central object, and aS, eS and ΩS are Saturn’s semi-major axis, eccentricity and angular velocity, respectively (MS01). Once within the 1:2 resonance, Saturn’s runaway migration breaks. This happens because Saturn approaches the outer edge of Jupiter’s gap; thus the disk inside its orbit starts to be partially depleted and consequently Saturn’s coorbital mass deficit becomes smaller that the planet’s own mass (Masset and Papaloizou, – 8 – 2003). After this change in migration regime, Saturn’s inward motion continues at a slower, approximately constant rate. After 280 initial orbital period, the migration of Jupiter stops and then it reverses. As they move in opposite directions, Jupiter and Saturn are eventually captured in their mutual 2:3 MMR, at t = 350. After this event, the two planets migrate outward in parallel. The eccentricity of Saturn stabilizes around 0.012 and that of Jupiter around 0.003. It is worth noticing that the ratio of semi major axes of the two planets does not correspond to a 2:3 ratio of the Keplerian orbital periods. It rather corresponds to a 5:8 ratio. However, by looking at the behavior of the resonant angles, we have checked that the planets are really captured in the 2:3 MMR. The gravity of the disk displaces the mean motion resonances relative to the unperturbed keplerian location. As we said in the introduction, it is an unsolved issue whether the initial evolution of Saturn, with its runaway migration and fast passage across the 1:2 MMR, plays a role in the subsequent dynamics. For instance, MS01 argue that the capture in the 2:3 MMR is favored by Saturn’s eccentricity being enhanced during the previous 1:2 MMR crossing. Our understanding of planet formation is too limited to assess with confidence where Saturn formed. However, there is an emerging view that Saturn might have accreted its atmosphere (and therefore acquired the bulk of its mass) when it was already quite close to Jupiter. In fact, the immediate neighborhood of Jupiter’s gap, being a local maximum of the disk’s surface density, acts as an accumulation point of dust and small planetesimals (Haghighipour and Boss, 2003), and thus appears as a sweet spot for the growth of Saturn’s core. Moreover, Saturn’s core, independently of its formation location, might have suffered Type I migration until it was halted at the edge of Jupiter’s gap, which acts as a planet trap (Masset et al., 2006), or in a mean motion resonance with Jupiter (Thommes, 2005). Thus, we believe that it is important to verify whether the MS01 mechanism can still work – 9 – if Saturn is released in the proximity of Jupiter. Fig. 2 shows the result of a simulation that differs from the previous one only for the initial location of Saturn (now at r = 1.4). As one sees, after a short range migration, the planet is trapped into the 2:3 MMR, and then the evolution is the same as in the previous simulation. Thus, the migration reversal found in MS01 does not depend on the history of the previous migration. For this reason, and because of the arguments described above in favor of a close formation of Saturn, in the following simulations we will always release Saturn at a distance of 1.4 (which corresponds to the typical position of the edge of Jupiter’s gap). 3. Dynamics of Jupiter and Saturn as a function of the disk properties and simulation parameters To explore how the dynamics of Jupiter and Saturn is affected by the main parameters of the problem, we use, from now on, the numerical scheme described in Crida et al. (2007). In this scheme, the disk is represented using a system of 2D and 1D grids. The main portion of the disk, in which the planets evolve, is represented with a 2D grid in polar coordinates, as usual. The origin of the coordinates is the barycentre of the system. The inner part of the disk (ranging from the inner physical radius, e.g. the X-wind truncation radius at a few tenths of AU, to the inner boundary of the 2D grid) and the outer part of the disk (ranging from the outer boundary of the 2D grid to the physical outer edge, e.g. the photo-dissociation radius at hundreds of AU) are represented with a 1D grid. The 1D grids have open outflow boundaries at the inner and outer physical edges, and exchange information with the 2D grid for the definition of realistic, time-dependent boundary conditions of the latter. The algorithm for the interfacing between the 1D and 2D grids is driven by the requirement that the angular momentum of the global system (the disk in the – 10 – 2D section, plus the disk in the 1D section plus the planet-star system) is conserved. With this approach, the global viscous evolution of the disk and the local planet-disk interactions are both well described and the feedback of one on the other can be properly taken into account. Because the migration of giant planets depends on the global evolution of the disk, this code provides more realistic results than the usual algorithms, in which the evolution of the considered portion of the disk depends crucially on the adopted (arbitrary) boundary conditions. For more information and accuracy tests we refer the reader to Crida et al. (2007). In all simulations presented below, the 2D grid is as before (from 0.3 to 5, with 282x325 resolution). The inner 1D grid starts at r = 0.016 and the outer 1D grid ends at r = 40 (which corresponds to about 200 AU in our units). They have the same radial resolution as the 2D grid. The initial surface density profile of the disk is of type Σ(r) = 3× 10−4 exp(−r2/53), as illustrated with a dash-dotted line in Fig. 4, and is derived from the analysis of Guillot and Hueso (2006) for a disk evolving under the collapse of new matter onto the plane from the proto-stellar cloud, viscous evolution and photo-evaporation. The viscosity is assumed constant, for simplicity (we have verified, as MS01, that an α prescription for the viscosity would not change the results significantly, as the planets are very close to each other, although it can affect the global evolution of the disk). Conversely to what we did in the previous section, we first let the planets evolve in the disk for 8,000 Jupiter orbits without feeling the disk’s perturbations, assuming an aspect ratio H/r = 3% and a viscosity ν = 10−5.5 (in our units, see above). This viscosity at r = 1 corresponds to α = 3.5 × 10−3 in a Shakura and Sunyaev (1973) prescription. This simulation allows the planets to sculpt the disk, opening gaps around their orbits, and it sets a new surface density profile of the disk. When we do simulations with different disk parameters, we start from this profile, and let the planets evolve for additional 400 – 11 – Jovian orbits still without feeling the disk’s perturbations, so that the disk profile adapts to the new situation. Only at this point we release the planets, letting them evolve under the effects of the disk and of their mutual perturbations. This procedure allows us to avoid possible spurious initial migrations, that might occur if the initial gas distribution is inconsistent with the presence of the planets. 3.1. Dependence on the disk aspect ratio In a first series of runs, we have fixed the value of the viscosity (ν = 10−5.5, in our units), and we have studied the evolution of Jupiter and Saturn as a function of the disk aspect ratio H/r. MS01 found that the aspect ratio simply changes the outward migration speed, by a quantity proportional to (H/r)−3. We find that the value of the aspect ratio can have a much more important impact on the dynamical evolution. As Fig. 3 shows, if the aspect ratio is small (3 – 4%), the evolution is similar to that previously considered. When released, Jupiter starts to migrate outward, while Saturn moves inward. After locking in the mutual 2:3 mean motion resonance, both planets move outward. In these cases, the migration is indeed faster if the disk is thinner, as found in MS01. However, for thicker disks, the evolution changes qualitatively. If the aspect ratio is 5%, we find a quasi-stationary solution. After locking in the 2:3 MMR, both Jupiter and Saturn essentially do not migrate any more. Actually, Jupiter moves outward by only 1.5% in 2000 orbits. To our knowledge, this is the first quasi-stationary solution ever found for a system of giant planets in a fully evolving disk. If the disk thickness is increased to 6%, both planets migrate inward, even after being captured into the 2:3 MMR. This migration is very slow, compared to that of an isolated Jupiter in the same disk. This sequence of behaviours relative to aspect ratio also suggests that, in a flaring disk, the planets might migrate until they find a position in the disk with the ‘good’ local aspect ratio that allows – 12 – them not to migrate any more. The reason of this parametric dependence of the evolution on H/r is quite clear if one looks at the gas density profile at the moment when the planets are released (Fig. 4). As explained in Crida et al. (2006), the disk aspect ratio governs the width and the depth of the gaps opened by the planets. Therefore, if H/r is large, there is more gas at the location of Saturn (i.e. just outside Jupiter’s orbit) than in the case where H/r is small. Consistently, there is slightly less gas inside of Jupiter’s orbit (for r < 0.7), because less material has been removed from the common gap formed by the two planets. As MS01 correctly pointed out, the direction of migration of Jupiter depends on the balance of the torques that the planet receives respectively from the disk inside its orbit (which pushes the planet outward) and and from the disk outside its orbit (which pushes the planet inward). In the case of an isolated planet, the torque from the outer disk is typically stronger, so that the planet migrates toward the Sun. But in this case, because the presence of Saturn depletes partially the outer disk, this torque is weakened. Obviously, it is weakened more if the gap at Saturn’s position is deeper, namely if the disk aspect ratio is smaller, as visible in Fig. 4. Thus, if the aspect ratio is small enough, the torque received by Jupiter from the inner disk dominates that from the outer disk, and the planet migrates outward, feeling a net positive torque. Indeed, this is what we see happening in Fig. 3. The direction of migration of Jupiter determines the subsequent evolution of both planets, once they are locked in resonance. The planets have to move in parallel to preserve the resonant configuration. Therefore there is a competition between the net positive torque received by Jupiter and the net negative torque received by Saturn from the disk. Because these torques are monotonic functions of the planets’ masses, and Jupiter is 3 times heavier than Saturn, in general the positive torque received by Jupiter dominates and the two planets move outward. In the case with H/r = 5%, however, the net torque felt by Jupiter – 13 – is close to zero, due to the specific density profile of the disk, and can be effectively canceled out by Saturn’s torque. Thus, a non-migrating evolution is achieved after the planets lock in resonance. For completeness and sake of clarity, in the remaining part of this sub-section we elaborate on some considerations already reported in sect. 2.4 of MS01. The principle of Type II migration is that, once a planet opens a gap, it positions itself inside the gap in order to balance the torques received from the inner and the outer parts of the disk. Then, locked into this equilibrium position, it is forced to follow the slow, global viscous evolution of the disk (Lin and Papaloizou, 1986), the latter described by the equations in Lynden-Bell and Pringle (1974). One could expect that the Jupiter-Saturn system should evolve in the same way. The outer migration of the pair of planets should approach Saturn to the outer edge of its gap, until Saturn feels a stronger torque that counterbalances the one received by Jupiter. In this situation the outward migration should stop, and the two planets should start to evolve towards the Sun, together with the disk. This, apparently, does not happen. For the disk parameters that we explore in this work, Saturn is not massive enough to open a clean gap (see Fig. 4). Thus, the conditions for a proper Type II migration are never fulfilled (see Crida and Morbidelli, 2007, for a discussion on Type II migration). If Saturn’s radial migration is not the same as the natural radial motion of the gas, new material flows into its gap. However, the gaps of Jupiter and Saturn overlap, so that material flowing from the outer disk into the coorbital region of Saturn, after experiencing half of a horse-shoe trajectory relative to Saturn, can also perform half of a horse-shoe trajectory relative to Jupiter. The net result is a flow of matter from the outer part of the disk, through the Jupiter-Saturn common gap, into the inner part of the disk. To illustrate this process, Fig. 6 shows the surface density profile of the disk in the simulation with H/r = 3%, at various times. Notice how, in first approximation, the Jupiter-Saturn gap simply ‘shifts’ through the disk. As the planets move outward, the disk is rebuilt inside the orbit of Jupiter and – 14 – the surface density at the bottom of the gap increases as well. Both features are diagnostic of a mass flow through the planets system. In fact, in the code of Crida et al. (2007) that we use, the boundary conditions cannot act as a source of mass. Thus, an increase of the surface density in the inner part of the disk is possible only if there is an influx of mass from the outer disk. The flow of gas has several effects. First, it unlocks the planets from the disk, allowing them to move against the gas stream. Second, it has positive feedbacks on the outer migration of the planets by (i) exerting a corotation torque on them, as it passes through their horseshoe regions and (ii) refurbishing the inner part of the disk, which exerts the positive torque on Jupiter discussed above. The signature of this feedback is well visible in the simulation with H/r = 3% in Fig. 3: the outward migration rate accelerates exponentially, which implies that there is a positive net torque that increases with the migration speed. At this point, one might wonder whether the motion of the planets is dominated by the torque felt by Jupiter from the inner disk, or by the corotation torque exerted by the gas flowing through the planets’ orbits. The flow of gas through the orbits of Saturn and Jupiter is the same; the size of the horseshoe regions of the two planets (and hence the magnitude of the corotation torque felt by each planet) is proportional to Mγp for some γ < 1; thus, the effect on the migration rate ȧp of the planet is proportional to M (γ−1) namely it is larger for a lighter planet. So, if the corotation torque dominated the evolution of the planets, Saturn would be extracted from the resonance and would migrate away from Jupiter. As long as this does not happen (as in Fig. 3), the corotation torque cannot be the dominant force driving the planets’ migration. For a further indication that the corotation torque is weaker than the torque felt by Jupiter from the inner disk (Lindblad torque), we have done another simulation, still with – 15 – H/r = 3% and ν = 10−5.5, but with an initial gas density reduced by a factor of 2. The Lindblad torque scales with the gas density. Conversely, the corotation torque does not scale simply with the gas density because a component of it depends on the the radial speed of the planet relative to the gas, which in turn also depends on the gas density (Masset and Papaloizou, 2003). Thus, if the Lindblad torque dominates, we expect the planets to have the same evolution, just a factor of 2 slower. If the corotation torque dominates, the change in the dynamics can not be trivially reduced to a simple scaling on time. Fig. 5 shows the result. The black curves show the evolution of Jupiter and Saturn in the nominal gas disk. The grey curves show the evolution of the planets in a disk with half the initial density. In plotting this second pair of curves, the time-span measured relative to the release time (400 orbits) has been divided by two. The black and grey curves superpose almost perfectly. This suggests that the Lindblad torque is stronger than the corotation torque. 3.2. Dependence on the simulation’s technical parameters Before proceeding further with our exploration of the dynamical evolution of Jupiter and Saturn, we check the impact of some technical parameters used in the simulation: specifically the smoothing length for the gravitational potential and the grid resolution for the disk. As usual in 2 dimensional hydro-dynamical simulations, the equations of motions are regularized in the vicinity of the planet by modifying the gravitational potential energy U(∆) = −(Mpm)/∆ into Uρ(∆) = −(Mpm)/ ∆2 + ρ2, where ∆ is the distance between the planet of mass Mp and a fluid element of mass m, and ρ is called the smoothing length. The choice of an appropriate value for the smoothing length is the subject of a vast debate. Essentially, two recipes are used: either ρ is chosen proportionally to the Hill radius of the – 16 – planet, or proportionally to the local thickness of the disk. In the simulations presented above, our choice of ρ was equal to 60% of the planet’s Hill radius RH = ap(Mp/3) 1/3, where ap is the semi major axis of the planet and Mp is normalized relative to the mass of the star. We have decided to redo the simulation with H/r = 3%, adopting ρ = 0.7H , where H is the thickness of the disk at the distance of the planet (namely 0.03 ap). In this case, the new value of ρ for Saturn and Jupiter is, respectively, 73% and 50% of those previously adopted. The new simulation is compared with the previous one in Fig. 7. Because the simulations do not start exactly in the same way, for a more meaningful comparison we have plotted the evolution of the planets only from the time T0 at which Saturn starts its outward migration (this time is slightly different in the two simulations) and we have renormalized the semi major axes of the planets by the semi major axis of Jupiter at T0. As one sees, the difference is not very big. Using the new value of ρ leads to a slightly faster migration. The reason is that Saturn opens a wider and deeper gap in the new simulation, because the smaller value of the smoothing length is equivalent to an enhancement of its gravitational potential. As we have seen before, a deeper gap at Saturn’s location increases the unbalance of the torques exerted on Jupiter from the inner and the outer parts of the disk, and hence leads to a faster outward migration speed. We have performed all the simulations of Fig. 3 with the new prescription of the smoothing length. None of the simulations changes significantly. In particular we still find a quasi-stationary, non-migrating evolution in the case with H/r = 5% (Jupiter now migrates outward only by 0.5% in 2,000 orbits), and an inward migration in the case of H/r = 6%. Because the choice of ρ based from the local thickness of the disk seems to us somewhat more physically motivated, we will adopt this prescription in all the simulations presented further in this paper. Whatever the choice of ρ above (0.6RH or 0.7H), the smoothing length is always a big – 17 – fraction of the planet’s Hill radius. Thus, it is interesting to explore what would happen if we chose a much shorter smoothing length. In Fig. 8 we compare the simulation with H/r =4% and ρ = 0.6RH (black curves, already illustrated in Fig. 3) with one with the same disk parameters, but ρ = 0.25RH (grey curves) and one with the same prescription of ρ but where we have nullified the torques exerted on the planet(s) by the gas inside their respective Hill spheres (light grey curves). The exclusion of the torques from the regions neighboring the planets is never implemented in all other simulations presented in this paper. As one sees, the planets’ migration rates depend quite strongly on the adopted prescription for smoothing and torque calculation. This is because, if ρ is small, Saturn opens a deeper gap at its location, which enhances the unbalance of the torques felt by Jupiter. What is important, however, is that in all cases the migration is outward. This, once again, shows the robustness of the MS01 mechanism. The resolution of the grid used to represent the disk can also have, in principle, an important impact on the evolution of the system. In particular it can affect the corotation torque that, as we have seen, plays a role in the outward migration of the planets. To test the effects of the grid resolution, we have repeated the simulation with H/r = 3% and ρ = 0.7H , increasing by a factor of two both the radial and azimuthal resolutions of the 2D grid, and the radial resolution of the 1D grids. The new simulation is also plotted in Fig. 7. As one sees, the difference with respect to the simulation with our nominal resolution is negligible. Given the computational cost of the high resolution simulation, we will continue to use 282×325 cells in the 2D grid in the subsequent experiments. Another technical issue concerns the initial condition for the gas distribution. As we said above, we start from a gas profile carved by the planets on fixed orbits in a simulation spanning 8,000 periods at r = 1, with H/r=3%, ν = 10−5.5 and ρ = 0.6RH . However, when we change the parameters of the simulation, we only wait for additional 400 orbital periods – 18 – before letting the planets free to evolve. This second delay might not be long enough for the gas to respond to the new conditions, possibly introducing artefacts in the subsequent planet dynamics. To check if this is indeed the case, we plot in Fig. 9 two simulations, for H/r =4%,ν = 10−5.5 and ρ = 0.7H . In one simulation (black curves) the planets have been released after 400 orbits, as usual; in the second simulation, the planets have been released after 5,000 orbits. The evolutions after the release time match so perfectly that, in order to see the two sets of curves we had to downshift the gray ones by 1%! Thus, we conclude that our relatively short rlelease time of 400 orbits does not introduce significant artefacts. 3.3. Dependence on the disk viscosity We have done a series of simulations, changing the value of the viscosity, from ν = 10−6 to 2 × 10−5 in our units. The disk aspect ratio is 4% in all simulations. Thus, at r = 1, these viscosities correspond to α ranging from 6.25× 10−4 to 1.25× 10−2, in a Shakura and Sunyaev (1973) prescription. As usual, Saturn starts at r = 1.4 and Jupiter at r = 1. The results are illustrated in Fig. 10. For a viscosity ν = 10−6, Jupiter migrates outward after it has been released. Saturn initially migrates inward and, after being locked in the 2:3 MMR with Jupiter, the two planets migrate outward in parallel. For a viscosity ν = 5 × 10−6, the evolution is qualitatively similar. The outward migration speed, however, is faster than in the previous case. The reason is that the inner edge of Jupiter’s gap is further away from the planet in the ν = 10−6 case than in the ν = 5 × 10−6 case (see Fig. 11), so that the positive torque felt from the inner disk is weaker in the first case. For a viscosity ν = 10−5, Saturn –when released– has some erratic motion, which is slightly outward, on average, until T = 1200. During this time-span, Jupiter, which is also – 19 – migrating outward, approaches Saturn. Eventually Saturn has a short inward migration and is captured in the 2:3 MMR with Jupiter, and the two planets migrate outward together. Their common outward migration is slower than in the previous cases. If the viscosity is increased to 2 × 10−5, as soon as released Saturn migrates outward. Jupiter in the meantime migrates inward. The mutual 1:2 MMR is crossed at T = 880. The eccentricity enhancement that results from this resonance crossing breaks Saturn’s outward migration. The planet starts a ‘normal’ inward migration, at a rate comparable to that of Jupiter. The two planets are close to the 1:2 MMR, but not locked in it. The resonant angles are in fact in circulation. The reason for the initial behavior of Saturn in these two simulations is most likely due to the corotation torque. As Fig. 11 illustrates, for these values of the viscosity there is quite a large amount of gas at Saturn’s location, and the outer edge of Saturn’s gap is very close to the planet. More importantly, Jupiter’s gap becomes shallower. This reveals that, even before that the planets are released, there is an important flow of gas from the outer part of the disk, through the planetary orbits, towards the inner part of the disk. This flow exerts a corotation torque on each planet, which, as we discussed above, has stronger effects on Saturn. Putting together these results with those of sect. 3.1, we conclude that the mechanism of MS01 works for a large range of values of aspect ratio and viscosity of the disk. Whenever the disk is enough thin and of low viscosity, Jupiter and Saturn can have a common outward migration, once locked in the 2:3 MMR. Finding a quasi-stationary solution, however, is more delicate. If the disk is relatively thick (5% and, presumably, more), a quasi-stationary solution can be found for some value of the viscosity. Conversely, if the disk is thin (4% or, presumably, less), a quasi-stationary solution may not be found. The reason is that, if the disk’s aspect ratio is decreased, in principle the viscosity needs to be increased in order to maintain a density at Saturn’s location that is sufficiently large to exert on Jupiter a torque that counter-balances the one received by the planet from the inner disk. This larger – 20 – viscosity, however, tends to destabilize Saturn, because it generates a stronger flow that exerts an more important corotation torque on the planet. 4. The effect of mass accretion onto the planets In all previous simulations, the mass of the planets was kept constant with time. Lubow et al. (1999) and Kley (1999) showed that the accretion of mass by Jovian or sub-Jovian planets is non negligible for most values of the disk’s parameters. The investigation of the effects of mass accretion onto the planets is therefore interesting. Mass accretion exerts additional torques onto the planets and breaks the flow of the gas across the planetary orbits. So, in principle it could modify the dynamics significantly. MS01 already explored the effect of mass accretion onto Jupiter, and found that it is negligible even from the quantitative point of view. Here we consider also the effect of mass accretion onto Saturn, which might have a larger impact on the dynamics. Our understanding on how planet accretion proceeds, and how it stops, is still too vague to be able to assert a priori which planet should have had a more important mass growth rate. As MS01, we have implemented mass accretion onto the planets following the recipe of Kley (1999). It consists in removing a fraction of the material in the Hill sphere of the planet and adding it to the mass of the planet. The amount which is removed in one time-unit is imposed as an input parameter. More specifically, we apply the input removal rate in the inner Hill sphere (extended up to 45% of the Hill radius RH); we apply 2/3 of the removal rate in the region from 0.45 to 0.75 RH and no removal rate in the region beyond 0.75 RH . We have done 6 simulations, with three removal rates applied to Saturn only or both Jupiter and Saturn. The removal rates, (expressed as fraction of mass removed in the unit of time, which is 1/2π of the initial Jupiter’s orbital period) are 0.1, 1 and 5, as in Kley – 21 – (1999). All the simulations started from an intermediate state achieved in the simulation with H/r = 3%, ν = 10−5.5 (equivalent to α = 3.5× 10−3) and ρ = 0.7H (already presented in Fig. 7), precisely after a time corresponding to 900 initial Jovian orbital periods after the release of the planets. Figure 12 shows the result in the case of an accretion rate of 1 applied on Saturn only and compares it with the nominal simulation that we started from, where no accretion was allowed. We notice that the outward migration rate of Saturn and Jupiter (up to 1700 orbital periods) is significantly smaller. During this time, the eccentricity of Saturn is larger than in the case without accretion (∼ 0.05 instead of ∼ 0.02), which means that Saturn is offering a stronger resistance to the outward push exerted by Jupiter through the 2:3 MMR. This is most likely due to the fact that Saturn is growing in mass, so that the negative torque that it receives from the outer part of the disk increases. In fact, from t = 900 to t = 1700, Saturn doubles it mass, in an essentially linear mode. At t = 1700 the dynamical evolution changes abruptly. The mass growth of Saturn is accelerated, so that the planet reaches one Jupiter mass at t = 1800. This abrupt flow of mass onto the planet, essentially from the outer disk, exerts a strong positive torque. Therefore Saturn is extracted from the 2:3 MMR with Jupiter and runs away from it. Once separated from Jupiter, ‘Saturn’ starts an inward, Type II-like migration, despite of the positive torque due to the accretion of gas, which is still ongoing. The simulation where the mass of Jupiter is also allowed to grow, is essentially identical to the one presented in Fig. 12. We note in passing that during the linear growth regime, while the mass of Saturn doubles, the mass of Jupiter increases by only 15%. This shows that neglecting the growth of Saturn while allowing the growth of Jupiter is not justified. We also remark that the growth of the planets does not stall until they reach a mass of several Jupiter masses. This stresses the unsolved problem of explaining the final masses of – 22 – the giant planets of the solar system (and of extra-solar systems in general). The simulations with a smaller (0.1) or larger (5) removal rate parameter behave essentially like that presented in Fig. 12. Obviously, during the linear mass growth regime, the deviation with respect to the nominal simulation without mass accretion is smaller in the first case and larger in the second case. Even in the case with a removal rate of 5, though, we observe an outer migration of Jupiter of Saturn. This implies that this kind of dynamical evolution is robust with respect to the accretion rate, unless the latter is very high. 5. Generic two-planet dynamics: dependence on the individual masses and mass ratio Although this paper is devoted to the evolution of Jupiter and Saturn, it is interesting to do a quick exploration of how the dynamics changes with the masses of the planets. We have done three simulations, all with H/r = 5%, ν = 10−5.5 (corresponding to α = 1.25 × 10−3) and ρ = 0.7H (these parameters corresponds to the quasi-stationary solution for the Jupiter-Saturn system): the first one assumes that the inner planet has the mass of Saturn and the outer one has the mass of Jupiter; the second one assumes both masses are equal to one Jupiter mass; the third simulation multiplies the masses of the real planets by a factor of three. The first two simulations give no surprises. As we explained in sect. 3.1, the outward migration is possible only if the inner planet is more massive than the outer one. Otherwise the balance between the positive torque felt by the inner planet and the negative torque felt by the outer planet is in favor of the latter one. In fact, in both the first and the second simulation the planets migrate inward. Initially, the outer planet migrates faster than the – 23 – inner one, so that the two planets get captured in the 2:3 MMR after some time. The third simulation is the most interesting. In this case the mass ratio is the same as in the Jupiter-Saturn case, favoring an outward migration. However, because the outer planet is more massive than Saturn, it may be more difficult to unlock the evolution of the planetary system from the evolution of the gas, which favors an inward migration. So, the result of this experiment is not evident a priori. Fig. 13 shows the outcome. When the planets are released (as usual after 400 orbital periods of the inner planet), the inner planet starts to move outward as expected. The outer planet remains essentially on the spot. Before that a resonant configuration is achieved, the planets destabilize each other, because their separation corresponds to less than 3 mutual Hill radii (a mutual Hill radius is defined as [(a1 + a2)/2][(M1 +M2)/3] 1/3, where a1, a2 are the semi major axes and M1, M2 the masses). As a result of this instability, at t = 600 the outer planet is propelled outward on an orbit with eccentricity equal to 0.25, and the inner planet is kicked inward, onto an orbit with eccentricity equal to 0.1. Because of the large masses of the planets, the two gaps still partially overlap. Therefore, the inner planet feels a net positive torque, and the outer planet a net inward torque and, at t = 700–800, they start to migrate in converging directions. During this time, their orbital eccentricities are damped down to less than 0.05. At t = 1000 the planets are captured in their mutual 1:2 MMR. For a while after the resonant capture, the two planets move outward, but then eventually they stop, in a sort of quasi-stationary configuration. Our interpretation is that the gap of the outer planet is much more impermeable to the gas flow than in Saturn’s case. Consequently, under the push felt from the inner planet, the outer planet simply approaches the edge of its gap and modifies its profile until the torque that it receives from the outer disk can counterbalance the torque from the inner planet. This stops the migration. In conclusion, the mechanism proposed in MS01 is not necessarily specific to our – 24 – solar system. It can apply to extra-solar planetary systems but only at given, stringent conditions: (i) the outer planet has to be significantly less massive than the inner one and (ii) the planets have to be locked in a resonance characterized by an orbital separation that is sufficiently small to allow the overlapping of the respective gaps. All of the 20 multi-planets extra-solar systems discovered so far should have suffered a wide range migration, as suggested by the close proximity of the planets to the central star (typically, the inner planet is within 1.5–2 AU and the outer planet within 4 AU). So, we should expect that the mechanism of MS01 did not work in these systems. In fact, in 13 cases criterion (i) is not fulfilled. In the remaining cases the planets are too separated, with ratios of orbital periods larger than 3, so that it is unlikely that they have ever been locked in resonances with small orbital separation in the past. We predict that extra-solar systems satisfying both conditions (i) and (ii) will be discovered in the future, when the observation time-span will become long enough to allow the detection of distant planets that did not migrate significantly. 6. Possible ratios of orbital periods of Jupiter and Saturn In all the simulations reported above, as well as in those of MS01, whenever Jupiter and Saturn are in a configuration that prevents their migration towards the Sun, they are locked in the 2:3 MMR. This supports the idea, proposed in Tsiganis et al. (2005) and Gomes et al. (2005), that - at the end of the gas disk phase- the system of the giant planets in our solar system was very compact (i.e. characterized by small separations between the planets’ orbits). However, from the quantitative point of view, our results do not support directly the initial conditions adopted in Tsiganis et al. and Gomes et al. The initial ratio between the orbital periods of Saturn and Jupiter in that model was 1.8–1.9. The exact value is not important, but it is required that it is close to 2, so that Saturn can cross the – 25 – 1:2 MMR with Jupiter in ∼ 650 My (the time of the Late Heavy Bombardment) due to its interaction with the remaining planetesimal disk. If, at the end of the gas disk phase, the ratio of orbital periods had been close to 1.5, it is unlikely that this would have happened (unless a very massive planetesimal disk is assumed, but this would lead to other problems concerning the evolution of Uranus and Neptune). Therefore, in this section we explore different ways to reconcile the MS01 mechanism with the initial conditions of the LHB model. In principle, it is not necessary that Saturn and Jupiter are locked in the 2:3 MMR in order to prevent their inward migration. Other resonances, characterized by a larger ratio of orbital periods, may work, provided that the gaps formed in the disk by the two planets are wide enough to overlap. This would give a constraint on the maximal viscosity and scale height of the disk for each chosen resonant configuration. Reality, however, is not that simple, because the resonances located between the 2:3 and 1:2 MMR are much thinner than first order resonances and they may be characterized by unstable motion. So, the possibility of capture and permanence of the planets in these resonances is not guaranteed, a priori. We have done a series of 5 simulations, starting Saturn at a distance of 1.5 (Jupiter being initially at 1, as usual, so that the initial ratio of the orbital periods is 1.84), in disks with aspect ratio of 3.5% and viscosities in the range 1–3×10−6. In all simulations we have observed only captures in the 3:5 MMR, which led to a quasi-stationary evolution or a slow outward migration of the giant planets. However, in all cases, once captured in the resonance, the eccentricity of Saturn grew above 0.1 in about 150 initial Jovian orbital periods. This led to an instability of the planetary motion, which eventually led to a phase of violent scattering among the planets. Thus, we conclude that resonances of order larger than 1, located in between the 2:3 and 1:2 MMR are not viable for a long phase of – 26 – quiescent, non-migrating evolution. They either don’t capture the planets, or lead to an unstable motion after a short timescale. We have also done two simulations with Saturn initially at a distance of ∼ 1.65 (initial ratio with Jupiter’s orbital period of ∼ 2.1), in a disk with H/r = 3% and viscosity of 5 × 10−6. In both cases we have obtained capture in the 1:2 MMR, and a subsequent quasi-stationary evolution of the semi major axes of the two planets. This shows that the passage across the resonance without capture observed in MS01 (and in Fig. 1 above) was due to the fact that Saturn was migrating very fast. Our initial conditions and the low viscosity of the disk allow a slower migration and a more gradual growth of the eccentricity, which favor capture. Once captured in the resonance, despite the eccentricities of the planets are not negligible, the orbital evolution of the planets looks stable. We have not found any obvious way of extracting the planets from the resonance after some time, and delivering them on orbits with orbital period ratio smaller than 2. So, we doubt that a capture in the 1:2 MMR during the gas disk phase may be compatible with the initial conditions of the LHB model. Finally, we have studied the possibility that Saturn is extracted from the 2:3 MMR with Jupiter, after a long phase of quasi-stationary evolution, and is transported to larger semi major axis, approaching the 1:2 MMR. A first idea is that, as the surface density of the disk decreases during the disk dissipation phase, the planetary motion might become unstable so that the planets push each other onto more widely separated orbits. We have rapidly discarded this possibility, because a ‘stability map’ shows that the Jupiter-Saturn system at low eccentricity is stable if the ratio of the orbital periods is larger than 1.45 (Gayon, private communication). A second idea is suggested by the simulation presented in Fig. 12. The simulation should be considered only at a qualitative level for several reasons: accretion was started – 27 – when Saturn already had one Saturn’s mass, so that the final mass of the planet is larger than the real one; the prescription used for mass accretion was ad-hoc and idealized. Nevertheless, the simulation shows that rapid accretion of mass onto a planet exerts a positive torque that can extract the planet from the resonance. For instance, in Fig. 12 the ratio of orbital periods of Saturn and Jupiter at t − T0 = 1800 is 1.85, consistent with the initial conditions of the LHB model. Of course, once the planets are extracted from the resonance, their orbital evolution is no longer at equilibrium, and migration is resumed. Thus, to advocate a final position of the planets close to the 1:2 MMR, one has to assume that the disk disappeared ‘at the right time’. Our understanding of planetary growth is still too poor to draw definite conclusions. However, the moderate mass of Saturn may be an indication that its rapid growth was indeed aborted by the disappearance of the disk (Pollack et al., 1996). Notice however that, if the growth of Saturn is really as rapid as the simulation shows (0.4 Jupiter masses in 700 orbits), the nebula has to dissipate on a timescale of ∼ 10, 000 y, otherwise Saturn would have become too massive. A further, possibly more promising idea, concerns the evolution of the viscosity of the disk. As we have seen in sect. 3.3, the MS01 mechanism works only if the viscosity is sufficiently small. If the viscosity exceeds some value, Saturn can be extracted from the resonance in a runaway migration mode (see Fig. 10). Thus, it is interesting to explore the dynamics of Jupiter and Saturn in the case of a disk whose viscosity increases with time. In principle, there are a few reasons to believe that the disk’s viscosity might grow towards the end of the disk’s lifetime. If the origin of viscosity is MHD turbulence (Balbus and Hawley, 1991), the viscosity depends on the ionization of the disk. A sufficiently massive disk is optically thick, so that the radiation from the star cannot penetrate in the disk and the gas is not ionized. Thus, a dead zone can exist inside the disk, at a typical distance from a few to a few tens of AU, where MRI turbulence is not sustained and therefore the viscosity is very small (Gammie, 1996). The giant planets might very well have formed in such a dead – 28 – zone. When the disk starts to disappear, the radiation of the star can penetrate deeper into the disk, ionizing the disk on the mid-plane at larger heliocentric distance. The dead zone is re-activated, which causes an important enhancement of the local viscosity. Moreover, dust tends to have chemical bonds with the ions, subtracting them from the gas. Thus, even if undergoing the ionizing effect of the stellar radiation, a disk might not exhibit MHD turbulence if a sufficient amount of dust is present (Ilgner and Nelson, 2006a,b). As time passes, most of the dust is accreted in planetesimals, and therefore cannot subtract ions as efficiently as before. Therefore, a late disk should be increasingly coupled to the magnetic field and be characterized by a more violent turbulence and stronger viscosity. Motivated by these considerations, we have designed the following experiment. We considered the simulation with Jupiter initially at r = 1, Saturn at r = 1.4 and a disk with H/r = 5% and ν = 10−5.5 (corresponding to α = 1.25 × 10−3), performed assuming a smoothing length ρ = 0.7H . In this simulation, after capture in the mutual 2:3 MMR, Jupiter and Saturn exhibit a remarkable quasi-stationary solution (see Fig. 14 up to t = 3200). At t = 3200 we started to increase the viscosity of the disk, at the rate of 10−9 per unit of time (we remind that in our units the orbital period at r = 1 is 2π). This rate is totally arbitrary, and not justified by any astrophysical considerations. As a consequence of the increase in viscosity, the gaps formed by the planets become narrower and overlap more marginally. Thus, the shape of the gap formed by Jupiter becomes more symmetric with respect to the position of the planet, so that the torque received by Jupiter from the outer part of the disk starts to dominate over that from the inner part of the disk. Consequently Jupiter starts to migrate towards the Sun. The migration rate increases with increasing viscosity. As the resonance with Jupiter moves inward, Saturn also migrates towards the Sun, but at a smaller rate. In fact, the flow of gas from the outer disk towards Jupiter exerts a corotation torque on Saturn, slowing down its inward migration. This extracts Saturn from the 2:3 MMR. As the viscosity increases, the corotation torque becomes stronger, – 29 – and eventually Saturn starts an outward runaway migration. At t = 7200 Saturn is very close to the 1:2 MMR with Jupiter, as required in the initial conditions of the LHB model. The eccentricities of Jupiter and Saturn are very low, less than 0.005 and 0.01 respectively, which is also consistent with the LHB model. At that time, the viscosity of the disk is ν = 3.1 × 10−5. Given that the aspect ratio is 5%, this viscosity would correspond to a value of α ∼ 10−2, which is still reasonable. Obviously, to support the initial conditions of the LHB model, one has to assume that the disk disappears at that time. If this were not the case, and the viscosity kept growing, Saturn would cross the 1:2 MMR with Jupiter. Notice that, overall, Jupiter has an inward migration that covers only 20% of its initial heliocentric distance. Thus, in this scenario, to justify its current position, Jupiter should have formed at about 6.5 AU (and Saturn at about 8.5 AU, to end up, more or less, at the same position). These ranges of migration are moderate, and do not violate, a priori, any of the constraints imposed by the current solar system architecture. Again, we think that this simulation should be considered only at a qualitative level. Our knowledge of the evolution of the disk close to its disappearance is too approximated to be able to build a realistic simulation. Fig. 14 is presented simply to show that it is possible, in principle, to release the planets on non-resonant orbits after that they have spent most of the disk lifetime on resonant, non-migrating ones. Obviously, making the bridge between the formation of the planets, their dynamics in the gas disk, and their subsequent evolution in the planetesimal disk remains an open, crucial problem that goes beyond the scopes of this work. 7. Summary In this paper we have analyzed in detail, by performing many numerical simulations, the mechanism proposed by Masset and Snellgrove (2001) to explain why Jupiter and – 30 – Saturn did not migrate towards the Sun. The simulations have been done with a new simulation scheme (Crida et al., 2007), that is particularly suitable to study the migration of the giant planets. We confirmed that, if Jupiter and Saturn are locked into their mutual 2:3 MMR and the disk’s viscosity and aspect ratio are sufficiently small, the planets do not migrate toward the Sun. The mechanism is robust with respect to grid resolution used for the disk, the smoothing length used for the regularization of the gravitational potential, and the accretion of mass onto the planets. In most cases, the planets migrate outward, which is not a viable evolution in our solar system, because it would imply that Jupiter was in the asteroid belt in the past. However, there is a range of values of viscosity and disk’s scale height such that, once in resonance, the planets have a quasi-stationary evolution during which their semi major axes remain practically constant. We argue that Jupiter and Saturn actually followed this kind of evolution. In general terms for a pair of planets, a quasi-stationary solution can be found only if the outer planet is significantly less massive than the inner one, and if the planets are locked in a resonance characterized by a small orbital separation, so that the gaps opened by the planets in the disk can overlap. We find that these conditions are not satisfied by any known extra-solar system of multiple planets. This is consistent with these planets having suffered a significant migration, that brought them close to the parent star where they could be discovered. We predict that systems similar to the Jupiter-Saturn case in terms of mass ratio and separation will be discovered only when it will be possible to detect distant planets that did not migrate substantially. The results of this paper support the view, proposed in Tsiganis et al. (2005) and Gomes et al. (2005), that the giant planets of the solar system, at the end of the gas disk era, were on orbits with small mutual separation. However, from the quantitative point of view, supporting the initial conditions adopted in the model of Tsiganis et al. and Gomes et – 31 – al. is problematic. We suggest that a late fast growth of Saturn’s mass or, more likely, a late enhancement of the viscosity towards the end of the disk’s lifetime, could have extracted Saturn from the 2:3 resonance with Jupiter and driven it close to the 1:2 resonance. We supported this scenario with simulations, but which are nevertheless qualitative, given our limited knowledge of process of planet growth and of disk disappearance. We are grateful to Frederic Masset for his suggestions and for a careful reading of this manuscript. We also thank the two anonymous reviewers for their constructive suggestions. We are grateful to the National Program of Planetology for support. REFERENCES Balbus, S. A., Hawley, J. F. 1991. A powerful local shear instability in weakly magnetized disks. I - Linear analysis. II - Nonlinear evolution. Astrophysical Journal 376, 214-233. Bate, M. R., Lubow, S. H., Ogilvie, G. I., Miller, K. A. 2003. Three-dimensional calculations of high- and low-mass planets embedded in protoplanetary discs. Monthly Notices of the Royal Astronomical Society 341, 213-229. Crida, A., Morbidelli, A., Masset, F. 2006. On the width and shape of gaps in protoplanetary disks. Icarus 181, 587-604. Crida, A., Morbidelli, A., Masset, F. 2007. Simulating planet migration in globally evolving disks. Astronomy and Astrophysics 461, 1173-1183. Crida, A., Morbidelli, A., 2007. Cavity opening by a giant planet in a protoplanetary disk and effects on planetary migration. MNRAS, under revision. – 32 – Gammie, C. F. 1996. Layered Accretion in T Tauri Disks. Astrophysical Journal 457, Goldreich, P., Tremaine, S. 1980. Disk-satellite interactions. Astrophysical Journal 241, 425-441. Gomes, R., Levison, H. F., Tsiganis, K., Morbidelli, A. 2005. Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets. Nature 435, 466-469. Guillot, T., Hueso, R. 2006. The composition of Jupiter: sign of a (relatively) late formation in a chemically evolved protosolar disc. Monthly Notices of the Royal Astronomical Society 367, L47-L51. Haghighipour, N., Boss, A. P. 2003. On Gas Drag-Induced Rapid Migration of Solids in a Nonuniform Solar Nebula. Astrophysical Journal 598, 1301-1311. Hayashi C. 1981. Structure of the solar nebula, growth and decay of magnetic fields and effects of magnetic and turbulent viscosities on the nebula. Prog. Theor. Phys. Suppl., 70, 35-53. Ilgner, M., Nelson, R. P. 2006a. On the ionisation fraction in protoplanetary disks. I. Comparing different reaction networks. Astronomy and Astrophysics 445, 205-222. Ilgner, M., Nelson, R. P. 2006b. On the ionisation fraction in protoplanetary disks. II. The effect of turbulent mixing on gas-phase chemistry. Astronomy and Astrophysics 445, 223-232. Kley, W. 1999. Mass flow and accretion through gaps in accretion discs. Monthly Notices of the Royal Astronomical Society 303, 696-710. Kley, W., D’Angelo, G., Henning, T. 2001. Three-dimensional Simulations of a Planet Embedded in a Protoplanetary Disk. Astrophysical Journal 547, 457-464. – 33 – Levison H. F., Morbidelli, A., Gomes, R. and Tsiganis, K. 2007 Origin of the structure of the Kuiper Belt during a Dynamical Instability in the Orbits of Uranus and Neptune. Icarus, submitted. Lin, D. N. C., Papaloizou, J. 1979. Tidal torques on accretion discs in binary systems with extreme mass ratios. Monthly Notices of the Royal Astronomical Society 186, 799-812. Lin, D. N. C., Papaloizou, J. 1986. On the tidal interaction between protoplanets and the protoplanetary disk. III - Orbital migration of protoplanets. Astrophysical Journal 309, 846-857. Lubow, S. H., Seibert, M., Artymowicz, P. 1999. Disk Accretion onto High-Mass Planets. Astrophysical Journal 526, 1001-1012. Lynden-Bell, D., Pringle, J. E. 1974. The evolution of viscous discs and the origin of the nebular variables.. Monthly Notices of the Royal Astronomical Society 168, 603-637. Malhotra, R. 1993. The Origin of Pluto’s Peculiar Orbit. Nature 365, 819. Masset, F. 2000a. FARGO: A fast eulerian transport algorithm for differentially rotating disks. Astronomy and Astrophysics Supplement Series 141, 165-173. Masset, F. S. 2000b. FARGO: A Fast Eulerian Transport Algorithm for Differentially Rotating Disks. ASP Conf. Ser. 219: Disks, Planetesimals, and Planets 219, 75. Masset, F., Snellgrove, M. 2001. Reversing type II migration: resonance trapping of a lighter giant protoplanet. Monthly Notices of the Royal Astronomical Society 320, L55-L59. – 34 – Masset, F. S., Papaloizou, J. C. B. 2003. Runaway Migration and the Formation of Hot Jupiters. Astrophysical Journal 588, 494-508. Masset, F. S., Morbidelli, A., Crida, A., Ferreira, J. 2006. Disk Surface Density Transitions as Protoplanet Traps. Astrophysical Journal 642, 478-487. Masset, F. S. 2006. Planet-disk interactions. In Tidal Interactions in Composite Systems M.-J. Goupil and J.-P. Zahn (eds) EAS Publications Series,in press. Morbidelli, A., Levison, H. F., Tsiganis, K., Gomes, R. 2005. Chaotic capture of Jupiter’s Trojan asteroids in the early Solar System. Nature 435, 462-465. Papaloizou, J., Lin, D. N. C. 1984. On the tidal interaction between protoplanets and the primordial solar nebula. I - Linear calculation of the role of angular momentum exchange. Astrophysical Journal 285, 818-834. Pollack, J. B., Hubickyj, O., Bodenheimer, P., Lissauer, J. J., Podolak, M., Greenzweig, Y. 1996. Formation of the Giant Planets by Concurrent Accretion of Solids and Gas. Icarus 124, 62-85. Rice, W. K. M., Wood, K., Armitage, P. J., Whitney, B. A., Bjorkman, J. E. 2003. Constraints on a planetary origin for the gap in the protoplanetary disc of GM Aurigae. Monthly Notices of the Royal Astronomical Society 342, 79-85. Shakura, N. I., Sunyaev, R. A. 1973. Black holes in binary systems. Observational appearance.. Astronomy and Astrophysics 24, 337-355. Thommes, E. W. 2005. A Safety Net for Fast Migrators: Interactions between Gap-opening and Sub-Gap-opening Bodies in a Protoplanetary Disk. Astrophysical Journal 626, 1033-1044. – 35 – Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H. F. 2005. Origin of the orbital architecture of the giant planets of the Solar System. Nature 435, 459-461. Varnière, P., Blackman, E. G., Frank, A., Quillen, A. C. 2006. Planets Rapidly Create Holes in Young Circumstellar Disks. Astrophysical Journal 640, 1110-1114. Ward, W. R. 1986. Density waves in the solar nebula - Differential Lindblad torque. Icarus 67, 164-180. Ward, W. R. 1997. Protoplanet Migration by Nebula Tides. Icarus 126, 261-281. – 36 – Figure captions Fig. 1 An illustration of the dynamical evolution described in MS01. The black and grey curves show the evolutions of the semi major axes of Jupiter and Saturn respectively. Capture in the 2:3 MMR occurs when the migration of Saturn is reversed. Fig. 2 The same as Fig. 1, but for an initial location of Saturn close (but not into) the 2:3 MMR with Jupiter. Fig. 3 The evolution of Jupiter (lower set of curves starting at r = 1) and Saturn (upper set of curves, starting at r = 1.4). Different grey levels refer to different aspect ratios, as labeled. The viscosity is independent of radius and equal to 10−5.5 in all cases. The planets are released after 400 orbits. Fig. 4 The dash-dotted curve shows the initial density profile of the disk, adopted in all simulations. The solid curves show the density profile corresponding to the moment when the planets are released, at 1 and 1.4 respectively for Jupiter and Saturn. Different grey levels refer to different aspect ratios, as labeled. The viscosity is independent of radius and equal to 10−5.5 in all cases. Fig. 5 The evolution of Jupiter and Saturn in two simulations, both with H/r=3% and ν = 10−5.5. The black curves refer to the simulation adopting the initial disk surface density shown in Fig. 5. The grey curves adopt an initial surface density profile that has been divided by a factor of 2. The planets are released in the two cases after about 400 orbital periods at r = 1. The gray curves are plotted after rescaling the time as t′ = (t − 400)/2. The fact that gray and balck curves overlap shows that the evolution is the same, but the migration speed is reduced proportionally with the mass of the disk. – 37 – Fig. 6 The surface density of the disk in the simulation with H/r = 3%. Different grey levels refer to different times, labeled in unit of initial Jupiter’s orbital period. The planets are released at time T = 400. Fig. 7 The evolution of Jupiter and Saturn in three simulations, with H/r = 3% and ν = 10−5.5. The simulation plotted in black and labeled ‘Hill’ is the one already shown in Fig. 3. The simulation reported in grey and labeled ‘H’ adopts a different prescription for the smoothing length, which is now imposed equal to 70% of the local thickness of the disk. The simulation plotted in light grey and labeled ’High res.’ is the same as the latter simulation, but with azimuthal and radial grid resolutions increased by a factor of 2. Time is measured relative to the instant T0 when Saturn starts to migrate outward. The semi major axes of the planets are normalized relative to the semi major axis of Jupiter at T0. This allows a more direct comparison among the three simulations. Fig. 8 The evolution of Jupiter and Saturn in two simulations, with H/r = 4%, ν = 10−5.5. The simulation represented by black curves, labelled ‘0.6 Hill’, is the one already shown in Fig. 3, obtained adopting a smoothing length ρ = 0.6RH. That represented by grey curves, labelled ‘0.25 Hill’ has been obtained with ρ = 0.25RH. Finally, the simulation represented by light grey curves, labelled ‘Excl. Hill’ has also been obtained with ρ = 0.25RH, but excluding the torques exerted on the planets by the gas inside their Hill spheres. Fig. 9 The evolution of Jupiter and Saturn in two simulations, with H/r = 4%, ν = 10−5.5 and ρ = 0.7RH. In the simulation plotted in black and labeled ‘Nominal’, the planets have been released on free-to-evolve orbits after 400 orbital periods at r = 1. In the simulation reported in grey and labeled – 38 – ‘Late release’ the planets have been released after 5,000 orbits. Notice that the grey curves have been shifted downwards by 1%, in order to avoid a perfect overlap with the black curves. Therefore, evolution of the planets after the release time is the same. Fig. 10 The evolution of Jupiter (lower set of curves starting at r = 1) and Saturn (upper set of curves, starting at r = 1.4). Different grey levels refer to different viscosities, as labeled. The aspect ratio H/r is equal to 4% in all cases. The planets are released after 400 initial Jovian orbits. Fig. 11 The density profiles of the disk at the moment when the planets are released. Different grey levels refer to different viscosities, as labeled. The aspect ratio is 4% in all simulations. Fig. 12 The grey curves show the evolutions of Saturn (starting at 1.4) and Jupiter (starting at 1) in the simulation with H/r = 3%, ν = 10−5.5 and ρ = 0.7H . Time is counted in initial Jovian orbital periods, from the instant when the planets are released. The black curves show how the evolutions of Saturn and Jupiter change, starting from t = 900, in the case where Saturn (but not Jupiter) is allowed to accrete mass with a removal rate of 1. Fig. 13 The evolution of two planets with 3 Jupiter masses (starting at r = 1) and 1 Jupiter masses (starting at r = 1.4). The disk aspect ratio is 3% and the viscosity is 10−5.5, independent of radius. A smoothing length equal to 70% of the local disk’s height is used. Fig. 14 The black solid curves show the evolution of Jupiter and Saturn. As usual, the planets start at r = 1 and 1.4 respectively, and are released after 400 initial Jovian orbital periods. The disk scale heigh is 5% and the viscosity is 10−5.5. After that the planets – 39 – lock in the 2:3 MMR (at t ∼ 900 the planets’ semi major axes remain substantially constant. The grey curves show the location of the 2:3 and 1:2 MMR with Jupiter, according to Kepler law. Notice that the semi major axis of Saturn is slightly larger than that corresponding to the 2:3 MMR in the Kepler approximation, due to the effects of the disk’s gravity. At t = 3200 (marked by a vertical dashed line), the viscosity of the disk is increased at a rate of 6.28 × 10−9 per orbital period. This eventually forces Jupiter to migrate inward and extracts Saturn from the 2:3 MMR. The simulation is stopped when Saturn reaches the vicinity of the 1:2 MMR with Jupiter. – 40 – Fig. 1.— – 41 – Fig. 2.— – 42 – Fig. 3.— – 43 – Fig. 4.— – 44 – Fig. 5.— – 45 – Fig. 6.— – 46 – Fig. 7.— – 47 – Fig. 8.— – 48 – Fig. 9.— – 49 – Fig. 10.— – 50 – Fig. 11.— – 51 – Fig. 12.— – 52 – Fig. 13.— – 53 – Fig. 14.— Introduction The Masset–Snellgrove mechanism Dynamics of Jupiter and Saturn as a function of the disk properties and simulation parameters Dependence on the disk aspect ratio Dependence on the simulation's technical parameters Dependence on the disk viscosity The effect of mass accretion onto the planets Generic two-planet dynamics: dependence on the individual masses and mass ratio Possible ratios of orbital periods of Jupiter and Saturn Summary

0704.1211

Kinetic Theory for Binary Granular Mixtures at Low-Density

7 Kinetic Theory for Binary Granular Mixtures at Low-Density Vicente Garzó Departamento de F́ısica, Universidad de Extremadura, E-06071 Badajoz, Spain [email protected] Many features of granular media can be modelled as a fluid of hard spheres with inelastic collisions. Under rapid flow conditions, the macroscopic be- havior of grains can be described through hydrodynamic equations. At low- density, a fundamental basis for the derivation of the hydrodynamic equations and explicit expressions for the transport coefficients appearing in them is pro- vided by the Boltzmann kinetic theory conveniently modified to account for inelastic binary collisions. The goal of this chapter is to give an overview of the recent advances made for binary granular gases by using kinetic theory tools. Some of the results presented here cover aspects such as transport properties, energy nonequipartition, instabilities, segregation or mixing, non-Newtonian behavior, . . . . In addition, comparison of the analytical results with those ob- tained from Monte Carlo and molecular dynamics simulations is also carried out, showing the reliability of kinetic theory to describe granular flows even for strong dissipation. 1 Introduction Granular systems have attracted the attention of the physics community in the past few years, in part because the behavior of these systems under many conditions exhibit a great similarity to ordinary fluids [1]. These conditions include rapid, dilute flows where the dominant transfer of momentum and energy is through binary collisions of the grains. The main difference from ordinary fluids is the absence of energy conservation, leading to both obvious and subtle modifications of the usual macroscopic balance equations as well as the constitutive equations for the irreversible fluxes. However, in spite of the utility of hydrodynamics to describe rapid granular flows, there are still some open questions about its domain of validity and the associated constitutive equations appearing in the hydrodynamic equations [2]. To isolate the effects of such collisional dissipation from other important properties of granular media, an idealized microscopic model system is usually http://arxiv.org/abs/0704.1211v1 2 Vicente Garzó considered: a system composed by smooth hard spheres with inelastic colli- sions. As in the elastic case, the collisions are specified in terms of the change in relative velocity at contact but with a decrease in the magnitude of the nor- mal component measured by a positive coefficient of restitution α ≤ 1. This parameter distinguishes the granular fluid (α < 1) from the ordinary fluid (α = 1). Given that the hard sphere system with elastic collisions has been widely studied for both equilibrium and non-equilibrium statistical mechanics [3], it is tempting to apply the same methods for the case of inelastic collisions. However, some care is warranted in translating properties of ordinary fluids to granular fluids. In this presentation, a kinetic theory description based on the Boltzmann kinetic equation (which applies at sufficiently low density) will be considered as the appropriate tool to study granular flows from a microscopic point of view. Although many efforts have been devoted in the past few years to the understanding of granular fluids, the derivation of the form of the constitu- tive equations with explicit expressions for the transport coefficients is still a subject of interest and controversy. The conditions to obtain a hydrodynamic description are expected to be similar to those for normal fluids. For a given initial state there are two stages of evolution. First, during the kinetic stage there is rapid velocity relaxation to a “universal” velocity distribution that depends on the average local density, temperature, and flow velocity. Subse- quently, the hydrodynamic stage is described through a slower evolution of these local hydrodynamic fields as they approach uniformity. The solution to the Boltzmann equation in this second stage is said to be normal, where all space and time dependence of the distribution function occurs through the macroscopic hydrodynamic fields. The Chapman–Enskog method [4] provides a constructive means to obtain an approximation to such a solution for states whose spatial gradients are not too large. In this case, the explicit form of this normal solution is given as a perturbation expansion in the spatial gradients of the fields. This solution is then used to evaluate the fluxes in the macroscopic balance equations in terms of these gradients. To lowest order the balance equations become the granular Euler equations while to second order they are the granular Navier–Stokes equations. In carrying out this analysis, explicit forms for the transport coefficients are obtained as functions of the coefficient of restitution and other parameters of the collision operator. In this general context, the study of hydrodynamics for granular gases is the same as that for ordinary fluids. The derivation of hydrodynamics from the inelastic Boltzmann equation has been widely covered in the case of a monodisperse gas where the particles are of the same mass and size. As for elastic collisions, the transport coeffi- cients are given in terms of the solutions of linear integral equations [5, 6], which are approximately solved by using Sonine polynomial expansions. The estimates for the transport coefficients provided by the Sonine solution com- pare in general quite well with both direct Monte Carlo simulation (DSMC) of the Boltzmann equation and molecular dynamics (MD) simulation of the Kinetic Theory for Binary Granular Mixtures at Low-Density 3 gas, even for relatively strong degrees of dissipation [7, 8, 9, 10]. This good agreement supports the formal theoretical analysis and the claim that hydro- dynamics is not limited to nearly elastic particles [11, 12, 13]. Nevertheless, a real granular system is generally characterized by some degrees of polydispersity in density and size, which leads to phenomena very often observed in nature and experiments, such as separation or segregation. Needless to say, the study of granular mixtures is much more complicated than for a monodisperse gas since not only the number of transport coeffi- cients in a multicomponent system is higher than that of a single gas, but also they depend on parameters such as masses, sizes, composition as well as several independent coefficients of restitution αij . Due to these difficulties, studies for multicomponent gases are more scarce in the literature. Many of the previous attempts [14] to derive hydrodynamics from kinetic theory were carried out in the quasi-elastic limit where the equipartition of energy can be considered as an acceptable assumption. In addition, according to this level of approximation, the inelasticity is only accounted for by the presence of a sink term in the energy balance equation, so that the expressions for the transport coefficients are the same as those obtained for ordinary fluids. However, the theoretical prediction of the failure of energy equipartition in multicomponent granular gases [15] has been confirmed by computer simulations [16], and even observed in real experiments [17]. Although the possibility of nonequipartition was already pointed out many years ago [18], it has not been until recently that a systematic study of the effect of nonequipartition on the Navier–Stokes hydrodynamic equations has been carefuly carried out [19, 20]. These new equations and associated trans- port coefficients provide a somewhat more stringent test of the analysis since the parameter space is much larger. As in the monodisperse case, explicit ex- pressions for the transport coefficients requires also to consider Sonine poly- nomial expansions. The numerical accuracy of this Sonine expansion has been confirmed by comparison with Monte Carlo simulations of the Boltzmann equation in the cases of the shear viscosity [21] and the tracer diffusion [22] coefficients. Exceptions to this agreement are extreme mass or size ratios and strong dissipation, although these discrepancies between theory and simula- tion diminish as one considers more terms in the Sonine polynomial approxi- mation [22]. The explicit knowledge of the Navier-Stokes transport coefficients allows quantitative application of the nonlinear hydrodynamic equations to a num- ber of interesting problems for granular mixtures, such as to quantify the violation of the Einstein relation [23, 24] or the Onsager reciprocal relations [20], the stability analysis of the homogeneous cooling state [20], and segrega- tion induced by a thermal gradient [25]. In all the cases, the analysis clearly shows the important role played by the inelasticity in the different physical situations. The analogy between rapid granular flow and ordinary fluids can be also extended to many other transport situations. A particularly simple case, al- 4 Vicente Garzó lowing detailed analysis even in far from equilibrium conditions is the simple or uniform shear flow (USF) problem. Macroscopically, it is characterized by uniform density and temperature and a constant mean velocity profile. This is a well-known non-equilibrium problem widely studied, for both granular monodisperse [13, 26, 27] and ordinary gases [28]. Nevertheless, the nature of this state is quite different in each system. While for elastic fluids the temper- ature increases monotonically in time due to viscous heating, a steady state is possible for granular media when the effect of the viscous heating is exactly compensated by the dissipation in collisions. Thus, in the steady state, there is an intrinsic connection between the shear field and dissipation so that the collisional cooling sets the strength of the velocity gradient. As a consequence, the USF state is inherently non-Newtonian and the rheological properties of the system cannot be obtained from the Navier-Stokes description, at least for finite dissipation [29]. The aim of this chapter is to offer a short review of recent results ob- tained for binary granular mixtures from the Boltzmann kinetic theory. It is structured as follows. The Boltzmann kinetic equation for a granular binary mixture and its associated macroscopic balance equations are introduced in Sec. 2. Section 3 deals with the solutions to the Boltzmann equation for homo- geneous states in the free cooling case as well as when the mixture is heated by an external thermostat. The Chapman–Enskog method around the local version of the homogenous distributions obtained in Sec. 2 is applied in Sec. 3 to get the form of the Navier–Stokes hydrodynamic equations. Theoretical re- sults for the diffusion and shear viscosity transport coefficients are compared with simulation data in Sec. 5, while the Einstein and the Onsager relations for granular mixtures are analyzed in Secs. 6 and 7, respectively. The dispersion relations for the hydrodynamic equations linearized about the homogeneous cooling state are obtained in Sec. 8, showing that the homogeneous reference state is unstable to long wavelength perturbations. The conditions for sta- bility are identified as functions of the wave vector, the dissipation, and the parameters of the mixture. Segregation due to thermal diffusion is studied in Sec. 9 by using the Navier–Stokes description. A new criterion for segregation is found that is consistent with recent experimental results. Section 10 deals with the USF problem for a granular mixture. Finally, the paper is closed in Sec. 11 with a discussion of the results presented here. Before ending this section, I want to remark that the present account is a personal perspective based on the author’s work and that of his collaborators so that no attempt is made to include the extensive related work of many others in this field. The references given are selective and apologies are of- fered at the outset to the many other important contributions not recognized explicitly. Kinetic Theory for Binary Granular Mixtures at Low-Density 5 2 Boltzmann Kinetic Equation for Binary Mixtures of Inelastic Hard Spheres Consider a binary granular mixture composed by smooth inelastic disks (d = 2) or spheres (d = 3) of massesm1 andm2, and diameters σ1 and σ2. The inelasticity of collisions among all pairs is characterized by three independent constant coefficients of restitution α11, α22, and α12 = α21, where αij ≤ 1 is the coefficient of restitution for collisions between particles of species i and j. Since the spheres are assumed to be perfectly smooth, only the translational degrees of freedom of grains are affected by dissipation. In the low density regime, a simultaneous interaction of more than two particles is highly unlike and so can be neglected. Consequently, in a dilute gas the interactions among the particles reduce to a succession of binary collisions. At this level of de- scription, all the relevant information on the state of the system is contained in the one-body velocity distribution functions fi(r,v; t) (i = 1, 2) defined so that fi(r,v; t)drdv is the most probable (or average) number of particles of species i which at time t lie in the volume element dr centered at the point r and moving with velocities in the range dv about v. For an inelastic gas, the distributions fi(r,v; t) (i = 1, 2) for the two species satisfy the coupled nonlinear Boltzmann equations [30, 31] ∂t + v · ∇+ fi(r,v, t) = Jij [v|fi(t), fj(t)] , (1) where the Boltzmann collision operator Jij [v|fi, fj ] is Jij [v1|fi, fj] = σ dσ̂Θ(σ̂ · g12)(σ̂ · v12) α−2ij fi(r,v 1, t)fj(r,v 2, t)− fi(r,v1, t)fj(r,v2, t) . (2) In Eq. (2), d is the dimensionality of the system, σij = (σi + σj) /2, σ̂ is an unit vector along the line of centers, Θ is the Heaviside step function, and v12 = v1 −v2 is the relative velocity. The primes on the velocities denote the initial values {v′1,v 2} that lead to {v1,v2} following a binary (restituting) collision: 1 = v1 − µji 1 + α−1ij (σ̂ · v12)σ̂, 2 = v2 + µij 1 + α−1ij (σ̂ · v12)σ̂, (3) where µij ≡ mi/ (mi +mj). In addition, Fi denotes an external conservative force acting on species i (such as a gravity field) and Fi is an operator repre- senting a possible effect of an external nonconservative forcing which injects energy into the system to compensate for the energy dissipated by collisional cooling. This type of force acts as a thermostat that tries to mimics a thermal bath. Some explicit forms for the operator Fi will be chosen later. 6 Vicente Garzó The relevant hydrodynamic fields for the mixture are the number densities ni, the flow velocity u, and the temperature T . They are defined in terms of moments of the velocity distribution functions fi as dvfi(v) , (4) dvvfi(v) , (5) nT = p = niTi = dvV 2fi(v) , (6) where V = v − u is the peculiar velocity, n = n1 + n2 is the total number density, ρ = m1n1 + m2n2 is the total mass density, and p is the pressure. Furthermore, the third equality of Eq. (6) defines the kinetic temperatures Ti for each species, which measure their mean kinetic energies. The collision operators conserve the particle number of each species and the total momentum, but the total energy is not conserved: dvJij [v|fi, fj ] = 0 , (7) dvvJij [v|fi, fj] +mj dvvJji[v|fj , fi] = 0 , (8) dvV 2Jij [v|fi, fj] = −dnTζ . (9) In Eq. (9), ζ is identified as the total “cooling rate” due to collisions among all species. It measures the rate of energy loss due to dissipation. At a kinetic level, it is also convenient to introduce the “cooling rates” ζi for the partial temperatures Ti. They are defined as ζij = − dniTi dvV 2Jij [v|fi, fj], (10) where the second equality defines the quantities ζij . The total cooling rate ζ can be written in terms of the partial cooling rates ζi as ζ = T−1 xiTiζi, (11) where xi = ni/n is the mole fraction of species i. From Eqs. (7)–(9), the macroscopic balance equations for the number den- sities ni, the total momentum density ρu and the energy density (d/2)nT can be obtained. They are given, respectively, by [19] Kinetic Theory for Binary Granular Mixtures at Low-Density 7 Dtni + ni∇ · u+ ∇ · ji = 0 , (12) ρDtu+∇ · P = niFi , (13) DtT − ∇ · ji ∇ · q+ P : ∇u− Fi · ji = −(ζ − ξ)T . (14) In the above equations, Dt = ∂t + u · ∇ is the material derivative, ji = mi dvV fi(v) (15) is the mass flux for species i relative to the local flow, dvVV fi(v) (16) is the total pressure tensor, and dv V 2V fi(v) (17) is the total heat flux. On the right-hand side of the temperature equation (14), the source term ξ (measuring the rate of heating due to the external thermostat) is given by ξ = − dvV 2Fifi(v). (18) In the balance equations (12)–(14) it is assumed that the external driving thermostat does not change the number of particles of each species or the total momentum, i.e., ∫ dvFifi(v) = 0, (19) dv v Fifi(v) = 0. (20) The utility of the balance equations (12)–(14) is limited without further specification of the fluxes and the cooling rate, which in general have a com- plex dependence on space and time. However, for sufficiently large space and time scales, one expects that the system reaches a hydrodynamic regime in which all the space and time dependence is given entirely through a functional dependence on the six hydrodynamic fields ni, u, and T . The corresponding 8 Vicente Garzó functional dependence of ji, P, and q on these fields are called constitutive equations and define the transport coefficients of the mixture. The primary feature of a hydrodynamic description is the reduction of the description from many microscopic degrees of freedom to a set of equations involving only six local fields. At a kinetic level, the constitutive equations are obtained when one admits the existence of a normal solution to the Boltzmann equation where the velocity distribution functions depend on r and t only through their functional dependence on the fields, namely, fi(r,v1, t) = fi[v1|ni(r, t), T (r, t),u(r, t)]. (21) This normal solution is generated by the Chapman–Enskog method [4] con- veniently adapted to dissipative dynamics. Since the method is based on an expansion around the local version of the homogeneous state, let us charac- terize it before considering inhomogeneous solutions. 3 Homogeneous States In this Section we are interested in spatially homogeneous isotropic states. In this case, we assume that the magnitude of the conservative external forces is at least of first order in the spatial gradients (i.e., Fi = 0), so that Eq. (1) becomes (∂t + Fi) fi = Jij [fi, fj ]. (22) For elastic collisions (αij = 1) and in the absence of external forcing (Fi = 0), it is well known that the long-time solution to (22) is a Maxwellian distribution for each species at the same temperature T . However, if the particles collide inelastically (αij < 1) and Fi = 0, a steady state is not possible in uniform situations since the temperature decreases monotonically in time. This state is usually referred to as the homogeneous cooling state (HCS). In this situation, since ni is uniform and u = 0, the normal (hydrodynamic) solution to fi requires that all its time dependence occurs only through the temperature T (t). Consequently, fi(v, t) must be of the form [15] fi(v, t) = niv 0 (t)Φi (v/v0(t)) , (23) where v0(t) = 2T (t)(m1 +m2)/ (m1m2) is a thermal speed defined in terms of the temperature T (t) of the mixture. The balance equations (12)–(14) to this order become ∂txi = ∂tu = 0, and T −1∂tT = −ζ, where the cooling rate ζ is determined by Eq. (9). In addition, from Eqs. (10) and (22) (when Fi = 0) one can derive the time evolution for the temperature ratio γ = T1(t)/T2(t): ∂t ln γ = ζ2 − ζ1. (24) Kinetic Theory for Binary Granular Mixtures at Low-Density 9 The fact that the distributions fi depend on time only through T (t) necessarily implies that the temperature ratio γ must be independent of time and so, Eq. (24) gives the condition ζ1 = ζ2 = ζ. (25) In the elastic case, where fi is a Maxwellian distribution, the above condition yields T1(t) = T2(t) = T (t) and the energy equipartition applies. However, in the inelastic case, the equality of the cooling rates leads to different values for the partial temperatures, even if one considers the Maxwellian approximation to fi. Nevertheless, the constancy of γ assures that the time dependence of the distributions is entirely through T (t), and in fact the partial temperatures can be expressed in terms of the global temperature as T1(t) = 1 + x1(γ − 1) T (t), T2(t) = 1 + x1(γ − 1) T (t). (26) Just as for the single gas case [30, 32], the exact form of Φi has not yet been found, although a good approximation for thermal velocities can be obtained from an expansion in Sonine polynomials [15]. In the leading order, Φi is given e−θiv ∗4 − (d+ 2)θiv d(d+ 2) Here, v∗ ≡ v/v0, m−1j , (28) and γi = Ti/T . The coefficients ci (which measure the deviation of Φi from the reference Maxwellian) are determined consistently from the Boltzmann equation. The explicit form of the approximation (27) provides detailed pre- dictions for the temperature ratio T1/T2 (through calculation of the cooling rates) and for the cumulants ci as functions of the mass ratio, size ratio, com- position and coefficients of restitution [15]. The numerical accuracy of this truncated Sonine expansion has been confirmed by comparison with Monte Carlo [33] and MD [34] simulations. As said in the Introduction, the existence of different temperatures for each species has been observed in real experiments of driven steady states. These states are achieved from external forces that do work at the same rate as collisional cooling. In experiments this is accomplished by vibrating the system so that it is locally driven at walls. Far from these walls a steady state is reached whose properties are expected to be insensitive to the details of the driving forces. Due to the technical difficulties involved in incorporating oscillating boundary conditions, it is usual to introduce external forces (or thermostats) acting locally on each particle. These forces are represented by the operator Fi in Eq. (22) and depend on the state of the system. Two 10 Vicente Garzó types of thermostats have been usually considered in the literature. One is a deterministic thermostat widely used in nonequilibrium MD simulations for ordinary fluids [35, 36]. The force is similar to a Stokes law drag force, linear in the velocity, but with the opposite sign so that it heats rather than cools the system. In this case, Fi is given by [21, 37] Fifi(v) = · [vfi(v)], (29) where the friction constant in the force has been adjusted to get a constant temperature in the long-time limit. It must be remarked that the correspond- ing Boltzmann equation (22) for this Gaussian thermostat force is formally identical with the Boltzmann equation in the HCS (i.e., with Fi = 0) when both equations are written in terms of the reduced distributions Φi(v ∗) [37]. In particular, the dependence of γ on the parameters of the system is the same with and without the Gaussian thermostat. A second method of driving the system is by means of a stochastic Langevin force representing Gaussian white noise [38]. The corresponding op- erator Fi has a Fokker–Planck form [32] Fifi(v) = − fi(v), (30) where for simplicity the covariance of the stochastic acceleration has been taken to the same for each species [39, 40]. This requirement gives the steady state condition ζ2. (31) The cooling rates ζi are no longer equal, in contrast to the HCS, and the de- pendence of γ on the control parameters is different as well [34]. The procedure for determining the temperature ratio and the cumulants ci is the same as in the HCS state since the steady state distribution Φi can also be represented as an expansion of the form (27) and the coefficients are now determined from the solution to the Boltzmann equation (22). The condition (31) gives the corresponding equation for the temperature ratio. Figure 1 illustrates the differences between the HCS and the stochastic steady state at the level of the temperature ratio T1/T2. We have considered mixtures constituted by spheres (d = 3) of the same material [and so, αij = α, and m1/m2 = (σ1/σ2) 3] and equal volumes of large and small particles [i.e., x2 = (σ1/σ2) 3x1]. Here, for the sake of simplicity, the cooling rates have been analytically estimated by using Maxwellians (namely, by taking ci = 0) for the distributions fi(v): ζij → 4π(d−1)/2 njµjiσ θi + θj ×(1 + αij) (1 + αij) θi + θj . (32) Kinetic Theory for Binary Granular Mixtures at Low-Density 11 1.0 1.5 2.0 2.5 3.0 Fig. 1. Temperature ratio T1/T2 versus the size ratio σ1/σ2 for αij ≡ α = 0.78 in the case of mixtures constituted by particles of the same material and equal total volumes of large and small particles. The lines are the kinetic theory results in (a) the stochastic driving case and (b) the free cooling case, while the points refer to MD simulations [41]. Simulation data recently obtained from MD simulations in agitated mixtures have also been included [41]. The experimental value of the (common) coef- ficient of restitution is α = 0.78. While a good agreement between kinetic theory and MD simulations is found when the gas is assumed to be driven by the stochastic thermostat, significant discrepancies appear in the undriven (HCS) case, especially as the size ratio σ1/σ2 increases. These results con- trast with the comparison made by Brey et al. [42] for agitated mixtures in the tracer limit (x1 → 0), where the predictions of the temperature ratio from kinetic theory based on the condition ζ1 = ζ2 compare quite well with MD simulations. However, for the cases studied in Ref. [42] the conditions (25) and (31) yield quite similar results for the dependence of T1/T2 on the parameters of the system. The good agreement found in Fig. 1 between MD simulations for agitated mixtures and kinetic theory suggests that the stochastic driving condition can be considered as a plausible first approximation for qualitative comparisons with experimental results [17]. 4 Navier–Stokes Hydrodynamic Equations We consider now a spatially inhomogeneous state created either by initial preparation or by boundary conditions. We are interested in a hydrodynamic description where the state of the system is completely specified through their hydrodynamic fields. This implies that the latter dominate over other excita- 12 Vicente Garzó tions for times large compared to the mean free time and for wavelengths large compared to the mean free time. The hydrodynamic regime is characterized by the existence of a normal solution to the Boltzmann equation which can be explicitly obtained by means of the Chapman–Enskog method [4]. For small spatial variations, the functional dependence (21) of the normal solution can be made local in space and time through an expansion in gradients of the fields: fi = f i + ǫ f i + · · · , (33) where each factor of ǫ means an implicit gradient of a hydrodynamic field. The reference distribution function f i (r,v, t) is the local version of the ho- mogeneous distribution (23), namely, it is obtained from the homogeneous distribution by replacing the temperature, densities and flow velocity by their nonequilibrium values: i (r,v, t) = ni(r, t)v 0 (T (r, t))Φi (V/v0(T (r, t))) , (34) where V = v−u(r, t). The time derivatives of the fields are also expanded as ∂t = ∂ t + ǫ∂ t + · · · . The coefficients of the time derivative expansion are identified from the balance equations (12)–(14) with a representation of the fluxes and the cooling rate in the macroscopic balance equations as a similar series through their definitions as functionals of fi. This is the usual Chapman–Enkog method for solving kinetic equations [4, 28, 43]. Nevertheless, the complexity introduced by the energy dissipation in collisions has led to the introduction by some authors [14] of some additional approximations, restricting the validity of most of the results to the small inelasticity limit. Only very recently, explicit expressions for the fluxes to first order in the gradients as explicit functions of the coefficients of restitution have been obtained [19, 21, 44]. To Navier–Stokes order, the constitutive equations for mass, momentum, and heat fluxes are given, respectively, by 1 = − m1m2n D∇x1 − Dp∇p− D′∇T + χ1iFi, j 2 = −j kℓ = p δkℓ − η ∇ℓuk +∇kuℓ − δkℓ∇ · u , (36) (1) = −T 2D′′∇x1 − L∇p− λ∇T + κiFi. (37) The transport coefficients in these equations are the diffusion coefficient D, the pressure diffusion coefficient Dp, the thermal diffusion coefficient D ′, the mobility coefficient χij , the shear viscosity η, the Dufour coefficient D ′′, the pressure energy coefficient L, the thermal conductivity λ, and the coefficient κi. These coefficients are defined as Kinetic Theory for Binary Granular Mixtures at Low-Density 13 D = − dvV ·A1, (38) Dp = − dvV · B1, (39) D′ = − dvV · C1, (40) χij = dvV · E ij , (41) η = − (d− 1)(d+ 2) dvVV : Di, (42) D′′ = − dv V 2V ·Ai, (43) L = − dv V 2V · Bi, (44) λ = − dv V 2V · Ci, (45) dv V 2V · Eij . (46) Here, Ai(V), Bi(V), Ci(V), Di(V), and Eij(V) are functions of the peculiar velocity and the hydrodynamic fields. They obey the following set of coupled linear integral equations: ζ (T∂T + p∂p) + F 1 + L1 A1 +M1A2 = A1 + × (pB1 + TC1) , (47a) ζ (T∂T + p∂p) + F 2 + L2 A2 +M2A1 = A2 + × (pB2 + TC2) , (47b) ζ (T∂T + p∂p) + F 1 + L1 − 2c B1+M1B2 = B1+ C1, (48a) ζ (T∂T + p∂p) + F 2 + L2 − 2c B2+M2B1 = B2+ C2, (48b) 14 Vicente Garzó ζ (T∂T + p∂p) + F 1 + L1 − C1+M1C2 = C1− B1, (49a) ζ (T∂T + p∂p) + F 2 + L2 − C2+M2C1 = C2− B2, (49b) ζ (T∂T + p∂p) + F 1 + L1 D1 +M1D2 = D1, (50a) ζ (T∂T + p∂p) + F 2 + L2 D2 +M2D1 = D2, (50b) ζ (T∂T + p∂p) + F 1 + L1 E11 +M1E21 = E11, (51a) ζ (T∂T + p∂p) + F 1 + L1 E12 +M1E22 = E12. (51b) Here, we have introduced the linearized Boltzmann collision operators L1X = − J11[f 1 , X ] + J11[X, f 1 ] + J12[X, f , (52) M1X = −J12[f 1 , X ], (53) with a similar definition for L2 andM2. In addition, c ζ = ζ (0) in the undriven case while c ζ = 0 in the driven case, where ζ (0) is given by Eq. (9) to zeroth order, i.e., ζ(0) = − dvV 2Jij [v|f i , f j ] . (54) In Eqs. (47a)–(51b) we have also introduced the operators i X = · (VX) , (Gaussian thermostat), (55a) i X = − X, (stochastic thermostat), (55b) and the quantities Ai(V) = − V, (56) Bi(V) = − i V + , (57) Ci(V) = V, (58) Kinetic Theory for Binary Granular Mixtures at Low-Density 15 Di(V) = V I, (59) Eij(V) = − δij − . (60) Here, I is the unit tensor in d dimensions and ρi = mini is the mass density of species i. Upon writing Eqs. (47a)–(51b) use has been made of the fact that there is no contribution to the cooling rate at this order, i.e., ζ(1) = 0. As a consequence, F i = 0. The property ζ (1) = 0 is special of the low density Boltzmann kinetic theory (since f i does not contain any contribution proportional to ∇ ·u), but such terms occur at higher densities [45, 46]. Note that in the particular case of the gravitational force Fi = mig (where g is the gravity acceleration), the combination m1E11+m2E12 = 0. This leads to Eij = 0, and so there are no contributions to the mass and heat fluxes coming from the external conservative forces. In summary, the solutions to the Boltzmann equations to first order in the spatial gradients are given by [19] fi = f i +Ai · ∇x1 +Bi · ∇p+ Ci · ∇T +Di : ∇u+ Eij · Fj . (61) The solution to zeroth-order is obtained from Eq. (22) while the functions {Ai,Bi,Ci,Di,Eij} are determined from the integral equations (47a)–(51b). Once these equations are solved, the Navier-Stokes transport coefficients are obtained from Eqs. (38)–(46) and the mass, momentum, and heat fluxes are explicitly known. These fluxes, together with the macroscopic balance equa- tions (12)–(14), provide the closed set of Navier-Stokes order hydrodynamic equations for a granular binary mixture. All these results are still formally exact and valid for arbitrary values of the coefficients of restitution. However, explicit expressions for the Navier–Stokes transport coefficients require to solve the integral equations (47a)–(51b). Accurate approximations for {Ai,Bi,Ci,Di,Eij} may be obtained using low order truncation of ex- pansions in a series of Sonine polynomials. The polynomials are defined with respect to a Gaussian weight factor whose parameters are chosen such that the leading term in the expansion yields the exact moments of the entire dis- tribution with respect to 1, v, and v2. The procedure is similar to the one followed for elastic collisions [4] and yields explicit expressions for the trans- port coefficients in terms of the parameters of the mixture [19, 20, 44]. 5 Comparison with Monte Carlo Simulations As said before, the expressions derived for the Navier Stokes transport coef- ficients are obtained by considering two different approximations. First, since the deviation of f i from its Maxwellian form is quite small in the region of 16 Vicente Garzó thermal velocities, one uses the distribution (27) as a trial function for f Second, one only considers the leading terms of an expansion of the distri- bution f i in Sonine polynomials. Both approximations allow one to offer a simplified kinetic theory for a granular binary mixture. To assess the de- gree of accuracy of these predictions, one resorts to numerical solutions of the Boltzmann equation, such as those obtained from the Direct simulation Monte Carlo (DSMC) method [47]. Although the method was originally devised for normal fluids, its extension to granular gases is straightforward [37]. In this Section we provide some comparisons between theory and numerical solutions of the Boltzmann equation by means of the DSMC method in the cases of the diffusion coefficient D (in the tracer limit) and the shear viscosity coefficient η of a heated gas. Let us study each coefficient separately. 5.1 Tracer Diffusion Coefficient We consider a free granular mixture (Fi = 0) in which one of the components of the mixture (say, for instance, species 1) is present in tracer concentration (x1 → 0). In this situation the diffusion coefficient of impurities in a granular gas undergoing homogeneous cooling state can be measured in simulations from the mean square displacement of the tracer particle after a time interval t [48]: 〈|r(t) − r(0)|2〉 = , (62) where |r(t) − r(0)| is the distance travelled by the impurity from t = 0 until time t. The relation (62) written in appropriate dimensionless variables to eliminate the time dependence of D(t) can be used to measure by computer simulations the diffusion coefficient [8, 22]. If the hydrodynamic description (or normal solution in the context of the Chapman–Enskogmethod) applies, then the diffusion coefficientD(t) depends on time only through its dependence on the temperature T (t). Dimensional analysis shows that D(t) ∝ T (t). In this case, after a transient regime, the reduced diffusion coefficient D∗ = (m1m2/ρ)D(t)ν0(t)/T (t) achieves a time-independent value. Here, ν0(t) = nσ 12 v0(t) ∝ T (t) is an effective collision frequency for hard spheres. The fact that D∗ reaches a constant value for times large compared with the mean free path is closely related with the validity of a hydrodynamic description for the system. In addition, as has been recently shown [49], the dependence of D∗ on the mass ratio m1/m2 and the coefficient of restitution α12 is only through the effective mass m∗1 = m1 + (m1 +m2)(1 − α12)/(1 + α12). The dependence of D∗ on the common coefficient of restitution αij ≡ α is shown in Fig. 2 in the case of hard disks (d = 2) for three different sys- tems. The symbols refer to DSMC simulations while the lines correspond to the kinetic theory results obtained in the first Sonine approximation [19, 44]. MD results reported in Ref. [8] when impurities and particles of the gas are Kinetic Theory for Binary Granular Mixtures at Low-Density 17 0.6 0.7 0.8 0.9 1.0 2.5 ω=µ µ=1/4 Fig. 2. Plot of the reduced diffusion coefficient D∗ as a function of the (common) coefficient of restitution α for binary mixtures with ω = µ in the case of a two- dimensional system (d = 2). Here, ω ≡ σ1/σ2 and µ ≡ m1/m2. The symbols are computer simulation results obtained from the mean square displacement and the lines are the theoretical results obtained in the first Sonine approximation. The DSMC results correspond to µ = 1/4 (•), µ = 4 (◦) and µ = 1 (⋄). Molecular dynamics results reported in Ref. [8] for µ = 1 (△) have also been included. mechanically equivalent have also been included. We observe that in the latter case MD and DSMC results are consistent among themselves in the range of values of α explored. This good agreement gives support to the applicability of the inelastic Boltzmann equation beyond the quasielastic limit. It is apparent that the agreement between the first Sonine approximation and simulation results is excellent when impurities and particles of the gas are mechanically equivalent and when impurities are much heavier and/or much larger than the particles of the gas (Brownian limit). However, some discrepancies be- tween simulation an theory are found with decreasing values of the mass ratio µ ≡ m1/m2 and the size ratio ω ≡ σ1/σ2. These discrepancies are not easily observed in Fig. 2 because of the small magnitude of D∗ for µ = 1/4. For these systems, the second Sonine approximation [22] improves the qualita- tive predictions over the first Sonine approximation for the cases in which the gas particles are heavier and/or larger than impurities. This means that the Sonine polynomial expansion exhibits a slow convergence for sufficiently small values of the mass ratio µ and/or the size ratio ω. This tendency is also present in the case of elastic systems [50]. 18 Vicente Garzó 5.2 Shear Viscosity Coefficient of a Heated Gas The shear viscosity η is perhaps the most widely studied transport coefficient in granular fluids. This coefficient can be measured in computer simulations in the special hydrodynamic state of uniform shear flow (USF). At a macro- scopic level, this state is characterized by constant partial densities ni, uni- form temperature T , and a linear flow velocity profile u1,k = u2,k = akℓrℓ, akℓ = aδkxδℓy, a being the constant shear rate. In this state, the temperature changes in time due to the competition between two mechanisms: on the one hand, viscous heating and, on the other hand, energy dissipation in collisions. In addition, the mass and heat fluxes vanish by symmetry reasons and the (uniform) pressure tensor is the only nonzero flux of the problem. The relevant balance equation is that for temperature, Eq. (14), which reduces to ∂tT + aPxy = −(ζ − ξ)T, (63) where Pxy = dVVxVyfi(V) (64) is the xy-element of the pressure tensor. For a granular fluid under USF and in the absence of a thermostatting force (ξ = 0), the energy balance equation (63) leads to a steady state when the viscous heating effect is exactly balanced by the collisional cooling. This situation will be analyzed in Sec. 10. However, if for instance the mixture is heated by the Gaussian thermostat (29) (with v → V), then the viscous heating still prevails so that the temperature increases in time. In this case, the collision frequency ν0(t) ∝ T (t) also grows with t and hence the reduced shear rate a∗(t) = a/ν0(t) (which is the relevant nonequilibrium parameter of the problem) monotonically decreases in time. Under these conditions, the system asymptotically achieves a regime described by linear hydrodynamics and the (reduced) shear viscosity η∗ = [ν0(t)/nT (t)]η(t) can be measured as η∗ = − lim P ∗xy , (65) where P ∗xy = Pxy/nT . This procedure allows one to identify the shear viscosity of a granular mixture excited by the Gaussian external force (29) and compare it with the predictions given by the Chapman–Enskog method. In Fig. 3, we plot the ratio η∗(α)/η∗(1) versus the mass ratiom1/m2 in the case of hard spheres (d = 3) for σ1/σ2 = 1, x1 = , and three different values of the (common) coefficient of restitution αij ≡ α. Here, η ∗(1) refers to the elastic value of the shear viscosity coefficient. Again, the symbols represent the simulation data obtained by numerically solving the Boltzmann equation [21], while the lines refer to the theoretical results obtained from the Boltz- mann equation in the first Sonine approximation. We see that in general the Kinetic Theory for Binary Granular Mixtures at Low-Density 19 0 2 4 6 8 10 η*(α)/η Fig. 3. Plot of the ratio η∗(α)/η∗(1) as a function of the mass ratio m1/m2 for σ1/σ2 = n1/n2 = 1 and three different values of the (common) coefficient of resti- tution α: α = 0.9 (circles), α = 0.8 (squares), and α = 0.7 (triangles). The lines are the theoretical predictions and the symbols refer to the results obtained from the DSMC method. 1 2 3 4 5 η*(α)/η Fig. 4. Plot of the ratio η∗(α)/η∗(1) as a function of the size ratio σ1/σ2 for m1/m2 = 4, n1/n2 = 1 and three different values of the (common) coefficient of restitution α: α = 0.9 (circles), α = 0.8 (squares), and α = 0.7 (triangles). The lines are the theoretical predictions and the symbols refer to the results obtained from the DSMC method. 20 Vicente Garzó deviation of η∗(α) from its functional form for elastic collisions is quite impor- tant. This tendency becomes more significant as the mass disparity increases. The agreement between the first Sonine approximation and simulation is seen to be in general excellent. This agreement is similar to the one previously found in the monocomponent case [7, 10, 51]. At a quantitative level, the dis- crepancies between theory and simulation tend to increase as the coefficient of restitution decreases, although these differences are quite small (say, for instance, around 2% at α = 0.7 in the disparate mass case m1/m2 = 10). The influence of the size ratio on the shear viscosity is shown in Fig. 4 for m1/m2 = 4 and x1 = [21]. We observe again a strong dependence of the shear viscosity on dissipation. However, for a given value of α, the influence of σ1/σ2 on η ∗ is weaker than the one found before in Fig. 3 for the mass ratio. The agreement for both α = 0.9 and α = 0.8 is quite good, except for the largest size ratio at α = 0.8. These discrepancies become more significant as the dissipation increases, especially for mixtures of particles of very different sizes. In summary, according to the comparison carried out in Figs. 3 and 4, one can conclude that the agreement between theory and simulation extends over a wide range values of the coefficient of restitution, indicating the relia- bility of the first Sonine approximation for describing granular flows beyond the quasielastic limit. 6 Einstein Relation in Granular Gases The results presented in Section 5 give some support to the validity of the hydrodynamic description to granular fluids. However, in spite of this sup- port some care is warranted in extending properties of normal fluids to those with inelastic collisions. Thus, for elastic collisions, in the case of an impurity (tracer) particle immersed in a gas the response to an external force on the impurity particle leads to a mobility coefficient proportional to the diffusion coefficient. This is the usual Einstein relation [48], which is a consequence of the fluctuation-dissipation theorem. A natural question is whether the Ein- stein relation also applies for granular fluids. To analyze it, let us consider the tracer limit (x1 → 0) and assume that the current of impurities j 1 is only generated by the presence of a weak concentration gradient ∇x1 and/or a weak external field F1 acting only on the impurity particles. Under these conditions, Eq. (35) becomes 1 = −m1D∇x1 + χ11F1. (66) The Einstein ratio ǫ′ between the diffusion coefficient D and the mobility coefficient χ11 is defined as ǫ′ = m1x1 , (67) Kinetic Theory for Binary Granular Mixtures at Low-Density 21 where T ≃ T2 in the tracer limit. For elastic collisions, the Chapman–Enskog results yield ǫ′ = 1. However, at finite inelasticity the relationship between D and χ11 is no longer simple and, as expected, the Chapman–Enskog expres- sions for D and χ11 in the case of an unforced granular gas [23] clearly show that ǫ′ 6= 1. This means that the Einstein relation does not apply in granular gases. The deviations of the (standard) Einstein ratio ǫ′ from unity has three distinct origins: the absence of the Gibbs state (non-Gaussianity of the distri- bution function of the HCS), time evolution of the granular temperature, and the occurrence of different kinetic temperatures between the impurity and gas particles. The second source of discrepancy can be avoided if the system is driven by an external energy input to achieve a stationary state. With respect to the third reason of violation, this could also be partially eliminated if the temperature of the gas T is replaced by the temperature of the impurity T1 in the usual Einstein relation (67). This change yields the modified Einstein ratio ǫ = m1x1 T1χ11 . (68) As a consequence, the only reason for which ǫ 6= 1 is due to the non-Maxwellian behavior of the HCS distribution. Given that the deviations of the gas dis- tribution f 2 from its Maxwellian form are small [15], the discrepancies of ǫ from unity could be difficult to detect in computer simulations. This conclu- sion agrees with recent MD simulations [52] of granular mixtures subjected to the stochastic driving of the form (30), where no deviations from the (modi- fied) Einstein relation ǫ = 1 have been observed for a wide range of values of the coefficients of restitution and parameters of the system. To illustrate the influence of dissipation on the Einstein ratio more gener- ally, in Figs. 5 and 6 the Einstein ratio as given by (68) is plotted versus the coefficient of restitution α12 for σ1/σ2 = 1 and different values of the mass ra- tio m1/m2 and the coefficient of restitution α22. The results obtained by using the Gaussian thermostat (29) are shown in Fig. 5, while Fig. 6 corresponds to the results derived when the system is heated by the stochastic thermostat (30) [24]. We observe that in general ǫ 6= 1, although its value is very close to unity, especially in the case of the stochastic thermostat, where the deviations from the Einstein relation are smaller than 1%. However, in the case of the Gaussian thermostat the deviations from unity are about 8%, which could be detected in computer simulations. Figures 5 and 6 also show the fact that the transport properties are affected by the thermostat introduced so that the latter does not play a neutral role in the problem [51]. 7 Onsager’s Reciprocal Relations in Granular Gases In the usual language of the linear irreversible thermodynamics for ordinary fluids [53], the constitutive equations (35) and (37) for the mass flux and heat flux in the absence of external forces can be written as 22 Vicente Garzó 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 5. Plot of the modified Einstein ratio ǫ versus the coefficient of restitution α12 for the Gaussian thermostat in the cases: (a) α22 = α12, m1/m2 = 5 and σ1/σ2 = 1; (b) α22 = α12, m1/m2 = 0.5 and σ1/σ2 = 1; and (c) α22 = 0.5, m1/m2 = 10 and σ1/σ2 = 1. 0.0 0.2 0.4 0.6 0.8 1.0 (c) (b) Fig. 6. Plot of the Einstein ratio ǫ versus the coefficient of restitution α12 for the stochastic thermostat in the cases: (a) α22 = α12, m1/m2 = 5 and σ1/σ2 = 1; (b) α22 = α12, m1/m2 = 0.5 and σ1/σ2 = 1; and (c) α22 = 0.5, m1/m2 = 10 and σ1/σ2 = 1. Kinetic Theory for Binary Granular Mixtures at Low-Density 23 i = − − Liq − Cp∇p, (69) q = −Lqq∇T − − C′p∇p, (70) where q ≡ q (1) − and ( ∇ ln(xip), (72) µi being the chemical potential per unit mass. In Eqs. (69) and (70), the coefficients Lij are the so-called Onsager phenomenological coefficients and the coefficients Cp and C p can be expressed in terms of the transport coefficients associated with the heat and mass fluxes. For elastic fluids, Onsager showed [53] that time reversal invariance of the underlying microscopic equations of motion implies important constraints on the above set of transport coefficients, namely Lij = Lji, Liq = Lqi, Cp = C p = 0. (73) The first two symmetries are called reciprocal relations as they relate trans- port coefficients for different processes. The last two are statements that the pressure gradient does not appear in any of the fluxes, even though it is ad- mitted by symmetry. Even for a one component fluid, Onsager’s theorem is significant as it leads to a new contribution to the heat flux proportional to the density gradient [5]. Since there is no time reversal symmetry for granular fluids, Eqs. (73) cannot be expected to apply. However, since explicit expres- sions for all transport coefficients are at hand, the quantitative extent of the violation can be explored. To make connection with the expressions (35) and (37) for the mass and heat fluxes, respectively, it is first necessary to transform Eqs. (69)–(71) to the variables x1, p, T. Since ∇x1 = −∇x2, Eq. (72) implies (∇µ1)T − (∇µ2)T ∇x1 + (m2 −m1)∇ ln p . (74) The coefficients {Lij, Liq, Lqi, Lqq, Cp, C p} then can be easily obtained in terms of the Navier-Stokes transport coefficients introduced in Sec. 4. The result is [20] L11 = −L12 = −L21 = m1m2ρ1ρ2 D, L1q = ρTD ′, (75) Lq1 = −Lq2 = T 2ρ1ρ2 D′′ − Tρ1ρ2 (m2 −m1)D, (76) 24 Vicente Garzó Lqq = λ− m2 −m1 D′, (77) (m2 −m1)D, (78) C′p ≡ L− m2 −m1 T 2(m2 −m1)D ′′. (79) Onsager’s relation L12 = L21 holds since the diffusion coefficient D is symmetric under the change 1 ↔ 2 [19]. However, in general L1q 6= Lq1, Cp 6= 0, and C′p 6= 0 [20]. The Chapman-Enskog results [19] show that there are only two limit cases for which L1q−Lq1 = Cp = C p = 0: (i) the elastic limit (αij = 1) with arbitrary values of masses, sizes and composition and (ii) the case of mechanically equivalent particles with arbitrary values of the (common) coefficient of restitution α ≡ αij . Beyond these limit cases, Onsager’s relations do not apply. At macroscopic level the origin of this failure is due to the cooling of the reference state as well as the occurrence of different kinetic temperatures for both species. 8 Linearized Hydrodynamic Equations and Stability of the Homogeneous Cooling State As shown in Sec. 4, the Navier-Stokes constitutive equations (35)–(37) have been expressed in terms of a set of experimentally accessible fields such as the composition of species 1, x1, the pressure p, the mean flow field u, and the granular temperature T . In terms of these variables and in the absence of external forces, the macroscopic balance equations (12)–(14) become Dtx1 + n2m1m2 ∇ · j1 = 0 , (80) Dtp+ p∇ · u+ (∇ · q+ P : ∇u) = −ζp, (81) Dtu+ ρ −1∇ · P = 0 , (82) DtT − ∇ · ji (∇ · q+ P : ∇u) = −ζT . (83) When the expressions (35)–(37) for the fluxes and the cooling rate ζ → ζ(0) are substituted into the above exact balance equations (80)–(83) one gets a closed set of hydrodynamic equations for x1,u, T, and p. These are the Navier–Stokes hydrodynamic equations for a binary granular mixture: Dtx1 = n2m1m2 m1m2n D∇x1 + Dp∇p+ , (84) Kinetic Theory for Binary Granular Mixtures at Low-Density 25 (Dt + ζ) p+ p∇ · u = T 2D′′∇x1 + L∇p+ λ∇T ∇ℓuk +∇kuℓ − δkℓ∇ · u ∇ℓuk, (Dt + ζ) T + p∇ · u = − m2 −m1 m1m2n D∇x1 + T 2D′′∇x1 + L∇p+ λ∇T ∇ℓuk +∇kuℓ − δkℓ∇ · u ∇ℓuk, (86) Dtuℓ + ρ −1∇ℓp = ρ −1∇kη ∇ℓuk +∇kuℓ − δkℓ∇ · u . (87) For the chosen set of fields, n = p/T and ρ = p [(m1 −m2)x1 +m2] /T . These equations are exact to second order in the spatial gradients for a low density Boltzmann gas. Note that in Eqs. (84)–(87) the second order contributions to the cooling rate have been neglected. These second order terms have been calculated for a monocomponent fluid [5] and found to be very small relative to corresponding terms from the fluxes. Consequently, they have not been considered in the hydrodynamic equations (84)–(87). One of the main peculiarities of the granular gases (in contrast to ordinary fluids) is the existence of non-trivial solutions to the Navier–Stokes equations (84)–(87), even for spatially homogeneous states, ∂tx1H = 0, ∂tuH = 0, (88) [∂t + ζ (x1H , TH , pH)]TH = 0, [∂t + ζ (x1H , TH , pH)] pH = 0, (89) where the subscript H denotes the homogeneous state. Since the dependence of the cooling rate ζ (x1H , TH , pH) on x1H , TH , pH is known [15, 44], these first order nonlinear equations can be solved for the time dependence of the homogeneous state. The result is the familiar Haff cooling law for T (t) at constant density [1, 54]: TH(t) = TH(0) [1 + ζ(0)t/2] . (90) As said before, each partial temperature Ti(t) has the same time dependence but with a different value [15], T1H(t) = 1 + x1(γ − 1) TH(t), T2H(t) = 1 + x1(γ − 1) TH(t), (91) where γ = T1H(t)/T2H(t) is the time-independent temperature ratio. 26 Vicente Garzó Nevertheless, the homogeneous cooling state (HCS) is unstable to suffi- ciently long wavelength perturbations. For systems large enough to support such spontaneous fluctuations, the HCS becomes inhomogeneous at long times. This feature was first observed in MD simulations of free monocomponent gases [55]. In MD simulations the inhomogeneities may grow by the forma- tion of clusters, ultimately aggregating to a single large cluster [56]; if cluster growth is suppressed, a vortex field may grow to the system size where peri- odic boundary conditions can induce a transition to a state with macroscopic shear. The mechanism responsible for the growth of inhomogeneities can be understood at the level of the Navier–Stokes hydrodynamics, where linear stability analysis shows two shear modes and a heat mode to be unstable [5, 7, 54, 57]. The objective here is to extend this analysis to the case of a binary mixture. To do that, we perform a linear stability analysis of the nonlinear hydrody- namic equations (84)–(87) with respect to this HCS for small initial spatial perturbations. For ordinary fluids such perturbations decay in time according to the hydrodynamic modes of diffusion (shear, thermal, mass) and damped sound propagation [3, 58, 59]. For inelastic collisions, the analysis is for fixed coefficients of restitution in the long wavelength limit. As will be seen be- low, the corresponding modes for a granular mixture are then quite different from those for ordinary mixtures. In fact, an alternative study with fixed long wavelength and coefficients of restitution approaching unity yields the usual ordinary fluid modes. Consequently, the nature of the hydrodynamic modes is non uniform with respect to the inelasticity and the wavelength of the perturbation. Let us assume that the deviations δyα(r, t) = yα(r, t) − yHα(t) are small. Here, δyα(r, t) denotes the deviation of {x1,u, T, p} from their values in the HCS. If the initial spatial perturbation is sufficiently small, then for some initial time interval these deviations will remain small and the hydrodynamic equations (84)–(87) can be linearized with respect to δyα(r, t). This leads to a set of partial differential equations with coefficients that are independent of space but which depend on time. As in the monocomponent case [5, 57], this time dependence can be eliminated through a change in the time and space variables, and a scaling of the hydrodynamic fields. We introduce the following dimensionless space and time variables: dt′ν0H(t ′), s = ν0H(t) v0H(t) r, (92) where ν0H(t) is an effective collision frequency for hard spheres and v0H =√ 2TH(m1 +m2)/m1m2. Since {x1H ,uH , TH , pH} are evaluated in the HCS, then Eqs. (88) and (89) hold. A set of Fourier transformed dimensionless variables are then introduced as ρk(τ) = δx1k(τ) , wk(τ) = δuk(τ) v0H(τ) , θk(τ) = δTk(τ) TH(τ) , Πk(τ) = δpk(τ) pH(τ) Kinetic Theory for Binary Granular Mixtures at Low-Density 27 where δyαk ≡ {δx1k, δuk, δTk, δpk} is defined as δyαk(τ) = ds e−ik·sδyα(s, τ). (94) Note that here the wave vector k is dimensionless. In terms of the above variables, the transverse velocity components wk⊥ = wk − (wk · k̂)k̂ (orthogonal to the wave vector k) decouple from the other four modes and hence can be obtained more easily. They obey the equation + η∗k2 wk⊥ = 0, (95) where ζ∗ = ζH/ν0H and η, (96) where ρH = m1n1H +m2n2H . The solution for wk⊥(τ) reads wk⊥(τ) = wk⊥(0) exp[s⊥(k)τ ], (97) where s⊥(k) = ζ∗ − η∗k2. (98) This identifies d − 1 shear (transversal) modes. We see from Eq. (98) that there exists a critical wave number kc⊥ given by kc⊥ = . (99) This critical value separates two regimes: shear modes with k ≥ kc⊥ always decay while those with k < kc⊥ grow exponentially. The remaining modes are called longitudinal modes. They correspond to the set {ρk, θk, Πk, wk||} where the longitudinal velocity component (parallel to k) is wk|| = wk · k̂. These modes are the solutions of the linear equation ∂δzαk(τ) αβ + ikM αβ + k δzβk(τ), (100) where δzαk(τ) denotes now the four variables ρk, θk, Πk, wk|| . The matrices in Eq. (100) are given by (0) =  0 0 0 0 ζ∗ −ζ∗ 0 ζ∗ −ζ∗ 0 0 0 0 1  , (101) 28 Vicente Garzó (1) = 0 0 0 0 0 0 0 − 2 0 0 0 − d+2 0 0 − 1 x1µ+x2  , (102) (2) = −D∗ −x−11 D ′∗ −x−11 D ′′∗ − 2 λ∗ − 2 0 0 0 − 2 (d− 1)η∗  , (103) where 21 = −x1 D′′∗ − x1µ+ x2 , (104) x1µ+ x2 D′∗ − λ∗, (105) 23 = − x1µ+ x2 D∗p. (106) In these equations, µ = m1/m2, xi = niH/nH , and we have introduced the reduced Navier–Stokes transport coefficients1 D, D∗p = ρ2Hν0H m1m2n Dp, D ρ2Hν0H m1m2n D′, (107) D′′∗ = ν0HTH D′′, L∗ = L, λ∗ = λ. (108) The longitudinal modes have the form exp[sn(k)τ ] with n = 1, 2, 3, 4, where sn(k) are the eigenvalues of the matrix M(k) = M (0)+ ikM(1)+k2M(2), namely, they are the solutions of the quartic equation det |M− sI| = 0. (109) The solution to (109) for arbitrary values of k is quite intricate. It is instructive to consider first the solutions to these equations in the extreme long wave- length limit, k = 0. In this case, they are found to be the eigenvalues of the matrix of M(0): s(0)n = 0, 0,− . (110) Hence, at asymptotically long wavelengths (k = 0) the spectrum of the lin- earized hydrodynamic equations (both transverse and longitudinal) is com- prised of a decaying mode at −ζ∗/2, a two-fold degenerate mode at 0, and a d-fold degenerate unstable mode at ζ∗/2. Consequently, some of the solutions are unstable. The two zero eigenvalues represent marginal stability solutions, 1 Note that the definition for the reduced diffusion coefficient D∗ given here differs from the one introduced in Sec. 5.1. Kinetic Theory for Binary Granular Mixtures at Low-Density 29 0.0 0.1 0.2 0.3 0.4 0.5 -0.10 -0.05 Fig. 7. Dispersion relations for α = 0.9, x1 = 0.2, ω = 1 and µ = 4. while the negative eigenvalue gives stable solutions. For general initial per- turbations all modes are excited. These modes correspond to evolution of the fluid due to uniform perturbations of the HCS, i.e., a global change in the HCS parameters. The unstable modes are seen to arise from the initial perturbations wk⊥(0) or wk||(0). The marginal modes correspond to changes in the composition at fixed pressure, density, and velocity, and to changes in Πk−θk at constant composition and velocity. The decaying mode corresponds to changes in the temperature or pressure for Πk = θk. The unstable modes may appear trivial since they are due entirely to the normalization of the fluid velocity by the time dependent thermal velocity. However, this normal- ization is required by the scaling of the entire set of equations to obtain time independent coefficients. The real parts of the modes s⊥(k) and sn(k) is illustrated in Fig. 7 in the case of hard spheres (d = 3) for α ≡ αij = 0.9, σ1/σ2 = 1, x1 = 0.2, and m1/m2 = 4. The k = 0 values correspond to five hydrodynamic modes with two different degeneracies. The shear mode degeneracy remains at finite k but the other is removed at any finite k. At sufficiently large k a pair of real modes become equal and become a complex conjugate pair at all larger wave vectors, like a sound mode. The smallest of the unstable modes is that associated with the longitudinal velocity, which couples to the scalar hydrodynamic fields. It becomes negative at a wave vector smaller than that of Eq. (99) and gives the threshold for development of spatial instabilities. The results obtained here for mixtures show no new surprises relative to the case for a monocomponent gas [5, 54, 57], with only the addition of the stable mass diffusion mode. Of course, the quantitative features can be quite different since there are additional degrees of freedom with the parameter set {x1H ,m1/m2, σ1/σ2, αij}. Also, the manner in which these linear instabilities 30 Vicente Garzó are enhanced by the nonlinearities may be different from that for the one component case [60]. 9 Segregation in Granular Binary Mixtures: Thermal Diffusion The analysis of the linearized hydrodynamic equations for a granular binary mixture has shown that the resulting equations exhibit a long wavelength instability for d of the modes. These instabilities lead to the spontaneous for- mation of velocity vortices and density clusters when the system evolves freely. A phenomenon related with the density clustering is the separation or species segregation. Segregation and mixing of dissimilar grains is perhaps one of the most interesting problems in agitated granular mixtures. In some processes it is a desired and useful effect to separate particles of different types, while in other situations it is undesired and can be difficult to control. A variety of mechanisms have been proposed to describe the separation of particles of two sizes in a mixture of vertically shaken particles. Different mechanisms include void filling, static compressive force, convection, condensation, ther- mal diffusion, interstitial gas forcing, friction, and buoyancy [61]. However, in spite of the extensive literature published in the past few years on this subject, the problem is not completely understood yet. Among the different competing mechanisms, thermal diffusion becomes one of the most relevant at large shaking amplitude where the sample of macroscopic grains resembles a granular gas. In this regime, binary collisions prevail and kinetic theory can be quite useful to analyze the physical mechanisms involved in segregation processes. Thermal diffusion is caused by the relative motion of the components of a mixture because of the presence of a temperature gradient. Due to this motion, concentration gradients subsequently appear in the mixture producing diffu- sion that tends to oppose those gradients. A steady state is finally achieved in which the separation effect arising from thermal diffusion is compensated by the diffusion effect. In these conditions, the so-called thermal diffusion fac- tor Λij characterizes the amount of segregation parallel to the temperature gradient. In this Section, the thermal diffusion factor is determined from the Chapman–Enskog solution described before. To make some contact with experiments, let us assume that the binary granular mixture is in the presence of the gravitational field g = −gêz, where g is a positive constant and êz is the unit vector in the positive direction of the z axis. In experiments [41], the energy is usually supplied by vibrating horizontal walls so that the system reaches a steady state. Here, instead of considering oscillating boundary conditions, particles are assumed to be heated by the action of the stochastic driving force (30), which mimics a thermal bath. As said above, although the relation between this driven idealized method with the use of locally driven wall forces is not completely understood, it Kinetic Theory for Binary Granular Mixtures at Low-Density 31 must be remarked that in the case of boundary conditions corresponding to a sawtooth vibration of one wall the condition to determine the temperature ratio coincides with the one derived from the stochastic force [34]. The good agreement between theory and simulation found in Fig. 1 for the temperature ratio confirms this expectation. The thermal diffusion factor Λij (i 6= j) is defined at the steady state in which the mass fluxes ji vanish. Under these conditions, the factor Λij is given through the relation [62] − Λij∇ lnT = ∇xi, Λij + Λji = 0. (111) The physical meaning of Λij can be described by considering a granular binary mixture held between plates at different temperatures T (top plate) and T ′ (bottom plate) under gravity. For the sake of concreteness, we will assume that gravity and thermal gradient point in parallel directions, i.e., the bottom is hotter than the top (T ′ > T ). In addition, without loss of generality, we also assume that σ1 > σ2. In the steady state, Eq. (111) describes how the thermal field is related to the composition of the mixture. Assuming that Λ12 is constant over the relevant ranges of temperature and composition, integration of Eq. (111) yields = Λ12 ln , (112) where xi refers to the mole fraction of species i at the top plate and x i refers to the mole fraction of species i at the bottom plate. Consequently, according to Eq. (112), if Λ12 > 0, then x 1 < x1, while if Λ12 < 0, then x 1 > x1. In summary, when Λ12 > 0, the larger particles accumulate at the top of the sample (cold plate), while if Λ12 < 0, the larger particles accumulate at the bottom of the sample (hot plate). The former situation is referred to as the Brazil-nut effect (BNE) while the latter is called the reverse Brazil-nut effect (RBNE). The RBNE was first observed by Hong et al. [63] in MD simulations of vertically vibrated systems. They proposed a very simple segregation criterion that was later confirmed by Jenkins and Yoon [64] by using kinetic theory. More recently, Breu et al. [65] have experimentally investigated conditions under which the large particles sink to the bottom and claim that their ex- periments confirm the theory of Hong et al. [63] provided a number of condi- tions are chosen carefully. In addition to the vertically vibrated systems, some works have also focused in the last few years on horizontally driven systems showing some similarities to the BNE and its reverse form [66]. However, it is important to note that the criterion given in Ref. [63] is based on some drastic assumptions: elastic particles, homogeneous temperature, and energy equipartition. These conditions preclude a comparison of the kinetic theory derived here with the above simulations. Some theoretical attempts to assess the influence of non-equipartition on segregation have been recently published. Thus, Trujillo et al. [67] have derived 32 Vicente Garzó 0.5 0.6 0.7 0.8 0.9 1.0 Fig. 8. Phase diagram for BNE/RBNE for mixtures constituted by spheres (d = 3) of the same mass density and equal total volumes of large and small particles. The data points represent the MD simulation results [41] for α = 0.78 when convection is suppressed. Points below (above) the curve correspond to RBNE (BNE). an evolution equation for the relative velocity of the intruders starting from the kinetic theory proposed by Jenkins and Yoon [64], which applies for weak dissipation. They use constitutive relations for partial pressures that take into account the breakdown of energy equipartition between the two species. How- ever, the influence of temperature gradients, which exist in the vibro-fluidized regime, is neglected in Ref. [67] because it is assumed that the pressure and temperature are constant in the absence of the intruder. A more refined theory has recently been provided by Brey et al. [42] in the case of a single intruder in a vibrated granular mixture under gravity. The theory displayed in this section covers some of the aspects not accounted for in the previous theories [42, 64, 67] since it is based on a kinetic theory [19] that goes beyond the quasi-elastic limit [64, 67] and applies for arbitrary composition x1 (and so, it reduces to the results obtained in Ref. [42] when x1 → 0). This allows one to assess the influence of composition and dissipation on thermal diffusion in bi-disperse granular gases without any restriction on the parameter space of the system. To determine the dependence of the coefficient Λ12 on the parameters of the mixture, we consider a non-convecting (u = 0) steady state with only gradients along the vertical direction (z axis). In this case, the mass balance equation (12) yields j1 = j2 = 0, while the momentum equation (13) gives = −ρg. (113) To first order in the spatial gradients, the constitutive equation for the mass flux j1,z is given by Eq. (35), i.e., Kinetic Theory for Binary Granular Mixtures at Low-Density 33 j1,z = − m1m2n , (114) where the susceptibility coefficient χij = 0 in the particular case of the grav- itational force. The condition j1,z = 0 yields m1m2np m1m2p , (115) where use has been made of Eq. (113). Substitution of Eq. (115) into Eq. (111) leads to Λ12 = D′ −Dpg , (116) where n (∂T/∂z) < 0 (117) is the reduced gravity acceleration. Since the mutual diffusion coefficient D is positive [19, 44], the sign of Λ12 is determined by the sign of the quantity D′ −Dpg ∗. This result is general since it goes beyond the regime of density considered. 1 2 3 4 5 Fig. 9. Phase diagram for BNE/RBNE in three dimensions for αij = 0.7 and three values of composition: (a) x1 = 0, (b) x1 = 0.3, and (c) x1 = 0.7. Points below (above) each curve correspond to RBNE (BNE). To gain some insight into the explicit dependence of D′ and Dp on the parameter space of the system, one has to resort to a kinetic theory descrip- tion. For a low-density gas, the expressions of the coefficients D′ and Dp in the first-Sonine approximation are given by 34 Vicente Garzó 1 2 3 4 Fig. 10. Phase diagram for BNE/RBNE in three dimensions for x1 = and three values of the (common) coefficient of restitution: (a) α = 0.9, (b) α = 0.8, and (c) α = 0.5. Points below (above) each curve correspond to RBNE (BNE). D′ = 0, Dp = x2 + x1γ , (118) where ν is the (positive) collision frequency [44] 2π(d−1)/2 ) nσd−112 v0(1 + α12) θ1 + θ2 (x2µ21 + x1µ12) . (119) Given that the driving stochastic term does not play a neutral role in the transport, it must be remarked that the expressions for the transport coeffi- cients obtained in the driven case slightly differ from the ones derived in the free cooling case [19, 44]. Consequently, according to Eqs. (116) and (118), the sign of Λ12 is the same as that of the pressure diffusion coefficient Dp. The condition Λ12 = 0 (or equivalently, Dp = 0) provides the criterion for the transition from BNE to RBNE. Equation (118) shows that the sign of Dp is determined by the value of the control parameter . (120) This parameter gives the mean square velocity of the large particles relative to that of the small particles. Thus, when θ > 1 (θ < 1), the thermal diffusion factor is positive (negative), which leads to BNE (RBNE). The criterion for the transition condition from BNE to RBNE is θ = 1, i.e., . (121) Kinetic Theory for Binary Granular Mixtures at Low-Density 35 In the case of equal granular temperatures (energy equipartition), θ → µ−1 and so segregation is predicted for particles that differ in mass, no matter what their diameters may be [64]. It must be remarked that, due to the lack of energy equipartition, the condition θ = 1 is rather complicated since it involves all the parameter space of the system. In particular, even when the species differ only by their respective coefficients of restitution they also seg- regate when subject to a temperature gradient. This is a novel pure effect of inelasticity on segregation [25, 68]. On the other hand, the criterion (121) for the transition BNE⇐⇒RBNE is the same as the one found previously in Ref. [67] when αij is close to 1 and in Ref. [42] in the intruder limit case (x1 → 0). However, as said before, the results obtained here are more general since they cover all the range of the parameter space of the system. To illustrate size segregation driven by thermal diffusion, we consider mix- tures constituted by spheres (d = 3) of the same material and equal total volumes of large and small particles. In this case, m1/m2 = (σ1/σ2) 3 and x2/x1 = (σ1/σ2) 3. Figure 8 shows the phase diagram BNE/RBNE for this kind of systems. The data points represent the simulation results obtained by Schröter et al. [41] for α = 0.78 in agitated mixtures constituted by parti- cles of the same density. To the best of my knowledge, this is one of the few experiments in which thermal diffusion has been isolated from the remain- ing segregation mechanisms [61]. Our results show that, for a given value of the coefficient of restitution, the RBNE is dominant at small diameter ra- tios. However, since non-equipartition grows with increasing diameter ratio, the system shows a crossover to BNE at sufficiently large diameter ratios. This behavior agrees qualitatively well with the results reported in Ref. [41] at large shaking amplitudes, where thermal diffusion becomes the relevant segregation mechanism. At a quantitative level, we observe that the results are also consistent with the simulation results reported in [41] when periodic boundary conditions are used to suppress convection since they do not ob- serve a change back to BNE for diameter ratios up to 3 (see red squares in Fig. 11 of [41]). Although the parameter range explored in MD simulations is smaller than the one analyzed here, one is tempted to extrapolate the simula- tion data presented in Ref. [41] to roughly predict the transition value of the diameter ratio at α = 0.78 (which is the value of the coefficient of restitution considered in the simulations). Thus, if one extrapolates from the simulation data at the diameter ratios of σ1/σ2 = 2 and σ1/σ2 = 3, one sees that the transition from RBNE to BNE might be around σ1/σ2 = 10, which would quantitatively agree with the results reported in Fig. 8. Figure 8 also shows that the BNE is completely destroyed in the quasielastic limit (α ≃ 1). Let us now investigate the influence of composition on segregation. Figure 9 shows a typical phase diagram in the three-dimensional case for αij ≡ α = 0.7 and three different values of the mole fraction x1. The lines separate the regimes between BNE and RBNE. We observe that the composition of the mixture has significant effects in reducing the BNE as the concentration of larger particles increases. In addition, for a given value of composition, the 36 Vicente Garzó transition from BNE to RBNE may occur following two paths: (i) along the constant mass ratio m2/m1 with increasing size ratio σ1/σ2, and (ii) along the constant size ratio with increasing mass ratio m2/m1. The influence of dissipation on the phase diagrams BNE/RBNE is illustrated in Fig. 10 for d = 3 in the case of an equimolar mixture (x1 = ) and three values of the (common) coefficient of restitution α. We observe that the role played by inelasticity is quite important since the regime of RBNE increases significantly with dissipation. Similar results are found for other values of composition. In summary, thermal diffusion (which is the relevant segregation mech- anism in agitated granular mixtures at large shaking amplitudes) can been analyzed by the Boltzmann kinetic theory. This theory is able to explain some of the experimental and/or MD segregation results [41] observed within the range of parameter space explored. A more quantitative comparison in the dilute regime with MD simulations is needed to show the relevance of the Boltzmann equation to analyze segregation driven by a thermal gradi- ent. As said before, comparison with MD simulations in the tracer limit case (x1 → 0) [42] for a dilute gas has shown the reliability of the inelastic Boltz- mann equation to describe segregation. In this context, one expects that the same agreement observed before in the intruder case [42] is maintained when x1 is different from zero. 10 Steady States: Uniform Shear Flow In the preceding sections, the Navier–Stokes equations (constitutive equa- tions that are linear in the hydrodynamic gradients) have been shown to be quite useful to describe appropriately several problems in granular mixtures. However, under some circumstances large gradients occur and more complex constitutive equations are required. The need for more complex constitutive equations does not signal a breakdown of hydrodynamics [69], only a failure of the Navier–Stokes approximation [70]. Although in this case the Chapman- Enskog method can be carried out to second order in gradients (Burnett or- der), it is likely that failure of the Navier-Stokes description signals the need for other methods to construct the normal solution that are not based on a small gradient expansion. One of the most interesting problems in granular fluids is the simple or uniform shear flow (USF) [13, 26]. As described in Sec. 5, this state is charac- terized by uniform density and temperature and a simple shear with the local velocity field given by u1,x = u2,x = ay, uy = uz = 0, where a is the constant shear rate. The USF is a well-known nonequilibrium problem widely studied, for both granular [26, 27, 71, 72, 73] and conventional [28, 74] gases. However, the nature of this state is quite different in both systems since a steady state is achieved for granular fluids when viscous (shear) heating is compensated for by energy dissipation in collisions: Kinetic Theory for Binary Granular Mixtures at Low-Density 37 aPxy = − nTζ. (122) This steady state is what we want to analyze in this section. The balance equation (122) shows the intrinsic connection between the shear field and dis- sipation in the system. This contrasts with the description of USF for elastic fluids where a steady state is not possible unless an external thermostat is in- troduced [28]. Note that the hydrodynamic steady shear flow state associated with the condition (122) is inherently beyond the scope of the Navier–Stokes or Newtonian hydrodynamic equations [29]. The reason for this is the exis- tence of an internal mechanism, collisional cooling, that sets the strength of the velocity gradient in the steady state. For normal fluids, this scale is set by external sources (boundary conditions, driving forces) that can be controlled to admit the conditions required for Navier–Stokes hydrodynamics. In con- trast, collisional cooling is fixed by the mechanical properties of the particles making up the fluid. This observation is significant because it prevents the possibility of measuring the Newtonian shear viscosity for granular fluids in the steady USF [72, 73]. More generally, it provides a caution regarding the simulation of other steady states to study Navier–Stokes hydrodynamics when the gradients are strongly correlated to the collisional cooling [29]. From a microscopic point of view, the simple shear flow problem becomes spatially uniform in the local Lagrangian frame moving with the flow velocity u. In this frame [28, 75, 76], the velocity distribution functions adopt the form: fi(r,v) → fi(V), where Vk = vk − akℓrℓ is the peculiar velocity. Here, akℓ = aδkxδℓy. Under these conditions, the set of Boltzmann kinetic equations (with Fi = 0) for an isolated system reads − aVy fi(V) = Jij [V|fi, fj] , (i = 1, 2). (123) The most relevant transport properties in a shear flow problem are obtained from the pressure tensor P = P1 + P2, where Pi is the partial pressure tensor of the species i given by Pi,kℓ = mi dVVkVℓfi(V). (124) The trace of Pi defines the partial temperatures Ti as Ti = TrPi/dni. As said before, these temperatures measure the mean kinetic energy of each species. The elements of the pressure tensor Pi can be obtained by multiplying the Boltzmann equation (123) by miVV and integrating over V. The result akmPi,mℓ + aℓmPi,mk = Aij,kℓ, (125) where we have introduced the collisional moments Aij as 38 Vicente Garzó Aij,kℓ = mi dVVkVℓJij [V|fi, fj ]. (126) From Eq. (125), in particular, one gets the balance equation for the partial temperature Ti aPi,xy = − piζi, (127) where pi = niTi is the partial pressure of species i and ζi is defined by Eq. (10). According to Eq. (127), the (steady) partial temperature in the simple shear flow problem can be obtained by equating the viscous heating term a|Pi,xy| to the collisional cooling term (d/2)piζi. The determination of Aij requires the knowledge of the velocity distribu- tion functions fi. This is quite a formidable task, even in the monocomponent case [27]. However, as in the elastic case, one expects to get a good estimate of Aij by using Grad’s approximation [43]: f(V) → fi,M (V) Ci,kℓVkVℓ , (128) where fi,M is a Maxwellian distribution at the temperature of the species i, i.e., fi,M (V) = ni . (129) As happens in the case of homogeneous states, in general the three temper- atures T , T1, and T2 are different in the inelastic case. For this reason we choose the parameters in the Maxwellians so that it is normalized to ni and provides the exact second moment of fi. The Maxwellians fi,M for the two species can be quite different due to the temperature differences. This aspect is essential in a two-temperature theory and has not been taken into account in most of the previous studies [14, 71, 72]. The coefficient Ci can be identified by requiring the moments with respect to VV of the trial function (128) to be the same as those for the exact distribution fi. This leads to − I (130) With this approximation, the Boltzmann collisional moments Aij can be explicitly evaluated. The result is [77, 78] Aij = − 2π(d−1)/2 dΓ (d/2) mininjµjiσ (1 + αij) Tj − Ti (mj/mi)Ti + Tj 1− αij 1 + (miTj/mjTi) Ci − Cj 1 + (mjTi/miTj) 2(d+ 2) , (131) Kinetic Theory for Binary Granular Mixtures at Low-Density 39 where λij = 2µji Tj − Ti (mj/mi)Ti + Tj (2d+ 3− 3αij). (132) The partial cooling rates ζi can be easily obtained from Eqs. (10) and (131). 1.0 1.5 2.0 2.5 3.0 3.5 4.0 α=0.8 α=0.9 Fig. 11. Plot of the temperature ratio T1/T2 as a function of the size ratio σ1/σ2 = (m1/m2) 1/2 for a two-dimensional system in the case x1 = 1/2 and two different values of the (common) coefficient of restitution: α = 0.9 and α = 0.8. The solid lines are the theoretical predictions based on Grad’s solution, while the symbols refer to the DSMC results. The dashed lines correspond to the results obtained from the stochastic thermostat condition (31). Substitution of Eq. (131) into the set of equations (125) allows one to get the partial pressure tensor Pi in terms of the temperature ratio γ = T1/T2 and the parameters of the mixture. The temperature ratio can be obtained from Eq. (127) as x2ζ2P1,xy x1ζ1P2,xy . (133) When the expressions of Pi and ζi are used in Eq. (133), one gets a closed equation for the temperature ratio γ, that can be solved numerically. In Fig. 11 we plot γ versus the diameter ratio σ1/σ2 for a two-dimensional (d = 2) granular gas with x1 = 1/2 and two different values of α. The symbols refer to the simulation data obtained from the DSMC method [79]. Here, we have assumed that the disks are made of the same material, and hence αij = α and m1/m2 = (σ1/σ2) 2. The dependence of γ on σ1/σ2 obtained in the homogeneous steady state driven by the stochastic thermostat (30) is also included for comparison. It is clearly seen that the kinetic theory results based 40 Vicente Garzó on Grad’s solution agree very well with simulation data, even for quite large values of the size ratio. In addition, the thermostat results overestimate the simulation ones (especially for large mass ratio), showing that the properties of the system are not insensitive to the way at which the granular gas is driven. � �� ��� Fig. 12. Plot of the reduced pressure p∗ versus the mass ratio µ = m1/m2 for a two- dimensional system with σ1 = σ2, x1 = 1/2 and α = 0.9. The solid line corresponds to the theoretical predictions derived from Grad’s solution, the dotted line refers to the latter theory but using the expression of T1/T2 obtained from the stochastic thermostat condition (31), and the dashed line is the result obtained from Grad’s solution by assuming the equality of the partial temperatures (γ = 1). The symbols are the DSMC results. Let us now consider the transport coefficients. To analyze the rheologi- cal properties in the steady state, it is convenient to introduce dimensionless quantities. As usual [72], for a low-density gas we introduce the reduced pres- sure p∗ and the reduced shear viscosity η∗ as , (134) , (135) where η = −Pxy/a is the non-Newtonian shear viscosity, Pxy = P1,xy + P2,xy and ν = [π(d−1)/2/Γ (d/2)]nσd−112 v0. In Figs. 12 and 13, we plot p ∗ and η∗, respectively, as functions of the mass ratio µ = m1/m2 for an equal-size (σ1 = σ2) binary mixture of disks (d = 2) with x1 = 1/2 and α = 0.9. Kinetic Theory for Binary Granular Mixtures at Low-Density 41 � �� ��� Fig. 13. Plot of the reduced shear viscosity η∗ versus the mass ratio µ = m1/m2 for a two-dimensional system with σ1 = σ2, x1 = 1/2 and α = 0.9. The solid line corresponds to the theoretical predictions derived from Grad’s solution, the dotted line refers to the latter theory but using the expression of T1/T2 obtained from the stochastic thermostat condition (31), and the dashed line is the result obtained from Grad’s solution by assuming the equality of the partial temperatures (γ = 1). The symbols are the DSMC results. We have also included the predictions for p∗ and η∗ given by the kinetic theory but taking the expression of γ derived when the system is driven by the stochastic thermostat [40]. We observe again in both figures an excellent agreement between the Boltzmann theory based on Grad’s solution and the DSMC results, even for very disparate values of the mass ratio. With respect to the influence of energy nonequipartition, Fig. 12 shows that p∗ presents a non-monotonic behaviour with the mass ratio whereas the theoretical predic- tions with the equipartition assumption monotonically increase with µ. In the case of the shear viscosity, as seen in Fig. 13, both theories (with and with- out energy nonequipartition) predict a non-monotonic dependence of η∗ on µ. However, at a quantitative level, the influence of energy nonequipartition is quite significant over the whole range of mass ratios considered. The non- monotonic dependence of p∗ and η∗ on µ obtained here from the Boltzmann kinetic theory also agrees qualitatively well with MD simulations carried out for bidisperse dense systems [72]. Thus, for instance, the minimum values of p∗ and η∗ are located close to µ = 10 in both dilute and dense cases. Moreover, the predictions for the transport properties given from the present theory by taking the stochastic thermostat expression of γ are quite close to those obtained from the actual value of γ, especially for large mass ratios. 42 Vicente Garzó 11 Summary and Concluding Remarks The primary objective of this review has been to derive the Navier–Stokes hy- drodynamic equations of a binary mixture of granular gases from the (inelas- tic) Boltzmann kinetic theory. The Chapman–Enskog method [4, 43] is used to solve the Boltzmann equation up to the first order in the spatial gradients and the associated transport coefficients are given in terms of the solutions of a set of linear integral equations. These equations have been approximately solved by taking the leading terms in a Sonine polynomial expansion. Compar- ison with controlled numerical simulations in some idealized conditions shows quite a good agreement between theory and simulation even for strong dis- sipation. This supports the idea that the hydrodynamic description (derived from kinetic theory) appears to be a powerful tool for analysis and predictions of rapid flow gas dynamics of polydisperse systems [12]. The reference state in the Chapman–Enskog expansion has been taken to be an exact solution of the uniform Boltzmann equation. An interesting and important result of this solution [15] is that the partial temperatures (which measure the mean kinetic energy of each species) are different. This does not mean that there are additional degrees of freedom since the partial temperatures can be expressed in terms of the global temperature. This is confirmed by noting that Haff’s cooling law [1] (in the free cooling case) is the hydrodynamic mode at long wavelengths and MD simulations confirm that the global temperature dominates after a transient period of a few collision times [34]. In this case, only the global temperature should appear among the hydrodynamic fields. Nevertheless, the species temperatures play a new and interesting secondary role [20]. For an ordinary (molecular) gas, there is a rapid velocity relaxation in each fluid cell to a local equilibrium state on the time scale of a few collisions (e.g., as illustrated by the approach to Haff’s law). Subsequently, the equilibration among cells occurs via the hydrodynamic equations. In each cell the species velocity distributions are characterized by the species temperatures. These are approximately the same due to equiparti- tion, and the hydrodynamic relaxation occurs for the single common temper- ature [43]. A similar rapid velocity relaxation occurs for granular gases in each small cell, but to a universal state different from local equilibrium and one for which equipartition no longer occurs. Hence, the species temperatures Ti are different from each other and from the overall temperature T of the cell. Nev- ertheless, the time dependence of all temperatures (in the free cooling case) is the same in this and subsequent states, i.e., they are proportional to the global temperature. This implies that the species temperatures do not provide any new dynamical degree of freedom at the hydrodynamic stage. However, they still characterize the shape of the partial velocity distributions and affect the quantitative averages calculated with these distributions. The transport coefficients for granular mixtures therefore have new quantitative effects aris- ing from the time independent temperature ratios for each species [19]. This view contrasts with some recent works [80], where additional equations for Kinetic Theory for Binary Granular Mixtures at Low-Density 43 each species temperature have been included among the hydrodynamic set. However, as mentioned before, this is an unnecessary complication, describing additional kinetics beyond hydrodynamics that is relevant only on the time scale of a few collisions. Another important issue discussed here has been the applicability of the Navier–Stokes transport coefficients since their expressions are not restricted to weak inelasticity [12]. However, the Navier–Stokes hydrodynamic equations themselves may or may not be limited with respect to inelasticity, depending on the particular states studied. The Chapman–Enskog method assumes that the relative changes of the hydrodynamic fields over distances of the order of the mean free path are small. In the case of ordinary fluids this can be controlled by the initial or boundary conditions. For granular gases the situ- ation is more complicated since in some cases (e.g., steady states such as the simple shear flow problem [29]) the boundary conditions imply a relationship between dissipation and gradients so that both cannot be chosen indepen- dently. In these cases, the Navier–Stokes approximation only holds for nearly elastic particles. However, the transport coefficients characterizing the Navier– Stokes hydrodynamic equations are nonlinear functions of the coefficients of restitution, regardless the applicability of those equations. In spite of the above cautions, the Navier–Stokes approximation is ap- propriate and accurate for a wide class of flows. One group refers to spatial perturbations of the homogeneous cooling state (HCS) for an isolated sys- tem. Both MD and DSMC simulations [7] have confirmed the dependence of the Navier–Stokes transport coefficients on the coefficient of restitution, and application of the Navier–Stokes hydrodynamics with these coefficients to describe cluster formation has also been confirmed quantitatively [60]. The same kinetic theory results apply to driven systems as well. This is so since the reference state is a local HCS whose parameters vary throughout the sys- tem to match the physical values in each cell. Examples include application of Navier–Stokes hydrodynamics from kinetic theory to symmetry breaking and density/temperature profiles in vertical vibrated gases, for comparison with simulation [81]. Similar comparison with Navier–Stokes hydrodynamics of the latter and of supersonic flow past a wedge in real experiments has been given [82, 83], showing both qualitative and quantitative agreement. In summary, the Navier–Stokes equations with the constitutive equations presented here remain an important and useful description for a wide class of granular flow, although more limited than for normal gases. The explicit knowledge of the transport coefficients and the cooling rate allows one to make some applications of the Navier–Stokes hydrodynamic equations. One of them has been to obtain the linear hydrodynamic equa- tions for small perturbations of the homogenous cooling state. The resulting equations exhibit a long wavelength instability for three of the modes. This is quite similar to the case of a monocomponent granular gas [5, 54, 57], and in fact the same modes are unstable here. The additional diffusion mode for two species behaves as for a normal fluid. 44 Vicente Garzó On the other hand, the constitutive equations for the mass and heat fluxes of a granular binary mixture differ from those obtained for ordinary fluids [4]. This is because the usual restrictions of irreversible thermodynamics no longer apply. These restrictions include Onsager’s reciprocal relations among various transport coefficients and the extent to which these are violated has also been shown here. Another application of the Navier–Stokes equations has been to assess the violation of the Einstein relation between the diffusion and mobility coefficients. In the undriven case, the analysis shows that this violation is due to three independent reasons [23]: the absence of the Gibbs state, the cooling of the reference state, and the occurrence of different temperatures for the particle and surrounding fluid. However, when the mixture is subjected to stochastic driving, a modified Einstein relation suggested by recent MD simulations [52] has also been analyzed. In this case, the results show that the deviations of the (modified) Einstein ratio from unity are in general very small (less than 1%), in agreement with MD simulations [52]. Thermal diffusion becomes the relevant segregation mechanism in agitated granular mixtures at large shaking amplitudes. In these conditions, the use of the Boltzmann kinetic theory for low-density gases appears justified to under- stand the influence of thermal gradient on segregation phenomena. The ther- mal diffusion factor in a heated granular mixture has been explicitly evaluated from the Chapman–Enskog solution to the Boltzmann equation. The results show that the criterion for the transition Brazil-nut effect ⇐⇒reverse Brazil- nut effect is provided by the control parameter θ = m2T1/m1T2 [25, 42, 67]. Given that the energy equipartition is broken, the condition θ = 1 is quite complex since it involves all the parameters of the system: composition, masses, sizes, and coefficients of restitution. The Boltzmann kinetic theory results agree qualitatively well with recent MD simulations [41] within the range of parameter space analyzed. The hydrodynamic description also seems to be justified in the case of steady states that are inherently beyond the scope of the Navier–Stokes hy- drodynamic equations. The reason for this non-Newtonian behavior is the existence of an internal mechanism, collisional cooling, that sets the scale of the spatial gradients in the steady state. For ordinary fluids, this scale can be externally controlled by external sources so that the conditions for Navier– Stokes hydrodynamics apply. On the other hand, for granular gases, collisional cooling is fixed by the mechanical properties of the particles of the system and so the gas can depart from the Navier–Stokes description. One well-known ex- ample of steady states is the simple or uniform shear flow (USF). However, in spite of the extensive prior work on USF for granular fluids [27, 72, 73], the in- herent non-Newtonian character of this state has not been conveniently taken into account. In fact, MD simulations of steady USF have been used for gran- ular fluids to measure the Newtonian or Navier–Stokes shear viscosity. The results derived here from Grad’s solution and DSMC simulations show that USF is an ideal testing ground for the study of rheology since any choice of the shear rate and the coefficients of restitution αij will provide non-Newtonian Kinetic Theory for Binary Granular Mixtures at Low-Density 45 effects. It is one of the fascinating features of granular fluids that phenom- ena associated with complex fluids are more easily accessible than for simple atomic fluids [12, 84]. Hydrodynamics derived from hard-sphere models have found widespread use in the description of numerous industrial processes involving solid parti- cles. Of particular relevance are high-speed, gas-solid flows as found in pneu- matic conveyors (of ores, chemicals, grains, etc.) and fluidized beds (for fluid catalytic cracking, power generation, granulation of pharmaceutical powders, synthesis of fine chemicals like titania, etc.). Such descriptions are now stan- dard features of commercial and research codes. Those codes rely upon accu- rate transport properties and a first order objective is to assure this accuracy from a careful theoretical treatment. As shown in this review, the price of this approach, in contrast to more phenomenological approaches, is an increasing complexity of the expressions as the systems become more complex. The analysis carried out in this presentation has been focused on mix- tures in the dilute regime, where the collisional transfer contributions to the transport coefficients are neglected and only their kinetic contributions are considered. A further step is to develop a theory for moderately dense gran- ular mixtures. This will provide a fundamental basis for the application of hydrodynamics under realistic conditions. Possible extension of the present Boltzmann kinetic theory to higher densities can be done in the context of the Enskog kinetic equation [43]. Preliminary results [85] have been restricted to the uniform shear flow state to get directly the shear viscosity coefficient of a heated granular mixture. The extension of this study [85] to states with gra- dients of concentration, pressure, and temperature is somewhat intricate due to subtleties associated with the spatial dependence of the pair correlations functions considered in the revised Enskog theory. A future work is to ex- tend the results derived for moderately dense mixtures of smooth elastic hard spheres [86] to inelastic collisions. This would allow us to assess the influence of density on the different problems addressed in this review. Of course, the precise expressions for transport coefficients in this case will be even more complex than for a dilute gas due to the expanded parameter space. However, this complexity is not a problem for implementation in a code. As shown along this overview, granular mixtures exhibit a wide range of interesting phenomena for which the Navier–Stokes hydrodynamic equations can be considered as an accurate and practical tool. However, due to their complexity, many of their features are not fully understood. Kinetic theory and hydrodynamics (in the broader sense) can be expected to provide some insight into the understanding of such complex materials. Acknowledgments I want to acknowledge J. W. Dufty, J. M. Montanero, and A. Santos in their roles as collaborators and critics for much of the material discussed here. 46 Vicente Garzó Partial support of the Ministerio de Ciencia y Tecnoloǵıa (Spain) through Grant No. FIS2004-01399 (partially financed by FEDER funds) and from the European Community’s Human Potential Programme HPRN-CT-2002-00307 (DYGLAGEMEM) is also acknowledged. References 1. P. K. Haff, J. Fluid Mech. 134, 187 (1983). 2. L. P. Kadanoff, Rev. Mod. Phys. 71, 435 (1999). 3. See for instance, J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). 4. S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press, Cambridge, 1970). 5. J. J. Brey, J. W. Dufty, C. S. Kim, and A. Santos, Phys. Rev. E 58, 4638 (1998). 6. J. J. Brey and D. Cubero, in Granular Gases, edited by T. Pöschel and S. Luding (Lectures Notes in Physics, Springer, Vol. 564, 2001), pp. 59–78. 7. J. J. Brey, M. J. Ruiz-Montero, and D. Cubero, Europhys. Lett. 48, 359 (1999). 8. J. J. Brey, M. J. Ruiz-Montero, D. Cubero, and R. Garćıa-Rojo, Phys. Fluids 12, 876 (2000). 9. J. J. Brey and M. J. Ruiz-Montero, Phys. Rev. E 70, 051301 (2004). 10. J. M. Montanero, A. Santos, and V. Garzó, in Rarefied Gas Dynamics 24, edited by M. Capitelli (American Institute of Physics, Vol. 762, 2005), pp. 797–802; preprint cond-mat/0411219. 11. J. W. Dufty, J. Phys.: Condens. Matt. 12, A47 (2000). 12. J. W. Dufty, in Recent Research Develop. in Stat. Phys. 2, 21 (2002) and preprint cond-mat/0108444. 13. I. Goldhirsch, Annu. Rev. Fluid Mech. 35, 267 (2003). 14. J. T. Jenkins and F. Mancini, Phys. Fluids A 1, 2050 (1989); P. Zamankhan, Phys. Rev. E 52, 4877 (1995); B. Arnarson and J. T. Willits, Phys. Fluids 10, 1324 (1998); J. T. Willits and B. Arnarson, ibid. 11, 3116 (1999); M. Alam, J. T. Willits, B. Arnarson, and S. Luding, ibid. 14, 4085 (2002); B. Arnarson and J. T. Jenkins, ibid. 16, 4543 (2004); D. Serero, I. Goldhirsch, S. H. Noskowicz, and M.-L. Tan, J. Fluid Mech. 554, 237 (2006). 15. V. Garzó and J. W. Dufty, Phys. Rev. E 60, 5706 (1999). 16. See for instance, J. M. Montanero and V. Garzó, Gran. Matt. 4, 17 (2002); A. Barrat and E. Trizac, ibid. 4, 57 (2002); S. R. Dahl, C. M. Hrenya, V. Garzó, and J. W. Dufty, Phys. Rev. E 66, 041301 (2002); R. Pagnani, U. M. B. Marconi, and A. Puglisi, Gran. Matt. 66, 051304 (2002); D. Paolotti, C. Cattuto, U. M. B. Marconi, and A. Puglisi, ibid. 5, 75 (2003); P. Krouskop and J. Talbot, Phys. Rev. E 68, 021304 (2003); H. Wang, G. Jin, and Y. Ma, ibid. 68, 031301 (2003); J. J. Brey, M. J. Ruiz-Montero, and F. Moreno, Phys. Rev. Lett. 95, 098001 (2005); Phys. Rev. E 73, 031301 (2006); M. Schröter, S. Ulrich, J. Kreft, J. B. Swift, and H. L. Swinney, Phys. Rev. E 74, 011307 (2006). 17. R. D. Wildman and D. J. Parker, Phys. Rev. Lett. 88, 064301 (2002); K. Feitosa and N. Menon, ibid. 88, 198301 (2002); M. Schröter, S. Ulrich, J. Kreft, J. B. Swift, and H. L. Swinney, Phys. Rev. E 74, 011307 (2006). 18. J. Jenkins and F. Mancini, J. Appl. Mech. 54, 27 (1987). http://arxiv.org/abs/cond-mat/0411219 http://arxiv.org/abs/cond-mat/0108444 Kinetic Theory for Binary Granular Mixtures at Low-Density 47 19. V. Garzó and J. W. Dufty, Phys. Fluids 14, 1476 (2002). 20. V. Garzó, J. M. Montanero, and J. W. Dufty, Phys. Fluids 18, 083305 (2006). 21. J. M. Montanero and V. Garzó, Phys. Rev. E 67, 021308 (2003). 22. V. Garzó and J. M. Montanero, Phys. Rev. E 69, 021301 (2004). 23. J. W. Dufty and V. Garzó, J. Stat. Phys. 105, 723 (2001). 24. V. Garzó, Physica A 343, 105 (2004). 25. V. Garzó, Europhys. Lett. 75, 521 (2006). 26. C. S. Campbell, Annu. Rev. Fluid Mech. 22, 57 (1990). 27. In the case of a monocomponent system, see for instance, C. K. K. Lun, S. B. Savage, D. J. Jeffrey, and N. Chepurniy, J. Fluid Mech. 140, 223 (1984); J. T. Jenkins and M. W. Richman, ibid. 192, 313 (1988); C. S. Campbell, ibid. 203, 449 (1989); M. A. Hopkins and H. H. Shen, ibid. 244, 477 (1992); P. J. Schmid and H. K. Kytömaa, ibid. 264, 255 (1994); C. K. K. Lun and A. A. Bent, ibid. 258, 335 (1994); I. Goldhirsch and M. L. Tan, Phys. Fluids 8, 1752 (1996); N. Sela, I. Goldhirsch, and S. H. Noskowicz, ibid. 8, 2337 (1997); J. J. Brey, M. J. Ruiz-Montero, and F. Moreno, Phys. Rev. E 55, 2846 (1997); C.-S. Chou and M. W. Richman, Physica A 259, 430 (1998); C.-S. Chou ibid. 287, 127 (2000); 290, 341 (2001); J. M. Montanero, V. Garzó, A. Santos, and J. J. Brey, J. Fluid Mech. 389, 391 (1999); A. Astillero and A. Santos, Phys. Rev. E 72, 031309 (2005). 28. V. Garzó and A. Santos, Kinetic Theory of Gases in Shear Flows. Nonlinear Transport (Kluwer Academic, Dordrecht, 2003). 29. A. Santos, V. Garzó, and J. W. Dufty, Phys. Rev. E 69, 061303 (2004). 30. A. Goldshtein and M. Shapiro, J. Fluid Mech. 282, 75 (1995). 31. J. J. Brey, J. W. Dufty, and A. Santos, J. Stat. Phys. 87, 1051 (1997). 32. T. P. C. van Noije and M. H. Ernst, Gran. Matt. 1, 57 (1998). 33. J. M. Montanero and V. Garzó, Gran. Matt. 4, 17 (2002). 34. S. R. Dahl, C. M. Hrenya, V. Garzó, and J. W. Dufty, Phys. Rev. E 66, 041301 (2002). 35. D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Academic Press, London, 1990). 36. W. G. Hoover, Computational Statistical Mechanics (Elsevier, Amsterdam, 1991). 37. J. M. Montanero and A. Santos, Gran. Matt. 2, 53 (2000). 38. D. R. M. Williams and F. C. McKintosh, Phys. Rev. E 54, R9 (1996). 39. C. Henrique, G. Batrouni, and D. Bideau, Phys. Rev. E 63, 011304 (2000). 40. A. Barrat and E. Trizac, Gran. Matt. 4, 52 (2002). 41. M. Schröter, S. Ulrich, J. Kreft, J. B. Swift, and H. L. Swinney, Phys. Rev. E 74, 011307 (2006). 42. J. J. Brey, M. J. Ruiz-Montero, and F. Moreno, Phys. Rev. Lett. 95, 098001 (2005); Phys. Rev. E 73, 031301 (2006). 43. J. Ferziger and H. Kaper, Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam, 1972). 44. V. Garzó and J. M. Montanero, preprint cond-mat/0604552. 45. V. Garzó and J.W. Dufty, Phys. Rev. E 59, 5895 (1999). 46. J. F. Lutsko, Phys. Rev. E 72, 021306 (2005). 47. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation Monte Carlo of Gas Flows (Clarendon, Oxford, 1994). 48. J. A. McLennan, Introduction to Nonequilibrium Statistical Mechanics (Prentice- Hall, New Jersey, 1989). http://arxiv.org/abs/cond-mat/0604552 48 Vicente Garzó 49. A. Santos and J. W. Dufty, Phys. Rev. Lett. 97, 058001 (2006). 50. M. López de Haro and E. G. D. Cohen, J. Chem. Phys. 80, 408 (1984). 51. V. Garzó and J. M. Montanero, Physica A 313, 336 (2002). 52. A. Barrat, V. Loreto, and A. Puglisi, Physica A 334, 513 (2004). 53. S. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics (Dover, New York, 1984). 54. N. Brilliantov and T. Pöschel, Kinetic Theory of Granular Gases (Oxford Uni- versity Press, Oxford, 2004). 55. I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619 (1993); I. Goldhirsch, M.L. Tan, and G. Zanetti, J. Sci. Comput. 8, 1 (1993). 56. S. Luding and H.J. Herrmann, Chaos 9, 673 (1999). 57. V. Garzó, Phys. Rev. E 72, 021106 (2005). 58. P. Résibois and M. de Leener, Classical Kinetic Theory of Fluids (John Wiley, NY, 1977). 59. J. P. Boon and S. Yip, Molecular Hydrodynamics (Dover, NY, 1980). 60. J. J. Brey, M. J. Ruiz-Montero, and D. Cubero, Phys. Rev. E 60, 3150 (1999). 61. See, for instance, A. Kudrolli, Rep. Prog. Phys. 67, 209 (2004). 62. J. M. Kincaid, E. G. D. Cohen, and M. López de Haro, J. Chem. Phys. 86, 963 (1983). 63. D. C. Hong, P. V. Quinn, and S. Luding, Phys. Rev. Lett. 86, 3423 (2001). 64. J. T. Jenkins and D. K. Yoon, Phys. Rev. Lett. 88, 194301 (2002); D. K. Yoon and J. T. Jenkins, Phys. Fluids 18, 073303 (2006). 65. A. P. J. Breu, H. M. Ensner, C. A. Kruelle, and I. Rehberg, Phys. Rev. Lett. 90, 014302 (2003). 66. T. Schautz, R. Brito, C. A. Kruelle, and I. Rehberg, Phys. Rev. Lett. 95, 028001 (2005) and references therein. 67. L. Trujillo, M. Alam, and H. J. Herrmann, Europhys. Lett. 64, 190 (2003). 68. D. Serero, I. Goldhirsch, S. H. Noskowicz, and M.-L. Tan, J. Fluid Mech. 554, 237 (2006). 69. M. L. Tan and I. Goldhirsch, Phys. Rev. Lett. 81, 3022 (1998). 70. J. W. Dufty and J. J. Brey, Phys. Rev. Lett. 82, 4566 (1999). 71. J. T. Willits and B. Arnarson, Phys. Fluids 11, 3116 (1999); M. Alam, J. Willits, B. Arnarson, and S. Luding, ibid. 14, 4085 (2002). 72. M. Alam and S. Luding, Gran. Matt. 4, 139 (2002); J. Fluid Mech. 476, 69 (2003); Phys. Fluids 17, 063303 (2005). 73. M. Alam and C. M. Hrenya, Phys. Rev. E 63, 0613018 (2001); R. Clelland and C. M. Hrenya, ibid. 65, 031301 (2002); S. R. Dahl, R. Clelland, and C. M. Hrenya, Phys. Fluids 14, 1972 (2002); S. R. Dahl, R. Clelland, and C. M. Hrenya, Powder Tech., 138, 7 (2003); S. R. Dahl and C. M. Hrenya, Phys. Fluids 16, 1 (2004); H. Iddir, H. Arastoopour, and C. M. Hrenya, Powder Tech. 151, 117 (2005). 74. See for instance, Nonlinear Fluid Behavior, edited by H. J. M. Hanley (North- Holland, Amsterdam, 1983). 75. A. W. Lees and S. F. Edwards, J. Phys. C 5, 1921 (1972). 76. J. W. Dufty, A. Santos, J. J. Brey, and R. F. Rodŕıguez, Phys. Rev. A 33, 459 (1986). 77. J. M. Montanero and V. Garzó, Physica A 310, 17 (2002). 78. V. Garzó, Phys. Rev. E 66, 021308 (2002). 79. V. Garzó and J. M. Montanero, Gran. Matt. 5, 165 (2003). Kinetic Theory for Binary Granular Mixtures at Low-Density 49 80. L. Huilin, D. Gidaspow, and E. Manger, Phys. Rev. E 64, 061301 (2001); M.F. Ramahan, J. Naser, and P. J. Witt, Powder Tech. 138, 82 (2003); J. E. Galvin, S. R. Dahl, and C. M. Hrenya, J. Fluid Mech. 528, 207 (2005). 81. J. J. Brey, M. J. Ruiz-Montero, F. Moreno, and R. Garćıa-Rojo, Phys. Rev. E 63, 061305 (2001); ibid. 65, 061302 (2002) 82. X. Yang, C. Huan, D. Candela, R. W. Mair, and R. L. Walsworth, Phys. Rev. Lett. 88, 044301 (2002); C. Huan, X. Yang, D. Candela, R. W. Mair, and R. L. Walsworth, Phys. Rev. E 69, 041302 (2004). 83. C. Bizon, M. D. Shattuck, J. B. Swift, and Harry L. Swinney, Phys. Rev. E 60, 4340 (1999); E. C. Rericha, C. Bizon, M. D. Shattuck, and H. L. Swinney, Phys. Rev. Lett. 88, 014302 (2002). 84. I. Goldhirsch, in Granular Gases, edited by T. Pöschel and S. Luding (Lectures Notes in Physics, Springer, Vol. 564, 2001), pp. 79–99. 85. V. Garzó and J. M. Montanero, Phys. Rev. E 68, 041302 (2003). 86. M. López de Haro, E. G. D. Cohen, and J. M. Kincaid, J. Chem. Phys. 78, 2746 (1983). Kinetic Theory for Binary Granular Mixtures at Low-Density Vicente Garzó

0704.1212

Optical carrier wave shocking: detection and dispersion

arXiv:0704.1212v1 [physics.optics] 10 Apr 2007 CSHOCK-II [email protected] Optical carrier wave shocking: detection and dispersion P. Kinsler, S.B.P. Radnor, J.C.A. Tyrrell, and G.H.C. New Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom. (Dated: November 20, 2018) Carrier wave shocking is studied using the Pseudo-Spectral Spatial Domain (PSSD) technique. We describe the shock detection diagnostics necessary for this numerical study, and verify them against theoretical shocking predictions for the dispersionless case. These predictions show Carrier Envelope Phase (CEP) and pulse bandwidth sensitivity in the single-cycle regime. The flexible dispersion management offered by PSSD enables us to independently control the linear and nonlinear dispersion. Customized dispersion profiles allow us to analyze the development of both carrier self- steepening and shocks. The results exhibit a marked asymmetry between normal and anomalous dispersion, both in the limits of the shocking regime and in the (near) shocked pulse waveforms. Combining these insights, we offer some suggestions on how carrier shocking (or at least extreme self-steepening) might be realised experimentally. I. INTRODUCTION The self-steepening of an optical pulse envelope was first studied by DeMartini et al. in 1967 [1], and is a well-known phenomenon associated with self-phase mod- ulation (SPM). Surprisingly however, the possibility of self-steepening of the optical carrier wave1 was considered even earlier in a 1965 paper by Rosen [2], who showed that, for a third order χ(3) nonlinearity (and under suit- able conditions), a shock (or field discontinuity) develops in a finite distance. This latter phenomenon received lit- tle attention for more than 30 years, until it was revisited in the 1990s by Moloney et al. [3, 4], who performed Fi- nite Difference Time Domain (FDTD) simulations of the process. Three dimensional FDTD simulations of carrier shocking have recently been performed by Trillo et al. In the present paper, we investigate carrier wave shock- ing in χ(3) nonlinear materials where SPM is accompa- nied by the generation of (odd) higher harmonics. For the dispersionless case we have generalized earlier pre- dictions based on the method of characteristics (MOC) to allow for arbitrary initial waveforms. This allows us, for example, to predict the carrier envelope phase (CEP) sensitivity of the shocking distance for optical pulses, as well as the dependence on pulse length. Our primary interest in this paper is the effect of linear dispersion on carrier shock formation. Since no analytic solutions exist in this case, we are forced to rely on nu- merical simulations. In any discussion of carrier wave shocking, it is important to distinguish between three related concepts: the physical system, the mathemati- cal model, and the numerical model. The mathematical model is an approximation to the physical system, while ∗Electronic address: [email protected] 1 Note that in this paper we use the term “carrier” to denote all the oscillations of the field, and do not use its other sense, i.e. that of fixed-frequency oscillations as used in a envelope and carrier representation of a pulse. the numerical model is an approximation to the mathe- matical model. Actual discontinuities occur only in the mathematical model, and it is these to which the idea of shock formation refers. In the physical system, disconti- nuities are prevented by phenomena not included in the mathematics, while numerical codes inevitably fail as a mathematical discontinuity is approached. The indicator of an imminent “shock” is the rapidly increasing gradi- ent (steepening) of the optical carrier. In the numerical model, we see numerical symptoms generated by extreme self-steepening, and these correlate with the onset of the discontinuities in the mathematical model. Under these circumstances, it has been necessary for us to develop a quantitative numerical test of “shock formation”. We have found the most satisfactory diagnostic to be “Local Discontinuity Detection” (LDD), which gives results that are in good agreement with theoretical MOC predictions based on the mathematical model. LDD provides a clear numerical measure of the rapid steepening that precedes the appearance of a discontinuity in the mathematical representation. We continue to use the term “shock” to describe the situation where a mathematical shock is im- minent. In our simulations, we exploit the flexibility in disper- sion management offered by the Pseudo-Spectral Spatial Domain (PSSD) technique [7] to study carrier shock for- mation for a range of simple dispersion profiles, and de- termine the degree of phase mismatch that carrier shock- ing can tolerate. As expected, we find that shocking oc- curs when the nonlinearity dominates the linear disper- sion. However, it emerges that the process is asymmet- ric, with anomalous dispersion being far more conducive to shocking than normal dispersion. Hence, the fact that the LDD scheme does not detect carrier shocks in simula- tions involving (normally dispersive) fused silica, even at powers equal to its damage threshold, is neither surpris- ing nor necessarily discouraging. After all, anomalously dispersive materials could potentially be engineered. Fur- ther, our results also relate to how one might perform carrier shaping (as opposed to carrier steepening), a pro- cess that has some interesting applications. 1 Kinsler-RTN-2007 http://arxiv.org/abs/0704.1212v1 mailto:[email protected] mailto:[email protected] CSHOCK-II [email protected] After briefly describing our simulation methods in section II, we consider dispersionless shocking and the method of characteristics in III, and our LDD shock de- tection scheme in IV. Then we discuss the effect of disper- sion on carrier shocking in V, followed by our numerical results in VI. In VII we consider the potential relevance to experimental detection of carrier steepening and/or shocks. Finally, in VIII, we present our conclusions. II. SIMULATION METHODS The PSSD method [7, 8] offers significant advantages over the traditional FDTD and Pseudospectral Time- Domain (PSTD) [9] techniques for modeling the propa- gation and interaction of few-cycle pulses. Run times are generally faster, and PSSD also offers far greater flexi- bility in the handling of dispersion. Whereas FDTD and PSTD [9] propagate fields E(z), H(z) forward in time, PSSD propagates fields E(t), H(t) forward in space. It is important to keep this difference in mind when com- paring our results to those in [9]. Under PSSD, the en- tire time-history (and therefore frequency content) of the pulse is known at any point in space, so arbitrary disper- sion incurs no extra computational penalty. In contrast, the FDTD or PSTD approaches use convolutions incor- porating time-response models for dispersion. We apply the PSSD algorithm to two representations of the field and source-free Maxwell’s equations in non- magnetic media; the first uses the E and H fields, and the second the directional fields ~G±(t) = αr(t)∗ ~Ex(t)± βr ~Hy(t) [10]. Here the αr, βr include the (linear) per- mittivity and permeability of the material (i.e. ǫ(t), µ). These G± fields enable us to rewrite Maxwell’s equations, and efficiently separate out the relevant forward-going part of the field. For an instantaneous χ(3) nonlinearity, the equations for E and H in the 1D (plane wave) limit are dHy(t; z) ǫ0ǫr(t) ∗ Ex(t; z) + χ (3)Ex(t; z) dEx(t; z) [µ0Hy(t; z)] . (2) The G± field simulations usually assume G− = 0, and as a result contain only forward traveling components. The forward-only wave equation for G+ is dG+(t; z) βrαr(t) ∗G +(t; z) + βrχ (3)E(t; z)3 where it is most straightforward to calculate the non- linear term by reconstructing E(t; z) from G+(t; z) in the frequency domain using Ẽ(ω; z) = G̃+(ω; z)/2α̃r(ω), since G− = 0. Notice the similarity between eqns. (1) and (3), but that eqn. (3) propagates the field in a single first order equation, rather than two. Typical array sizes used in pulse simulations were N = 214 covering a time window T = 200fs, and (spatial) propagation steps were dz = 0.4cT/N ≈ 0.9nm. We en- sured the stability of our integration using Orszag’s 2/3 rule [11], which involves setting the upper part of the spectral range to zero. It is worth noting that changing this cut-off, either by adjusting its position, or by using a smoothed (rather than step-like) filter, made little dif- ference to test simulations. The pulse profile used as an initial condition was E(t) = E0 sin(ω1t+ φ) sech(0.28ω1t/τ), (4) where our standard parameters were ω1 = 2.356 × 1015rad/s (i.e. λ = 800nm) with τ = 0.93̇. Such pulses are rather short (since the number of cycles inside the intensity FWHM is τ), but in fact the shocking distance is only weakly dependent on the pulse width, with signif- icant variation only appearing for pulses of a few cycles or less. We also performed CW simulations, for which we mod- eled just a single cycle of the carrier. assisted by the pe- riodic nature of the discrete Fourier transform. For these we used array sizes of N = 210, and the time window was set by the period of the field oscillations. Our default value of nonlinear strength was χ(3)E20 = 0.02, which is comparable to that in fused silica at an in- tensity of 0.7×1014 W/cm2; our χ(3) parameter is equiv- alent to η in Rosen [2], and a in Gilles et al.[4]. We use an instantaneous nonlinearity, since our primary interest is linear dispersion, and that is a far more significant effect than the nonlinear time response. III. CARRIER WAVE SHOCKING As an introduction to the process of carrier wave shock- ing, fig. 1 shows the profile of a pulse propagating in a dispersionless χ(3) medium, just before a shock occurs. The nonlinearity gives rise to a nonlinear index of refrac- tion n2E 2, the effect of which is to increase the effective refractive index in the more intense regions of the pro- file. This reduces the phase velocity at the peak of each oscillation with respect to the rest of the waveform, and causes the slope on the trailing edges to increase dramat- ically. The effects seen in fig. 1 are associated with the gen- eration of third and higher harmonics, although the har- monic components are not particularly strong even when a shock is about to occur. As an example, fig. 2 shows how the harmonics build up as shocking is approached. The ω4 scaling of the intensity spectrum in the figure ex- aggerates the contribution of the higher orders, and has been chosen for illustrative purposes. Notice that the profile becomes nearly flat (i.e. the spectrum falls off as the 4th power of the frequency) just before shocking is registered at around 4.3µm. If we write E = A(t) [sin(ω1t) + γ cos(3ω1t+ ψ)] , (5) 2 Kinsler-RTN-2007 mailto:[email protected] CSHOCK-II [email protected] 5 10 15 20 time(fs) FIG. 1: The profile of a few-cycle optical pulse just prior to shocking in the dispersionless limit. The larger oscillations in the centre of the pulse undergo more self-steepening than those in the wings. The standard pulse parameters were used. FIG. 2: Development of the heights of the scaled harmonic peaks as a pulse approaches the (LDD) shocking distance of 4.3µm in the dispersionless case. Each line perpendicular to the ω/ω1 axis corresponds to the (scaled) contribution from that spectral peak. Note that the viewpoint has been rotated so that the contribution from the fundamental is to the right. The initial pulse contained about 33 cycles (τ = 33). we find that an appropriate choice of A(t), with γ = 0.1 and ψ = 0, gives us a passable match to fig. 1. Note that this choice of ψ corresponds to the phase of third harmonic generation (THG) under index matched conditions, i.e. where n0 = n(ω1) = n(ω3). Rosen’s original paper [2] used the MOC to predict the formation of a value discontinuity in the field at cer- tain points within the profile. If the displacement of the dispersionless medium is written D = ǫ0 E + χ(1)E + χ(3)E3 , (6) he showed that the wave equation for E is 1 + χ(1) ) ∂2E + χ(3) , (7) and that the associated equation governing the charac- teristic lines of E is + v(E) = 0. (8) Here, the velocity v(E) is given by v(E) = (ǫr + 3χ(3)E2)1/2 , (9) where ǫr = 1 + χ (1) = n0 2 is the (relative) dielectric constant and n0 the linear refractive index. v − dv FIG. 3: Method of Characteristics. Two points A and B on the field profile, separated initially by a time difference dt travel at different speeds v − dv and v, and meet at point C. Using eqn. (9) along with the construction shown in fig. 3, we can derive a simple formula for the distance to shocking. The figure shows two characteristics AC and BC, originating from points A and B, and converging towards a shock at C after a distance of L. The intensity at A is higher than at B, so the speed associated with AC (represented by its gradient) is lower than that of BC. This means that at C, the field has two values, and a discontinuity has formed. From the geometry of the figure, it is easy to show that (c/n0) , (10) where t, v = c/n0, and L = vt are respectively time, speed and distance. On the other hand, differentiating eqn. (9) leads to 3cχ(3) n2o + 3χ (3)E2 , (11) and this combined with eqn. (10) yields 1 + 3χ(3)E2/n2o 3χ(3)(−dE2/dt) 1 + 8n2E2/no (−dE2/dt) , (13) where n2 = 3χ (3)/8no is the material parameter deter- mining the intensity induced refractive index shift as 2. For a given profile, a shock will occur first at the point where −dE2/dt reaches its negative extremum. 3 Kinsler-RTN-2007 mailto:[email protected] CSHOCK-II [email protected] We can therefore define the shocking distance as Lshock = 1 + 8n2E2/no (−dE2/dt) (−dE2/dt) for 8n2E 2/no ≪ 1.(15) This formula is more general than that of either Rosen [2] or Gilles et al. [4]. Notice in particular that the parameter that controls the shock behaviour is not the gradient of the field (dE/dt), but that of the field squared (dE2/dt). For a sinusoidal initial waveform E = E0 sin(ω1t+ φ), it is easy to show that shocks form at ω1t = −π/4 + jπ − φ, (16) where j is an integer, and that the shocking distance is Lshock = 3ω1χ(3)E 4ω1n2E . (17) The approach is readily extended to pulsed waveforms of the type defined in eqn. (4). Analytical results can be derived in the new situation in which eqn. (16) be- comes a transcendental equation. However, the results are cumbersome and it is simpler to scan the profile nu- merically to determine the shocking parameters. It turns out that the shocking distance for short pulses exhibits an interesting sensitivity to the carrier envelope phase φ. Whilst in the CW case all locations defined by eqn. (16) were equivalent, for pulses, the one nearest the peak of the envelope has a shorter Lshock than the others. A set of results is displayed in fig. 4 where the shock- ing distance is plotted as a function of the carrier phase φ for sech profiles with different pulse widths τ . The dotted line is for the case of a very broad pulse for which Lshock is given by eqn. (17). The sharp peaks mark a curve crossings where the shock location switches from one point on the sinusoid to another. Notice that the range of shocking distances increases for shorter pulses, and that, unsurprisingly, the lowest values occur when φ is around π/2 i.e. when the pulse has a cosine form. One very important point to note from fig. 4 is that for pulses containing more than a few cycles, the dependence of the shocking distance on pulse width is very weak. We will see this message repeated later in section V, with shock regions being similar for both single cycle (τ = 1) pulses and CW (τ = ∞) fields. IV. SHOCK DETECTION Since optical shock formation is directly associated with regions of increasingly steep field gradient, any nu- merical scheme, however sophisticated, is bound to fail at some point in the process. We therefore want to recognize when a shock is imminent, not only to avoid numerical FIG. 4: MOC shocking distances as a function of phase, for pulses as in eqn. (4), allowing for different pulse lengths. We include the 1/2 cycle τ = 0.56 results to emphasize the trend; but even for a long pulse (τ = 28), a small peak can be seen just below φ/π = 0.01. The peak position is weakly τ dependent. The + signs denote LDD shocking distances obtained from simulations of the τ = 0.93̇ case. problems, but as a means to estimate the distance at which a discontinuity would occur in the mathematical model. One obvious symptom of impending numerical failure is loss of energy conservation [7]. However, this does not give an indication at the first instance of a shock form- ing, but rather signals the accumulated effect of multiple small numerical failures from many shocked regions. A more physical strategy is to search for regions where the field gradient dE/dt is large and increasing rapidly, and to use this to predict the shocking point. A useful variant, suggested by the MOC calculation in the previ- ous section, is to use the value of −dE2/dt instead of the gradient. Overall, we find that the best method is Local Discon- tinuity Detection (LDD), which is similar to techniques used in other fields (see e.g. [12]). As the shock regime is approached, narrow shoulders with associated points of inflection appear within the regions of rapidly increasing gradient. The procedure is therefore to scan the field pro- file for the maximum gradient (of either E or E2) and, if it occurs near a point of inflection, an incipient shock is registered. An example of a pulse that has just triggered the LDD diagnostic is shown in fig 5. The LDD method requires two parameters. The first determines the time scale used in determining whether a point of inflection exists. For this, we pick the scale set by our temporal grid and insist that the field gradients calculated at three adjacent grid points have opposing signs: either up-down-up, or down-up-down. The sec- ond determines the maximum range allowed between the maximum gradient and the point of inflection, and our default value for this was 10 grid points. In our simula- tions, we see that the position of the first detected shock depends only weakly on this range. We can easily mini- mize the small sensitivity to these parameters by holding 4 Kinsler-RTN-2007 mailto:[email protected] CSHOCK-II [email protected] them fixed throughout any given set of simulations. As a result, we have found LDD to be a sensitive and reliable method of shock detection. FIG. 5: The profile of a few-cycle optical pulse at the point of shocking. The larger oscillations in the centre of the pulse undergo more self-steepening than those in the wings. (a) The whole pulse, with the LDD carrier shock circled (◦). (b) An expanded view of the shock region, with its point of inflection (at ∼ 9.3fs) being very close to the steepest gradient (at ∼ 9.1fs). The standard pulse parameters were used. Eqn. (17) predicts that the shocking distance should increase linearly with the refractive index n0. We use this to test the LDD diagnostic in fig. 6, where the ana- lytical formula is compared with the results of numerical simulations. This figure shows close agreement between prediction (using eqn. (15)) and simulation, where the approximation causes the MOC prediction to be reduced by less than 0.1µm at n0 = 1. We see similar agreement between simulations using the LDD method and the car- rier phase sensitivity shown in fig. 4. The presence of small systematic differences (as on e.g. fig. 4, 6) can be easily understood, since the LDD diagnostic is (strictly speaking) a test of the numerics, and is not a direct test for the presence of a physical shock or mathematical dis- continuity. V. THE EFFECT OF DISPERSION ON SHOCKING The primary purpose of this paper is to understand the principles of carrier shock formation in the presence of dispersion. Some simple ideas about the role of dis- persion can be understood from eqn.(5) using the insight FIG. 6: The LDD shocking distance as a function of refractive index, comparing the approximate MOC prediction from eqn. (15) to PSSD simulations (+) for pulses with τ = 0.93̇. Other PSSD simulation results from independent codes give very similar results to those shown on the graph. from eqns.(10) – (15), i.e. that the rate of change of E2 is the critical factor in shock development, rather than that of E. If only the fundamental and third harmonic components are considered (as in eqn.(5)), the effective refractive indices are ∆nNL:1 = n1 + n2 1 + 3γ2 I1, (18) ∆nNL:3 = n3 + n2 3 + γ2 I1. (19) Evidently, the relative phase velocity of the two waves is affected by both linear and nonlinear dispersion, so the phase ψ in eqn.(5) will vary accordingly as the pulse prop- agates. Fig. 7, which shows how dE2/dt varies with ψ, suggests that shocking is likely to be exacerbated when ψ is small and positive, but moderated when ψ is negative. Broadly speaking, the former case will be promoted by anomalous dispersion and the latter by normal disper- sion; however, the process is clearly complicated, since it involves time-dependent phase shifts between the waves and the interplay of linear and nonlinear dispersion. Of course, since carrier shocking relies on the establishment and maintenance of specific phase relationships between a set of harmonics, it must be expected that strong dis- persion of either sign will disrupt the shock formation process. On the other hand, the simple argument that has been offered suggests that shocking may be tolerant to a degree of anomalous dispersion, but not to a simi- lar amount of normal dispersion. In general, therefore, a graph of shocking signature versus refractive index mis- match might be expected to exhibit a shock region where nonlinearity dominates dispersion (displaced in the direc- tion of anomalous dispersion), surrounded by a shock- free region where dispersion dominates the nonlinearity. Moreover, if a dominant coherence length LC can be de- fined, it is reasonable to expect shocking to occur when this exceeds the characteristic SPM length (LSPM ). As we shall see, all these features are borne out by the nu- merical results. 5 Kinsler-RTN-2007 mailto:[email protected] CSHOCK-II [email protected] FIG. 7: The effect of different time-lags between fundamental and third harmonic on the steepness of the pulse. The field profile is as in eqn. (5), with A(t) = A0 and γ = 1/16. (a) Scaled maximum values of Mj = Max(dE j/dt) as a function of third harmonic phase offset ψ. We can see that the maxima of dE/dt and dE2/dt occur at different ψ. (b) Position on the pulse θj = ω1tj of the maxima plotted in the top frame. Note that the kinks in (a) and (b) occur for offsets that give the longest shocking distances, not the shortest. In our numerical simulations, we make extensive use of model dispersion curves, which enable refractive in- dex differences between harmonics to be freely controlled, and lead to uncomplicated boundaries between the shock and no-shock regimes. The dispersion profiles shown in fig. 8 duly contain either refractive index steps ∆n at the midpoints between successive odd harmonics, or smooth gradients δ that provide a similar net change. In the simplest option, there is a single step at 2ω1, in which case the dominant coherence length is clearly LC = π/ |k3 − 3k1| = πc/3ω1 |n3 − n1|; we have also tried a single step at 4ω1, which has a shorter LC . In all cases, when n increases with frequency, the situation corresponds broadly to normal dispersion, while decreas- ing n corresponds to anomalous dispersion. The step size (or gradient) is chosen to be comparable to values in fused silica at ω1 = 1.5rad/fs, where the refractive index differences between the lower harmonics are ∆n1,3 ≈ 0.06 and ∆n3,5 ≈ 0.12[4]. While we consider the case of fused silica itself in Section VII, the results based on the model dispersion characteristics are invaluable for understand- 2 n∆ ω1 ω3 ω5 ω7 FIG. 8: Types of refractive index profile used. The solid lines show a single refractive index step midway between ω1 and ω3, so that the fundamental is always phase mismatched from its higher harmonics, although they remain perfectly phase matched with each other. The dashed lines show multiple re- fractive index steps, each of the same size, and always midway between subsequent harmonics. The dotted lines show a lin- ear refractive index gradient which gives the same mismatch between subsequent harmonics as the multi-stepped case. We do not show lines for the case of a single step at 4ω1 to avoid cluttering the figure. ing the essential principles of carrier shock formation in dispersive media. VI. RESULTS AND DISCUSSION We will now analyze our numerical simulations of car- rier shocking in the presence of dispersion on the basis of the principles discussed above. Results are included for both a CW wave and for a single-cycle pulse. The CW case gives slightly wider shocking regions, but the differ- ences are minor. This is because the effect of the pulse envelope on the field amplitudes and gradients of the cen- tral carrier oscillation is small, except when considering sub-cycle pulses. In the results we present, the energy conservation and LDD measures reveal the presence of sharp boundaries between the shocking and non-shocking regions. Results for single refractive index steps at 2ω1 and 4ω1 are presented in figs. 9 and 10 respectively, whereas fig. 11 has a step midway between all harmonics. In both single stepped cases, a useful coherence length can be defined on the basis of the mismatch between the fun- damental and the harmonic just above the step. In the multi-stepped case, there is no easy way to define a dom- inant coherence length. Fig. 9(a) shows the pulse profiles in the anomalous (negative) and normal (positive) single step cases, for the smallest step size at which shocking did not occur. The first obvious difference between the profiles is the oppo- site walk-off direction of the third harmonic, although the third harmonic contribution is hard to see in the anoma- lous case. The second is that the pulse profiles exhibit 6 Kinsler-RTN-2007 mailto:[email protected] CSHOCK-II [email protected] −20 −15 −10 −5 0 time(fs) −0.1 −0.05 0 0.05 0.1 ∆ n1,3 −0.1 −0.05 0 0.05 0.1 (b) X FIG. 9: Carrier shocking for CW and pulsed (τ = 0.93̇) cases with a single refractive index step at 2ω1. Frame (a) compares chirped pulses at a 40µm propagation distance from the non- shock region immediately outside of the shocked region; the upper curve (X) is for the negative step, the lower (Y) for the positive step. Frame (b) shows the correlation in the region where LC . LSPM between energy conservation failure (logarithmic right hand scale, dots) and the LDD detected shocking distance (left hand scale, solid line). The dotted line shows the LDD results for a CW field (τ = ∞). distinctly different characteristics according to the sign of the dispersion. Narrow spikes are visible in the anoma- lous case, whereas profiles with more rounded maxima occur for normal dispersion. A possible interpretation is suggested by fig. 7, where we saw that anomalous dis- persion tends to create regions of higher gradient. The shocking region in fig. 10 has a similar outline to that of fig. 9(b) except that it is slightly narrower, as expected from the coherence length discussion above. However, the shocking region in fig. 11 is much nar- rower, especially given the change in the ∆n scale. This is because the multi-stepped nature of the refractive in- dex leads to a correspondingly large range of coherence lengths, with shorter ones corresponding to those span- ning several steps. Since the shocking region in the multi- step case has reduced in size by a factor of two or three, a reasonable inference might be that the dominant co- herence length results from interaction over two or three refractive index steps. In realistic media, the refractive index will vary smoothly with frequency, and the group velocity can dif- ferent from the phase velocity. We can approximate this situation most simply using a refractive index gradient −0.1 −0.05 0 0.05 0.1 ∆ n3,5 −0.1 −0.05 0 0.05 0.1 FIG. 10: Carrier shocking for CW and pulsed (τ = 0.93̇) cases with a single refractive index step at 4ω1; showing the correlation in the region where LC . LSPM between energy conservation failure (logarithmic right hand scale, dots) and the LDD detected shocking distance (left hand scale, solid line). The dotted line shows the LDD results for a CW field (τ = ∞). −0.05 0 0.05 ∆ n1,3,5... −0.05 0 0.05 FIG. 11: Carrier shocking for CW and pulsed (τ = 0.93̇) cases with multiple refractive index steps; showing the correlation in the region where LC . LSPM between energy conserva- tion failure (logarithmic right hand scale, dots) and the LDD detected shocking distance (left hand scale, solid line). Note the narrower range of ∆n as compared to the previous two graphs. The dotted line shows the LDD results for a CW field (τ = ∞). rather than a series of steps. The results using the LDD method in this case can be seen on fig. 12, where now we also vary the strength of the nonlinearity. As in the previous cases, we see a well defined shocking regime that is asymmetric about the non-dispersive case. Detailed examination shows that the curves in fig. 12 exhibit a marked similarity, and can be brought into near perfect coincidence by applying the scaling χ(3) → χ(3)/m, L → mL, and ∆n → ∆n/m. We also get a comparable similarity for each of figs. 9,10,11, when sim- ulations at nonlinear strengths of χ(3)E20 = 0.01 and 0.04 are added to the results. This demonstrates that the character of the shocking is dominated by the sign of inter-harmonic phase velocity differences, not by the local group velocity dispersion at each harmonic. Thus anoma- lous (normal) dispersion is primarily interesting because 7 Kinsler-RTN-2007 mailto:[email protected] CSHOCK-II [email protected] it gives a negative (positive) refractive index shift be- tween successive harmonics. We leave the complicated (and far more subtle) effects of group velocity differences or dispersion for later work. FIG. 12: Shocking distance vs mismatch for weak refractive index gradients δ, and a range of nonlinearities. This shows an abrupt cut-off at positive ∆n, but a relatively gradual one at negative ∆n. The initial pulses are identical to those in fig. 9. No CW results are shown, since the dispersion experienced by the field would be identical to the multi-stepped case shown in fig. 11. An important feature of all the results is the pro- nounced asymmetry, with shocking persisting much fur- ther into the anomalous dispersion regime; we have even seen it for the case of a weakly parabolic refractive index. We can quantify this asymmetry by considering lin- ear and nonlinear contributions to the phase matching of the harmonics (i.e. linear and nonlinear refractive index shifts), as described at the beginning of section V. In the simple single step case shown on fig. 9(b), the 1st–3rd harmonic phase shift will dominate, because it applies to the two most intense spectral components. We can calcu- late the SPM-induced refractive index shift between the fundamental and third harmonic with eqns. (18) and (19) to be ∆nNL:1−∆nNL:3 = −0.011, since χ (3)E2 = 0.02 at the peak of the carrier oscillations, the refractive index is n20 = 2, and 2n2E 1 = 2 × 3χ (3)E2/8n0. This is roughly comparable to the offset of the shocking region, which is centred at about ∆n1,3 ≃ −0.014. We cannot ex- pect perfect agreement, since the calculations ignore the role of higher harmonic generation and depletion of the fundamental. Whilst a simple calculation is reasonably successful in this single-step case, it cannot be applied for a realistic medium – or indeed to the situation shown in fig. 12. There, the effects of dispersion, higher har- monic generation, and nonlinear refractive index shifts are inextricably intertwined. To summarize, we have demonstrated that shocking is strongly dependent on the interplay between LC and LSPM , with shorter LC ’s (increasing ∆n’s) decreasing the likelihood of shocking. We have also deduced the reasons for the strong asymmetry of the shocking region. VII. APPLICATIONS AND EXPERIMENTS In sections I and IV, we discussed how imminent car- rier shocking might be recognised computationally. In considering whether shocking might be detectable exper- imentally, we must now decide how it might manifest itself in the laboratory. A mathematical discontinuity is clearly not a physical possibility, even if Rosen [2] did manage to accommodate it theoretically, albiet at the expense of energy conservation. In practice, the increas- ing field gradients (and spectral broadening) that precede a shock will inevitably engender new physical processes that will limit the steepening. Indeed, we have already seen this happening in the previous two sections, where dispersion has been seen to frustrate the self-steepening process; the next barrier would be the time-scale of the nonlinear response. FIG. 13: Numerically predicted shocking distances Lshock in fused silica. These are the MOC predicted distances for the waveform, assuming the dispersion was (abruptly) ne- glected, the shortest distance shown is about 1.9µm. The damage threshold for fused silica was taken to be Pthreshold = 50TW/cm2. Although there is no question of a mathematical dis- continuity being observed in an experiment, the recent advances in the measurment of optical pulse profiles (see e.g. [13]), suggest that it might be possible to observe carrier steepening. Indeed, it has already been pre- dicted [4] that noticeable steepening effects could occur for pulses propagating in fused silica. Unfortunately, in our own simulations of this process, the LDD shock de- tection was not triggered at any realistic pulse intensity. Although visible steeping did occur, at no point was a seriously distorted waveform approached, and the effects would have been milder if we had included the finite re- sponse time of the nonlinearity. The results presented in fig. 13 show the steepest gradient recorded as a func- tion of distance and pulse intensity for these simulations. The third harmonic coherence length for these parame- ters is about 7µm, and we can attribute the regular vari- ation with distance seen in the figure to the third har- monic component aligning with successive oscillations of the fundamental as the waves move across each other. 8 Kinsler-RTN-2007 mailto:[email protected] CSHOCK-II [email protected] The key reason why incipient shocking was not detected is because the pulse frequency lies in a region of weak normal dispersion, whereas we have shown in section V that a region of weak anomalous dispersion (assuming it could be achieved) would be favourable. The current interest in media with tailored dispersion characteristics [14, 15, 16] raises the interesting question of whether it might be possible to engineer a material that maximize self-steepening. A major stumbling block would be the need for control over many harmonic orders; however, for a narrow-band pulse, only the dispersion characteristics close to the harmonics would be relevant, which might perhaps make the technical challenge less formidable. The recognition that dispersion control can enhance (or reduce) carrier self-steepening suggests other appli- cations if we widen our horizons to encompass the more general idea of carrier shaping. In this case, we would exploit both nonlinearity and dispersion control to opti- mize the shape of the carrier oscillations for a particular experiment. Applications such as high harmonic genera- tion (e.g. [17]) might well benefit from suitably designed carrier wave modulation. VIII. CONCLUSIONS In this paper we have investigated carrier shock for- mation, developed criteria for detecting its onset in nu- merical simulations, and shown how it is influenced by a range of parameters, particularly dispersion. We have also obtained remarkable agreement between numerical simulations and theoretical predictions of the shocking distance in the dispersionless limit, and shown that the process is sensitive to both CEP and pulse duration. Although we have confirmed that shocking occurs in a narrow parameter range, this is far from being the whole picture. In particular, there is a distinct asymmetry be- tween the anomalous and normal dispersion regimes. The former leads to shocking signatures such as the appear- ance of narrow spikes as the higher harmonics interfere on the steepening part of the pulse profile. In contrast, normal dispersion creates no such features, and the pulse profiles have a rather blunt appearance. The asymmetry arises from the effect of the nonlinear refractive index on the dispersion induced phase mismatch. The conclusion to be drawn from our results is clear: if they could be engineered, materials with wide regions of weakly anomalous dispersion are much better candidates for generating steep, shock-like field profiles than those (such as silica) with a weak normal dispersion. Detecting incipient shock formation in materials like fused silica is likely to be a near impossible task, given the constraints imposed by their damage thresholds. [1] F. DeMartini, C.H. Townes, T.K. Gustafson, P.L. Kelly, Phys. Rev. 164, 312 (1967). [2] G. Rosen, Phys. Rev. 139, A539 (1965). [3] R.G. Flesch, A. Pushkarev, and J.V. Moloney, Phys. Rev. Lett. 76, 2488 (1996). [4] L. Gilles, J.V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999). [5] S.Trillo, G. Millot, C. Conti, ECLEO 2006, paper EB1- 05-Mon. [6] A. M. Kamchatnov, R. A. Kraenkel, B. A. Umarov, Physi. Rev. E66, 036609 (2002). [7] J.C.A. Tyrrell, P. Kinsler, G.H.C. New, J.Mod.Opt. 52, 973 (2005). [8] B. Fornberg, “A Practical Guide to Pseudospectral Meth- ods”, Cambridge University Press (1996). [9] L. Gilles, S.C. Hagness, and L. Vázquez, J. Comp.Phys. 161 379 (2000). [10] P. Kinsler, S.B.P. Radnor, G.H.C. New, Phys. Rev. A. 72, 063807 (2005). [11] “Numerical Recipes in C”, W.H. Press et al., (CUP, 1992). [12] H.-G.Pagendarm, B.Seitz, in “Scientific Visualization - Advanced Software Techniques” (EllisHorwood 1993), p159. [13] E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Bal- tuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, F. Krausz Sci- ence 305, 1267 (2004). [14] Photonic Bandgap Materials, (Ed: C. M. Soukoulis), (Kluwer, Dordrecht, The Netherlands 1996). [15] Photonic Crystals, (Eds: J. D. Joannopoulos, R. D. Meade, J. N. Winn), (Princeton Academic Press, Prince- ton, NJ 1995). [16] J.K. Ranka, R.S. Windeler, A.J. Stentz, Opt. Lett. 25, 25-27 (2000). [17] I.P. Christov, M.M. Murnane, H.C. Kapteyn, Phys. Rev. Lett. 78, 1251 (1997). 9 Kinsler-RTN-2007 mailto:[email protected]

0704.1213

Spin-polarized transport through weakly coupled double quantum dots in the Coulomb-blockade regime

Spin-polarized transport through weakly coupled double quantum dots in the Coulomb-blockade regime Ireneusz Weymann1, ∗ Department of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland (Dated: October 29, 2018) We analyze cotunneling transport through two quantum dots in series weakly coupled to external ferromagnetic leads. In the Coulomb blockade regime the electric current flows due to third-order tunneling, while the second-order single-barrier processes have indirect impact on the current by changing the occupation probabilities of the double dot system. We predict a zero-bias maximum in the differential conductance, whose magnitude is conditioned by the value of the inter-dot Coulomb interaction. This maximum is present in both magnetic configurations of the system and results from asymmetry in cotunneling through different virtual states. Furthermore, we show that tunnel magnetoresistance exhibits a distinctively different behavior depending on temperature, being rather independent of the value of inter-dot correlation. Moreover, we find negative TMR in some range of the bias voltage. PACS numbers: 72.25.Mk, 73.63.Kv, 85.75.-d, 73.23.Hk I. INTRODUCTION Transport properties of double quantum dots (DQDs) have recently attracted much interest.1,2,3,4,5,6,7,8,9,10,11,12,13 This is because these structures are ideal systems to study the fundamental interactions between electrons and spins.14,15,16,17,18 Double quantum dots exhibit a variety of interesting effects, such as for example current rectification due to the Pauli spin blockade,1,10,11,12,13 negative differential conductance,3 formation of molecular states,7 spin pumping,8,9 etc. Furthermore, double quantum dots are also being considered for future applications in quantum computing.19 Experimentally, such systems may be realized for example in lateral and vertical semiconduc- tor quantum dots.1,3,5,20 Another implementation of DQDs are single wall metallic carbon nanotubes with top gate electrodes, which enable changing of charge on each dot separately, as well as the intrinsic DQD parameters.7,21,22,23 The goal of this paper is to analyze transport prop- erties of DQDs weakly coupled to ferromagnetic leads in the Coulomb blockade regime. When the leads are ferromagnetic, transport strongly depends on the mag- netic configuration of the system, giving rise to tunnel magnetoresistance (TMR), spin accumulation, exchange field, etc.24,25,26,27,28,29,30,31,32 In the Coulomb blockade regime the electric current flows due to higher-order tun- neling processes (cotunneling), while the first-order tun- neling processes (sequential tunneling) are exponentially suppressed.31,33,34,35 The problem of spin-polarized co- tunneling has been so far addressed mainly in the case of single quantum dots.31,32,36,37,38,39,40,41 For example, it was shown that tunnel magnetoresistance exhibits dis- tinctively different behavior depending on the number of electrons on the dot.31 Moreover, the zero-bias anomaly was found in the differential conductance when magnetic moments of the leads form antiparallel configuration.36 Another interesting behavior was predicted for quantum dots coupled to ferromagnetic leads with non-collinear alignment of magnetizations – the exchange field was found to increase the differential conductance for certain non-collinear configurations, as compared to the parallel one.32,42 On the other hand, it was shown experimentally for nonmagnetic systems that the conductance of quan- tum dots in the cotunneling regime may serve as a handle to determine the spectroscopic g-factor.43 In the case of double quantum dots considered in this paper, in the Coulomb blockade regime the electric cur- rent flows due to third-order tunneling processes, while the single-barrier second-order processes together with third-order processes determine the double dot occupa- tion probabilities. Assuming that double quantum dot is occupied by two electrons in equilibrium, one on each dot, we calculate the differential conductance G and tun- nel magnetoresistance TMR. We show that differential conductance exhibits a maximum at the zero bias. We further distinguish two different mechanisms leading to this new zero-bias anomaly. The first one is an asym- metry in cotunneling through different virtual states of the DQD system, which leads to an enhancement of G at zero bias. Such asymmetry is induced by a finite value of the inter-dot Coulomb interaction. This mechanism is rather independent of magnetic configuration of the system. The second mechanism leading to the zero-bias maximum in differential conductance is the interplay be- tween spin accumulation and third-order tunneling pro- cesses carrying the current. This mechanism does depend on the magnetic configuration of the system and, as we show in the sequel, is found to be more efficient in the antiparallel configuration. We also analyze the behavior of TMR and show that the TMR exhibits a maximum at zero bias, which strongly depends on the temperature. Furthermore, the TMR may become negative in some range of the bias voltage. Finally, we note that there are several experimental re- alizations of single quantum dots attached to ferromag- http://arxiv.org/abs/0704.1213v1 netic leads,44,45,46,47,48,49,50,51,52,53,54 while experimental data on spin-polarized transport through double quan- tum dots is lacking. We believe that the results presented in this paper will be of assistance in discussing future ex- periments. II. MODEL AND METHOD The schematic of a double quantum dot coupled to fer- romagnetic leads is shown in Fig. 1. It is assumed that the magnetizations of the leads are oriented collinearly, so that the system can be either in the parallel or antipar- allel magnetic configuration. The Hamiltonian Ĥ of the DQD system is given by, Ĥ = ĤL+ĤR+ĤDQD+ĤT. The first two terms describe noninteracting itinerant electrons in the leads, Ĥj = εjkσc cjkσ for the left (j = L) and right (j = R) lead, where εjkσ is the energy of an electron with the wave vector k and spin σ in the lead j, and c (cjkσ) denotes the respective creation (anni- hilation) operator. The double dot is described by the Hamiltonian ĤDQD = j=L,R εjnjσ + j=L,R Ujnj↑nj↓ + U ′ nLσnRσ′ , (1) with njσ = d jσdjσ , where d jσ (djσ) is the creation (an- nihilation) operator of an electron with spin σ in the left (j = L) or right (j = R) quantum dot, and εj is the cor- responding single-particle energy. The Coulomb interac- tion on the left (right) dot is described by UL (UR). The last part of ĤDQD corresponds to the inter-dot Coulomb correlation, whose strength is given by U ′. As we are in- terested in the low bias voltage regime where the system is in the Coulomb blockade, it is justifiable to assume that the energy level of each dot is independent of the bias voltage. For the sake of clarity of further discussion we also assume εL = εR ≡ ε and UL = UR ≡ U . We note that in a general case, the exchange interac- tion between spins in the two dots may lead to the forma- tion of singlet and triplet states.7 However, this exchange interaction was found to be rather small as compared to the other energy scales,1 and thus, following previous the- oretical works,6,55 we will neglect it. Tunneling processes between the two dots and elec- trodes are described by the Hamiltonian, ĤT = j=L,R djσ + h.c. LσdRσ + h.c. where tj denotes the tunnel matrix elements between the jth lead and the jth dot, and t describes the hopping be- tween the two quantum dots. Coupling of the jth dot to the jth lead can be expressed as Γσj = 2π|tj | 2ρσj , with ρσj being the spin-dependent density of states of the cor- responding lead. With the definition of the spin polar- LL U+ε RR U+ε FIG. 1: (color online) Schematic of a double quantum dot coupled to ferromagnetic leads. The magnetic moments of the leads can form either parallel or antiparallel configuration. The system is symmetrically biased. ization of lead j, pj = (ρ j − ρ j )/(ρ j + ρ j ), the cou- pling can be expressed as, Γ j = Γj(1 ± pj), with Γj = (Γ j + Γ j )/2. Here, Γ j and Γ j describe the coupling of the jth dot to the spin-majority and spin- minority electron bands of lead j, respectively. As re- ported in Ref. 43, typical values of the coupling strength are of the order of tens of µeV. In the following, we as- sume symmetric couplings, ΓL = ΓR ≡ Γ/2, and equal spin polarizations of the leads, pL = pR ≡ p. In this paper we analyze spin-dependent transport through double quantum dot in the case of the Coulomb blockade regime. We assume that in equilibrium each dot is singly occupied, so that there are two electrons in the DQD system. This transport regime can be real- ized for example in lateral quantum dots3,5,20 or in single wall carbon nanotubes with top gate electrodes.7,21,22,23 In such devices by changing the respective gate voltages one can tune the charge on each dot separately and also change the strength of the coupling t between the two dots. Furthermore, we also note that in DQDs the on- level interaction U is usually larger than the inter-dot interaction U ′. In the case where the DQD is doubly occupied and U > U ′, the system can be in four differ- ent states |χ〉, namely | ↑↑〉 = | ↑〉| ↑〉, | ↑↓〉 = | ↑〉| ↓〉, | ↓↑〉 = | ↓〉| ↑〉, | ↓↓〉 = | ↓〉| ↓〉, where the first (sec- ond) ket corresponds to the left (right) dot. The occu- pation of the other two-particle states |d0〉 = | ↑↓〉|0〉 and |0d〉 = |0〉| ↑↓〉 is suppressed due to large on-level interaction on the dots. In the Coulomb blockade the charge fluctuations are suppressed and the system is in a well-defined charge state. As a consequence, all tunneling processes leading to a change of the DQD charge state are exponentially suppressed. The current can thus flow due to higher- order tunneling processes (cotunneling) through virtual states in the double quantum dot.31,33,34,35 The lowest- order processes which give a dominant contribution to electric current flowing through the DQD structure in the case of Coulomb blockade are the third-order tunneling processes. Generally, the rate for an nth-order (n ≥ 2) tunneling from lead j to lead j′ associated with a change of the double dot state from χ into χ′ is given by35 jj′ = v1,v2,...,vn−1 |HT|Φv1〉〈Φv1 |HT|Φv2〉 × · · · × 〈Φvn−1 |HT|Φ (εi − εv1)(εi − εv2)× · · · × (εi − εvn−1) δ(εi − εf ) , (3) where εi and εf denote the energies of initial and final states and |Φ j 〉 is the state of the system with an electron in the lead j and the double dot in state |χ〉, while |Φvn〉 denotes a virtual state of the DQD system and εvn its en- ergy. From the above expression one can determine the third-order (n = 3) tunneling rates that give the main contribution to electric current. We note that there are also tunneling events that do not affect the DQD charge state but can have an influence on transport. These are the second-order processes which take place through a single tunnel barrier, either left or right. Such single- barrier processes contribute to the electric current in an indirect way, namely by changing the occupation prob- abilities and this way the current. The rate of single- barrier second-order cotunneling is given by Eq. (3) for n = 2 and j = j′. It is also worth noting that among dif- ferent higher-order tunneling events one can distinguish the elastic (non-spin-flip) and inelastic (spin-flip) pro- cesses. The former ones change the state of the double dot (χ 6= χ′), while the latter ones do not (χ = χ′). L R L R L RL Rσ FIG. 2: (color online) Examples of possible tunneling pro- cesses through double quantum dot in the Coulomb blockade regime. The third-order process (a) from left to right lead, where | ↑↓〉 is the initial state (1) and | ↑↑〉 is the final state (7), takes place via five virtual states (2)-(6). This process contributes directly to the current flowing through the sys- tem. The second-order process through the left barrier (b), with | ↑ σ〉 (| ↓ σ〉) being the initial (1) [finite (4)] state, where σ =↑, ↓, affects the occupations of the double quantum dot. This process takes place via two virtual states (2)-(3). Examples of possible processes in the case of the Coulomb blockade regime are shown in Fig. 2. The upper part of the figure presents a third-order process from the left to right lead which contributes to electric current. This is an inelastic process which leads to a change of the double dot state from | ↓↑〉 to | ↑↑〉. It takes place through five virtual states, as sketched in Fig. 2a. On the other hand, the bottom part of Fig. 2 displays a single- barrier second-order process, occurring via two virtual states. This process does not contribute to electric cur- rent but affects the DQD occupation probabilities. The process shown in Fig. 2b takes place through the left barrier and changes the double dot state from | ↑ σ〉 into | ↓ σ〉, with σ =↑, ↓. To make the discussion more trans- parent, in Appendix we present the explicit formulas for the rates corresponding to the two processes shown in Fig. 2. By calculating all the second-order single-barrier and third-order rates one can determine the occupation prob- abilities from the following stationary master equation jj + γ (3)χ→χ jj + γ , (4) where Pχ denotes the probability for the double dot to be in state |χ〉. The occupations are fully determined with the aid of the normalization condition, Pχ = 1. The third-order current flowing through the system from the left to right lead is then given by I = e LR − γ (3)χ→χ . (5) We note that generally the use of the master equation approach may lead to wrong results in the regime where the level renormalization effects or the effects due to ex- change field become important, i.e. close to resonance or for noncollinear magnetic configurations.29,30,31,32 How- ever, in the following we consider only the case of deep Coulomb blockade and collinear configurations, which justifies the employed approach.36,39,40 III. RESULTS AND DISCUSSION We first present the results for cotunneling through double quantum dots coupled to nonmagnetic leads, p = 0. Next, we analyze the case when the leads are fer- romagnetic (p > 0) and the system can be either in parallel or antiparallel magnetic configuration. The cur- rent flowing through the system depends then on the magnetic configuration giving rise to tunnel magnetore- sistance. The TMR is qualitatively defined as24,25,28 TMR = IP/IAP−1, where IP (IAP) is the current flowing through the system in the parallel (antiparallel) magnetic configuration. A. DQD coupled to nonmagnetic leads The differential conductance G of the DQD coupled to nonmagnetic leads as a function of the bias voltage for several values of the inter-dot interaction parameter U ′ is shown in Fig. 3. In the case of negligible U ′, the differen- tial conductance exhibits a smooth parabolic dependence on the bias voltage. However, for a finite inter-dot cor- relation, there is a maximum in G at zero bias. As one can see in the figure, the magnitude of this maximum in- creases with increasing U ′. Furthermore, when increasing U ′, the minimum of G at V = 0 splits into two minima, separated by the zero-bias peak. In order to understand the mechanism leading to such behavior, we note that in the spinless case, p = 0, the occupation probabilities do not depend on the applied voltage and are equal to 1/4, i.e. each of the four DQD states, | ↑↑〉, | ↑↓〉, | ↓↑〉, | ↓↓〉, is equally occupied. Fur- thermore, the single-barrier second-order processes which provide a channel for spin relaxation in the dots do not have any influence on transport either. As a consequence, the zero-bias maximum results from intrinsic dependence of third-order tunneling rates on the value of the inter- dot correlation parameter U ′. The electric current flows through the DQD system due to third-order processes which involve correlated tunneling through virtual states of the system. More specifically, these virtual states include the single-particle states, |σ0〉 and |0σ〉, the two-particle states, |0d〉 and |d0〉, and the three-particle states, |σd〉 and |dσ〉. In equilibrium, the energy of these virtual states is respec- tively given by ε1 = ε, ε2 = 2ε+U , and ε3 = 3ε+U+2U On the other hand, the energy of the initial state is εi = 2ε+U ′. Consequently, the resolvents that determine the rates, see Eq. (3), are given by (εi − ε1) −1 = (ε+U ′)−1, (εi−ε2) −1 = (U ′−U)−1, (εi−ε3) −1 = (−ε−U −U ′)−1. The third-order tunneling processes take place via two consecutive virtual states. Thus, the rate is propor- tional to the product of two resolvents, depending on virtual states being involved in a process. Generally, one can distinguish four different contributions – the first one involves two single-particle states, (ε + U ′)−2, the second one involves one single-particle and one two- -6 -4 -2 0 2 4 6 U' = 0 U' = Γ U' = 2Γ U' = 3Γ FIG. 3: (color online) Differential conductance as a function of the bias voltage for different inter-dot interaction parame- ter U ′, as indicated in the figure. The other parameters are kBT = 0.1Γ, ε = −10Γ, U = 20Γ, t = Γ, and p = 0. particle state, [(ε+U ′)(U ′−U)]−1, the third contribution comes from one two-particle and one three-particle state, [(U − U ′)(ε + U + U ′)]−1, and the last one involves two three-particle states, (ε + U + U ′)−2. After a crude es- timation, one can see from the above formulas that by increasing U ′, the contribution coming from the first two resolvents is increased, the third one is roughly constant, while that of the last resolvent is decreased. This gener- ally leads to an asymmetry of cotunneling through dif- ferent virtual states. Such asymmetry gives rise to an enhancement of the conductance through the system by increasing the rate of processes occurring via one-particle and two-particle DQD states. As a result, with increas- ing U ′, a maximum develops in the differential conduc- tance at the zero bias, see Fig. 3. On the other hand, for a given value of U ′, the differential conductance de- creases with increasing the bias voltage and reaches a minimum at |eV | ≈ 2U ′. At this bias voltage the effect of finite inter-dot Coulomb interaction is compensated by the transport voltage, and the differential conductance reaches minimum, which is present on both sides of the zero-bias anomaly. Moreover, although the position of the two minima depends on the inter-dot correlation, its value is rather independent of U ′, see Fig. 3. When considering the case of linear response, zero tem- perature and negligible inter-dot correlation, the mini- mum value of the differential conductance can be ap- proximated by the following formula e2t2Γ2 ε(ε+ U) (ε+ U)2 . (6) For the parameters assumed to calculate Fig. 3, from the above formula one finds, G = 0.45× 10−3 e2/h, which is in good agreement with numerical results. -6 -4 -2 0 2 4 6 1.0 b U' = 0 U' = Γ U' = 2Γ U' = 3Γ FIG. 4: (color online) Bias dependence of the differential con- ductance in the parallel (a) and antiparallel (b) magnetic con- figurations for different inter-dot interaction parameter U ′, as indicated. The other parameters are kBT = 0.1Γ, ε = −10Γ, U = 20Γ, t = Γ, and p = 0.5. B. DQD coupled to ferromagnetic leads If the leads are ferromagnetic (p 6= 0), the single- barrier second-order processes start to influence trans- port by affecting the DQD occupation probabilities. Transport characteristics are then a result of the inter- play between processes driving the current and processes leading to spin relaxation in the dots. First, we note that the rate of single-barrier processes is proportional to tem- perature, while that of third-order processes depends on the applied bias voltage, see Eqs. (A1) and (A2). This will give rise to interesting phenomena, depending on the relative ratio of the second-order and third-order pro- cesses, as will be discussed in the following. In Fig. 4 we show the bias dependence of the differ- ential conductance for the parallel and antiparallel mag- netic configurations of the system for several values of the inter-dot interaction parameter U ′. First of all, it can be seen that the value of G at the zero bias increases with increasing the inter-dot correlation. This is a general fea- ture which is present in both magnetic configurations of the system and gives rise to the zero-bias maximum, see Fig. 4a and b. The mechanism leading to such behavior was already discussed in the nonmagnetic case, i.e. a fi- nite value of U ′ results in increased cotunneling through one-particle and two-particle virtual states, which in turn leads to an enhancement of the differential conductance at the zero bias. Another feature visible in the case of ferromagnetic leads is that even for negligible U ′ there is a small max- imum in G at the zero bias, irrespective of magnetic configuration of the system. This maximum bears a re- semblance to the zero-bias anomaly found in the case of single quantum dots.36 However, in single quantum dots the maximum is present only in the antiparallel configu- ration, while in the case of double quantum dots, inter- estingly, the zero-bias peak is present in both magnetic configurations, see Fig. 4a and b. In order to understand this behavior we note that when there is a finite bias voltage applied to the system, a nonequilibrium spin ac- cumulation can build up in the DQD. More precisely, for positive bias voltage in the parallel configuration one ob- serves unequal occupation of singlet states, P|↓↑〉 > P|↑↓〉, while triplets are roughly equally occupied (no spin ac- cumulation), P|↑↑〉 ≈ P|↓↓〉. On the other hand, in the antiparallel configuration there is unequal occupation of triplet states (spin accumulation), P|↓↓〉 > P|↑↑〉, whereas singlets are equally occupied, P|↓↑〉 ≈ P|↑↓〉. It is fur- ther interesting to realize that for positive bias voltage main contribution to the current comes from third-order tunneling processes having the initial state | ↑↓〉 for the parallel and | ↑↑〉 for the antiparallel magnetic configura- tion. Thus, with increasing the bias voltage (V > 0), the contribution coming from those processes is decreased, leading to a decreased conductance. As a consequence, one observes a maximum at the zero bias even in the case of U ′ = 0, see Fig. 4. The zero-bias maximum in differential conductance is therefore a result of superposition of two different ef- fects. The first one concerns the asymmetry of cotun- neling through virtual states, which is induced by a fi- nite value of the inter-dot Coulomb interaction. Whereas the second one is associated with unequal occupation of the corresponding DQD states, which results from spin- dependent tunneling rates. When the DQD is coupled to ferromagnetic leads, an important role is played by the single-barrier second- order processes – they do not contribute to the current, but lead to the spin relaxation in the double dot system. In order to gain more intuitive understanding of the dis- cussed phenomena, in the following we present a crude quantitative analysis of the processes determining trans- port behavior. When considering the low temperature limit and assuming U = −2ε, U ′ = 0, the rate of single- barrier second-order processes can be approximated by39 |σχ〉→|σ̄χ〉 4kBTΓ . (7) On the other hand, we note that generally the fastest third-order processes are the ones leading to the change of the dot state from |σσ̄〉 into |σ̄σ〉. With the same assumptions as made above, one can approximate the rate of such processes by the following formula |σσ̄〉→|σ̄σ〉 jj′ ≈ 16|eV |t2Γ2 . (8) -6 -4 -2 0 2 4 6 0.6 b T = 0.2Γ T = 0.1Γ T = 0.05Γ T = 0.02Γ T = 0.01Γ FIG. 5: (color online) Bias dependence of the differential con- ductance in the parallel (a) and antiparallel (b) magnetic con- figurations for different temperatures and for U ′ = 2Γ. The other parameters are the same as in Fig. 4. The above expressions show explicitly that the relative ratio of both processes depends on the internal system parameters as well as the temperature and applied bias voltage. Furthermore, one can now roughly estimate the bias voltage at which the corresponding second-order and third-order processes become comparable, it is given by |eV | ≈ . (9) This formula will be helpful in discussing the temperature dependence of transport characteristics. The influence of temperature on the bias dependence of differential conductance in both magnetic configurations is shown in Fig. 5. One can see that with increasing thermal energy, the width of the zero-bias peak is in- creased, while the maximum value of G for V = 0 stays rather unchanged. This is due to the fact that by raising the temperature, one increases the role of single-barrier second-order processes, see Eq. (7), giving rise to faster spin relaxation. Spin relaxation in turn leads to a de- crease in the spin accumulation induced in the system.39 Therefore, the temperature effects on the differential con- ductance are more visible in the antiparallel configuration than in the parallel one. By decreasing T , the relative role of second-order processes is decreased, which leads to larger spin accumulation, P|↓↓〉 > P|↑↑〉. This in turn gives rise to an increased and more robust drop of the differential conductance with the bias voltage, see for ex- ample the curves for kBT = 0.2Γ and kBT = 0.01Γ in -6 -4 -2 0 2 4 6 -0.04 U' = 0 U' = Γ U' = 2Γ U' = 3Γ FIG. 6: (color online) Bias dependence of the TMR for differ- ent inter-dot interaction parameter U ′ and for kBT = 0.1Γ. The other parameters are the same as in Fig. 4. Fig. 5. As a consequence, with decreasing temperature, the value of the differential conductance at the minimum is decreased and the width of the zero-bias peak becomes smaller – the two minima in G appear at smaller bias voltage. This is due to the fact that the relative ra- tio of the second-order and third-order processes changes with changing T and, consequently, the bias voltage at which the rates of these two processes are comparable is changed, see Eq. (9). The dependence of the differential conductance on temperature in the parallel configuration is less pronounced than in the antiparallel configuration because for the parallel configuration the single-barrier spin-flip processes only slightly affect the DQD occupa- tions. This results from the fact that in the parallel configuration there is a left-right symmetry between the couplings to the spin-majority and spin-minority electron subbands.36 We also note that in the spinless case discussed in pre- vious subsection the single-barrier second-order processes do not affect transport in any way, and the occupations of all DQD states are equal. Therefore, the differential conductance only slightly depends on temperature. In Fig. 6 we present the TMR as a function of the bias voltage for several values of the inter-dot correla- tion parameter. First of all, it can be seen that for low bias voltages tunnel magnetoresistance is only slightly af- fected by the inter-dot interaction. This is due to the fact that the asymmetry in tunneling through virtual states induced by finite value of U ′ changes transport charac- teristics in both magnetic configurations in a similar way, see Fig. 4. As a consequence, the TMR, which reflects the difference between the parallel and antiparallel mag- netic configuration, is roughly independent of the value of inter-dot correlation. Another interesting feature visible in Fig. 6 is the sign change of the TMR – with increasing the bias voltage, tunnel magnetoresistance decreases from a maximum at the zero bias to a minimum, at which TMR changes sign and becomes negative. At this bias voltage conductance in the parallel configuration is smaller than in the an- -6 -4 -2 0 2 4 6 T = 0.2Γ T = 0.1Γ T = 0.05Γ T = 0.02Γ T = 0.01Γ FIG. 7: (color online) Tunnel magnetoresistance as a function of the bias voltage for different temperatures and for U ′ = 2Γ. The other parameters are the same as in Fig. 4. tiparallel configuration. This seemingly counterintuitive fact can be understood when one takes into account the effect of second-order processes giving rise to spin re- laxation. As already mentioned, in the parallel configu- ration one finds, P|↑↓〉 6= P|↓↑〉, while in the antiparallel configuration one has, P|↑↑〉 6= P|↓↓〉. Spin relaxation pro- cesses decrease the spin accumulation in the antiparallel configuration, which leads to an enhancement of the dif- ferential conductance, see Fig. 5b. On the other hand, in the parallel configuration the DQD occupations only slightly depend on second-order processes. As a conse- quence, if the spin relaxation processes are sufficiently fast, γ(2) |σχ〉→|σ̄χ〉 jj & γ (3)|σσ̄〉→|σ̄σ〉 jj′ , one observes negative TMR effect. In Fig. 7 we display the TMR effect as a function of the bias voltage for different temperatures. First of all, one can see that TMR exhibits a nontrivial depen- dence on temperature. This is because by changing T , one effectively changes the amount of processes leading to spin relaxation which affect spin accumulation and, thus, conductance in the antiparallel configuration. For low temperatures, second-order processes are suppressed and TMR becomes positive in the whole range of the bias voltage with a minimum at the zero bias, see the curve for kBT = 0.01Γ in Fig. 7. On the other hand, for higher temperatures the rate of single-barrier second-order pro- cesses is increased, which gives rise to two minima in the TMR separated by the zero-bias maximum, see Figs. 6 and 7. Moreover, at these minima TMR changes sign and becomes negative. We note that the negative TMR was also observed in single quantum dots in the limit of fast spin relaxation in the dot.39 Finally, we present analytical formulas approximat- ing tunnel magnetoresistance in the most characteristic transport regimes. For |eV | ≫ kBT , the TMR can be expressed as TMR = 1− p2 53− 3p2(1 + 7p2 − p4) (5 + 3p2)(3 − p2)2 , (10) where we have assumed the symmetric Anderson model for each dot, U = −2ε, and U ′ = 0. This formula ap- proximates the TMR in the zero temperature limit, i.e. in the absence of second-order processes. On the other hand, the linear response TMR calculated with the same assumptions can be approximated by TMR = 1− p2 13 + 3p4 (9 + p2)(5 + 3p2) . (11) IV. CONCLUDING REMARKS We have considered cotunneling transport through double quantum dots in series weakly coupled to fer- romagnetic leads. In the Coulomb blockade regime the current flows through the system due to third-order tun- neling processes. We have also taken into account the single-barrier second-order processes which do not con- tribute to the current but affect the DQD occupation probabilities. We have shown that the differential conductance ex- hibits a maximum at the zero bias, irrespective of mag- netic configuration of the system. This anomalous be- havior results from the superposition of two different ef- fects. The first effect is associated with asymmetry of cotunneling through different virtual states which can be induced by the inter-dot Coulomb interaction. The second mechanism results from the interplay of single- barrier second-order processes leading to spin relaxation and the third-order tunneling processes contributing to the current. The first mechanism does not depend on the value of spin polarization of the leads, the second one, on the contrary, results from the spin dependency of tunneling rates. We have also analyzed the temperature dependence of transport characteristics. By changing thermal energy, one effectively changes the rate of the second-order pro- cesses, i.e. the amount of spin relaxation processes. We have shown that the width of the zero-bias maximum in the differential conductance increases with increasing temperature. This effect is most visible in the antiparal- lel configuration, which is due to the fact that in the an- tiparallel configuration spin relaxation decreases the spin accumulation induced in the DQD system, while occupa- tions in the parallel configuration only slightly depend on the spin relaxation. Furthermore, we have also shown that TMR exhibits a nontrivial dependence on temperature. For low tem- peratures, the TMR exhibits a minimum at the zero bias. However, for higher temperatures this minimum splits into two minima separated by a maximum at the zero bias. At the these minima tunnel magnetoresistance changes sign and becomes negative. Acknowledgments We acknowledge discussions with J. Barnaś. This work, as part of the European Science Foundation EU- ROCORES Programme SPINTRA, was supported by funds from the Ministry of Science and Higher Educa- tion as a research project in years 2006-2009 and the EC Sixth Framework Programme, under Contract N. ERAS- CT-2003-980409, and the Foundation for Polish Science. APPENDIX A: EXAMPLES OF COTUNNELING RATES In the following we present the explicit formulas for the third-order and second-order tunneling rates correspond- ing to processes shown in Fig. 2a and b. To determine the rate γ(3) |↓↑〉→|↑↑〉 LR one needs to find the initial and final energies of the whole process, as well as the energies of the virtual states, as sketched in Fig. 2a. Then, by cal- culating the respective energy differences and plugging them into Eq. (3), one finds |↓↑〉→|↑↑〉 dωf+(ω)f−(ω + µL − µR) (ω + µL − ε− U − U ′)2 U − U ′ ω + µL − ε− U − U ′ ω + µL − ε− U ′ ,(A1) where f+ is the Fermi function and f− = 1 − f+. On the other hand, the single-barrier second-order rate for the process shown in Fig. 2b can be found in a similar way. This rate is given by |↑σ〉→|↓σ〉 dωf+(ω)f−(ω) ω + µL − ε− U − U ′ ω + µL − ε− U ′ . (A2) We note that in the simplest approximation38 for the Coulomb blockade regime one can pull out the resolvents in front of the integrals. Then, one arrives at the follow- ing integral, dωf+(ω)f−(ω+ ξ), where ξ = µL − µR or ξ = 0, correspondingly, which can be easily calculated.35 As a consequence, one can see that the rate of single- barrier second-order processes is proportional to temper- ature T , whereas that of the third-order processes de- pends on the bias voltage V . ∗ Electronic address: [email protected] 1 K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Science 297, 1313 (2002). 2 W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev. Mod. Phys. 75, 1 (2003). 3 H. W. Liu, T. Fujisawa, T. Hayashi, and Y. Hirayama, Phys. Rev. B 72, 161305 (2005). 4 V. N. Golovach and D. Loss, Phys. Rev. B 69, 245327 (2004). 5 D. T. McClure, L. DiCarlo, Y. Zhang, H.-A. Engel, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. 98, 056801 (2007). 6 B. Wunsch, M. Braun, J. König, and D. Pfannkuche, Phys. Rev. B 72, 205319 (2005). 7 M. R. Gräber, W. A. Coish, C. Hoffmann, M. Weiss, J. Furer, S. Oberholzer, D. Loss, and C. Schönenberger, Phys. Rev. B 74, 075427 (2006). 8 E. Cota, R. Aguado, and G. Platero, Phys. Rev. Lett. 94, 107202 (2005). 9 B. L. Hazelzet, M. R. Wegewijs, T. H. Stoof, and Yu. V. Nazarov, Phys. Rev. B 63, 165313 (2001). 10 J. Fransson, M. Rasander, Phys. Rev. B 73, 205333 (2006). 11 J. Fransson, Nanotechnology 17, 5344 (2006). 12 J. Inarrea, G. Platero and A. H. MacDonald, cond-mat/0609323 (unpublished). 13 B. Muralidharan and S. Datta, cond-mat/0702161 (unpub- lished). 14 S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka, and D. M. Treger, Science 294, 1488 (2001). Semiconductor Spintronics and Quantum Computation, ed. by D.D. Awschalom, D. Loss, and N. Samarth (Springer, Berlin 2002). 16 S. Maekawa and T. Shinjo, Spin Dependent Transport in Magnetic Nanostructures (Taylor & Francis 2002). 17 I. Zutic, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 18 S. Maekawa, Concepts in Spin Electronics (Oxford 2006). 19 D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 20 A. C. Johnson, J. R. Petta, and C. M. Marcus, M. P. Han- son and A. C. Gossard, Phys. Rev. B 72, 165308 (2005). 21 H. I. Jorgensen, K. Grove-Rasmussen, J. R. Hauptmann, P. E. Lindelof, Appl. Phys. Lett. 89 232113 (2006). 22 S. Sapmaz, C. Meyer, P. Beliczynski, P. Jarillo-Herrero, Leo P. Kouwenhoven, Nano Lett. 6 (7), 1350 (2006). 23 M. R. Gräber, M. Weiss, C. Schönenberger, Semicond. Sci. Technol. 21, 64 (2006). 24 M. Julliere, Phys. Lett. A 54, 225 (1975). 25 J. Barnaś and A. Fert, Phys. Rev. Lett. 80, 1058 (1998). 26 S. Takahashi and S. Maekawa, Phys. Rev. Lett. 80, 1758 (1998). 27 B. R. Bu lka, Phys. Rev. B 62, 1186 (2000). mailto:[email protected] http://arxiv.org/abs/cond-mat/0609323 http://arxiv.org/abs/cond-mat/0702161 28 W. Rudziński and J. Barnaś, Phys. Rev. B 64, 085318 (2001). 29 J. König and J. Martinek, Phys. Rev. Lett. 90, 166602 (2003). 30 M. Braun, J. König, J. Martinek, Phys. Rev. B 70, 195345 (2004). 31 I. Weymann, J. König, J. Martinek, J. Barnaś, and G. Schön, Phys. Rev. B 72, 115334 (2005). 32 I. Weymann, J. Barnaś, cond-mat/0611447 (to be pub- lished in Phys. Rev. B) 33 D. V. Averin and A. A. Odintsov, Phys. Lett. A 140, 251 (1989); D. V. Averin and Yu. V. Nazarov, Phys. Rev. Lett. 65, 2446 (1990). 34 K. Kang and B. I. Min, Phys. Rev. B 55, 15412 (1997). 35 D. V. Averin and Yu. V. Nazarov, in Single Charge Tunnel- ing, edited by H. Grabert and M. Devoret (Plenum, New York 1992). 36 I. Weymann, J. Barnaś, J. König, J. Martinek, and G. Schön, Phys. Rev. B 72, 113301 (2005). 37 S. Braig and P. W. Brouwer, Phys. Rev. B 71, 195324 (2005). 38 I. Weymann, J. Barnaś, Eur. Phys. J. B 46, 289 (2005). 39 I. Weymann, J. Barnaś, Phys. Rev. B 73, 205309 (2006). 40 I. Weymann, Europhys. Lett. 76, 1200 (2006). 41 F. M. Souza, J. C. Egues, and A. P. Jauho, cond-mat/0611336 (unpublished). 42 J. Fransson, Europhys. Lett. 70, 796 (2005). 43 A. Kogan, S. Amasha, D. Goldhaber-Gordon, G. Granger, M.A. Kastner, and H. Shtrikman, Phys. Rev. Lett. 93, 166602 (2004). 44 Y. Chye, M. E. White, E. Johnston-Halperin, B. D. Ger- ardot, D. D. Awschalom, and P. M. Petroff, Phys. Rev. B 66, 201301(R) (2002). 45 M. M. Deshmukh and D. C. Ralph, Phys. Rev. Lett. 89, 266803 (2002). 46 H. B. Heersche, Z. de Groot, J. A. Folk, L. P. Kouwen- hoven, and H. S. J. van der Zant, A. A. Houck, J. Labaziewicz, and I. L. Chuang, Phys. Rev. Lett. 96, 017205 (2006). 47 L. Y. Zhang, C. Y. Wang, Y. G. Wei, X. Y. Liu, and D. Davidović, Phys. Rev. B 72, 155445 (2005). 48 K. Tsukagoshi, B. W. Alphenaar, and H. Ago, Nature 401, 572 (1999). 49 B. Zhao, I. Mönch, H. Vinzelberg, T. Mühl, and C. M. Schneider, Appl. Phys. Lett. 80, 3144 (2002); J. Appl. Phys. 91, 7026 (2002). 50 A. Jensen, J. R. Hauptmann, J. Nygard, and P. E. Linde- lof, Phys. Rev. B 72, 035419 (2005). 51 S. Sahoo, T. Kontos, J. Furer, C. Hoffmann, M. Gräber, A. Cottet, and C. Schönenberger, Nature Physics 1, 102 (2005). 52 A. N. Pasupathy, R. C. Bialczak, J. Martinek, J. E. Grose, L. A. K. Donev, P. L. McEuen, and D. C. Ralph, Science 306, 86 (2004). 53 A. Bernand-Mantel, P. Seneor, N. Lidgi, M. Munoz, V. Cros, S. Fusil, K. Bouzehouane, C. Deranlot, A. Vaures, F. Petroff, and A. Fert, Appl. Phys. Lett. 89, 062502 (2006). 54 K. Hamaya, S. Masubuchi, M. Kawamura, and T. Machida, M. Jung, K. Shibata, and K. Hirakawa, T. Taniyama, S. Ishida and Y. Arakawa, Appl. Phys. Lett. 90, 053108 (2007). 55 J. Aghassi, A. Thielmann, M. H. Hettler, G. Schön, Appl. Phys. Lett. 89, 052101 (2006); Phys. Rev. B 73, 195323 (2006). http://arxiv.org/abs/cond-mat/0611447 http://arxiv.org/abs/cond-mat/0611336

0704.1214

Possible origin of Larson's lows

Possible origin of Larson’s lows A.W. Zaharow Ufa state petrotechnical university, Ufa, Russia∗ (Dated: April 10, 2007) It was found that approximately constant column densities of giant molecular clouds (Larson’s low) can be explained as cloud existence condition in external (galac- tic) gravitational field. This condition can be also applied to objects (clumps and cores) embedded into the cloud and its gravitational field. Derived existence condi- tion do not rely on any internal dynamic of a cloud and embedded objects. 1. Introduction Giant molecular clouds (GMCs) play a crucial role in the star formation process ([1],[2],[3],[4],[5]). GMCs are complex objects with masses ∼ 105−6M⊙, diameters ∼ 50pc and average densities nH2 ∼ 10 2cm−3 (e.g. [6]). GMCs are generally gravitationally bound and may contain several sites of star formations. Equilibrium of self-gravitating gas was the- oretically investigated in many works (e.g. [4],[9] and references in these papers). Internal structure of GMS is usually very complicated. The inhomogeneous structure of cloud could be described as a set of descrete clumps ([7]). These clumps themselves contain dense cores with densities n ∼ 103−5cm−3. The one point of view at this time is that this cloud structure is due to supersonic turbulence. Remarkably enough, the properties of cloud complexes are rather simply interrelated. Total masses, mean densities and average velosity dispersions vary with sizes (effective radii) roughly as M ∝ R2,ρ ∝ R−1, σ ∝ R1/2 (Larson’s lows) ([8]). In this paper I propose simple hypothesis to explain relation between masses and sizes of clumps and cores, embedded into clouds. ∗Electronic address: [email protected] http://arxiv.org/abs/0704.1214v1 mailto:[email protected] 2. Larson’s lows Let us briefly consider some of the most salient characteristics of GMCs summarized by Larson ([8]). See alsow ([2]). The first relation is the line width-size relation: molecular clouds are supersonically turbulent with line widths ∆v that increase as a power of size, ∆v ∝ Rp. Larson himself estimated that p ≃ 0.38. Subsequent work has distinguished between the relation valid for a collection of GMCs and that valid within individual GMC. Within low-mass cores, Caselli & Myers ([10]) found that the nonthermal velocity dispersion σnt ≃ 0.55R pc km s −1 , (2.1) which is near to relation σ ∝ R1/2 . (2.2) Larson’s second result was that GMCs and clumps within them are gravitationally bound. It implies that σ2 ∝ GM/R . (2.3) This relation is result of virial equilibrium. His third conclusion was that all GMCs have about the same column density N ≃ const . (2.4) Column density is defined as N ∝ M/R2 ∝ nR . (2.5) As Larson pointed out, only two of these conclusions are independent: any one of them can be derived from other two. Opposite to the second conclusion the first and the second Larson’s lows have no evident explanation. I think this explanation may be as follows. Embedded clumps and cores moves through the cloud in its gravitational field and are subject of turbulent motion and various accelerations. If their own gravitational field is not enough to hold on these objects they must rapidly decay. It is easy to derive necessary condition for confinement gaseous body with mass M and radius R in external gravitational field of cloud with mass Mcl and radius Rcl. The variation of self gravitational potential ∆φ must be greater then that of the cloud on size R. Bearing in mind that ∆φ ≃ GM/R (2.6) ∆φcl ≃ R GM/Rcl ≃ GMR/R cl (2.7) we get GM/R ≥ GMclR/R cl (2.8) or equivalently M/R2 ≥ Mcl/R cl . (2.9) Inequality (2.9) is the main result of this work. It is important for undestanding the third Larson’s low (2.4). Inequality (2.9) is strong only for very massive and compact objects within the cloud which are hard to generate in turbulent motion. So inequality can not be strong for almost all objects. Further, in expression (2.7) we did not take into account placement of object within the cloud. Summarizing all we can generalize third Larson’s low in the form Mcl(r) , (2.10) where r is the dinstance of object from the center of cloud, Mcl(r) is the mass of cloud within r, C is the non-dimensional constant of order one. As far as clouds itself are concerned they move in galactic gravitational field through the interstellar gaseous media (ISM) and we can get for them analogous condition Mgal(r) , (2.11) where r is the dinstance of cloud from the center of galactic, Mgal(r) is the mass of galactic within r. 3. Summary We have seen that it is possible to understand the third Larson’s low and some proper- ties of molecular clouds and objects within them in terms of their existence conditions in external gravitational field. In particular, we have seen possible reason why the clouds have approximately constant column densities. I have to stress now that conditions (2.10) and (2.11) do not rely on any internal dynamic of cloud. It will be also interesting to examine relations (2.10) and (2.11) with astronomical data sets. [1] McLaughlin Dean E. , Pudritz Ralph E. A model for the internal structure of molecular cloud cores. arXiv:astro-ph/9605018. [2] McKee Christopher F. The dynamical structure and evolution of giant molecular clouds. arXiv:astro-ph/9901370. [3] Williams Jonathan P., Blitz Leo, McKee Christopher F. The structure and evolution of molec- ular clouds: From clumps to cores to the IMF. arXiv:astro-ph/9902246. [4] McKee Christopher F. , Holliman John H.,II Multi-pressure polytropes as models for the struc- ture and stability of molecular clouds. I. Theory. arXiv:astro-ph/9903213. [5] Padoan Paolo, Nordlund Ake, Kritsuk Alexei G., Norman Michael L., Shing Pak Li Two regimes of turbulent fragmentation and the stellar IMF from primordial to present day star formation arXiv:astro-ph/0701795. [6] Blitz L. 1993. Giant molecular clouds. In Protostars and Planets III, eds E.H. Levy and J.I. Lunine, (Tucson: Univ. of Arizona Press), pp 125-161. [7] Blitz L., Stark A. A. 1986. Detection of clump and interclump gas in the Rosette molecular cloud complex. Astrophys. J. Lett. 300:L89-L93 [8] Larson R. B., 1981, MNRAS, 194, 809. [9] Zaharow A. W. Gravitational stability of finite massive bodies. arXiv:astro-ph/0610450. [10] Caselli P., Myers P. C. 1995, ApJ, 446, 665. http://arxiv.org/abs/astro-ph/9605018 http://arxiv.org/abs/astro-ph/9901370 http://arxiv.org/abs/astro-ph/9902246 http://arxiv.org/abs/astro-ph/9903213 http://arxiv.org/abs/astro-ph/0701795 http://arxiv.org/abs/astro-ph/0610450 Introduction larson's lows summary References

0704.1215

Double Neutron Stars: Evidence For Two Different Neutron-Star Formation Mechanisms

Double Neutron Stars: Evidence For Two Different Neutron-Star Formation Mechanisms E.P.J. van den Heuvel Astronomical Institute “Anton Pannekoek” and Center for High Energy Astrophysics, University of Amsterdam, The Netherlands, and Kavli Institute for Theoretical Physics, University of California, Santa Barbara, USA Abstract. Six of the eight double neutron stars known in the Galactic disk have low orbital eccentricities (< 0.27) indicating that their second-born neutron stars received only very small velocity kicks at birth. This is similar to the case of the B-emission X-ray binaries, where a sizable fraction of the neutron stars received hardly any velocity kick at birth (Pfahl et al. 2002). The masses of the second-born neutron stars in five of the six low-eccentricity double neutron stars are remarkably low (between 1.18 and 1.30M⊙). It is argued that these low-mass, low-kick neutron stars were formed by the electron-capture collapse of the degenerate O-Ne-Mg cores of helium stars less massive than about 3.5M⊙, whereas the higher-mass, higher kick-velocity neutron stars were formed by the collapses of the iron cores of higher initial mass. The absence of low-velocity single young radio pulsars (Hobbs et al. 2005) is consistent with the model proposed by Podsiadlowski et al. (2004), in which the electron-capture collapse of degenerate O-Ne-Mg cores can only occur in binary systems, and not in single stars. Keywords: stars: neutron — stars: magnetic fields — stars: relativistic — pulsars: general PACS: 04.40.Dg, 97.60.Jd, 97.60.Gb THE BIRTH KICK VELOCITIES OF NEUTRON STARS Pfahl et al. (2002) discovered the existence of a separate class of B-emission X-ray binaries (abbreviated here as Be/X-ray binaries) with wide orbits of low eccentricity (< 0.25). The systems in this class tend to have relatively low X-ray luminosities (< 1034 ergs/s). A well-known example is X-Per, in which the neutron star has an almost circular orbit with a period of 250 days. About half of all Be/X-ray binaries with known orbits appear to belong to this class and the relatively low X-ray luminosities of these sources imply that these systems are on average considerably nearer to us than the high-eccentricity Be/X-ray binaries (which during outbursts can reach a luminosity of 1038 ergs/s). Therefore, as Pfahl et al. (2002) pointed out, the systems in the low- eccentricity class probably form the bulk of the Be/X-ray binary population, since the known numbers of sources in both classes are about the same. These authors pointed out that the neutron stars in the low-eccentricity systems cannot have received a kick velocity at their birth exceeding 50 km/s. Until the discovery of this class of X-ray binaries it was generally thought that all neutron stars receive a high kick velocity at their birth, of order at least a few hundred km/s (see e.g.: Lyne and Lorimer 1994; Hansen and Phinney 1997, Hobbs et al.2005). Often a Maxwellian distribution is used to represent the observed distribution of pulsar velocities, and the characteristic velocity of these Maxwellians is typically around 300 – 400 km/s (Hansen and Phinney 1997). http://arxiv.org/abs/0704.1215v2 A recent very detailed study by Hobbs et al. (2005) of the accurately determined proper motions of 233 radio pulsars shows that there is no room for a separate popu- lation of low-velocity single pulsars. Particularly, these authors found that the velocity distribution of young pulsars (age < 3 million years) is very well represented by a single Maxwellian with a characteristic velocity of about 400 km/s, and there is no evidence for a bimodal velocity distribution as had been argued by Cordes and Chernoff (1998). On the other hand, Pfahl et al. (2002) showed, by means of population synthesis cal- culations that include the evolution of binaries and the presence of birth kicks imparted to neutron stars, that with the assumption of only one Maxwellian with a high character- istic velocity (several hundred km/s) one can reproduce the high-eccentricity population of the Be/X-ray binaries, but one totally fails to reproduce the presence of a large popu- lation of systems with low eccentricities. They convincingly showed that the only way in which both the observed high-e and the low-e populations of the Be/X-ray binaries can be reproduced is: by assuming that there are two distinct populations of neutron stars: one population that receives hardly any kick velocity at birth (vk < 50 km/s) and another which receives the “canonical” high velocity kick of order several hundreds of km/s at birth. DOUBLE NEUTRON STARS AND THE LOW KICK VELOCITY NEUTRON STAR POPULATION At present 9 double neutron stars are known, 8 of them in the galactic disk and one in a globular cluster (see Stairs 2004, Lorimer et al. 2006). The eight systems in the galactic disk are listed in table 1. As the table shows, the double neutron stars tend to have very narrow orbits. They are the later evolutionary products of wide high-mass X- ray binary systems with orbital periods > 100 days (van den Heuvel and Taam 1984), mostly B-emission X-ray binaries (for an alternative view, see Brown 1995). When the massive star in such a system has expanded to become a red giant, its envelope engulfs the neutron star, causing this star to spiral down into this envelope, reducing its orbital separation by several orders of magnitude. The large energy release due to friction and accretion during this spiral-in process is expected to cause the hydrogen-rich envelope of the giant to be expelled such that a very close binary remains, consisting of the helium core of the giant together with the neutron star (van den Heuvel and Taam 1984; Dewi and Pols 2003). (Depending on the orbital separation at the onset of spiral in, the helium core itself may already be (somewhat) evolved and possibly contain already some C and O in its core). [In Be/X-ray systems that started out with orbital periods < 100 days the neutron star spirals in so deeply that it most probably merges with the core of the giant, and so no binary will be left; e.g. see Taam 1996]. Due to the large frictional and tidal effects during spiral in the orbit of the system is expected to be perfectly circular. The helium star generates its luminosity by helium burning, which produces C and O, and subsequently by carbon burning, producing Ne and Mg. If the helium star has a mass in the range 1.6 to 3.5 M⊙ (corresponding to a main- sequence progenitor in the range 8 to 11 (±1) M⊙, the precise limits of this mass range depending on metallicity and on the assumed model for convective energy transport; Sugimoto and Nomoto 1980; Miyaji et al. 1980; Podsiadlowski et al. 2004) it will during carbon burning develop a degenerate O-Ne-Mg core, surrounded by episodic C- and He-burning shells (e.g. Nomoto 1984, Habets 1986). When such a degenerate core develops, the envelope of the helium star begins to expand, causing in a binary the onset of mass transfer by Roche-lobe overflow (Habets 1986; Dewi and Pols 2003). Roche- lobe overflow leads to the formation of an accretion disk around the neutron star and accretion of matter with angular momentum from this disk will cause the spin frequency of the neutron star to increase. Therefore one expects that during the later evolution of these helium stars of relatively low mass the first-born neutron star in the system will be “spun up” to a short spin period. This neutron star had already a long history of accretion: first when it was in a wide binary with an early-type (presumably Be) companion; subsequently during the spiral-in phase into the envelope of its companion and now as companion of a Roche-lobe overflowing helium star. Since all binary pulsars which had a history of mass accretion (so-called “recycled” pulsars; Radhakrishnan and Srinivasan 1982) tend to have much weaker magnetic fields than normal single pulsars, it is thought that accretion in some way causes a weakening of the surface dipole magnetic field of neutron stars (Taam and van den Heuvel 1986) and several theories have been put forward to explain this accretion-induced field decay (Bisnovatyi-Kogan and Komberg 1974; see Bhattacharya and Srinivasan 1995 for a review; Zhang 1998 and Cumming 2004). With a field weakened to about 1010 Gauss (as observed in the recycled components of the double neutron stars (see table 1), and an Eddington-limited accretion rate of helium of ∼ 4× 10−8 M⊙/yr, a neutron star can be spun-up to a shortest possible spin period of a few tens of milliseconds (Smarr and Blandford 1976, Srinivasan and van den Heuvel 1982). When the helium star finally explodes as a supernova, the second neutron star in the system is born. This is a newborn neutron star without a history of accretion and is therefore expected to resemble the “normal” strong-magnetic field single radio pulsars (Srinivasan and van den Heuvel 1982), which have typical surface dipole magnetic fields strengths of 1012 – 1013 Gauss. This theoretical expectation has been beautifully confirmed by the discovery of the double pulsar systems PSRJ0737- 3039AB, which consists of a recycled pulsar (star A) with a very rapid spin (P = 23 ms) and a weak magnetic field (7× 109 G) and a normal strong-magnetic-field (1.2× 1012 G) pulsar (star B) with a “normal” pulse period of 2.8 sec (Burgay et al. 2003, Lyne et al. 2004; see table 1). The explosive mass loss in the second supernova has made the orbit eccentric and since the two neutron stars are basically point masses, tidal effects in double neutron star systems will be negligible and there will be no tidal circularization of the orbit. (On timescales of tens of millions of years the orbits may circularize by a few percent due to the emission of gravitational waves in the shortest-period system of PSRJ0737-3039, but in all the other double neutron stars this is a negligible effect, except in the final stages of spiraling together; see e.g. Shapiro and Teukolsky 1983). In case of spherically symmetric mass ejection in the supernova there is a simple relation between the orbital eccentricity and the amount of mass ∆Msn ejected in the supernova: e = ∆Msn/(Mns1 +Mns2) (1) where Mns1 and Mns2 are the masses of the first- and the second-born neutron stars. The “conventional” kick velocities of neutron stars of about 400 km/s (Hobbs et al.2005) are quite similar to the orbital velocities of the neutron stars in close double neutron stars such as the Hulse-Taylor binary pulsar PSRB1913+16 (Porb = 7.75 hours). Therefore, a kick velocity of this order produces a major disturbance of the orbit and – unless it is imparted in a very specific direction – will in general impart a large eccentricity to the orbit, of order 0.5 or more. The Hulse-Taylor binary pulsar has a large eccentricity e = 0.617 and the same is true for the system PSRJ1811-1736 (e = 0.828), which indeed might be due to such large kick velocities. However, as table 1 shows, very surprisingly all of the other 6 double neutron stars in the galactic disk have very small orbital eccen- tricities, in the range 0.088 to 0.27. Such eccentricities are the ones which one expects from the pure sudden mass loss effects in the supernova explosion, given by equation (1), but not in case a randomly directed kick velocity of order 400 km/s is imparted to the second-born neutron star at birth. [In particular, the small orbital eccentricities of the two relatively wide double neutron stars PSRJ1518+4909 and PSRJ1829+2456 are impossible to reconcile with high kick velocities]. Furthermore, Dewi et al. (2005) and van den Heuvel (2005) have pointed out that the relation between spin period of the recycled neutron star and orbital eccentricity observed in double neutron star systems (Faulkner et al. 2005) can only be understood if the second-born neutron stars in these systems received a negligible velocity kick in their birth events. Interestingly, also the Hulse-Taylor binary pulsar PSRB1913+16 and PSRJ1811-1736 fit this relation, which suggests that also their high orbital eccentricities were purely due to the effects of the sudden mass loss in the second supernova. And indeed, since their first-born neutron stars are quite strongly recycled, they must have had a quite extended episode of disk accretion. This implies an extended episode of stable Roche-lobe overflow from the helium star progenitor of the second-born neutron star. And this in turn suggests that these helium stars had a degenerate O-Ne-Mg core, as only the development of such cores causes the envelopes of helium stars to expand. It thus appears that the second-born neutron stars in these 6 low-eccentricity systems belong to the same “kick-less” class as the neutron stars in the low-eccentricity class of Be/X-ray binaries (van den Heuvel 2004, 2005, 2006). The same holds for the young strong-magnetic-field pulsar in the eccentric radio-pulsar binary PSRJ1145-6545 which has a massive white dwarf as a companion (Kaspi et al. 2000; Bailes et al. 2003; Bailes 2005). The orbital eccentricity of 0.172 of this binary shows that the neutron star was the last-born object in the system (Portegies Zwart and Yungelson 1999, Tauris and Sennels 2000; formation of a white dwarf cannot introduce an orbital eccentricity). The low value of its eccentricity would be hard to understand if the neutron star received the canonical 400 km/s kick at its birth. TABLE 1. Double neutron star binaries and the eccentric-orbit white-dwarf neutron star system J1145-6545. References: (1) Lyne et al. (2004); (2) Nice et al. (1996); (3) Stairs (2004); (4) Faulkner et al. (2005); (5) Champion et al. (2004); (6) Bailes (2005); (7) Lorimer et al. (2006). Pulsar Spin Porb Compan. Pulsar Sum of Bs Name Per. e Mass Mass masses Ref (ms) (d) (M⊙) (M⊙) (M⊙) (10 10 G) J0737- 22.7 0.10 0.088 1.250(5) 1.337(5) 2.588(3) 0.7 (1) 3039A J0737- 2770 0.10 0.088 1.337(5) 1.250(5) 2.588(3) 1.2× 102 (1) 3039B J1518+ 1.05 1.56 4904 40.9 8.63 0.249 (+0.45) (+0.13) 2.62(7) 0.1 (2) (-0.11) (-0.45) B1534+ 37.9 0.42 0.274 1.3452(10) 1.3332(10) 2.678(1) 1 (3) J1756- 28.5 0.32 0.18 1.18(3) 1.40(3) 2.574(3) 0.54 (4) J1811- 1.11 1.62 1736 104 18.8 0.828 (+0.53) (+0.22) 2.60(10) 1.3 (3) (-0.15) (-0.55) J1829+ 1.27 1.30 2456 41.0 1.18 0.139 (+0.11) (+0.05) 2.53(10) ∼ 1 (5) (-0.07) (-0.05) J1906+ 144.1 0.165 0.085 — — 2.61(2) 1.7× 102 (7) B1913+ 59 0.33 0.617 1.3873(3) 1.4408(3) 2.8281(1) 2 (3) J1145- 394 0.20 0.172 1.00(2) 1.28(2) 2.288(3) ∼ 102 (6) THE MASSES OF THE SECOND-BORN NEUTRON STARS IN THE DOUBLE NEUTRON STAR SYSTEMS AND IN PSRJ1145-6545 In the eccentric white-dwarf/neutron-star system of PSRJ1145-6545 the mass of the neutron star is known from the measurement of relativistic effects to be 1.28(2) M⊙ (Bailes 2005). Also in two of the low-eccentricity double neutron stars the masses of both stars are accurately known from measured relativistic effects (see Stairs 2004): (i) in PSR J0737-3039 the second-born neutron star has MB = 1.250(3) M⊙ and the first- 1 The number within parentheses indicates the 95% confidence uncertainty of the last digit; the total mass of the system is 2.30 M⊙ and the mass of the white dwarf is at least one solar mass. born one has MA = 1.330(3) M⊙ (Lyne et al. 2004). (ii) in PSR J1756-2251 the second-born neutron star has a mass of 1.18(3) M⊙ and the first-born one a mass of 1.40(3) M⊙ (Faulkner et al. 2005). In most of the other double neutron stars the masses of the stars are not yet accurately known, but in two of the other low-eccentricity systems the second-born neutron stars must be less massive than 1.30 M⊙ for the following reasons. In all double neutron star systems the relativistic parameter that can be measured most easily is the General Relativistic rate of periastron advance, which directly yields the sum of the masses of the two neutron stars (e.g. see Stairs 2004). In the low-eccentricity systems of PSR J1518+4904, PSR J1829+2456 and PSR J1906+0746 the resulting sum of the masses turns out to be 2.62, 2.53 and 2.61M⊙, respectively. The individual masses of the neutron stars in these systems are still rather poorly determined, but in the first two of these three systems the already crudely determined other relativistic parameters indicate that the second-born neutron star has the lowest mass of the two (see references in van den Heuvel 2004). As in these systems the sum of the masses is around 2.60 M⊙, the second- born neutron stars in these two systems cannot be more massive than 1.30 M⊙. Thus we find that in at least four of these six systems the second-born neutron star has a low mass, in the range 1.18 to 1.30 M⊙ and belongs to the low-kick category. And the same holds for the second-born neutron star in the low-eccentricity white-dwarf- neutron-star binary PSR J1145-6545, which has a mass of only 1.28 M⊙. Also in the system of PSR J1909+0746 the masses of the neutron stars cannot differ much from 1.30 M⊙. We thus see that in at least five cases a low (or no) kick velocity is correlated with a low neutron star mass of on average around 1.25 (± 0.06) M⊙. A neutron star of 1.25 M⊙ corresponds to a pre-collapse mass of about 1.44 M⊙, as during the collapse the gravitational binding energy of the neutron star of about 0.20 M⊙ (slightly depending on the assumed equation of state of neutronized matter) is lost in the form of neutrinos. So apparently the cores, which collapsed to these second-born neutron stars, had a mass very close to the Chandrasekhar mass. FORMATION MECHANISMS OF NEUTRON STARS AND POSSIBLE RESULTING KICKS It is long known (Miyaji et al. 1980, Sugimoto and Nomoto 1980) that there are two basically different ways in which neutron stars are expected to form, i.e.: (i) In stars which originated in the main-sequence mass range between 8 and about 11 (±1) M⊙, which in binaries produce helium stars in the mass range 1.6 to 3.5 M⊙ (Habets 1986, Dewi and Pols 2003), the O-Ne-Mg core which forms during carbon burning becomes degenerate and when its mass approaches the Chandrasekhar mass, electron captures on Mg and Ne cause the core to collapse to a neutron star. Since these stars did not reach Oxygen- and Silicon burning, the baryonic mass of the neutron star, which forms in this way, is expected to be purely determined by the mass of the collapsing degenerate core, which is the Chandrasekhar mass. The gravitational mass of this neutron star is then the Chandrasekhar mass minus the gravitational binding energy of the neutron star, which is about 0.20 M⊙. Thus a neutron star with a mass of about 1.24 M⊙ is expected to result. (ii) In stars initially more massive than 11 (±1)M⊙, the O-Ne-Mg core does not become degenerate and these cores proceed through Oxygen and Silicon burning to form an iron core. When the mass of this iron core exceeds a critical value it collapses to form a neutron star. The precise way in which here neutrino transport during core bounce and shock formation results in a supernova explosion is not yet fully understood. It appears that first the shock stalls and then several hundreds of milliseconds later, is revitalized. Some fall back of matter from the layers surrounding the proto neutron star is expected to occur (see Fryer 2004) such that the neutron star that forms may be substantially more massive than the mass of the collapsing Fe-core. In fact there are two expected mass regimes for the resulting neutron stars: for stars with initial main-sequence masses in the range 11 (±1) M⊙ to 19 M⊙ the collapsing cores are expected to be about 1.3 M⊙, whereas for stars more massive than 19 M⊙ the collapsing iron core is expected to have a mass > 1.7 M⊙ (Timmes et al. 1996), leading to the formation of neutron stars with (gravitational) masses > 1.6 M⊙. Taking some fall-back of matter into account, the neutron stars formed from these types of iron cores may be expected to have gravitational masses > 1.3 M⊙ and > 1.7 M⊙, respectively. The fact that the pre-collapse masses of the low-mass, low-kick neutron stars were very close to the Chandrasekhar limit suggests that these neutron stars are the result of the electron-capture collapse of the degenerate O-Ne-Mg cores of helium stars that originated in the mass range 1.6 to 3.5 M⊙ (initial main-sequence mass in the range 8 to 11 (±1) M⊙). Can one understand why such neutron stars would not receive a birth kick whereas those formed by the collapse of an iron core would? While in the past neutron-star kicks generally were ascribed to asymmetric neutrino emission (e.g. Burrows and Hayes 1996), in recent years the ideas have shifted towards hydrodynamic instabilities during the explosion. For example, Scheck et al. (2004, 2005) found large- scale hydrodynamic instabilities to develop in the layers surrounding the proto neutron star during the explosion of a 15 M⊙ star with a collapsing iron core, which imparted velocities up to 1000 km/s to the neutron star. On the other hand, for collapsing O-Ne- Mg cores, Kitaura et al. (2006) did not find large neutron-star velocities. This is ascribed to the facts that (a) here the ejecta mass in the immediate vicinity of the proto neutron star is very small, and (b) the explosion of the O-Ne-Mg core by neutrino heating occurs very fast (much faster than for iron cores, where the development of the explosion takes hundreds of milliseconds), not allowing hydrodynamic instabilities to develop. It thus appears that a difference in the purely hydrodynamic effects during these very different types of explosions may explain the differences in the kick velocities of the resulting neutron stars. WHY ARE THERE NO LOW-VELOCITY SINGLE PULSARS? Podsiadlowski et al. (2004) recently argued that single stars in the mass range 8 to 11 (±1) M⊙ do not produce neutron stars, for the following reason. These stars produce helium cores in the mass range 1.6 to 3.5 M⊙, but when they ascend the Asymptotic Giant Branch (AGB), their convective envelope during the “dredge-up” phase penetrates the helium layers surrounding their degenerate O-Ne-Mg cores, and erodes these helium layers away. Therefore the degenerate cores of these stars can no longer grow by helium shell burning. These stars lose their envelopes due to the heavy wind mass loss during the AGB phase, and are expected to leave behind their degenerate O-Ne-Mg cores as white dwarfs. Only single stars more massive than about 11 (±1) M⊙ will leave neutron stars, formed in this case by iron core collapse. As argued above, these neutron stars will be of the high-kick class, so all single neutron stars are expected to be high-velocity objects, as is indeed observed (Hobbs et al. 2005). On the other hand, as argued by Podsiadlowski et al. (2004), an 8 to 11 (±1) M⊙ star in an interacting binary system cannot reach the AGB, as already before reaching that very extended phase, it will in a binary have lost its hydrogen envelope by Roche-lobe overflow. Therefore, in binaries such stars will leave helium stars with masses in the range 1.6 to 3.5 M⊙, which will evolve to e-capture core collapse, which according to our above-described model leaves a low-velocity neutron star. One therefore expects these low-velocity neutron stars to only be born in binary systems. CONCLUSIONS The combination of observations indicating that: (i) among the Be/X-ray binaries and the double neutron stars there is a substantial group with low orbital eccentricities, indicating that their last-born neutron stars received hardly any velocity kick at birth, (ii) the low-kick second-born neutron stars in the double neutron star systems have a low mass, ∼ 1.25 M⊙, and (iii) the absence of low-velocity neutron stars in the young radio pulsar population can be consistently explained if the low-mass low-kick neutron stars originate from the electron-capture collapse of the degenerate O-Ne-Mg cores of stars that started out with main-sequence masses in the range ∼ 8−11 M⊙, while the high- kick-velocity neutron stars originated from the iron-core collapses of stars that started out with masses in excess of ∼ 11 M⊙. Such an explanation is fully consistent with the model proposed by Podsiadlowski et al. (2004) according to which neutron star formation by electron-capture collapse can only occur in interacting binaries and not in single stars. ACKNOWLEDGMENTS This research was supported in part by the National Science Foundation under Grant No. PHY99-07949. I am grateful to the Leids Kerkhoven-Bosscha Fonds and the Nether- lands Research School for Astronomy NOVA for providing financial support enabling me to take part in the Cefalu Conference. REFERENCES 1. Bailes M., Ord S.M., Knight H. and Hotan A.W., 2003, ApJ, 595, L49 2. Bailes M., 2005, in “Binary radio Pulsars” (ed. F.A. Rasio and I.H. Stairs), ASP Conf. Series 328, 33 3. Bhattacharya D., and Srinivasan G., 1995, in : “X-ray Binaries” (ed. W.H.G. Lewin, J.A. van Paradijs and E.P.J. van den Heuvel), Cambridge Univ. Press, 495 4. Bisnovatyi-Kogan G.S., and Komberg B.V., 1974, Soviet Astron. 18, 217 5. Brown G.E., 1995, ApJ, 440, 270 6. Burgay M., D’Amico N., Possenti A., et al., 2003, Nature, 426, 531 7. Burrows A., and Hayes J., 1996, Phys. Rev. Letters, 76, 352 8. Champion D.J., Lorimer D.R., McLaughlin M.A., Cordes J.M., and Taylor J.H., 2004, MNRAS, 350, 9. Cordes J., and Chernoff D.F., 1998, ApJ, 505, 315 10. Cumming A., 2004, in: “Binary Pulsars” (editors: F.A. Rasio and I.H. Stairs), ASP Conf. Series, 328, 11. Dewi J.D.M., and Pols O.R., 2003, MNRAS, 344, 629 12. Dewi J.D.M., Podsiadlowski P., and Pols O., 2005, MNRAS, 363, L71 13. Ergma E., and Antipova J., 1999, A&A, 343, L45 14. Faulkner A.J., Kramer M., Lyne A.G., et al., 2005, ApJ, 618, L119 15. Fryer C.L., 2004, Astrophys. and Space Science Library Vol. 302, Kluwer Acad. Publ. Dordrecht, 16. Habets G.M.H.J., 1986, A&A, 167, 61 17. Hansen B.M.S., and Phinney E.S., 1997, MNRAS, 291, 569 18. Hartman J.W., 1997, A&A, 322, 127 19. Hobbs G., Lorimer D.R., Lyne A.G., and Kramer M., 2005, MNRAS, 360, 974 20. Kaspi V.M., Lyne A.G., and Manchester R.N., 2000, ApJ, 543, 321 21. Kitaura F.S., Janka H.-Th., and Muller E., 2006, A&A, 450, 345 22. Lorimer D.R., Stairs I.H., Freire P.C., et al., 2006, ApJ, 640, 428 23. Lyne A.G., and Lorimer D.R., 1994, Nature, 369, 127 24. Lyne A.G., Burgay M., Kramer M., et al., 2004, Science, 303, 1153 25. Miyaji S., Nomoto K., Yokoi K., Sugimoto D., 1980, PASJ, 32, 303 26. Nice D.J., Sayer R.W., and Taylor J.H. 1996, ApJ, 466, L87 27. Nomoto K., 1984, ApJ, 277, 791 28. Pfahl E., Rappaport S., Podsiadlowski P., and Spruit H., 2002, ApJ, 574, 364 29. Podsiadlowski P., Langer N., Poelarends A.J.T., Rappaport S., Heger A., and Pfahl E., 2004, ApJ, 612, 1044 30. Portegies Zwart S.F., and Yungelson L.R., 1999, MNRAS, 309, 26P 31. Radhakrishnan V., and Srinivasan G., 1982, Current Science 51, 1096 32. Scheck L., Plewa T., Janka H.-T., Mueller E., 2004, Phys. Rev. Letters 92, id. 011103 33. Scheck L., Kifonidis H., Janka H.-Th., and Muller E., 2006, A&A, 457, 963 34. Shapiro S.L., and Teukolsky S.A., 1983, “Black Holes, White Dwarfs and Neutron Stars”, New York, Wiley-Interscience, 645 35. Smarr L.L., and Blandford R.D., 1976, ApJ, 207, 574 36. Srinivasan G., and van den Heuvel E.P.J., 1982, A&A, 108, 143 37. Stairs I.H., 2004, Science, 304, 547 38. Sugimoto D., and Nomoto K., 1980, Space Sci. Rev., 25, 155 39. Taam R.E., 1996, in “Compact Stars in Binaries” (editors J. van Paradijs, E.P.J. van den Heuvel and E. Kuulkers), Proc. IAU Symp. 165, Kluwer Acad. Publ., Dordrecht, 3 40. Taam R.E., and van den Heuvel E.P.J., 1986, ApJ, 305, 235 41. Tauris T. M., and Sennels T., 2000, A&A, 355, 236 42. Timmes F.X., Woosley S.E., Weaver T.A., 1996, ApJ, 457, 834 43. Van den Heuvel E.P.J., 2004, in: Proc. 5th INTEGRAL Workshop, (eds. V. Schoenfelder, G. Lichti and C. Winkler), ESA SP-552 (Noordwijk, ESA Publ. Div. ESTEC), 185 44. Van den Heuvel E.P.J., 2005, in: "The Electromagnetic Spectrum of Neutron Stars" (eds. A. Baykal et al.), Springer, Netherlands, 191 45. Van den Heuvel E.P.J., 2006, Advances in Space Research, 38, 2667 46. Van den Heuvel E.P.J., and Taam R.E., 1984, Nature, 309, 235 47. Zhang C.M., 1998, A&A, 330, 195 The birth kick velocities of neutron stars Double neutron stars and the low kick velocity neutron star population The masses of the second-born neutron stars in the double neutron star systems and in PSRJ1145-6545 Formation mechanisms of neutron stars and possible resulting kicks Why are there no low-velocity single pulsars? Conclusions

0704.1216

Large Scale Self-Similar Skeletal Structure of the Universe

Revelation of the Sun Self-Similarity Skeletal Structures Large-Scale Self-Similar Skeletal Structure of the Universe. V. A. Rantsev-Kartinov INF RRC "Kurchatov Institute", Moscow, Russia, [email protected] 1. Introduction The given paper is natural continuation of a series of papers (published earlier [1a, b]) about of a revealing of self-similar skeletal structures (SSSS) in various types of plasma up to space scales. Research by author of the SSSS begin from the analysis of images of various types of plasma by means of a method multilevel dynamic contrasting (ММDC), developed and described earlier [1c, d]. The analysis of images by this method is carried out by imposing of various computer maps of contrasting on the image of plasma received by the various methods and in anyone spectral ranges. Results of the given analysis of a modern database of images of space objects are shown, that the topology of the revealed space structures is identical to those which have been already found out and described earlier in a wide range of physical environments , the phenomena and scales [1-2]. The typical SSSS consists of separate identical blocks which are linked together to form a network. Two types of such blocks are found: (i) coaxially tubular structures (CTS) with internal radial bonds, and (ii) cartwheel-like structures (CWS) , located either on an axle or in the edge of CTS block. The large-scale skeletal structures of the Universe (SSU) have a whole series of remark- able properties which have been also described before[1c-1f]. So, a long filaments consist of straight (“rigid”) nearly identical CTS blocks which is joined flexibly similarly to joints in a skeleton. It is assumed such joints may be realized due to stringing of the individual CTS blocks on common flow of the magnetic field which penetrates the whole such filament, and itself the CTS blocks are an interacting magnetic dipoles with micro-dust skeletons, which are immersed into plasma. Here, the result of the analysis which was been received (with the de- scription of sequence of the lead operations) by means of the MMDC of maps of the Redshift Surveys of galaxies and quasars which have allowed to reveal large-scale structures of the mentioned above topologies are given. 2. Observations of similarity of a structuring in a very broad range of length scales 2.1. Cartwheel-like structures in the range 10-5 cm – 1023 cm. Here, we will try to draw a bridge between laboratory experiments and space with pre- senting on consideration of a short gallery of cartwheel-like structures, which are probably the most inconvenient objects for universally describing in the entire range of observed space scales. In a laboratory electric discharges [3,4] and of respective dust deposits [5], the cart- wheels are located either in the butt-ends of a tubes or on an «axle-tree» filament, or as a separate block (the smallest cartwheels are of diameter less than 100 nm (see Figs. 2 and 3 in Ref. 5)). So, similar structures of different scales are found in the following typical examples: (i) a big icy particles of a hails (Fig. 1A), (ii) a fragment of tornado (Fig. 1B), (iii) a super- nova remnant (Fig. 1C). mailto:[email protected] Fig. 1. The cartwheel-like structures at different length scales. A, Big icy particles of a hail of diameter 4.5 cm (a), 5 cm (b), and 5 cm (c). The original images are taken from Ref. [6]. The frame in the left lower part of the image (a) is contrasted separately to show an elliptic image of the edge of the radial directed tubular structure. The entire structure seems to contain a number of similar radial blocks. A distinct coaxial structure of the cartwheel type is seen in the central part of image (b). Image (c) shows strong radial bonds between the central point and the «wheel». B, Top section: A fragment of the photographic image [7] of a massive tor- nado of estimated size of some hundred meters in diameter. Bottom section: A fragment of the top image shows the cartwheel whose slightly elliptic image is seen in the center. The cartwheel seems to be located on a long axle-tree directed down to the right and ended with a bright spot on the axle’s edge (see its additionally contrasted image in the left corner insert on the bottom image). C, «A flaming cosmic wheel» of the supernova remnant E0102-72, with «puzzling spoke-like structures in its interior», which is stretched across forty light-years in Small Magellan Cloud, 190,000 light-years from Earth (.../snrg/e0102electricbluet.tif [8]). The radial directed spokes are ended with tubular structures seen on the outer edge of the cartwheel. The inverted (and additionally contrasted) image of the edge of such a tubule (marked with the square bracket) is given in the left corner insert (note that the tubule’s edge itself seems to possess a tubular block, of smaller diameter, seen on the bottom of the insert). Note that the cosmic wheel’s skeleton (Fig. 1C) tends to repeat the structure of the icy cartwheel (Fig. 1A) up to details of its constituent blocks. In particular, some of radial di- rected spokes are ended with a tubular structure seen on the outer edge of the cartwheel. Moreover, in the edge cross-section of this tubular structure, the global cartwheel of the icy particle contains a smaller cartwheel whose axle is directed radial (see left lower window in Fig. 1A). Thus, there is a trend toward self-similarity (the evidences for such a trend in tubu- lar skeletons found in the dust deposits are given in Ref. [5]). Note that the images of Fig. 1 are processed with MMDC [1c,d]. As a rule, the structuring revealed with the help of this method may then be easily recognized in the original, non-processed images (especially, for properly magnified high-resolution images). Thus, the cosmic wheel’s skeleton tends to repeat the structure of the icy cartwheel up to details of its constituent blocks. One may obviously add to the last item of this list namely galaxy - "Cartwheel", which have 150,000 light-years in diameter and 500 million light-years on a distance from Earth in constellation of Sculptor [9]. Fig. 2. “Cartwheel galaxy" [9], Located 500 million light-years away in the constellation Sculptor, 150,000 light- years across ~ 1023 cm, the galaxy looks like a cartwheel. The galaxy's nucleus is the bright object in the center of the image; the spoke-like structures are wisps of material connecting the nucleus to the outer ring of young stars. Fig. 3. The schematic image of structures such as " cartwheel" is given here. Thus, the cosmic wheel's skeleton tends to repeat the structure of the cartwheel itself up to details of its constituent blocks as in the icy cartwheel. The wheel-like supernova remnant G11.2-0.3 which have 40 light-years in diameter and 25,000 light-years away in the constellation Sagittarius [10] a two-ring coaxial structure, with the inner ring of one light-year in diameter. Fig. 4. G11.2-0.3: A supernova remnant with a central pulsar, located in the constellation of Sagittarius [10], 40 light years across and 25,000 light-years away. Here it is applied MMDC which has allowed to reveal CTS blocks from which the design of given CW (this space object) is made. Fig. 5. The Crab nebula which is 6,000 light-years away in the constellation Taurus [11]. The Crab Nebula is the remnant of a supernova explosion that was seen on Earth in 1054 AD. It is 6000 light-years from Earth, with the inner ring of one light year (1018 cm) in diame- ter. At the center of the bright nebula is a rapidly spinning neutron star, or pulsar that emits pulses of radiation 30 times a second. Here it is well visible, that the given structure represents system of tu- bular structures which are telescopic enclosed each into other. At the attentive analysis of the image a radial connections which are characteristic for structure such as CWS are looked through. 2.3. The signs of skeletal structuring at cosmological lengths, up to 1027 cm. With increasing length scales, the self-illumination of the skeletal network in its certain, critical points continues working but the respective dramatic decrease of the average density of hot radiating baryonic matter leads to observing of exclusively dim dotted imprints of skeletons, like e.g. mysterious dotted images of arcs and circles / ellipses. Note that the typi- cal blocks of skeletons, the cartwheels on an axle and the tubs with the central rod and the cartwheel in the butt-ends are both the dendrites (which are corresponded to examples of the fractal dust deposits - the skeletons composed of tubular nano-fibers [5,12]). Here it is necessary to note, the blocks of the common network of the Universe (taking into attention of very big distances) can be revealed only in places of infringement of its tightness, i.e. there where takes place breaks up and tears up of network filaments which show a strong luminescence of these areas. Therefore the surface of lengthy filamentary blocks of the Universe either is absolutely not visible, or nevertheless is looked through hardly due to its illumination in places of breakings up of filaments of suitable blocks which surface is formed by these filaments. It appears, that butt-ends of large-scale blocks of the Universe are congestions of galaxies, and galaxies play a role of separate points of these blocks depicting contours. Thus, here the hypothesis is actually offered, that many galaxies can be presented as butt-ends of filamentary tubular blocks which size will be coordinated to scale of galaxies. Cooperating galaxies in that case are butt-ends of the broken blocks or tears of their connections formed in result of cosmology accidents (collisions between filaments and their tears up as a result of tension). The CWS are the most interesting and complex observable blocks in the Universe, and also they are the most typical blocks of the observable fractal the structures of which are diffi- cultly to confuse with any another. If such structure is well oriented in a flatness of a shearing, then (at condition of a corresponding statistics) the structure clearly becomes apparent be- cause the basic massif of points of this structure is fitted to a rim of a wheel, to its axis and ra- dial spokes, making (on the square) half of area of whole wheel. It allows to identify precisely its under such circumstances. It is theoretically difficult to explain topology CWS by means of magnetic hydrodynamics and the theory of construction of fractals in open systems. The mechanism [1a,b] of construction of the revealed by us topology of fractals spontaneously gathering at formation of electric breakdown at the presence of elementary blocks of a dust, which have tendency to forming structures (for example, as carbon nanotubes or a similar structures but of other elements and chemical compounds) have been earlier considered. The sequence of generations CWS right up to the size ~ 1023 cm already has been shown earlier [1a,b]. It appears that at largest observable lengths the more or less definite examples of distinctive topology, similar to that of Fig. 1, may be found only in the redshift surveys of thin slices of space (the redshift surveys are believed to provide a three dimensional distribution of galaxies, which may give, in particular, the side-on view on a thin conical slice of space). The original data are taken from three different projects, the two-degree-field The result of the analysis means of ММDC [7] of maps of redshifts by together with the image of a kind of structures on an exit of a radial spokes of the CWS at their passage through its rim is given on a next figure. Fig. 6. A fragment of similar distribution of the galaxies [13] (20,000 galaxies (for redshifts Z < 0.3, i.e. at distances L up to 2.5 billion light-years away) 1.5° thick slice is cantered at dec- lination -45° in the South Galactic Pole strip, see red points in the colored image at http://www.astro.ucla.edu/~wright/lcrs.html). The left border of the cone crosses the left hand side of the figure at a distance ~ 1.5 109 light years. Thickening of the spots and subsequent smoothing of the image gives a circle and straight radial filaments. Fig. 7. Here the same fragment, as on Fig. 6 is resulted. The left part of figure corresponds to the initial data, average - the same initial data, but all points are increased on the area twice, the right part of figure gives the final image after of Gauss smoothing of intensity distribution of points and carrying out of the correlation analysis by means of MMDC for an establish- ment of connections between separate points. Fig. 8. Here the right fragment of Fig. 7 is increased. Fig. 9. A fragment of similar distribution of the galaxies but some smaller size [13] (1.5° thick slice is centred at declination -6° in the North Galactic Pole strip, see green points in the coloured image at http://www.astro.ucla.edu/~wright/lcrs.html)). In particular, as the most significant discovery, the author in detail describes the frame structure revealed at it such as CWS in scale ~ 1.5*1027 cm that makes about 10 % of the scale observable Universe (~1.5 billion light years). The comparative analysis of its structure with similar observable structures of smaller scales is carried out to show their absolute topo- logical identity (Fig. 1). Let's look now, what structures can be revealed in database of Galaxy Redshift Survey (2dFGRS) [14], which plotted in the redshift space the distribution of some 140,000 galaxies (for redshifts Z <0.3, i.e. at distances L up to 2.5 billion light-years away. Fig.10. A fragment of the projected distribution [14] of the galaxies in the South Galactic Pole strip (4° thick slice is centred at declination -27.5°), as a function of redshift Z and right ascension. The lower border of the cone reaches the bottom of the figure at Z ~ 0.027 (or, equivalently, at a distance ~ 2.7 108 light years). The slight increase of spots’ size in original image at .../Public/Pics/2dFzcone.gif [14] gives elliptic image of a circular (or at least, an arc- like) structure. Despite the structuring seen in this figure is obviously less reliable than that in Fig. 1, the correlation revealed makes it reasonable to suggest an extrapolation of our hy- pothesis farther to cosmological scales. Fig.11. An other fragment of the projected distribution [14] of the galaxies. Here the structure such as CWS with its axis located on an axis of this figure is looked through. The rim of this structure is represented as a pentagon. Very easily it is possible to track structure of connection of separate blocks of the common structure and its interweaving into structure of the big scales, i.e., its connections with others, external in relation to it, structures. Fig.12. One more fragment of the same projected distribution [14] of the galaxies. Here the structure such as CWS with its axis located on an axis of this figure is also looked through. The given structure here is well recognized as well as a perfectly visible connections of it with other blocks of the general structure. The kind and subtleties of connections and structure of separate blocks is well looked through. Fig.13. One more fragment of the same projected distribution [14] of the galaxies. Here in the lower figure for recognition of its kind it is isolated CTS with structure such as CWS at a for- ward butt-end of it. Quasars are the space objects farthest from us. Therefore the structure of their spatial dis- tribution can tell to us about structure of the Universe during very far times. The database of these objects is still small, but, nevertheless, its analysis by means of the described method al- lows to allocate in an arrangement of quasars the same elements and blocks of a skeletal structures, as for more close galaxies. It can suggest an idea us, that the kind of structure of the universe during those times far from us differed from that structure which we notice and presently a little. Thus, the interpolation of visually correlated sequences of spots of such da- tabase (e.g., by means of thickening the spots and subsequent smoothing the image) often gives various skeletal structures (namely, arcs, rings, straight filaments, sometimes the frag- ments of tubules and cartwheels) of various size and declination with respect to the observer. From optimistic viewpoint, a substantial part of the images may be reduced to a superposition of skeletal structures. Fig. 14. A fragment of the 2dF Quasar Redshift Survey (2QZ) [15], which plotted in the red- shift space the distribution of some 20,000 quasars (Z < 3, i.e. L ~ 1.5 1010 light-years). Here an each reader can try to unite a separate arches and direct lines, which are designated by points - by quasars in an uniform structure or even into a separate blocks (CWS or CTS) of the corresponding sizes of the Universe. REFERENCES [1] A.B.Kukushkin, V.A.Rantsev-Kartinov, a) Phys.Lett. A, 306, 175, (2002); b) Science in Russia, 1, 42, (2004); c) Laser and Particle Beams, 16, 445,(1998); d) Rev.Sci.Instrum., 70, 1387,( 1999); e) Current Trends in International Fusion Research: Review and Assessment (Proc. 3rd Symposium, Washington D.C., March 1999), Ed. E. Panarella, NRC Research Press, Ottawa, Canada, 121, (2002); f) “Advances in Plasma Phys. Research”, (Ed. F. Gerard, Nova Science Publishers, New York), 2, 1, (2002). [2] B.N.Kolbasov, A.B.Kukushkin, V.A.Rantsev-Kartinov, et.set., Phys. Lett. A:a) 269, 363, (2000); b) 291, 447, (2001); c) Plasma Devices and Operations, 8 , 257, (2001). [3]. Kukushkin, A. B. & Rantsev-Kartinov, V. A. Wild cables in tokamak plasmas. in Proc. 27-th Eur. Phys. Soc. conf. on Plasma Phys. and Contr. Fusion, Budapest, Hungary, June 2000 (http:// epsppd.epfl.ch/Buda/pdf/p2_029.pdf; .../p2_028.pdf). [4] Kukushkin, A. B. & Rantsev-Kartinov, V. A. Long-lived filaments in fusion plasmas: re- view of observations and status of hypothesis of microdust-assembled skeletons. In Current Trends in International Fusion Research, Proc. 4 th Symposium, Washington D.C., 2001 (eds. C.D. Orth, E. Panarella, and R.F. Post) (NRC Research Press, Ottawa, Canada, 2002) (to be published); (see also preprint http://xxx.lanl.gov/pdf/physics/0112091). [5] Kolbasov, B. N., Kukushkin, A. B., Rantsev-Kartinov, V. A. & Romanov, P. V. Similarity of micro- and macrotubules in tokamak dust and plasma. Phys. Lett. A, 269, 363-367 (2000). [6] Australian Severe Weather, http://australiasevereweather.com/photography/ photos/1996/0205mb11.jpg; .../0205mb12.jpg. [7] National Oceanic & Atmospheric Administration Photo Library, Historic National Weather Service Collection, http://www.photolib.noaa.gov/historic/nws/images/big/wea00216.jpg. [8] Chandra X-Ray Observatory - Home. http://chandra.harvard.edu/photo/... [9] http://hubblesite.org/newscenter/newsdesk/archive/releases/1996/36/image/a [10] http://chandra.harvard.edu/photo/2001/1227/index.html [11] http://chandra.harvard.edu/photo/2002/0052/index.html [12] Kolbasov, B. N., Kukushkin, A. B., Rantsev-Kartinov, V. A. & Romanov, P. V. Phys. Lett. A, 291, 447-452 (2001). [13] Shectman, S. A. et. al. The Las Campanas redshift survey. Astrophys. J. 470, 172-188 (1996); [14] The 2dF Galaxy Redshift Survey, http://www.mso.anu.edu.au/2dFGRL/…// [15] a) The 2dF QSO Redshift Survey, http://www.2dfquasar.org/wedgeplot.html; http://hubblesite.org/newscenter/newsdesk/archive/releases/1996/36/image/a http://chandra.harvard.edu/photo/2001/1227/index.html http://chandra.harvard.edu/photo/2002/0052/index.html http://oposite.stsci.edu/pubinfo/ http://www.2dfquasar.org/wedgeplot.html

0704.1217

The Manin conjecture in dimension 2

THE MANIN CONJECTURE IN DIMENSION 2 T.D. BROWNING Contents 1. Introduction 2 1.1. Notation 5 1.2. The Manin conjectures 6 1.3. Degree 3 surfaces 10 1.4. Degree 4 surfaces 13 1.5. Degree > 5 surfaces 17 1.6. Universal torsors 19 2. Further heuristics 20 2.1. The lines on a cubic surface 21 2.2. Cubic characters and Jacobi sums 23 2.3. The Hardy–Littlewood circle method 25 3. The A1 del Pezzo surface of degree 6 34 3.1. Elementary considerations 35 3.2. The asymptotic formula 38 4. The D4 del Pezzo surface of degree 3 44 4.1. Elementary considerations 45 4.2. A crude upper bound 48 4.3. A better upper bound 50 References 55 Date: November 12, 2018. 2000 Mathematics Subject Classification. 11G35 (14G05, 14G10). http://arxiv.org/abs/0704.1217v1 2 T.D. BROWNING 1. Introduction The study of integer solutions to Diophantine equations is a topic that is almost as old as mathematics itself. Since its inception at the hands of Diophantus of Alexandria in 250 A.D., it has been found to relate to virtu- ally every mathematical field. The purpose of these lecture notes is to focus attention upon an aspect of Diophantine equations that has only crystallised within the last few decades, and which exhibits a fascinating interplay be- tween the subjects of analytic number theory and algebraic geometry. Suppose that we are given a polynomial f ∈ Z[x1, . . . , xn], and write Sf := {x = (x1, . . . , xn) ∈ Zn \ {0} : f(x) = 0} for the corresponding locus of non-zero solutions. There are a number of basic questions that can be asked about the set Sf . When is Sf non-empty? How large is Sf when it is non-empty? When Sf is infinite can we describe the set in some way? A lot of the work to date has been driven by trying to understand the situation for equations in only n = 2 or 3 variables. The last 50 years in particular has delivered a remarkable level of understanding concerning the arithmetic of curves. In stark contrast to this, the situation for equations in 4 or more variables remains a relatively untamed frontier, with only a scattering of results available. We will restrict attention to the study of Diophantine equations f = 0 for which the corresponding zero set Sf is infinite. The description that we will aim for is quantative in nature, the main goal being to understand how the counting function N(f ;B) := #{x ∈ Sf : |x| 6 B} (1.1) behaves, as B → ∞. Here, as throughout these lecture notes, |z| denotes the norm max16i6n |zi| for any z ∈ Rn. Aside from being intrinsically interesting in their own right, as we will see shortly, the study of functions likeN(f ;B) is often an effective means of determining whether or not the equation f = 0 has any non-trivial integer solutions at all. In many applications of the Hardy–Littlewood circle method, for example, one is able to prove that Sf is non-empty by showing that N(f ;B) > 0 for large enough values of B. In fact the method usually carries with it a proof of the fact that Sf is infinite. In the context of the circle method at least, it is useful to have a general idea of which polynomials f might have an infinite zero locus Sf . Suppose that f ∈ Z[x1, . . . , xn] has degree d > 1. Then for the vectors x ∈ Zn counted by N(f ;B), the values of f(x) will all be of order Bd. In fact a positive proportion of them will have exact order Bd. Thus the probability that a randomly chosen value of f(x) should vanish might be expected to be of order 1/Bd. Since the number of x to be considered has order Bn, this leads us to the following general expectation. Heuristic. When n > d we have Bn−d ≪ N(f ;B) ≪ Bn−d. (1.2) As a crude first approximation, therefore, this heuristic tells us that we might expect polynomials whose degree does not exceed the number of vari- ables to have infinitely many solutions. Unfortunately there are a number THE MANIN CONJECTURE IN DIMENSION 2 3 of things that can conspire to upset this heuristic expectation. First and foremost, local conditions will often provide a reason for N(f ;B) to be iden- tically zero no matter the values of d and n. By local obstructions we mean that the obvious necessary conditions for Sf to be non-empty fail. These are the conditions that the equation f(x) = 0 should have a real solution x ∈ Rn, and secondly, that the congruence f(x) ≡ 0 (mod pk) should be soluble for every prime power pk. When f is homogeneous we must take care to ignore the trivial solution x = (0, . . . , 0) in both cases. It is quite easy to construct examples that illustrate the failure of these local conditions. For example, when d is even, the equation x2d1 + · · ·+ x2dn = 0 doesn’t have any integer solutions, since it patently doesn’t have any real solutions. Let us now exhibit an example, due to Mordell [50], of a poly- nomial equation that fails to have integer solutions because it fails to have solutions as a congruence modulo a prime p. Let K be an algebraic number field of degree d, with ring of integers OK . Write N(y1, . . . , yd) := NK/Q(y1ω1 + · · · + ydωd) for the corresponding norm form, where ω1, . . . , ωd is a basis for K over Q. It is clear that N is a homogeneous polynomial of degree d, with coefficients in Z. Exercise 1. Let y ∈ Zn and let p be a rational prime such that the ideal (p) ⊂ OK is prime. Show that p | N(y) if and only if p | y. We define the homogeneous polynomial f1 := N(x1, . . . , xd) + pN(xd+1, . . . , x2d) + · · ·+ pd−1N(xd2−d+1, . . . , xd2), (1.3) which has degree d and d2 variables. We claim that the only integer solution to the equation f1(x) = 0 is the trivial solution x = 0. To see this we argue by contradiction. Thus we suppose there to be a vector x ∈ Zd2 such that f1(x) = 0, with gcd(x1, . . . , xd2) = 1. Viewed modulo p we deduce that p | N(x1, . . . , xd), whence p | x1, . . . xd by Exercise 1. Writing xi = pyi for 1 6 i 6 d, and substituting into the equation f1 = 0, we deduce that pd−1N(y1, . . . , yd) +N(xd+1, . . . , x2d) + · · ·+ pd−2N(xd2−d+1, . . . , xd2) = 0. But then we deduce in a similar fashion that p | xd+1, . . . , x2d. We may clearly continue in this fashion, ultimately concluding that p | x1, . . . , xd2 , which is a contradiction. This polynomial illustrates that for any d it is possible to construct examples of homogeneous polynomials in d2 variables that have no non-zero integer solutions. This fits with the facts rather well: when d = 2 we know from Meyer’s theorem that an indefinite quadratic form always has non-trivial solutions as soon as its rank is at least 5. Similarly, it is conjectured that 10 variables are always enough to ensure the solubility in integers of an arbitrary homogeneous cubic equation. So far we have only seen examples of polynomials f for which the zero locus Sf is empty. In this case the corresponding counting function N(f ;B) 4 T.D. BROWNING is particularly easy to estimate! There are also examples which show that N(f ;B) may grow in quite unexpected ways, even when n > d. An equation that illustrates excessive growth is provided by the polynomial f2 := x 1 − x2(xd−13 + · · · + x Here there are “trivial” solutions of the type (0, 0, a3, . . . , an) which already contribute ≫ Bn−2 to the counting function N(f ;B), whereas (1.2) predicts that we should have exponent n− d. It is also possible to construct examples of varieties which demonstrate inferior growth. Let n > d2 and choose any d2 linear forms L1, . . . , Ld2 ∈ Z[x1, . . . , xn] that are linearly independent over Q. Consider the form f3 := f1(L1(x1, . . . , xn), . . . , Ld2(x1, . . . , xn)), where f1 is given by (1.3). Then it is clear that N(f3;B) has the same order of magnitude as the counting function associated to the system of linear forms L1 = · · · = Ld2 = 0. Since these forms are linearly independent we deduce that N(f3;B) has order of magnitude B n−d2 , whereas (1.2) led us to expect an exponent n− d. We have seen lots of reasons why (1.2) might fail — how about some evidence supporting it? One of the most outstanding achievements in this direction is the following very general result due to Birch [4]. Theorem 1.1. Suppose f ∈ Z[x1, . . . , xn] is a non-singular homogeneous polynomial of degree d in n > (d−1)2d variables. Assume that f(x) = 0 has non-trivial solutions in R and each p-adic field Qp. Then there is a constant cf > 0 such that N(f ;B) ∼ cfBn−d, as B → ∞. Birch’s result doesn’t apply to either of the polynomials f2, f3 that we con- sidered above, since both of these actually have a rather large singular locus. Since generic homogeneous polynomials are non-singular, Birch’s result an- swers our initial questions completely for typical forms with n > (d− 1)2d. It would be of considerable interest to reduce the lower bound for n, but ex- cept for d 6 4 this has not been done. Theorem 1.1 is established using the circle method, and exhibits a common feature of all Diophantine problems successfully tackled via this machinery: the number of variables involved needs to be large compared to the degree. In particular, there is an obvious disparity between the range for n in Birch’s result and the range for n in (1.2). The main aim of these lecture notes is to discuss the situation when n is comparable in size with d. It turns out that phrasing things in terms of single polynomial equations is far too restrictive. It is much more satisfactory to work with projective algebraic varieties V ⊆ Pn−1. All of the varieties that we will work with are assumed to be cut out by a finite system of homogeneous equations, all of which are defined over Q. In line with the above, our main interest lies with those varieties for which we expect the set V (Q) = V ∩ Pn−1(Q) to be infinite. Let x = [x] ∈ Pn−1(Q) be a projective rational point, with x = (x1, . . . , xn) ∈ Zn chosen so that gcd(x1, . . . , xn) = 1. Then we define THE MANIN CONJECTURE IN DIMENSION 2 5 the height of x to be H(x) := |x|, where as usual |z| denotes the norm max16i6n |zi|. Given any subset U ⊆ V , we may then define the counting function NU (B) := #{x ∈ U(Q) : H(x) 6 B}, (1.4) for each B > 1. The main difference between this counting function and the quantity introduced in (1.1) is that we are now only interested in primitive integer solutions, by which we mean that the components of the vector x ∈ Zn should share no common prime factors. When the polynomial in (1.1) is homogeneous, this formulation has the advantage of treating all scalar multiples of a given non-zero integer solution as a single point. Recall the definition of the Möbius function µ : N → {0, 1}, which is given µ(n) = 0, if p2 | n for some prime p, 1, if n = 1, (−1)r, if n = p1 · · · pr for distinct primes p1, . . . , pr. The Möbius function is a multiplicative arithmetic function, and will play a very useful rôle in our work. Exercise 2. Let S ⊆ Zn be an arbitrary set. Show that #{x ∈ S : gcd(x1, . . . , xn) = 1} = µ(k)#{x ∈ S : k | xi, (1 6 i 6 n)}. We now have the tools with which to relate the counting function (1.4) to our earlier counting function N(f ;B) in (1.1), when U = V and V ⊂ Pn−1 is a hypersurface with underlying homogeneous polynomial f ∈ Z[x1, . . . , xn]. On noting that x and −x represent the same point in Pn−1, it follows from Exercise 2 that NV (B) = µ(k)N(f ;B/k). (1.5) When f is non-singular of degree d, with n > (d − 1)2d, it can be deduced from Theorem 1.1 that NV (B) ∼ c̃fBn−d, where c̃f = 12ζ(n− d) −1cf . Returning to the counting function (1.4), it is easy to check that NU (B) is bounded for each B, no matter what the choice of U and V . This fol- lows on combining the fact that NV (B) 6 NPn−1(B) with the self-evident inequalities NPn−1(B) 6 #{x ∈ Zn : |x| 6 B} 6 (2B + 1)n. In fact it is not so hard to establish an asymptotic formula for NPn−1(B). Exercise 3. Let n > 2. Use Exercise 2 to show that NPn−1(B) = Bn +On Bn−1(logB)bn where b2 = 1 and bn = 0 for n > 2. 1.1. Notation. Before embarking on the main thrust of these lecture notes, we take a moment to summarise some of the key pieces of notation that we will make use of. 6 T.D. BROWNING • A(x) = O(B(x)) means that there exists a constant c > 0 and x0 ∈ R such that |A(x)| 6 cB(x) for all x > x0. Throughout our work we will follow the convention that the implied constant is absolute unless explicitly indicated otherwise by an appropriate subscript. We will often use the alternative notation A(x) ≪ B(x) or B(x) ≫ A(x). • A(x) ≍ B(x) means A(x) ≪ B(x) ≪ A(x). • A(x) = o(B(x)) means limx→∞A(x)/B(x) = 0. • A(x) ∼ B(x) means limx→∞A(x)/B(x) = 1. • N = {1, 2, 3, ...} will denote the set of natural numbers. • Zn will denote the set of primitive vectors in Zn, and Zn∗ will denote the set of v ∈ Zn such that v1 · · · vn 6= 0. • |z| := max16i6n |zi|, for any vector z ∈ Rn. 1.2. The Manin conjectures. Around 1989 Manin initiated a program to relate the asymptotic behaviour of counting functions to the intrinsic geometry of the underlying variety, for suitable families of algebraic varieties. It is precisely this rich interplay between arithmetic and geometry that this set of lecture notes aims to communicate. Several of the varieties that we have looked at so far have many rational points, in the sense that NV (B) grows like a power of B. For such varieties it is natural to look at the quantity βV := lim logNV (B) assuming that this limit exists. In general we may consider βU for any Zariski open subset U ⊆ V . It is clear that βU gives a measure of “how large” the set U(Q) is, since we will have BβU−ε ≪ NU (B) ≪ BβU+ε for sufficiently large values of B and any ε > 0. The insight of Manin was to try and relate βU to the geometry of V via the introduction of a certain quantity α(V ). Before defining this quantity we will need some facts from algebraic geometry. The facts that we will need are summarised in more detail in the book of Hindry and Silverman [44, §A]. Assume that V ⊂ Pn−1 is non-singular and let Div(V ) be the free abelian group generated by finite formal sums of the shape D = nY Y , with nY ∈ Z and Y running over geometrically irreducible codimension 1 subvarieties of V . A divisor D ∈ Div(V ) is effective if nY > 0 for all Y , and D is said to be principal if D = Y ordY (f)Y = Df , say, for some rational function f ∈ C(V ). The intuitive idea behind the definition of the ordY function for a codimension 1 subvariety Y is that ordY (f) = k if f has a zero of order k along Y , while ordY (f) = −k if f has a pole of order k along Y . If f has neither a zero nor a pole along Y , then ordY (f) = 0. Since Df + Dg = Dfg and D1/f = −Df , the principal divisors form a subgroup PDiv(V ) of Div(V ). We define the geometric Picard group associated to V to be (V ) := Div(V )/PDiv(V ). A divisor class [D] ∈ PicQ(V ) is effective if there exists an effective di- visor in the class. One may also construct the geometric Néron–Severi THE MANIN CONJECTURE IN DIMENSION 2 7 group NSQ(V ), which is Div(V ) modulo a further equivalence relation called “algebraic equivalence”. When V is covered by curves of genus zero, as in all the cases of interest to us in these lecture notes, it turns out that NSQ(V ) = PicQ(V ). We illustrate the definition of PicQ(V ) by calculating it in the simplest possible case V = Pn−1. Lemma 1.1. We have Pic (Pn−1) = Z. Proof. An irreducible divisor on Pn−1 has the form Y = {F = 0} for some absolutely irreducible form F ∈ C[x1, . . . , xn]. For such a divisor, define the degree of Y to be degY = degF . Extend the definition of degree additively, so that nY deg Y. The map deg : Div(Pn−1) → Z is clearly a homomorphism, and to establish the lemma it will suffice to show that the kernel of this map is precisely the subgroup PDiv(Pn−1). To see this, we note that degDf = 0 for any rational function f = F1/F2. Indeed, the sum of the positive degree terms will be degF1, whereas the sum of the negative degree terms will be degF2, and this two degrees must coincide in order to have a well-defined rational function. Conversely, if D = n1Y1 + · · · + nkYk has degree zero, with Yi = {Fi = 0} for 1 6 i 6 k, then f = F 1 · · ·F k is a well-defined rational function on Pn−1 with Df = D. This completes the proof of the lemma. � Returning to the setting of arbitrary non-singular varieties V ∈ Pn−1, let H ∈ Div(V ) be a divisor corresponding to a hyperplane section. Further- more, let KV ∈ Div(V ) be the canonical divisor. This is a common abuse of notation: really KV refers to the class of Dω in PicQ(V ) for any differen- tial (dimV )-form ω of V . It would take us too far afield to include precise definitions of these objects here. We may now define the real number α(V ) := inf{r ∈ R : r[H] + [KV ] ∈ Λeff(V )}, where Λeff(V ) := {c1[D1] + · · ·+ ck[Dk] : ci ∈ R>0, [Di] ∈ NSQ(V ) effective} is the so-called effective cone of divisors. It does not matter too much if this definition is currently meaningless: the main thing is that α(V ) depends in an explicit way on the geometry of V over C. We now have the following basic conjecture due to Batyrev and Manin [1, Conjecture A]. Conjecture 1.1. For all ε > 0 there exists a Zariski open subset U ⊆ V such that βU 6 α(V ) + ε. A non-singular variety V ⊂ Pn−1 is said to be Fano if KV does not lie in the closure of the effective cone Λeff(V ) ⊂ NSQ(V )⊗Z R. This is equivalent to −KV being ample, and implies in particular that V is covered by rational curves. As an example, suppose that V is a complete intersection, with V = W1 ∩ · · · ∩ Wt for hypersurfaces Wi ⊂ Pn−1 of degree di. Then V is Fano if and only if d1+ · · ·+dt < n. With this in mind we have the following supplementary prediction. 8 T.D. BROWNING Conjecture 1.2. Assume that V is Fano and V (Q) is Zariski dense in V . Then there exists a Zariski open subset U ⊆ V such that βU = α(V ). We have α(V ) = n − d1 − · · · − dt when V is a non-singular complete intersection as above. In particular, when V is a hypersurface of degree d we may deduce from Theorem 1.1 that Conjecture 1.2 holds when n is sufficiently large in terms of d. It also holds for n > 3 when d = 2 (see Heath-Brown [37], for example). Finally we remark that Conjecture 1.2 holds for projective space. This follows from Exercise 3 and the fact that [KPn−1 ] = [−nH] in PicQ(Pn−1), whence α(Pn−1) = n. The title of these lecture notes suggests that we will focus our attention on the situation for varieties of dimension 2. Before doing so, let us consider the situation for curves briefly. For simplicity we will discuss only projective plane curves V ⊂ P2 of degree d. There is a natural trichotomy among such curves, according to the genus g of the curve. For curves with g = 0, otherwise known as rational curves, it is possible to show that NV (B) ∼ 2/d. This is in complete accordance with the Manin conjecture. It is an amusing exercise to check that such an asymptotic formula holds with d = 2 when V is given by the equation x21 + x 2 = x 3, for example. When g = 1 and V (Q) 6= ∅, the curve is elliptic and it has been shown by Néron [44, Theorem B.6.3] that NV (B) ∼ cV (logB)rV /2, where rV denotes the rank of V . Thus although there can be infinitely many points in V (Q), we see that the corresponding counting function grows much more slowly than for rational curves. Elliptic curves are not Fano, and so this is not covered by the Manin conjecture. However it does confirm Conjecture 1.1, since α(V ) = 0. When g > 2 the work of Faltings [28] shows that V (Q) is always finite, and so it does not make sense to study NV (B). Let us now concern ourselves with Fano varieties of dimension 2. We begin with some simple-minded numerics. Suppose that we are given a Fano variety V of dimension 2 and degree d, which is a non-singular complete intersection in Pn−1. Thus V = W1 ∩ · · · ∩Wt for hypersurfaces Wi ⊂ Pn−1 of degree di, and we assume that the intersection is transversal at a generic point of V . We are not interested in hyperplane sections of V , and so we will assume without loss of generality that di > 2 for each 1 6 i 6 t. Then the following inequalities must be satisfied: (1) d1 + · · ·+ dt < n, [Fano] (2) n− 1− t = 2, [complete intersection of dimension 2] (3) d = d1 · · · dt, [Bézout] (4) dt > · · · > d1 > 2. It follows that the only possibilities are (d; d1, . . . , dt;n; t) ∈ (2; 2; 4; 1), (3; 3; 4; 1), (4; 2, 2; 5; 2) These surfaces correspond to a quadric in P3, a cubic surface in P3, and an intersection of 2 quadrics in P4, respectively. We have already observed that the Manin conjecture holds for quadrics. Hence one would like to examine the latter two surfaces. In fact these are the most familiar examples of “del Pezzo surfaces”. We will see in §1.5 that not all del Pezzo surfaces are THE MANIN CONJECTURE IN DIMENSION 2 9 complete intersections, and so we have missed out on several surfaces in this analysis. Nonetheless, a substantial portion of these lecture notes will focus on cubic surfaces in P3 and intersections of 2 quadrics in P4. It is now time to give a formal definition of a del Pezzo surface. Let us begin with a discussion of non-singular del Pezzo surfaces. Let d > 3. Then a del Pezzo surface of degree d is a non-singular surface S ⊂ Pd of degree d, with very ample anticanonical divisor −KS. This latter condition is equivalent to the equality [−KS ] = [H] in PicQ(S), for a hyperplane section H ∈ Div(S). The facts that we will recall here are all established in the book of Manin [49], for example. It is well-known that del Pezzo surfaces S ⊂ Pd arise either as the quadratic Veronese embedding of a quadric in P3, which is a del Pezzo surface of degree 8 in P8 (isomorphic to P1 × P1), or as the blow-up of P2 along 9 − d points in general position, in which case the degree of S satisfies 3 6 d 6 9. We will meet the notion of “general position” when d = 3 in §2.1. Since [−KS ] = [H] in PicQ(S), we see that α(S) = 1 for non-singular del Pezzo surfaces of degree d. The geometry of del Pezzo surfaces is very beautiful and well-worth study- ing. However, to avoid straying from the main focus of these lecture notes, we will content ourselves with simply quoting the facts that are needed. One of the remarkable features of del Pezzo surfaces of small degree is that each such surface contains finitely many lines. The precise number of lines is recorded in Table 1. d number of lines Table 1. Lines on non-singular del Pezzo surfaces of degree d It turns out that dealing with del Pezzo surfaces of degree d gets easier as the degree increases. In these lecture notes we will focus our attention on the del Pezzo surfaces of degree d ∈ {3, 4, 5, 6}. It turns out that for del Pezzo surfaces of degree d, the geometric Picard group PicQ(S) is a finitely generated free Z-module, with PicQ(S) ∼= Z10−d. (1.6) This is established in Manin [49], where an explicit basis for the group is also provided (see §2.1 for a concrete example). Let K be a splitting field for the finitely many lines contained in S. The final invariant that we will need to introduce is the Picard group Pic(S) := Pic (S)Gal(K/Q) (1.7) of the surface. This is just the set of elements in PicQ(S) that are fixed by the action of the Galois group. Write ρS for the rank of Pic(S). Let U ⊂ S be the Zariski open subset formed by deleting the finitely many lines from S. Then we have the following [1, Conjecture C ′]. 10 T.D. BROWNING Conjecture 1.3. Suppose that S ⊂ Pd is a non-singular del Pezzo surface of degree d. Then there exists a non-negative constant cS,H such that NU (B) = cS,HB(logB) 1 + o(1) . (1.8) In these lecture notes this is what will commonly be termed as “the Manin conjecture”. Note that the exponent of B agrees with Conjecture 1.2, since α(S) = 1. Moreover the exponent of logB is at most 9 − d, since the geometric Picard group has rank 10 − d. We will develop some heuristics to support this power of logB in §2. The value of the constant cS,H has also received a conjectural interpretation at the hands of Peyre [51], an interpretation that has been extended by Batyrev and Tschinkel [2], and by Salberger [55]. There are a number of refinements to Conjecture 1.3 that are currently emerging, which we will not have space to discuss here. Some of these are discussed in more details in the author’s survey [14, §2], for example. One such refinement is that there should exist a polynomial P ∈ R[x] of degree ρS − 1, and a real number δ > 0, such that NU (B) = BP (logB) +O(B 1−δ). (1.9) One obviously expects the leading coefficient of P to agree with Peyre’s prediction, but there has so far been rather little investigation of the lower order terms. All of the del Pezzo surfaces that we have discussed so far have been non-singular. In the following section we will meet some singular ones. 1.3. Degree 3 surfaces. The del Pezzo surfaces S ⊂ P3 of degree 3 are the geometrically integral cubic surfaces in P3, which are not ruled by lines. In particular, this definition covers both singular and non-singular del Pezzo surfaces of degree 3. Given such a surface S defined over Q, we may always find an absolutely irreducible cubic form C ∈ Z[x1, x2, x3, x4] such that S is defined by the equation C = 0. In this section we will discuss the Manin conjecture in the context of cubic surfaces. Let us begin by considering the situation for non-singular cubic surfaces, for which one takes U ⊂ S to be the open subset formed by deleting the famous 27 lines. Peyre and Tschinkel [53, 54] have provided ample numerical evidence for the validity of the Manin conjecture for diagonal cubic surfaces. However we are still rather far away from proving it for any single example. The best upper bound available is NU (B) = Oε,S(B 4/3+ε), (1.10) due to Heath-Brown [38]. This applies when the surface S contains 3 copla- nar lines defined over Q, and in particular to the Fermat cubic surface x31 + x 2 = x 3 + x Heath-Brown [40] has extended the bound (1.10) to all non-singular cubic surfaces, subject to a natural conjecture concerning the size of the rank of elliptic curves over Q. The problem of proving lower bounds is somewhat easier. Under the as- sumption that S contains a pair of skew lines defined over Q, Slater and Swinnerton-Dyer [59] have shown that NU (B) ≫S B(logB)ρS−1, as pre- dicted by the Manin conjecture. This does not apply to the Fermat cubic THE MANIN CONJECTURE IN DIMENSION 2 11 surface, however, since the only skew lines contained in this surface are defined over Q( It turns out that much more can be said if one permits S to contain isolated singularities. For the remainder of this section let S ⊂ P3 be a geometrically integral cubic surface, which has only isolated singularities and is not a cone. Then there exists a unique “minimal desingularisation” π : S̃ → S of the surface, which is just a sequence of blow-up maps, and furthermore, that the asymptotic formula (1.8) is still expected to hold, with ρS now taken to be the rank of the Picard group of S̃. As usual U ⊂ S is obtained by deleting all of the lines from S. The classification of singular cubic surfaces S is a well-established subject, and can be traced back to the work of Cayley [18] and Schläfli [56] over a century ago. A contemporary classification of singular cubic surfaces has since been given by Bruce and Wall [16], over Q. Of course, if one is interested in a classification over the ground field Q, then many more singularity types can occur (see Lipman [48], for example). In Table 2 we have provided a classification table of the 20 singularity types over Q, including the number of lines that each surface contains. We will presently meet some explicit examples of cubic forms C ∈ Z[x1, x2, x3, x4] that typify some surface types. type # lines singularity i 21 A1 ii 16 2A1 iii 15 A2 iv 12 3A1 v 11 A1 +A2 vi 10 A3 vii 9 4A1 viii 8 2A1 +A2 ix 7 A1 +A3 x 7 2A2 xi 6 A4 xii 6 D4 xiii 5 2A1 +A3 xiv 5 A1 + 2A2 xv 4 A1 +A4 xvi 3 A5 xvii 3 D5 xviii 3 3A2 ix 2 A1 +A5 xx 1 E6 Table 2. Classification (over Q) of singular del Pezzo sur- faces of degree 3 in P3 The labelling of each singularity type corresponds to the “Dynkin dia- gram” that describes the intersection behaviour of the exceptional divisors obtained by resolving the singularities in the surface. For example, consider 12 T.D. BROWNING the cubic surface S1 = {x21x3 + x2x23 + x34 = 0}. (1.11) Up to isomorphism over Q this is the unique cubic surface of type xx in the table, and is discussed further in [34]. The process of resolving the singularity gives 6 exceptional divisors E1, . . . , E6 and produces the minimal desingularisation S̃1 of the surface S1. If L denotes the strict transform of the unique line on S1, then L,E1, . . . , E6 satisfy the intersection behaviour encoded in the Dynkin diagram E1 E3 E6 E5 E4 L There is a line connecting two divisors in this diagram if and only if they meet in S̃1. In what follows the reader can simply think of these Dynkin diagrams as a convenient way to label the surface type. It turns out, as discussed in [16], that some types of surfaces do not have a single normal form, but an infinite family. This happens precisely for the surfaces of type i, ii, iii, iv, v, vi and ix. By [16, Lemma 4] the type xii surface, with a D4 singularity, is the only surface that has more than one normal form, but not a family. In fact it has precisely two normal forms, given by S2 = {x1x2(x1 + x2) + x4(x1 + x2 + x3)2 = 0} (1.12) S3 = {x1x2x3 + x4(x1 + x2 + x3)2 = 0}. (1.13) That these equations actually define distinct surfaces can be seen by calcu- lating the corresponding Hessians in each case. Let S̃ denote the minimal desingularisation of any surface S from Table 2, and assume that all of its singularities and lines are defined over Q. In this case the surface is said to be split, and it follows that the Picard group of S̃ has maximal rank 7 by (1.6), since Pic(S̃) = Pic (S̃). For example, [L], [E1], . . . , [E6] provide a basis for Pic(S̃1). One would like to try and establish (1.8) for each such surface S, with ρS = 7. Several del Pezzo surfaces are actually special cases of varieties for which the Manin conjecture is already known to hold. Recall that a variety of dimension D is said to be toric if it contains the algebraic group variety GDm as a dense open subset, whose natural action on itself extends to all of the variety. The Manin conjecture has been established for all toric varieties by Batyrev and Tschinkel [3]. It can be checked that the surface representing type xviii is toric. In fact this particular surface has been studied by numerous authors, including la Bretèche [5], la Bretèche and Swinnerton-Dyer [11], Fouvry [29], Heath-Brown and Moroz [43], and Salberger [55]. Of the unconditional asymptotic formulae obtained, the most impressive is the first. This consists of an estimate like (1.9) for any δ ∈ (0, 1/8), with degP = 6. The next surface to have received serious attention is the Cayley cubic surface S4 = {x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 = 0}, THE MANIN CONJECTURE IN DIMENSION 2 13 which is the type vii surface in the table. Heath-Brown [42] has shown that there exist absolute constants A1, A2 > 0 such that A1B(logB) 6 NU (B) 6 A2B(logB) An estimate of precisely the same form has been obtained by the author [13] for the D4 surface S3 in (1.13). In both cases the lines in the surface are all defined over Q, so that the surfaces are split. Thus the corresponding Picard groups have rank 7 and the exponents of B and logB agree with Manin’s prediction. In this set of lecture notes we will establish an upper bound for the remaining D4 cubic surface S2 in (1.12). This will be carried out in §4 in two basic attacks. First we will give a completely self-contained account of the upper bound NU (B) = Oε(B 1+ε), for any ε > 0. Next, by making use of the work in [13], we will establish the following finer result. Theorem 1.2. Let S2 be given by (1.12). We have NU (B) ≪ B(logB)6. The cubic surface S2 contains the unique singular point [0, 0, 0, 1], together with the 6 lines xi = x4 = 0, x1 + x2 = xj = 0, xi = x1 + x2 + x3 = 0, (1.14) for distinct indices i ∈ {1, 2} and j ∈ {3, 4}. Thus the surface is split and it follows that Theorem 1.2 agrees with the Manin conjecture. Exercise 4. Check that (1.14) are all of the lines contained in S2. The final surface to have been studied extensively is the E6 cubic surface S1 that we discussed above. The figure below, which was constructed by Der- enthal, shows all the rational points of height 6 1000 on this surface. Recent joint work of the author with la Bretèche and Der- enthal [12] has succeeded in establishing the Manin conjecture for this surface. In fact an asymptotic for- mula of the shape (1.9) is achieved, with P of degree 6 and any δ ∈ (0, 1/11). It should be remarked that Dr. Michael Joyce has also established the Manin conjecture for S1 in his doctoral thesis, albeit with a weaker error term of O(B(logB)5). 1.4. Degree 4 surfaces. A quartic del Pezzo surface S ⊂ P4, that is defined over Q, can be recognised as the zero locus of a suitable pair of quadratic forms Q1, Q2 ∈ Z[x1, . . . , x5]. Again we do not stipulate that the surface should be non-singular. As usual let U ⊂ S denote the open subset formed by deleting all of the lines from S. Let us begin by discussing the situation for non-singular surfaces, where there are 16 lines to delete. The best result available is the estimate NU (B) = Oε,S(B 1+ε), valid for any quartic non- singular del Pezzo surface S ⊂ P4 containing a conic defined over Q. This 14 T.D. BROWNING result was established in an unpublished note due to Salberger in 2001. It would be interesting to see whether one could adapt the methods of [40] to show that NU (B) = Oε,S(B 5/4+ε) for any non-singular del Pezzo surface of degree 4, assuming the same hypothesis on the ranks of elliptic curves. As previously, it emerges that much more can be said if one permits S to contain isolated singularities. For the remainder of this section let S ⊂ P4 be a geometrically integral intersection of two quadric hypersurfaces, which has only isolated singularities and is not a cone, and let S̃ be the minimal desingularisation of S. Then the asymptotic formula (1.8) is still expected to hold, with ρS now taken to be the rank of the Picard group of S̃, and U ⊂ S obtained by deleting all of the lines from S. In particular, when S is split one always has ρS = 6. The classification of singular quartic del Pezzo surfaces can be extracted from the work of Hodge and Pedoe [45, Book IV, §XIII.11], where it is phrased in terms of the so-called Segre symbol. The Segre symbol of a matrix M ∈ M5(C) is defined as follows. If the Jordan form of M has Jordan blocks of sizes a1, . . . , an, with a1+ · · ·+an = 5, then the Segre symbol is the symbol (a1, . . . , an) with extra parentheses around the Jordan blocks with equal eigenvalues. Suppose that our quartic del Pezzo surface S is defined by a pair of quadric hypersurfaces, with underlying symmetric matrices A,B ∈ M5(Q). Then the Segre symbol of S is defined to be the Segre symbol associated to A−1B. A crucial property of the Segre symbol is that it does not depend on the choice ofA andB in the pencil of quadrics defining S. Since we are assuming that S is not a cone, one may always suppose that A,B are chosen so that A has full rank. To illustrate the calculation of the Segre symbol, let us consider the sur- face S defined by the pair of equations x1x2 + x3x4 = 0, x1x4 + x2x3 + x3x5 + x4x5 = 0. (1.15) Let A,B ∈ M5(Q) denote the underlying matrices of the first and second equations, respectively. Then A has rank 4, and so we replace it with A + 2B, which has full rank. A simple calculation reveals that the matrix (A+ 2B)−1B has Jordan form  0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1  This matrix has 4 Jordan blocks, one of size 2 and the rest of size 1. The eigenvalues associated to the different Jordan blocks are all different, and so it follows that the surface (1.15) has Segre symbol (2, 1, 1, 1). Exercise 5. Find any matrices A,B ∈ M5(Q) so that the corresponding surface xtAx = xtBx = 0 is non-singular. Show that the surface has Segre symbol (1, 1, 1, 1, 1). THE MANIN CONJECTURE IN DIMENSION 2 15 So far we have given a very easy way to check the isomorphism type of a given singular del Pezzo surface of degree 4. How do we match this up with a classification according to the singularity type, as in our discussion of cubic surfaces in Table 2? It turns out that up to isomorphism over Q, there are 15 possible singularity types for S. Over Q, Coray and Tsfasman [23, Proposition 6.1] have calculated the extended Dynkin diagrams for all of the 15 types, and Knörrer [47] has determined the precise correspondence between the singularity type and the Segre symbol. Table 3 is extracted from this body of work, and matches each possible singularity type with the Segre symbol, and the number of lines that the surfaces contains. type Segre symbol # lines singularity i (2,1,1,1) 12 A1 ii (2,2,1) 9 2A1 iii ((1,1),1,1,1) 8 2A1 iv (3,1,1) 8 A2 v ((1,1),2,1) 6 3A1 vi (3,2) 6 A1 +A2 vii (4,1) 5 A3 viii ((2,1),1,1) 4 A3 ix ((1,1),(1,1),1) 4 4A1 x ((1,1),3) 4 2A1 +A2 xi ((2,1),2) 3 A1 +A3 xii (5) 3 A4 xiii ((3,1),1) 2 D4 xiv ((2,1),(1,1)) 2 2A1 +A3 xv ((4,1)) 1 D5 Table 3. Classification (over Q) of singular del Pezzo sur- faces of degree 4 in P4 In general, given a particular Segre symbol, its not entirely straightfor- ward to determine explicit equations that define a singular del Pezzo surface of degree 4 having this symbol. Nonetheless in Table 4 we have done pre- cisely this for each Segre symbol that occurs. In doing so we have retrieved some of the calculations carried out by Derenthal [26]. An important feature of the table is that the surfaces recorded are split over Q. It remains a sig- nificant open challenge to establish the Manin conjecture for the 15 surfaces given in Table 4. This will furnish a proof of the Manin conjecture for the class of split singular del Pezzo surfaces of degree 4 that are defined over Q, and is undoubtedly a key stepping stone on the way towards a resolution of the conjecture for all del Pezzo surfaces. There is huge potential for further work in this area, and I hope that these lecture notes succeed in showing that analytic number theorists are well placed to make an important con- tribution. Exercise 6. Calculate the Segre symbol for each of the surfaces in Table 4, and check they match up with the correct singularity type in Table 3. 16 T.D. BROWNING type Q1(x) Q2(x) i x1x2 − x3x4 x1x4 − x2x3 + x3x5 + x4x5 ii x1x2 − x3x4 x1x4 − x2x3 + x3x5 + x25 iii x1x2 − x23 x1x3 − x2x3 + x4x5 iv x1x2 − x3x4 (x1 + x2 + x3 + x4)x5 − x3x4 v x1x2 − x23 x2x3 + x23 + x4x5 vi x1x2 − x3x4 x1x5 + x2x3 + x4x5 vii x1x2 − x3x4 x1x4 + x2x4 + x3x5 viii x1x4 − (x2 − x3)x5 (x1 + x4)(x2 + x3) + x2x3 ix x1x2 − x23 x23 − x4x5 x x1x2 − x23 x2x3 − x4x5 xi x1x4 − x3x5 x1x2 + x2x4 + x23 xii x1x2 − x3x4 x1x5 + x2x3 + x24 xiii x1x4 − x2x5 x1x2 + x2x4 + x23 xiv x1x2 − x23 x21 − x4x5 xv x1x2 − x23 x1x5 + x2x3 + x24 Table 4. Split surfaces representing the 15 singularity types Whereas they share the same singularity type, the surfaces of type vii and viii differ because in the former there are 5 lines, 4 of which pass through the singularity, whereas in the latter all 4 lines pass through the singularity. Similarly, an important difference between the surfaces of type ii and iii in Tables 3 and 4 is that for the surface of type ii, the line joining the two singularities is contained in the surface, whereas for the surface of type iii it is not. When the 2 singular points are defined over a quadratic extension of Q, the latter surface is called an Iskovskih surface. There is ample evidence available (see Coray and Tsfasman [23], for example) to the effect that Iskovskih surfaces are the most arithmetically interesting surfaces among the singular del Pezzo surfaces of degree 4. In fact they are the only such surfaces for which the Hasse principle can fail to hold. The main focus of these lecture notes is upon the situation for split singular del Pezzo surfaces, and so we will say no more about Iskovskih surfaces here. As usual, let S̃ denote the minimal desingularisation of any surface S from Table 4. Then the Picard group of S̃ has rank ρS = 6. The goal recorded above is to try and establish (1.8) for each S. As in the case of singular cubic surfaces several of the surfaces are actually special cases of varieties for which the Manin conjecture is already known to hold. Thus it can be shown that the surfaces representing types ix, x, xiv are all toric, so that (1.8) already holds in these cases by the work of Batyrev and Tschinkel [3]. In a very real sense these surfaces are the “easiest” to deal with in our list. Exercise 7. Show that NU (B) = Oε(B 1+ε) for the surfaces of type ix, x and xiv. It has also been shown by Chambert-Loir and Tschinkel [19] that the Manin conjecture is true for equivariant compactifications of the algebraic group G2a. Although identifying such surfaces in the table is not entirely routine, it transpires that the type xv surface (with a D5 singularity) is THE MANIN CONJECTURE IN DIMENSION 2 17 covered by this work. In joint work with la Bretèche, the author [8] has provided an independent proof of the Manin conjecture for this particular surface. In addition to obtaining a finer asymptotic formula of the shape given in (1.9), this work has provided a useful line of attack for several other singular del Pezzo surfaces. In Table 5 we have recorded a list of progress towards the final resolution of the Manin conjecture for the split singular del Pezzo surfaces of degree 4. We have included the relevant reference in the literature, and whether the result attained amounts to an asymptotic formula for the counting function, or an upper bound. We will not pay attention here to the quality of the error term in the asymptotic formula, but each upper bound is of the correct order of magnitude B(logB)5. There is still plenty left to do! type type of estimate achieved v upper bound [14] ix asymptotic formula [3] x asymptotic formula [3] xiii asymptotic formula [27] xiv asymptotic formula [3] xv asymptotic formula [8] Table 5. Summary of progress for the split singular del Pezzo surfaces of degree 4 It is also interesting to try and establish the Manin conjecture for singular del Pezzo surfaces of degree 4 that are not split over the ground field. In further joint work of the author with la Bretèche [9], the Manin conjecture is established for the surface x1x2 − x23 = 0, x21 + x2x5 + x24 = 0. This surface has a D4 singularity and is isomorphic over Q(i) to the surface of type xiii in Table 4. The Picard group of S̃ has rank 4 in this case, and an asymptotic formula of the shape (1.9) is obtained for any δ ∈ (0, 3/32), with P a polynomial of degree 3. 1.5. Degree > 5 surfaces. It turns out that all del Pezzo surfaces of degree d > 7 are toric [26, Proposition 8], and that all non-singular del Pezzo surfaces of degree d > 6 are toric. Thus (1.8) already holds in these cases by the work of Batyrev and Tschinkel [3]. For non-singular del Pezzo surfaces S ⊂ P5 of degree 5, the situation is rather less satisfactory. In fact there are very few instances for which the Manin conjecture has been established. The most significant of these is due to la Bretèche [7], who has proved the conjecture for the split non-singular del Pezzo surface S of degree 5, in which the 10 lines are all defined over Q. To be precise, if U ⊂ S denotes the open subset formed by deleting the lines from S, then la Bretèche shows that NU (B) = c0B(logB) log logB for a certain constant c0 > 0. This confirms Conjecture 1.3, since we have seen in (1.6) that Pic(S) ∼= Z5 for split non-singular del Pezzo surfaces 18 T.D. BROWNING of degree 5. The other major achievement in the setting of quintic del Pezzo surfaces is a result of la Bretèche and Fouvry [10], where the Manin conjecture is established for a surface that is not split, but contains lines defined over Q(i). So far we have only discussed the situation for non-singular del Pezzo surfaces of degree d > 5. Let us now turn to the singular setting. When d = 6 it emerges that there exist such surfaces that are not toric, and so are not covered by [3]. We will focus attention on the situation for del Pezzo surfaces of degree 6, following the investigation of Derenthal [26], where the degree 5 surfaces are also considered. In view of [23, Proposition 8.3], Table 6 lists all possible types of singular del Pezzo surfaces of degree 6. type # lines singularity i 4 A1 ii 3 A1 iii 2 2A1 iv 2 A2 v 1 A1 +A2 Table 6. Classification (over Q) of singular del Pezzo sur- faces of degree 6 As noted in [26, §5], the surfaces of type i, iii and v are all toric and so do not interest us here. Any singular del Pezzo surface of degree 6 can be realised as the intersection of 9 quadrics in P6. For example, the type iv surface is cut out by the system of equations x1x6 − x4x5 = x1x7 − x2x5 = x1x7 − x3x4 = x3x7 + x4x5 + x25 = x5x7 − x3x4 = x2x7 + x24 + x4x5 = x4x7 − x2x6 = x4x6 + x5x6 + x 7 = x2x3 − x1x4 + x1x5 = 0. (1.16) In this set of lecture notes we will establish the Manin conjecture for the type ii surface, which has the simplest possible singularity. When S ⊂ P6 is a split surface of type ii, then there is unimodular change of variables that takes S into the surface with equations x21 − x2x4 = x1x5 − x3x4 = x1x3 − x2x5 = x1x6 − x3x5 = x2x6 − x23 = x4x6 − x25 = x21 + x1x4 + x5x7 = x1x2 + x 1 + x3x7 = x1x3 + x1x5 + x6x7 = 0. (1.17) Let S̃ denote the minimal desingularisation of S. It follows from (1.6) that Pic(S̃) ∼= Z4 since S is split. We will establish the following result in §3. Theorem 1.3. Let S ⊂ P6 be the A1 surface given by (1.17). Then there exist constants c1, c2 > 0 such that NU (B) = c1B(logB) 3 + c2B(logB) B logB where THE MANIN CONJECTURE IN DIMENSION 2 19 σ∞ = 6 {u,t,v∈R: 0<u,ut2,uv2,|tv(t−v)|61} dtdudv. (1.18) Since Pic(S̃) has rank 4, the exponents of B and logB in this asymptotic formula are in complete agreement with Conjecture 1.3. Although we will not give details here, it turns out that the value of the constant c1 also confirms the prediction of Peyre [51] in this case. It is hoped that our proof of Theorem 1.3 will encourage other researchers to try their hand at proving asymptotic formulae for NU (B). With this in mind Exercise 8 is more of a research problem, and its resolution will therefore conclude the proof of the Manin conjecture for all split (non-singular or singular) del Pezzo surfaces of degree 6. Exercise 8. Establish an asymptotic formula for the type iv surface in Table 6, with underlying equations (1.16). 1.6. Universal torsors. Universal torsors were originally introduced by Colliot-Thélène and Sansuc [20, 21] to aid in the study of the Hasse principle and weak approximation for rational varieties. Since their inception it is now well-recognised that they also have a central rôle to play in proofs of the Manin conjecture for Fano varieties, and in particular, for del Pezzo surfaces. Let S ⊂ Pd be a del Pezzo surface of degree d ∈ {3, 4, 5, 6}, and let S̃ denote the minimal desingularisation of S if it is singular, and S̃ = S otherwise. Let E1, . . . , E10−d ∈ Div(S̃) be generators for PicQ(S̃), and let E×i = Ei \ {zero section}. Working over Q, a universal torsor above S̃ is given by the action of G10−dm on the map π : E×1 ×eS · · · ×eS E 10−d → S̃. A proper discussion of universal torsors would take us too far afield, and the reader may consult the survey of Peyre [52] for further details, or indeed the construction of Hassett and Tschinkel [34]. The latter outlines an alterna- tive approach to universal torsors via the Cox ring. Given the usual open subset U ⊂ S, the general theory of universal torsors ensures that there is a partition of U(Q) into a disjoint union of patches, each of which is in bijection with a suitable set of integral points on a universal torsor above S̃. The guiding principle behind the use of universal torsors is simply that they ought to be arithmetically simpler than the original variety. In our work it will suffice to think of universal torsors as “particularly nice parametri- sations” of rational points on the surface. The universal torsors that we encounter in these lecture notes all have embeddings as affine hypersurfaces of high dimension. Moreover, in each case we will show how the underlying equation of the universal torsor can be deduced in a completely elementary fashion, without any recourse to geometry whatsoever. The torsor equations we will meet all take the shape A+B + C = 0, for monomials A,B,C of various degrees in the appropriate variables. As in many examples of counting problems for higher dimensional varieties, one can occasionally gain leverage by fixing some of the variables at the 20 T.D. BROWNING outset, in order to be left with a counting problem for a family of small dimensional varieties. If one is sufficiently clever about which variables to fix first, one is sometimes left with a quantity that we know how to estimate — and crucially — whose error term we can control once summed over the remaining variables. As a concrete example, we note that Hassett and Tschinkel [34] have calculated the universal torsor for the cubic surface (1.11). It is shown that there is a unique universal torsor above S̃1, and that it is given by the equation 4s5 + y 2s2 + y 1s3 = 0, for variables y1, y2, yℓ, s1, s2, s3, sℓ, s4, s5, s6. One of the variables does not explicitly appear in this equation, and the torsor should be thought of as being embedded in A10. It turns out that the way to proceed here is to fix all of the variables apart from y1, y2, yℓ. One may then view the equation as a congruence y22s2 ≡ −y31s21s3 (mod s3ℓs24s5), in order to take care of the summation over yℓ. This is the approach taken in [12], the next step being to employ very standard facts about the number of integer solutions to polynomial congruences that are restricted to lie in certain regions. One if left with a main term and an error term, which the remaining variables need to be summed over. While the treatment of the main term is relatively routine, the treatment of the error term presents a much more serious obstacle. The universal torsors that turn up in the proofs of Theorems 1.2 and 1.3 can also be embedded in affine space as hypersurfaces. We will see in §3 that the approach discussed above also produces results for the del Pezzo surface of degree 6 considered in Theorem 1.3. In the proof of Theorem 1.2 in §4 our approach will be more obviously geometric, and we will actually view the equation as a family of projective lines, and also as a family of conics. We will then call upon techniques from the geometry of numbers to count the relevant solutions. 2. Further heuristics We have seen in §1.2, and in particular in the statement of Conjecture 1.3, that for any del Pezzo surface S ⊂ Pd one expects a growth rate like cSB(logB) A for the counting function NU (B). We have already given some motivation for the exponent of B in (1.2). The focus of the present section is to produce much more sophisticated heuristics than we have previously met. In particular we will gain an insight into the exponent of logB that appears in the Manin conjecture. For ease of presentation we restrict attention to non-singular diagonal cubic surfaces S ⊂ P3. Thus S = {a1x31 + a2x32 + a3x33 + a4x34 = 0}, (2.1) for a = (a1, . . . , a4) ∈ N4 such that gcd(a1, . . . , a4) = 1. Define P := {3} ∪ {p : p | a1a2a3a4}. (2.2) THE MANIN CONJECTURE IN DIMENSION 2 21 This is the set of primes p for which the reduction of S modulo p is singular. Before passing to the arithmetic of diagonal cubic surfaces, we will need to discuss some of their geometry. 2.1. The lines on a cubic surface. The facts that we will need in this section are explained in detail in the books of Hartshorne [33] and Manin [49]. In general, a non-singular cubic surface S ⊂ P3 is obtained by blowing up P2 along a collection of 6 points P1, . . . , P6 in general position. By general position we mean that no 3 of them are collinear and they do not all lie on a conic. The 27 lines on the surface arise in the following way. There are 6 exceptional divisors Ei above Pi, for 1 6 i 6 6, and 15 strict transforms Li,j of the lines going through precisely 2 points Pi, Pj , for 1 6 i < j 6 6. Finally there are the 6 strict transforms Qi of the conics going through all but one of the 6 points. If Λ is the strict transform of a line in P2 that doesn’t go through any of the Pi, then a basis of the geometric Picard group (S) is given by [Λ], [E1], . . . , [E6]. The remaining divisors may be expressed in terms of these elements via the relations [Li,j] = [Λ]− [Ei]− [Ej ], [Qi] = 2[Λ]− j 6=i [Ej ]. (2.3) The class of the anti-canonical divisor −KS is given by [−KS ] = 3[Λ] −∑6 i=1[Ej ], although we will not need this fact in our work. One can check that the hyperplane section has class −3[Λ] + i=1[Ej ] in PicQ(S), so that the cubic surface has very ample anticanonical divisor, as claimed in §1.2. When S takes the shape (2.1) it is not hard to write down the 27 lines explicitly. The calculations that we present below are based on those carried out by Peyre and Tschinkel [54, §2]. Fix a cubic root α (resp. α′, α′′) of a2/a1 (resp. a3/a1, a4/a1). We will assume that α ∈ Q if a2/a1 (resp. a3/a1, a4/a1) is a cube in Q. Put , β′ = , β′′ = We denote by θ a primitive cube root of one. Let i run over elements of Z/3Z. Then the 27 lines on the cubic surface (2.1) are given by the equations iαx2 = 0, iβx4 = 0, iαx2 = 0, i+1βx4 = 0, L′′i : iαx2 = 0, i+2βx4 = 0, iα′x3 = 0, iβ′x2 = 0, M ′i : iα′x3 = 0, i+1β′x2 = 0, M ′′i : iα′x3 = 0, i+2β′x2 = 0, iα′′x4 = 0, iβ′′x3 = 0, N ′i : iα′′x4 = 0, i+1β′′x3 = 0, N ′′i : iα′′x4 = 0, i+2β′′x3 = 0. Let K = Q(θ, α, α′, α′′). It is a Galois extension of Q, and in the generic case has degree 54 with Galois group Gal(K/Q) ∼= (Z/3Z)3 ⋊ Z/2Z. We need to equate these lines to the divisors Ei, Li,j , Qi that we met earlier. There is a certain degree of freedom in doing this, as discussed in 22 T.D. BROWNING [33, §V.4], but it turns out that the choice E1 = L0, E2 = L1, E3 = L2, E4 = M1, E5 = M 2, E6 = M Q1 = L 1, Q2 = L 2, Q3 = L Q4 = M0, Q5 = M 1, Q6 = M L1,2 = L 1 , L2,3 = L 2, L1,3 = L L4,5 = M 1 , L5,6 = M2, L4,6 = M L1,4 = N0, L1,5 = N1, L1,6 = N2, L2,4 = N 1, L2,5 = N 2, L2,6 = N L3,4 = N 2 , L3,5 = N 0 , L3,6 = N (2.4) is satisfactory. In assigning lines to E1, . . . , E6, all that is required is that they should all be mutually skew. Given any cubic surface of the shape (2.1), we now have the tools with which to compute the Picard group (1.7). In fact, from this point forwards the process requires little more than basic linear algebra. Let us illustrate the procedure by calculating the Picard group for a spe- cial case. Consider the Fermat surface S1 = {x31 + x32 + x33 + x34 = 0}. (2.5) In this case α = α′ = α′′ = β = β′ = β′′ = 1, in the notation above, and K = Q(θ) is a quadratic field extension. We wish to find elements of the geometric Picard group PicQ(S1) that are fixed by the action of Gal(K/Q). Thus we want vectors c = (c0, . . . , c6) ∈ Z7 such that (c0[Λ] + c1[E1] + · · ·+ c6[E6])σ = c0[Λ] + c1[E1] + · · ·+ c6[E6], (2.6) for every σ ∈ Gal(K/Q). Under the action of Gal(K/Q) it is not hard to check that Λ and E1 are fixed, that E2 and E3 are swapped, and that E4 (resp. E5, E6) is taken to L5,6 (resp. L4,5, L4,6). Using (2.3) one sees that the left hand side of (2.6) is equal to (c0 + c4 + c5 + c6)[Λ]+c1[E1] + c2[E3] + c3[E2] − (c5 + c6)[E4]− (c4 + c5)[E5]− (c4 + c6)[E6], in Pic (S1). Thus we are interested in the space of c ∈ Z7 for which c4 + c5 + c6 = 0, c2 − c3 = 0, c4 + 2c5 = 0, c4 + 2c6 = 0. This system of homogeneous linear equations in 7 variables has underlying matrix of rank 3. Thus the space of solutions has rank 7− 3 = 4, and so we may conclude that Pic(S1) ∼= Z4. In fact a little thought reveals that the 4 elements [Λ], [E1], [E2] + [E3],−2[E4] + [E5] + [E6], provide a basis for Pic(S1). Next, consider the surface S2 = {x31 + x32 + x33 + px34 = 0} (2.7) THE MANIN CONJECTURE IN DIMENSION 2 23 for a prime p. When p = 2 or 3, the arithmetic of this surface has been considered in some detail by Heath-Brown [36], who provides some numer- ical evidence to the effect that the corresponding counting function NU (B) should grow like cpB for a certain constant cp > 0. Exercise 9. Show that Pic(S2) ∼= Z. This calculation has also been carried out by Colliot-Thélène, Kanevsky and Sansuc [22, p. 12]. More generally, it is known that the Picard group of the surface (2.1) has rank 1 if and only if the ratio aσ(1)aσ(2) aσ(3)aσ(4) is not a cube in Q, for each permutation σ of (1, 2, 3, 4). This result is due to Segre [57]. 2.2. Cubic characters and Jacobi sums. Throughout this section let p be a prime. Recall that a (multiplicative) character on Fp = Z/pZ is a map χ : F∗p → C∗ such that χ(ab) = χ(a)χ(b) for all a, b ∈ F∗p. The trivial character ε is defined by the relation ε(a) = 1 for all a ∈ F∗p. It is convenient to extend the domain of definition to all of Fp by assigning χ(0) = 0 if χ 6= ε and ε(0) = 1. We begin by collecting together a few basic facts, all of which are estab- lished in [46, §8]. Lemma 2.1. Let p be a prime. Then the following hold: (1) Let χ be a character on Fp and let a ∈ F∗p. Then χ(1) = 1, χ(a) is a (p− 1)-th root of unity, and χ(a−1) = χ(a) = χ(a)−1. (2) For any character χ on Fp we have χ(a) = 0, if χ 6= ε, p, if χ = ε. (3) The set of characters on Fp forms a cyclic group of order p− 1. It follows from part (3) of Lemma 2.1 that χp−1 = ε for any character on Fp. We define the order of a character to be the least positive integer n such that χn = ε. In our work we will mainly be concerned with the characters of order 3. Let us turn briefly to the topic of generalised Jacobi sums. Given any characters χ1, . . . , χr on Fp, a Jacobi sum is a sum of the shape J0(χ1, . . . , χr) := t=(t1,...,tr)∈F t1+···+tr≡0 mod p χ1(t1) · · ·χr(tr). The key fact that we will need concerning these sums is that |J0(χ1, . . . , χr)| = 0, if χ1 · · ·χr 6= ε, (p − 1)pr/2−1, if χ1 · · ·χr = ε. (2.8) This is established in [46, §8.5]. Let p be a rational prime. We proceed to consider p as an element of the ring of integers Z[θ] associated to the quadratic field Q(θ) obtained by 24 T.D. BROWNING adjoining a primitive cube root of unity θ. It follows from basic algebraic number theory that p is a prime in Z[θ] if p ≡ 2 mod 3, whereas it splits as p = ππ if p ≡ 1 mod 3, where π is a prime in Z[θ]. When p ≡ 2 mod 3 the only cubic character on Fp is the trivial character ε. On the other hand, when p = ππ ≡ 1 mod 3 then there are precisely two non-trivial cubic characters χπ, χπ on Fp, where χω(·) = is the cubic residue symbol for any prime ω in Z[θ]. All of these facts are established in [46, §9]. It turns out that Jacobi sums can be used to give formulae for the number of solutions to appropriate equations over finite fields. Given any q ∈ N, let N(q) := #{x mod q : a1x31 + · · ·+ a4x34 ≡ 0 (mod q)}, (2.9) N∗(q) := # x mod q : 1 + · · ·+ a4x34 ≡ 0 (mod q), gcd(q, x1, . . . , x4) = 1 . (2.10) When p is a prime not belonging to the finite set of primes P defined in (2.2), we can write down a very precise expression for N(p). Thus it follows from [46, §8.7] that N(p) = p3 + χ1,χ2,χ3,χ4 1 )χ2(a 2 )χ3(a 3 )χ4(a 4 )J0(χ1, χ2, χ3, χ4) where the summation is over all non-trivial cubic characters χi : F p → C such that χ1χ2χ3χ4 = ε. Exercise 10. Let p 6∈ P be a prime, and let χ1, χ2, χ3, χ4 be non-trivial cubic characters on Fp such that χ1χ2χ3χ4 = ε. Deduce from (2.8) that J0(χ1, χ2, χ3, χ4) = p(p− 1). It follows from Exercise 10 that N∗(p) = N(p)− 1 = p3 + p(p− 1)δp(a)− 1. (2.11) for any prime p 6∈ P, where δp(a) := χ1,χ2,χ3,χ4 1 )χ2(a 2 )χ3(a 3 )χ4(a Let a ∈ F∗p and suppose that p splits as ππ. Then it will be useful to observe χπ(a) + χπ(a) = 2, if a is a cubic residue modulo π, −1, otherwise. (2.12) We have δp(a) = 0 when p ≡ 2 mod 3, since there are then no non-trivial cubic characters modulo p. When p ≡ 1 mod 3, with p = ππ 6∈ P, we have δp(a) =χπ (a1a2 (a1a2 (a1a3 (a1a3 (a1a4 (a1a4 THE MANIN CONJECTURE IN DIMENSION 2 25 Still with this choice of prime p, let νp(a) denote the number of indices i ∈ {2, 3, 4} for which the cubic character χπ( a1aiajak ) is equal to 1, with {i, j, k} a permutation of {2, 3, 4}. Then we may deduce from (2.12) that δp(a) = 0, if p ≡ 2 mod 3, 3νp(a)− 3, if p ≡ 1 mod 3, (2.13) when p 6∈ P. 2.3. The Hardy–Littlewood circle method. We are now ready to con- sider the counting function NU (B) that is associated to the diagonal cubic surface S ⊂ P3 given in (2.1). Our aim is to provide heuristic evidence in support of Manin’s original conjecture, and we will say rather little about the predicted value of the constant. The Hardy–Littlewood circle method is an extremely effective means of estimating counting functions associated to projective algebraic varieties, but it only works when the dimension of the variety is substantially larger than the degree. We have already seen evidence of this in the statement of Theorem 1.1, which is based on an application of the circle method. Although it has not been made to pro- duce asymptotic formulae for the counting functions associated to del Pezzo surfaces, in this section we will see how the Hardy–Littlewood method can still be used as a useful heuristic tool. The key idea is to consider only the contribution from the major arcs. We will simplify matters by applying the heuristic to count all of the rational points on S, rather than restricting attention to the open subset U . Although the details are formidable, it is in fact possible to obtain upper bounds for NU (B) using the circle method. Thus Heath-Brown [39] has shown that NU (B) = Oε,S(B 3/2+ε) under a certain hypothesis concerning the Hasse–Weil L-function associated to the surface. An interesting feature of this work is that the contribution from the rational points lying on rational lines in the surface is successfully separated out. When the surface contains no lines defined over Q, such as the surface given by (2.7) for example, one obviously has NU (B) = NS(B) +O(1). When S contains lines defined over Q there is a general consensus among people working on the circle method that the dominant contribution (ie. the contribution from the points on rational lines) should come from the minor arc integral. In what follows let e(z) := e2πiz for any z ∈ R. As usual, Z4 denotes the set of primitive vectors in Z4. The igniting spark in the Hardy–Littlewood circle method is the simple identity e(αn)dα = 1, if n = 0, 0, if n ∈ Z \ {0}. 26 T.D. BROWNING On taking into account the fact that x and −x represent the same point in projective space, and applying Exercise 2, we deduce that NS(B) = |x|6B e(α(a1x 1 + · · ·+ a4x34))dα |x|6B/k e(α(a1x 1 + · · ·+ a4x34))dα S(α)dα, say. Let us write P = B/k and IA(P ) := S(α)dα, for any bounded subset A ⊂ R. The cubic exponential sum S(α) can actually be rather large when α is well-approximated by a rational number with small denominator. For example, we clearly have S(0) = 24P 4 + O(P 3). The philosophy that underpins the Hardy–Littlewood method is that one expects S(α) to be small for values of α ∈ [0, 1] that are not well-approximated by rational numbers with small denominator. This is notoriously difficult to prove in general, and as indicated above, is expected to be false in the present setting! Our heuristic will be based on analysing IM(P ) for a suitable choice of “major arcs” M. We will not give full details here, the gaps being easily filled by consulting the relevant techniques in Davenport [25]. Let ε > 0 be a small parameter. Given a, q ∈ Z such that 1 6 a 6 q 6 P ε, gcd(a, q) = 1, (2.14) we define the interval M(a, q) := − P−3+ε, a + P−3+ε We take as major arcs the union q6P ε 16a6q gcd(a,q)=1 M(a, q). It is clear that M contains all the points in the interval [0, 1] that are well- approximated by rational numbers with small denominator. Exercise 11. Show that M is a disjoint union for ε < 1. The “minor arcs” are defined to be m := [0, 1] \M, and we will proceed under the assumption that the minor arc integral Im(P ) can be ignored. Actually we will also ignore the contribution that this term makes once it is summed up over values of k. In truth there will be several points in the argument where we will simply ignore subsidiary contributions. We will indicate all of these by an appearance of the word “error”. Thus, to begin with, we have NU (B) = µ(k)IM(P ) + error . THE MANIN CONJECTURE IN DIMENSION 2 27 Here we have made the further assumption that the contribution NS\U(B) from the points lying on lines in S arises in the minor arc integral. Let a, q ∈ Z such that (2.14) holds, and let α = a/q + z ∈ M(a, q). Let C(x) denote the diagonal cubic form in (2.1). We now break the sum into congruence classes modulo q, giving S(a/q + z) = r mod q e(aC(r)/q) x∈Z4∩[−P,P ]4 x≡r mod q e(zC(x)). (2.15) We would like to replace the discrete variable x by a continuous one in the inner sum, and the summation over x by an integral. For this we will appeal to the following general result. Lemma 2.2. Let P > 1, let a ∈ Zn and let r ∈ N such that r 6 P . Let F be a function on Rn all of whose first order partial derivatives exist and are continuous on R := [−P,P ]n. Define MF := sup 16i6n Then we have x∈Zn∩R x≡a mod r e(F (x)) = e(F (t))dt +O (Pn−1(1 + PMF ) Proof. Our proof of Lemma 2.2 is based on the Euler–Maclaurin summation formula [60, §I.0]. Let Bk(x) denote the kth Bernoulli polynomial, for k ∈ Z>0, and let s ∈ Z>0. Let A,B ∈ Z, with A < B. For any function f : R → C whose (s+1)-th derivative f (s+1) exists and is continuous on the interval [A,B], the Euler–Maclaurin summation formula states that A<n6B f(n) = f(t)dt+ (−1)k+1Bk+1(0) (k + 1)! f (k)(B)− f (k)(A) (−1)s (s+ 1)! Bs+1(t)f (s+1)(t)dt. (2.16) Let a ∈ Z and r ∈ N. We will apply this result with f0(x) = f(a+ rx) and , B0 = B − a Taking s = 0 in the Euler–Maclaurin formula, we therefore deduce that A<n6B n≡a mod r f(n) = f(t)dt− f(B0)− f(A0) (t− a f ′(t)dt, (2.17) since B1(x) = x− [x]− 12 . We are now ready to establish Lemma 2.2, which we will do by induction on n. Write Sn for the n-dimensional sum that is to be estimated. The case 28 T.D. BROWNING n = 1 of Lemma 2.2 follows from (2.17) with f(x) = e(F (x)). Assuming now that n > 2, we have y∈Z∩[−P,P ] y≡a1 mod r x2,...,xn e(G(x2, . . . , xn)), where G(x2, . . . , xn) = F (y, x2, . . . , xn), and the sum over x2, . . . , xn is over all integers in [−P,P ] such that xi ≡ ai mod r for 2 6 i 6 n.We may employ the induction hypothesis to estimate the inner sum in n − 1 variables. It therefore follows that y∈Z∩[−P,P ] y≡a1 mod r [−P,P ]n−1 e(G(t2, . . . , tn))dt2 · · · dtn (P (n−2)(1 + PMG) Now there are O(P/r) integers y in the interval [−P,P ] that are congruent to a1 modulo r, since r 6 P by assumption. Moreover, it is clear that MG 6 MF . Hence y∈Z∩[−P,P ] y≡a1 mod r f(y) +O (P (n−1)(1 + PMF ) where f(y) = [−P,P ]n−1 e(F (y, t2, . . . , tn))dt2 · · · dtn. The statement of Lemma 2.2 is now an easy consequence of (2.17). � It is clear from the proof of Lemma 2.2 that when F has partial derivatives to a higher order, one may obtain a much sharper estimate by including higher order terms in the Euler–Maclaurin summation formula. The present bound is satisfactory for our purposes, however. Returning to (2.15) we apply Lemma 2.2 with F (x) = zC(x), n = 4, a = r, r = q. In particular we have q 6 P ε 6 P , as required for the lemma. Furthermore, MF = MzC ≪ |z|P 2 6 P−1+ε for any α = a/q+z ∈ M(a, q). It follows that S(a/q + z) = q−4T (a, q)VP (z) +O(P 3+2ε) on the major arcs, where T (a, q) := r mod q e(aC(r)/q), VR(z) := [−R,R]4 e(zC(x))dx. (2.18) The set of major arcs has meas(M) = O(P−3+3ε). On carrying out the integration over z and the summation over a and q one is therefore led to THE MANIN CONJECTURE IN DIMENSION 2 29 the conclusion that IM(P ) = P q6P ε 16a6q gcd(a,q)=1 q−4T (a, q) |z|6P−3+ε V1(zP 3)dz +O(P 5ε) q6P ε 16a6q gcd(a,q)=1 q−4T (a, q) |z|6P ε V1(z)dz +O(P Define I(R) := |z|6R V1(z)dz = |z|6R [−1,1]4 e(zC(x))dxdz. It can be shown that I(R) is a bounded function of R, and furthermore, I(R) → I0 > 0 as R → ∞. A standard calculation reveals that the limit I0 is equal to 2σ∞, where σ∞ := dx1dx2dx3 1 + a2x 2 + a3x is the real density of solutions. Here the integral is over x1, x2, x3 ∈ [−1, 1] such that |(a1x31 + a2x32 + a3x33)/a4| 6 1. We now define S(R) := 16a6q gcd(a,q)=1 T (a, q). On bringing everything together, our investigation has so far succeeded in showing that NU (B) = σ∞ µ(k)PS(P ε) + error, (2.19) for a suitably small value of ε > 0, where P = B/k. Our task is now to examine the sum S(R), as R → ∞. Let us write Sq := 16a6q gcd(a,q)=1 T (a, q), where T (a, q) is the complete exponential sum defined in (2.18). Then S(R) = q6R q −4Sq. It turns out that Sq is a multiplicative function of q. This can be established along the lines of [25, Lemma 5.1]. Recall the definition (2.9) of N(q). We now come to the key relation between Sq and N(q) at prime power values of q. Exercise 12. Let p be a prime and let e > 1. Use Lemma 2.1 to show that Spe = p eN(pe)− p3+eN(pe−1). Let us for the moment ignore considerations of convergence, and consider the local factors e=0 p −4eSpe in the infinite product formula for S(∞). Now it follows from Exercise 12 that p−4eSpe = 1 + p−3eN(pe)− p3−3eN(pe−1) = p−3EN(pE), 30 T.D. BROWNING for any E > 1. Hence, formally speaking, we have S(∞) = p τp, where τp := lim p−3eN(pe). If S(R) was convergent, which it certainly is not in general, we could then conclude from (2.19) that NU (B) = Bσ∞ τp + error . (2.20) Arguing formally, we now replace the summation over k by its Euler product, concluding that NU (B) = Bσ∞ σp + error, (2.21) with σp := (1− 1/p)τp. Note that σp = lim p−3eN∗(pe), (2.22) in the notation of (2.10), which clearly follows from the observation that N∗(pe) = N(pe)− 8N(pe−3), for any e > 3. The estimate in (2.21) therefore gives a heuristic asymptotic formula for NU (B) which is visibly a product of local densities. Among other things, we have assumed that S(R) is convergent in formulating this heuristic. This is expected to be true for diagonal cubic surfaces whose Picard group has rank 1, but not in general. One can make the transition from (2.20) to (2.21) completely rigorous if the Hardy–Littlewood heuristic produces a leading term involving Pα with exponent α > 1. The outcome is that to go from counting points on the affine cone to counting projective points, one merely replaces N(pe) by N∗(pe). For α = 1, however, the “renormalization” procedure remains heuristic. Let ∗(R) := q−4S∗q , where S∗q := 16a6q gcd(a,q)=1 r mod q gcd(r,q)=1 e(aC(r)/q). It is easily checked that S∗q is a multiplicative function of q, and furthermore, that the corresponding version of Exercise 12 holds, relating S∗pe to N ∗(pe). In fact, formally speaking, one has p−4eS∗pe = with σp given by (2.22). Bearing all of this in mind we will proceed under the bold assumption that (2.19) can be replaced by NU (B) = Bσ∞S ∗(B) + error, (2.23) where we have taken ε = 1 in the expressions for S(Bε) and S∗(Bε). THE MANIN CONJECTURE IN DIMENSION 2 31 We now turn to a finer analysis of S∗(B), as B → ∞. Our task is to determine the analytic properties of the corresponding Dirichlet series F (s) := (2.24) for s = σ+it ∈ C. Armed with this analysis we will ultimately apply Perron’s formula to obtain an estimate forS∗(R). Using the multiplicativity of q−sS∗q we deduce that F (s) = σp(s), σp(s) := p−esS∗pe . (2.25) In examining F (s) it clearly suffices to ignore the value of the factors σp(s) at any finite collection of primes p. With this in mind we will try and determine σp(s) for p 6∈ P, where P is given by (2.2). Recall the definition (2.10) of N∗(q). Exercise 13. Let e > 1 and let p 6∈ P be a prime. Use Hensel’s lemma to show that N∗(pe) = p3e−3N∗(p). It therefore follows from Exercise 13 that σp(s) = 1 + peN∗(pe)− p3+eN∗(pe−1) = 1− 1 N∗(p) for any p 6∈ P. Hence (2.11) yields σp(s) = 1 + δp(a) − δp(a) where δp(a) is given by (2.13). We now pursue our analysis in the special case a = (1, 1, 1, 1) of the Fermat cubic surface (2.5). Now it is clear from (2.13) that δp(1, 1, 1, 1) = 0 if p ≡ 2 mod 3 and δp(1, 1, 1, 1) = 3νp(1, 1, 1, 1) − 3 = 6 if p ≡ 1 mod 3. Let λ : Z → C be the real Dirichlet character of order 2 defined by λ(n) := ), if 3 ∤ n, 0, otherwise, where (n ) is the Legendre symbol. Then we may write σp(s) = 1 + 3(1 + λ(p)) − 3(1 + λ(p)) 1− λ(p) pmin{σ−2,2σ−2} for any p 6∈ P. Let L(s, λ) denote the usual Dirichlet L-function associated to λ. When a = (1, 1, 1, 1) we have therefore succeeded in showing that F (s) = ζ(s− 3)3L(s− 3, λ)3G(s), (2.26) 32 T.D. BROWNING where G(s) is a function that is holomorphic and bounded on the half-plane σ > 7/2+ δ, for any δ > 0. For future reference we note that G(4) has local factors Gp(4) = (1− 1 )7(1 + 7 ), if p ≡ 1 mod 3, (1− 1 )4(1 + 1 )3(1 + 1 ), if p ≡ 2 mod 3. (2.27) Although we will not prove it here, it can be deduced from (2.22) and Hensel’s lemma that G3(4) = 3−3eN∗(3e) = . (2.28) We are now ready for our application of Perron’s formula, which we will apply in the following form. Lemma 2.3. Let F (s) = n=1 ann −s be a Dirichlet series with abscissa of absolute convergence σa. Suppose that x 6∈ Z and let c > σa. Then we have ∫ c+iT F (s) |an|n−c | log(x/n)| for any T > 1. Proof. Let c > 0. The lemma follows from the identity ∫ c+iT 1 +O(xc(T | log x|)−1), if x > 1, +O(cT−1), if x = 1, O(xc(T | log x|)−1), if 0 < x < 1, which is a straightforward exercise in contour integration. � In our case we have aq = S q and we are interested in the Dirichlet series F (s + 4), in the notation of (2.24). In order to apply Lemma 2.3 we will need an upper bound for this quantity. For our purposes the trivial upper bound aq ≪ q5 is sufficient. Thus the Dirichlet series F (s+ 4) is absolutely convergent for σ > 2. Taking c = 2 + ε for any ε > 0, we may deduce from Lemma 2.3 that ∗(B) = ∫ c+iT F (s+ 4) n1+ε| log(B/n)| for any T > 1 and any B 6∈ Z. It is not hard to see that the error term here is ≪ T−1Bc. We now apply Cauchy’s residue theorem to the rectangular contour C joining c′ − iT , c′ + iT , c+ iT and c− iT , where c′ = −1/2 + ε. The relation (2.26) implies that in this region F (s + 4)Bs/s has a unique pole at s = 0, and it is a pole of order 4. It has residue Ress=0 F (s+ 4)Bs L(1, λ)3G(4)P (logB) where P ∈ R[x] is a monic polynomial of degree 3. Putting all of this together we have therefore shown that ∗(B) = L(1, λ)3G(4)P (logB) +O(E(B)), (2.29) THE MANIN CONJECTURE IN DIMENSION 2 33 where E(B) = ( ∫ c′+iT c′−iT ∫ c−iT c′−iT ∫ c′+iT )∣∣∣H(s)3 ∣∣∣ds, for any T > 1, and where H(s) = ζ(s + 1)L(s + 1, λ). Here we have used the fact that G(s+ 4) is bounded on the half-plane ℜe(s) > c′. To make the analysis simpler, it will be convenient to proceed under the assumption that the Lindelöf hypothesis holds for ζ(s), and also for the Dirichlet L-function L(s, λ). This could be avoided at the cost of extra effort, but there seems no harm in supposing it here. Thus we may assume the bounds ζ(σ + it) ≪ε |t|ε, L(σ + it, λ) ≪ε |t|ε, for any σ ∈ [1/2, 1] and any |t| > 1. It therefore follows that H(σ + it) ≪ε |t|ε, if −1/2 6 σ 6 0, 1, if σ > 0, for any |t| > 1, which gives ∫ c−iT c′−iT ∣∣∣H(s)3 ∣∣∣ds ≪ε BσT−1+3εdσ ≪ε BcT−1+3ε. One obtains the same estimate for the contribution from the remaining horizontal contour. Turning to vertical integral, we find that ∫ c′+iT c′−iT ∣∣∣H(s)3 ∣∣∣ds ≪ Bc |H(1/2 + ε+ it)|3 1 + |t| dt ≪ε Bc (1 + |t|)3ε−1dt ≪ε Bc T 3ε, under the assumption of Lindelöf hypothesis. This shows that E(B) ≪ε BεT 3ε for any T > 1. Taking T sufficiently large, we therefore conclude from (2.29) ∗(B) = L(1, λ)3G(4)P (logB) +O(B−∆), for some ∆ > 0. We are now ready to return to the Hardy–Littlewood major arc analysis which led us to (2.23). Substituting in our estimate for S∗(B), we conclude NU1(B) ∼ c1B(logB)3 (2.30) where U1 ⊂ S1 is the usual open subset of the Fermat surface (2.5), and σ∞L(1, λ) 3G(4) Now it follows from the class number formula that L(1, λ) = π 3/9. Hence, on combining this with (2.27) and (2.28), our heuristic argument has led us 34 T.D. BROWNING to the expectation that (2.30) holds, with p≡1 mod 3 p≡2 mod 3 The exponents of B and logB in (2.30) agree with the Manin conjecture, since we have already seen in §2.1 that the Picard group of S1 has rank 4. It is interesting to compare our analysis with the work of Peyre and Tschinkel [54], who calculate the leading constant cPeyre in Peyre’s refine- ment [51] of the conjectured asymptotic formula for NU1(B). It turns out cPeyre = γ(S1)c1, with γ(S1) = 7/3. For a general non-singular cubic surface S ⊂ P3, the constant γ(S) is defined to be the volume γ(S) := e−〈−KS ,t〉dt. Thus, in general terms, γ(S) measures the volume of the polytope obtained by intersecting the dual of Λeff(S) with a certain affine hyperplane. In particular γ(S) ∈ Q for any non-singular cubic surface S, and γ(S) = 1 if and only if the corresponding Picard group has rank 1. Exercise 14. Let S2 denote the surface (2.7). Using a similar argument, show that one expects an asymptotic formula of the shape NS2(B) ∼ c2B for some constant c2 > 0. Check your answer with the heuristic formula obtained by Heath-Brown [36]. 3. The A1 del Pezzo surface of degree 6 In this section we will establish Theorem 1.3. Any line in P6 is defined by the intersection of 5 hyperplanes. It is not hard to see that the equations x1 = x2 = x3 = x5 = x6 = 0, x1 = x3 = x4 = x5 = x6 = 0, x3 = x5 = x6 = x1 + x4 = x1 + x2 = 0, (3.1) all define lines contained in the singular del Pezzo surface S given by (1.17). Table 6 ensures that these are the only lines contained in S. By definition U is the open subset of S on which none of these equations hold. We begin by establishing the following result. Lemma 3.1. We have NU (B) = 2M(B) +O(B), where M(B) denotes the number of x ∈ Z7 such that x21 − x2x4 = x1x5 − x3x4 = x1x3 − x2x5 = x1x6 − x3x5 = x2x6 − x23 = x4x6 − x25 = x21 − x1x4 + x5x7 = x21 − x1x2 − x3x7 = x1x3 − x1x5 + x6x7 = 0, (3.2) with gcd(x1, . . . , x7) = 1, 0 < |x1|, x2, x3, x4, |x5|, x6 6 B and |x7| 6 B. THE MANIN CONJECTURE IN DIMENSION 2 35 Proof. In view of the fact that x and −x represent the same point in P6, we NU (B) = #{x ∈ Z7 : |x| 6 B, (1.17) holds, but (3.1) does not}, where Z7 denotes the set of primitive vectors in Z7. We need to consider the contribution to the right hand side from points such that xi = 0, for some 1 6 i 6 7. Let us begin by considering the contribution from vectors x ∈ Z7 for which x1 = 0. But then the equations in (1.17) imply that x2x4 = 0. If x2 = 0, it is straightforward to check that either x satisfies the first system of equations in (3.1), or else x0 = x2 = x3 = x7 = 0, x4x6 = x Such points are therefore confined to a plane conic. We therefore obtain O(B) points overall with x1 = x2 = 0. If on the other hand x1 = x4 = 0, then a similar analysis shows that there are O(B) points in this case too. In view of the first equation in (1.17), the contribution from vectors x such that x2x4 = 0 is also O(B). Let us now consider the contribution from vectors x such that x3 = 0 and x1x2x4 6= 0. It is easily checked that the only such vectors have x5 = x6 = 0 and x1 + x4 = x1 + x2 = 0, and so must lie on a line contained in S. Finally, arguing in a similar fashion, we see that there are no points contained in S with x5x6 = 0 and x1x2x3x4 6= 0. We have therefore shown that NU (B) = #{x ∈ Z7 : x1 · · · x6 6= 0, |x| 6 B, (3.2) holds}+O(B). Here we have noted that there is an obvious unimodular transformation that takes the set of equations in (1.17) into (3.2). We would now like to restrict our attention to positive values of x1, . . . , x6. The equations for S imply that x2, x4, x6 all share the same sign. On ab- sorbing the minus sign into x1 there is a clear bijection between solutions to (3.2) with x2, x4, x6 < 0 and solutions with x2, x4, x6 > 0. We choose to count the former. Arguing similarly, by absorbing the minus signs into x1 and x7, we see that there is a bijection between the solutions to (3.2) with x3 < 0 and x2, x4, x6 > 0, and the solutions with x2, x3, x4, x6 > 0. Fixing our attention on the latter set of points, we therefore complete the proof of Lemma 3.1. � Let S̃ denote the minimal desingularisation of the surface S. By determin- ing the Cox ring associated to S̃, Derenthal [26] has calculated the universal torsor above S̃. In this setting it is defined by a single equation s1y1 − s2y2 + s3y3 = 0, (3.3) embedded in A7. In particular one of the variables does not appear explicitly in the equation. 3.1. Elementary considerations. As promised in §1.6, we proceed to show how NU (B) can be related to a count of the integer points on the corresponding universal torsor, which in this case is given by (3.3). Our deduction of this fact is completely elementary, and is based on an analysis of the integer solutions to the system of equations (3.2). It is still somewhat 36 T.D. BROWNING mysterious as to how or why this rather low-brow process should ultimately lead to the same outcome! Typical of the facts that we will employ is the following. Exercise 15. Show that the general solution of the equation xy = z2 is x = a2c, y = b2c, z = abc, with |µ(c)| = 1. Given s0 ∈ R and s = (s1, s2, s3),y = (y1, y2, y3) ∈ R3, define Ψ(s0, s,y) := max |s30s21s22s23|, |y1y2y3|, |s0s21y21|, |s0s22y22| . (3.4) We are now ready to record our translation of the problem to the universal torsor. Lemma 3.2. We have NU (B) = 2# (s0, s,y) ∈ Z7 : Ψ(s0, s,y) 6 B, (3.3) holds, s0, s1, s2, s3, y1 > 0, gcd(yi, s0sjsk) = 1, gcd(si, sj) = 1 +O(B), with i, j, k a permutation of 1, 2, 3 in the coprimality conditions. Proof. Let x ∈ Z7 be a primitive vector counted by M(B), as defined in the statement of Lemma 3.1. Combining the first equation in (3.2) with Exercise 15 we see that x1 = a1a2a4, x2 = a 2a1, x4 = a for integers a1, a2, a4 such that a1, a2 > 0 and |µ(a1)| = gcd(a1 gcd(a2, a4)2, x3, x5, x6, x7) = 1. Inserting this into the equation x2x6 = x 3 we deduce that a1a2 | x3, whence x3 = a1a2a3, x6 = a1a for a positive integer a3 such that |µ(a1)| = gcd(a1 gcd(a2, a4)2, a1a3 gcd(a2, a3), x5, x7) = 1. Substituting this into the equation x4x6 = x 5, we deduce that x5 = a1a3a4, |µ(a1)| = gcd(a1, x7) = gcd(a2, a3, a4, x7) = 1. (3.5) Note that the second equation in (3.2) implies that x1, x5 must share the same sign, which here is the sign of a4. The equations x1x5 = x3x4, x1x3 = x2x5 and x1x6 = x3x5 reveal no new information. Turning instead to the equation x21 = x1x4 − x5x7, we obtain 2a4 = a1a2a 4 − a3x7. (3.6) The coprimality conditions imply that a1 | a3. Moreover, we deduce from this equation that a2a4 | a3x7/a1. We may therefore write a2 = a23a27, a4 = a43a47, THE MANIN CONJECTURE IN DIMENSION 2 37 for integers a2i, a4i, with i = 3, 7, such that a2i, a43, |a47| > 0, and a1a23a43 | a3, a27a47 | x7. Thus there exist further integers b3, a7 with b3 > 0, such that a3 = a1a23a43b3, x7 = a27a47a7, with (3.5) and (3.6) becoming |µ(a1)| = gcd(a1, a27a47a7) = gcd(a23a27, a23a43b3, a43a47, a27a47a7) = 1, a23a27 = a43a47 − b3a7, respectively. The final two equations are redundant. Let us write d for the highest common factor of a23, a43, b3. Thus a23 = da 23, a43 = da 43, b3 = db for positive integers d, a′23, a 43, b 3. On making these substitutions the equa- tion remains the same, but with appropriate accents added, whereas the coprimality conditions become |µ(a1)| = gcd(da1, a27a47a7) = gcd(a′23a27, a′23a′43b′3, a′43a47, a27a47a7) = gcd(a′23, a 43, b Now any n ∈ N can be written uniquely in the form n = ab2 for a, b ∈ N such that |µ(a)| = 1. We may therefore make the change of variables (s0; s1, s2, s3; y1, y2, y3) = (a1d 2; a′23, a 43, b 3; a27, a47, a7). Bringing everything together, we have therefore established the existence of (s0, s,y) ∈ Z7 such that (3.3) holds, with s0, s1, s2, s3, y1 > 0, (3.7) gcd(s0, y1y2y3) = gcd(s1, s2, s3) = gcd(s1y1, s2y2, s1s2s3, y1y2y3) = 1. Note that y2 is automatically non-zero for s0, s,y satisfying the remaining conditions. Once combined with (3.3), it is easy to check that the latter coprimality conditions are equivalent to the conditions gcd(y1, s0s2s3) = gcd(y2, s0s1s3) = gcd(y3, s0s1s2) = 1, gcd(s1, s2) = gcd(s1, s3) = gcd(s2, s3) = 1, (3.8) that appear in the statement of the lemma. At this point we may summarise our argument as follows. Let T ⊂ Z7 denote the set of (s0, s,y) ∈ Z7 such that (3.3), (3.7) and (3.8) hold. Then for any primitive vector x counted by M(B), we have shown that there 38 T.D. BROWNING exists (s0, s,y) ∈ T such that   x1 = s0s1s2y1y2, x2 = s0s x3 = s 1s2s3y1, x4 = s0s x5 = s 2s3y2, x6 = s x7 = y1y2y3. Conversely, we leave it as an exercise to check that any (s0, s,y) ∈ T pro- duces a primitive point x ∈ Z7 such that (3.2) holds, with |x1|, x2, x3, x4, |x5|, x6 > 0. We may now conclude that M(B) is equal to the number of (s0, s,y) ∈ T such that i=1,2 |s0s1s2y1y2|, |s0s2i y2i |, |s20s1s2s3siyi|, |s30s21s22s23|, |y1y2y3| In view of the fact that |s0s1s2y1y2| = |s0s21y21| |s0s22y22 |, and furthermore, |s20s1s2s3siyi| = |s30s21s22s23| |s0s2i y2i |, it follows that this height condition is equivalent to Ψ(s0, s,y) 6 B for any (s0, s,y) ∈ T , where Ψ is given by (3.4). In summary we have therefore shown that M(B) is equal to the number of (s0, s,y) ∈ T such that Ψ(s0, s,y) 6 B. Once inserted into Lemma 3.1, this completes the proof of Lemma 3.2. � At first glance it might seem a little odd that the height restriction |s0s23y23 | 6 B doesn’t explicitly appear in the lemma. However, (3.3) implies that 0 < s1y1 = s2y2 − s3y3 for any (s0, s,y) ∈ T , whence the restriction Ψ(s0, s,y) 6 B is plainly equivalent to max{|s0s23y23|,Ψ(s0, s,y)} 6 B. We have preferred not to include it explicitly in the statement of Lemma 3.2 however. 3.2. The asymptotic formula. Our starting point is Lemma 3.2. Let T (B) denote the quantity on the right hand side that is to be estimated. Once taken together with (3.3), the height condition Ψ(s0, s,y) 6 B is equivalent to i=1,2 |s30s21s22s23|, |s0s2i y2i |, |y1y2(s1y1 − s2y2)/s3| Define X0 := (s30s21s22s23 , Xi := (s1s2s3B for i = 1, 2. Then the height conditions above can be rewritten as |X0| 6 1, |f1(y1)| 6 1, |f2(y2)| 6 1, |g(y1, y2)| 6 1, where fi(y) := X0 , g(y1, y2) := for i = 1, 2. In order to count solutions to the equation (3.3), our plan will be to view the equation as a congruence s1y1 − s2y2 ≡ 0 (mod s3), THE MANIN CONJECTURE IN DIMENSION 2 39 which has the effect of automatically taking care of the summation over y3. In order to make this approach viable we will need to first extract the coprimality conditions on the y3 variable. Define the set S := {(s0, s) ∈ N4 : gcd(si, sj) = 1, X0 6 1}, (3.9) with i, j generic indices from the set {1, 2, 3}. We now apply Möbius in- version, as in Exercise 2, in order to remove the coprimality condition gcd(y3, s0s1s2) = 1. Thus we find that T (B) = (s0,s)∈S k3|s0s1s2 µ(k3)#   y ∈ Z3 : gcd(y1, s0s2s3) = 1, gcd(y2, s0s1s3) = 1, y1 > 0, s1y1 − s2y2 + k3s3y3 = 0, |fi(yi)| 6 1, |g(y1, y2)| 6 1   Now it is clear that the summand vanishes unless gcd(k3, s1s2) = 1. Hence T (B) = (s0,s)∈S k3|s0 gcd(k3,s1s2)=1 µ(k3)Sk3(B), (3.10) where Sk3(B) := #   y1, y2 ∈ Z : gcd(y1, s0s2s3) = 1, gcd(y2, s0s1s3) = 1, y1 > 0, s1y1 ≡ s2y2 mod k3s3, |fi(yi)| 6 1, |g(y1, y2)| 6 1   Clearly Sk3(B) depends on the parameters s0 and s, in addition to k3 and B. We now turn to the estimation of Sk3(B), for which we need the following basic result. Exercise 16. Let b > a and q > 0. Show that #{n ∈ Z ∩ (a, b] : n ≡ n0 mod q} = +O(1). We will fix y2 and apply Exercise 16 to handle the summation over y1. Be- fore this we must use Möbius inversion to remove the coprimality condition gcd(y1, s0s2s3) = 1 from the summand. Thus we find that Sk3(B) = k1|s0s2s3 µ(k1)# y1, y2 ∈ Z : gcd(y2, s0s1s3) = 1, k1s1y1 ≡ s2y2 mod k3s3, |f1(k1y1)| 6 1, |f2(y2)| 6 1, |g(k1y1, y2)| 6 1, y1 > 0 In view of the other coprimality conditions, the summand plainly vanishes unless gcd(k1, k3s3) = 1. We may therefore write ρ ∈ Z for the (unique) inverse of k1s1 modulo k3s3, whence Sk3(B) = k1|s0s2 gcd(k1,k3s3)=1 µ(k1)Sk1,k3(B), (3.11) 40 T.D. BROWNING Sk1,k3(B) := y2∈Z: |f2(y2)|61 gcd(y2,s0s1s3)=1 y1 ∈ N : y1 ≡ ρs2y2 mod k3s3, |f1(k1y1)| 6 1, |g(k1y1, y2)| 6 1 An application of Exercise 16 now reveals that Sk1,k3(B) = y2∈Z: |f2(y2)|61 gcd(y2,s0s1s3)=1 (X1F1(X0, y2/X2) k1k3s3 +O(1) , (3.12) where F1(u, v) := {t∈R>0: |ut 2|,|tv(t−v)|61} We close this section by showing that once summed over all (s0, s, y2) ∈ N5, the error term in (3.12) makes a satisfactory overall contribution to the error term in Theorem 1.3. Using the fact that k|n |µ(k)| = 2ω(n), we find that this contribution is (s0,s)∈S 4ω(s0)2ω(s2)X2 = B1/2 (s0,s)∈S 4ω(s0)2ω(s2) s0,s1,s2∈N 4ω(s0)2ω(s2) s20s1s ≪ B logB. This is satisfactory for Theorem 1.3, and so we may henceforth ignore the error term in the above estimate for Sk1,k3(B). Define the arithmetic function φ∗(n) := where as is common convention the product is over distinct prime divisors of n. It will be useful to note that φ∗(mn) = φ∗(m)φ∗(n) φ∗(gcd(m,n)) , (3.13) for any m,n ∈ N. We must now sum over the variable y2, for which we will employ the following basic result. Exercise 17. Let I ⊂ R be an interval, let a ∈ N and let f : R → R>0 be a function that is continuously differentiable on I. Use (2.16) to show that n∈Z∩I gcd(n,a)=1 f(n) = φ∗(a) f(t)dt+O 2ω(a) sup |f(t)| We may now return to (3.12). Using Exercise 17 we deduce that Sk1,k3(B) = φ∗(s0s1s3)X1X2F2(X0) k1k3s3 (2ω(s0s1s3)X1 k1k3s3 , (3.14) THE MANIN CONJECTURE IN DIMENSION 2 41 where F2(u) := {t,v∈R: t>0, |ut2|,|uv2|,|tv(t−v)|61} dtdv. We must now estimate the overall contribution to NU (B) from the error term in this estimate, once summed up over the remaining variables. This gives (s0,s)∈S 4ω(s0)2ω(s2)2ω(s0s1s3)X1 ≪ B1/3 s0,s1,s2,s3∈N 8ω(s0)2ω(s1s2s3)s ≪ B logB, by summing over s2 6 B/(s30s 3). This is satisfactory for Theorem 1.3, and so we may henceforth ignore the error term in (3.14). As pointed out to the author by Régis de la Bretèche, it is easy to sharpen this error term to O(B) using the fact that φ∗ has constant average order. Now it is trivial to check that gcd(d,a)=1 φ∗(n) φ∗(gcd(a, n)) for any a, n ∈ N. Bringing together (3.10), (3.11) and (3.14), we conclude T (B) = (s0,s)∈S k3|s0 gcd(k3,s1s2)=1 µ(k3) φ∗(s0s2)φ ∗(s0s1s3) φ∗(gcd(k3s3, s0s2)) X1X2F2(X0) where S is given by (3.9). It is clear that gcd(k3s3, s0s2) = gcd(k3s3, s0). Let us define the arithmetic function ϑ(s0, s) = φ∗(s0s2)φ ∗(s0s1s3) φ∗(gcd(s0, s3)) k3|s0 gcd(k3,s1s2)=1 µ(k3) φ∗(gcd(k3, s0, s3)) φ∗(gcd(k3, s0)) when gcd(si, sj) = 1 for 1 6 i < j 6 3, and ϑ(s0, s) = 0 otherwise. It follows from (3.13) that ϑ(s0, s) = φ∗(s0s2)φ ∗(s0s1s3) φ∗(gcd(s0, s3)) p|gcd(s0,s3) p∤s1s2 p∤s1s2s3 (1− 2 = φ∗(s0s2)φ ∗(s0s1s3) p∤s1s2s3 (1− 2 = φ∗(s0)φ ∗(s1s2s3) p∤s1s2s3 42 T.D. BROWNING when gcd(si, sj) = 1 for 1 6 i < j 6 3. On recalling the definitions of X1,X2, we deduce that T (B) = B2/3 ∆(n)F2 (n/B)1/3 , (3.15) where ∆(n) := ϑ(s0, s) (s1s2s3)1/3 , (3.16) for any n ∈ N. We will use Perron’s formula to estimate n6B ∆(n), before combining it with partial summation to estimate (3.15). Consider the Dirichlet series D(s) := n=1 ∆(n)n −s. We have D(s+ 1/3) = s0,s1,s2,s3=1 ϑ(s0, s) s3s+10 s and it is straightforward to check that D(s+ 1/3) = p ap(s), with ap(s) = 1 + 3(1− 1/p) p2s+1(1− 1/p2s+1) + (1− 1/p)(1 − 2/p) p3s+1(1− 1/p3s+1) 3(1 − 1/p)2 p5s+2(1− 1/p2s+1)(1 − 1/p3s+1) . Hence D(s+ 1/3) = E1(s)E2(s), where E1(s) = ζ(2s+ 1) 3ζ(3s + 1) and E2(s) = D(s+ 1/3) ζ(2s+ 1)3ζ(3s+ 1) p4σ+2 (3.17) on the half-plane ℜe(s) > −1/2. In particular, E1(s) has a meromorphic continuation to all of C with a pole of order 4 at s = 0, and E2(s) is holomorphic and bounded on the half-plane ℜe(s) > −1/4. Let c = 1/3 + ε for any ε > 0, and let T > 1. Then an application of Lemma 2.3 reveals that ∆(n) = ∫ c+iT E1(s − 1/3)E2(s− 1/3) ds+Oε (B1/3+ε provided that B 6∈ Z. We apply Cauchy’s residue theorem to the rectangular contour C joining the points 1/6− iT , 1/6 + iT , c+ iT and c− iT . Now the residue of E1(s− 1/3)E2(s− 1/3)Bs/s at s = 1/3 is clearly Ress=1/3 E1(s− 1/3)E2(s− 1/3) E2(0) B1/3P (logB), for some monic polynomial P ∈ R[x] of degree 3. Define the difference E(B) = ∆(n)− E2(0) B1/3P (logB). Then it follows that E(B) ≪ε B1/3+ε (∫ 1/6+iT 1/6−iT ∫ c−iT 1/6−iT ∫ 1/6+iT )∣∣∣E1(s − 1/3) ∣∣∣ds, since E2(s−1/3) is holomorphic and bounded on the half-plane ℜe(s) > 1/6. THE MANIN CONJECTURE IN DIMENSION 2 43 We proceed to estimate the contribution from the horizontal contours. Recall the well-known convexity bounds ζ(σ + it) ≪ε |t|(1−σ)/3+ε, if σ ∈ [1/2, 1], |t|(3−4σ)/6+ε, if σ ∈ [0, 1/2], for any |t| > 1. A proof of these can be found in [60, §II.3.4], for example. It therefore follows that E1(σ − 1/3 + it) ≪ε |t|1−3σ+ε (3.18) for any σ ∈ [1/6, 1/3) and any |t| > 1. We may now deduce that ∫ c−iT 1/6−iT ∣∣∣E1(s− 1/3) ∣∣∣ds ≪ε BσT−3σ+εdσ B1/3+εT ε B1/6T ε T 1/2 (3.19) One obtains the same estimate for the contribution from the remaining horizontal contour. Turning to the vertical contour, (3.18) gives ∫ 1/6+iT 1/6−iT ∣∣∣E1(s− 1/3) ∣∣∣ds ≪ B1/6 |E1(−1/6 + it)| 1 + |t| dt ≪ B1/6 |t|1/2+ε 1 + |t| dt ≪ B1/6T 1/2+ε. Once combined with (3.19), we conclude that E(B) ≪ε B1/3+εT−1+ε +B1/6T 1/2+ε, for any T > 1. Taking T = B1/9 we obtain ∆(n) = E2(0) B1/3P (logB) +Oε(B 2/9+ε), for any ε > 0. We are now ready to complete the proof of Theorem 1.3. For this it suffices to combine the latter estimate with partial summation in (3.15), and then apply Lemma 3.2. In this way we deduce that NU (B) = 2T (B) +O(B logB) σ∞E2(0) BQ(logB) +Oε(B 8/9+ε) +O(B logB), for a further cubic monic polynomial Q ∈ R[x]. Here σ∞ = 6 F2(u)du is given by (1.18), and it follows from (3.17) that E2(0) = This therefore completes the proof of Theorem 1.3. 44 T.D. BROWNING 4. The D4 del Pezzo surface of degree 3 In this section we consider Manin’s conjecture for the cubic surface S2 = {[x1, x2, x3, x4] ∈ P3 : x1x2(x1 + x2) + x4(x1 + x2 + x3)2 = 0}, considered in (1.12) Let U2 ⊂ S2 be the open subset formed by deleting the lines (1.14) from S2. Our task is to estimate NU2(B). In doing so it will clearly suffice to establish the estimate for any surface that is obtained from S2 via a unimodular transformation. In view of this we will make the change of variables t1 = x1, t2 = x2, t3 = x1 + x2 + x3, t4 = −x4, which brings S2 into the shape t1t2(t1 + t2) = t 3t4, (4.1) and which we henceforth denote by S. The 6 lines on the surface (4.1) take the shape ti = tj = 0, tj = t1 + t2 = 0, where i denotes a generic element of the set {1, 2}, and j an element of {3, 4}. If U ⊂ S denotes the open subset formed by deleting these lines from the surface, then we have t3t4 = 0 for any [t] 6∈ U . It now follows that NU (B) = #{t ∈ Z4 : (4.1) holds, |t| 6 B, t3t4 6= 0}. As in the argument of Lemma 3.1, the factor 1 reflects the fact that t and −t represent the same point in P3. There is a clear symmetry between solutions such that t3 is positive and negative. Similarly, (4.1) is invariant under the transformation t1 = −z1, t2 = −z2, t3 = z3 and t4 = −z4. Thus we have NU (B) = 2#{t ∈ Z4 : (4.1) holds, |t| 6 B, t3, t4 > 1}, (4.2) for any B > 1. In the following section we will explicate the relation between (4.2) and counting integral points on the corresponding universal torsor. When it comes to the latter task, we will be led to consider the counting problem for rational points on plane curves of degree 1 and 2. The estimates that we require will need to be completely uniform in the coefficients of the equations defining the curves. Given any plane curve C ⊂ P2 of degree d > 1, that is defined over Q, let NC(B) := #{x ∈ C(Q) : H(x) 6 B}. As usual we write Zn for the set of primitive vectors in Zn, and Zn∗ for the set of vectors in Zn with no components equal to zero. We will restrict our at- tention to curves that are defined by diagonal ternary forms. We clearly have NC(B) = Md(a;B,B,B) for certain non-zero integers a1, a2, a3, where Md(a;B) := #{x ∈ Z3 : a1xd1 + a2xd2 + a3xd3, |xi| 6 Bi}, (4.3) and B = (B1, B2, B3). Let us begin with the situation for projective lines. The following result is due to Heath-Brown [35, Lemma 3]. THE MANIN CONJECTURE IN DIMENSION 2 45 Lemma 4.1. Let a ∈ Z3∗ and let Bi > 0. Then we have M1(a;B) ≪ 1 + B1B2B3 max |ai|Bi Lemma 4.1 shows that there are only O(1) rational points on lines of sufficiently large height. If one has a line L ⊂ P2 given by the equation a.x = 0, for a ∈ Z3∗ , then the height of L is simply defined to beH(L) := |a|. When L is an arbitrary line in Pn, which is defined over Q, there is still a very natural way of defining its height. The height of L is just the height of the rational point in the Grassmannian G(1, n) that corresponds to the line. We will not need this fact in our work. Let L ⊂ P2 be an arbitrary line defined over Q. Then it follows from Lemma 4.1 that NL(B) ≪ 1 + ≪ B2. This is essentially best possible, as can be seen by taking n = 2 in Exercise 3. Turning to curves of higher degree, we have the following result, which is a special case of a result due to the author and Heath-Brown [15, Corollary Lemma 4.2. Let a ∈ Z3∗ such that gcd(ai, aj) = 1, and let Bi > 0. Then we have M2(a;B) ≪ B1B2B3 |a1a2a3| τ(a1a2a3), where τ(n) := d|n 1 denotes the usual divisor function. In keeping with our discussion of lines, let us consider to what extent our estimate reflects the true growth rate ofNC(B), for a quadratic curve C ⊂ P2 that is defined by a diagonal equation with pairwise coprime coefficients. Recall the estimate τ(n) = Oε(n ε), that holds for any ε > 0. Writing ‖C‖ for the maximum modulus of the coefficients defining C, we deduce from Lemma 4.2 that NC(B) ≪ε ‖C‖εB. This should be compared with the work of Heath-Brown [41, Theorem 3] that shows NC(B) ≪d,ε B2/d+ε, for any irreducible plane curve C ⊂ P2 of degree d. Both Lemma 4.1 and Lemma 4.2 are established using the geometry of numbers. 4.1. Elementary considerations. We proceed to show how NU (B) can be related to a count of the integer points on the corresponding universal torsor. Our argument is in complete analogy to that presented in §3.1, although the individual steps differ somewhat. If S̃ denotes the minimal desingularisation of the surface S, then Derenthal [26] has calculated the universal torsor over S̃, it being embedded in A10 by a single equation s1u1y 1 + s2u2y 2 + s3u3y 3 = 0. (4.4) Note that one of the variables does not appear explicitly in the equation. We will need the following basic fact. 46 T.D. BROWNING Exercise 18. Let a, b ∈ N. Show that a | b2 if and only if a = uv2 for u, v ∈ N such that u is square-free and uv | b. Given v ∈ R and s,u,y ∈ R3, define Ψ(v, s,u,y) := max { |s1s2s3|, |u21u22u23v3y1y2y3| |s1u21u2u3v2y21|, |s2u1u22u3v2y22 | . (4.5) We are now ready to record our translation of the problem to the universal torsor. Lemma 4.3. We have NU (B) = 2#   (v, s,u,y) ∈ N4 × Z3 × N3 : u3 > 0, Ψ(v, s,u,y) 6 B (4.4) holds, |µ(u1u2u3)| = 1, gcd(s1s2s3, u1u2u3v) = 1, gcd(yi, yj) = 1, gcd(yi, sj , sk) = 1,   where i, j, k denote distinct elements from the set {1, 2, 3}. Proof. Let t ∈ Z4 be a vector such that (4.1) holds, with t3, t4 > 1. Write η14 = gcd(t1, t4), η24 = gcd(t2, t4/η14), η12 = gcd(t1/η14, t2/η24). Then η12, η14, η24 ∈ N and there exists z4 ∈ N and z1, z2 ∈ Z such that t1 = η12η14z1, t2 = η12η24z2, t4 = η14η24z4. Moreover, it is not hard to deduce that gcd(η12z1, η24z4) = gcd(η12z2, z4) = gcd(z1, z2) = 1, gcd(t3, η14, η12η24z2) = 1. Under this substitution the equation (4.1) becomes η312z1z2(η14z1 + η24z2) = t It follows that η312 | t23 in any given integer solution. Exercise 18 therefore implies that there exist u, v, z3 ∈ N such that |µ(u)| = 1 and η12 = uv 2, t3 = u 2v3z3, z1z2(η14z1 + η24z2) = uz We proceed to consider the effect of the divisibility condition z1z2 | uz23 that this equation entails. Recall that gcd(z1, z2) = gcd(z1, z4) = gcd(z2, z4) = 1. Since z1z2 | uz23 , there must exist u1, u2, u3, w1, w2, w3 ∈ Z such that w1, w2, w3, u3 > 0 and u = u1u2u3, z1 = u1w 1, z2 = u2w 2, z3 = w1w2w3. Here we have used the fact that if p is a prime such that p ∤ u and p | z1z2, then p must divide z1 or z2 to even order. Under these substitutions our equation becomes η14u1w 1 + η24u2w 2 = u3w THE MANIN CONJECTURE IN DIMENSION 2 47 Moreover, we will have the corresponding coprimality conditions gcd(u1u2u3vw1, η24z4) = gcd(u1u2u3vw2, z4) = gcd(u1w1, u2w2) = 1, (4.6) |µ(u1u2u3)| = 1, gcd(u21u22u23v3w1w2w3, η14, η24u1u22u3v2w22) = 1. (4.7) We now set s = (η14, η24, z4) and y = w, and replace (u1, u2, u3) by (−u1,−u2, u3). Tracing through our argument, one sees that we have made the transformation  t1 = −s1u21u2u3v2y21, t2 = −s2u1u22u3v2y22, t3 = u 3y1y2y3, t4 = s1s2s3. In particular, it is clear that the height condition |x| 6 B is equivalent to Ψ(v, s,u,y) 6 B, in the notation of (4.5). We now observe that under this transformation the equation (4.1) becomes (4.4), and the coprimality relations (4.6) and (4.7) can be rewritten gcd(s2s3, u1u2u3vy1) = gcd(s3, u1u2u3vy2) = gcd(u1y1, u2y2) = 1, |µ(u1u2u3)| = 1, gcd(s1, u1u2u3vy2 gcd(y3, s2)) = 1. We can combine these relations with (4.4) to simplify them still further. In fact, once combined with (4.4), we claim that they are equivalent to the conditions |µ(u1u2u3)| = 1, gcd(s1s2s3, u1u2u3v) = gcd(yi, yj) = gcd(yi, sj , sk) = 1, appearing in the statement of the lemma. To establish the forward impli- cation, it suffices to show that gcd(y1, y3) = gcd(y2, y3) = 1, the remaining conditions being immediate. But these two conditions follow on combining (4.4) with the fact that gcd(y1, s2u2y2) = 1. To see the reverse implication, the conditions are all immediate apart from gcd(y1, s2s3) = gcd(y2, s1s3) = gcd(u1, y2) = gcd(u2, y1) = 1. But each of these is an easy consequence of the assumed coprimality rela- tions, and (4.4). Finally, we leave it as an exercise to the reader to check that each (v, s,u,y) counted in the right hand side of Lemma 4.3 produces a primitive solution of (4.1) with t3, t4 > 1. This completes the proof of Lemma 4.3. � In what follows let us write i for a generic element of the set {1, 2, 3}. Fix a choice of v ∈ N and Si, Ui, Yi > 0, and write N = Nv(S;U;Y) (4.8) for the total contribution to NU (B) in Lemma 4.3 from s,u,y contained in the intervals Si/2 < si 6 Si, Ui/2 < |ui| 6 Ui, Yi/2 < yi 6 Yi. (4.9) Write S = S1S2S3, U = U1U2U3, Y = Y1Y2Y3. 48 T.D. BROWNING If N = 0 there is nothing to prove, and so we assume henceforth that the dyadic ranges in (4.9) produce a non-zero value of N . In particular we must S ≪ B U2Y ≪ B/v3, SiUUiY 2i ≪ B/v2. (4.10) In this set of lecture notes we will provide two upper bounds for NU (B). The object of our first bound is to merely establish linear growth, without worrying about the factor involving logB that we expect to see. By ignoring some of the technical machinery needed to get better bounds it is hoped that the overall methodology will be brought into focus. Later we will indicate how the expected upper bound can be retrieved with a little more work. 4.2. A crude upper bound. We begin by establishing linear growth for NU (B). Note that (4.10) forces the inequalities Si, Ui, Yi ≪ B. We proceed to establish the following upper bound. Lemma 4.4. We have NU (B) ≪ (logB)9 v6B1/3 Si,Ui,Yi>0 Nv(S;U;Y), where the maximum is over Si, Ui, Yi > 0 such that (4.10) holds. Proof. Our starting point is Lemma 4.3. It follows from (4.5) that v 6 B1/3 for any v, s,u,y that contributes to the right hand side. Let us fix a choice of v ∈ N such that v 6 B1/3, and cover the ranges for s,u,y with dyadic intervals. Thus for fixed integers σi, νi, ηi > 0, we write Si = 2 σi , Ui = 2 νi , Yi = 2 and consider the contribution from s,u,y in the range (4.9). But this is just N = Nv(S;U;Y). Now we have already seen that N = 0 unless (4.10) holds. Finally, since each Si, Ui, Yi is O(B), it follows that the number of dyadic intervals needed is O((logB)9). This completes the proof of the lemma. � We may now restrict our attention to bounding Nv(S;U;Y) for fixed values of Si, Ui, Yi > 0 such that (4.10) holds, and fixed v 6 B 1/3. In the arguments that follow it will be necessary to focus attention on primitive vectors s ∈ N3. To enable this we draw out possible common factors between s1, s2, s3, obtaining Nv(S;U;Y) = N ∗v (k−1S;U;Y). (4.11) Here N ∗v (S;U;Y) is defined as for Nv(S;U;Y) but with the extra condition that gcd(s1, s2, s3) = 1. Let us write S i = Si/k and S ′ = k−1S. Recall the equation (4.4) that we must count solutions to, which it will be convenient to denote by T , and which we will think of as defining a variety in P2 ×P2 ×P2, with homogeneous coordinates s,u,y. The key idea will be to count points on the fibres of projections π : T → P2 × P2. This amounts to fixing six of the variables and estimating the number of points on the resulting plane curve. Since this family of curves will vary with B, so it is vital to obtain bounds that are completely uniform in the coefficients of the defining equation. THE MANIN CONJECTURE IN DIMENSION 2 49 Let us begin by fixing the variables u,y, and estimating the corresponding number of vectors s. Now it follows from the coprimality conditions in Lemma 4.3 that gcd(u1y 1 , u2y 2, u3y 3) = 1. For fixed u,y, (4.4) defines a line in P2. We clearly have N ∗v (S′;U;Y) 6 M1(a;S in the notation of (4.3), with ai = uiy i . Since a is primitive, it therefore follows from Lemma 4.1 that N ∗v (S′;U;Y) ≪ k2 maxSiUiY ≪ UY + k−2S2/3U2/3Y 1/3. Here we have used the trivial lower bound max{a, b, c} > (abc)1/3, valid for any a, b, c > 0. Using (4.10) we conclude that N ∗v (S′;U;Y) ≪ UY + . (4.12) The second term here will be satisfactory from our point of view, but the first is disastrous, since we will run into trouble when it comes to summing over k in (4.11). It turns out that an altogether different bound is required to handle the contribution from really small values of S′. For this we will fix values of s,u in (4.4), and count points on the resulting family of conics. First we need to record the coprimality relation gcd(siui, sjuj) = 1, which we claim holds for any of the vectors s,u,y in which we are interested. But this follows on noting that gcd(si, sj) = 1 for any primitive vector s ∈ Z3 such that (4.4) holds and gcd(si, u1u2u3) = gcd(yi, sj, sk) = 1. We now have N ∗v (S′;U;Y) 6 M2(a;Y), in the notation of (4.3), with ai = siui. In particular ai is non-zero and gcd(ai, aj) = 1 in the statement of Lemma 4.2, whence N ∗v (S′;U;Y) ≪ k3/2Y 1/2 S1/2U1/2 2ω(s1s2s3u1u2u3). In view of the bounds S,U ≪ B, we clearly have 2ω(s1s2s3u1u2u3) ≪ε (s1s2s3u1u2u3)ε ≪ε (SU)ε ≪ε B2ε. Once inserted into our bound for N ∗v (S′;U;Y), and combined with (4.10), we deduce that N ∗v (S′;U;Y) ≪ε B2ε S1/2U1/2Y 1/2 SUB2ε B1+2ε k3/2v3/2 (4.13) Here the second term will provide a satisfactory contribution, and we will balance the first term with our earlier estimate (4.12). 50 T.D. BROWNING Note that k3/2v3/2 by (4.10). It therefore follows from (4.13) and (4.12) that N ∗v (S′;U;Y) ≪ε B1+2ε k3/2v Once inserted into (4.11), and then into the statement of Lemma 4.4, we may conclude that NU (B) ≪ε (logB)9 v6B1/3 B1+2ε k3/2v ≪ε B1+2ε(logB)10 ≪ε B1+3ε. Recall that NU2(B) 6 NU (B), where U ⊂ S is the open subset associated to the surface (4.1), and NU2(B) is the counting function associated to (1.12). On redefining the choice of parameter ε > 0, we have therefore established the following result. Theorem 4.1. We have NU2(B) ≪ε B1+ε, for any ε > 0. The reader will note that there many places in our argument where we have been wasteful. The most damaging has been in our use of the trivial bound 2ω(n) = Oε(n ε), in the deduction of (4.13). Using the fact that 2ω(n) has average order ζ(2)−1 log n, it is not particularly difficult to replace the Bε in Theorem 4.1 with a large power of logB. Exercise 19. By analysing the proof of Theorem 4.1, find an explicit value of A > 6 such that NU2(B) ≪ B(logB)A. In the next section we will be able to show that the value A = 6 is an admissible exponent, as claimed in Theorem 1.2. 4.3. A better upper bound. Crucial to the proof of Theorem 4.1 was an investigation of the density of integer solutions to the equation (4.4). It is in our treatment of this equation that we will hope to gain some saving. Lets put the problem on a more general footing. For any A,B,C ∈ R3>1, let M(A,B,C) denote the number of a,b, c ∈ Z3∗ such that a1b1c 1 + a2b2c 2 + a3b3c 3 = 0 (4.14) |ai| 6 Ai, |bi| 6 Bi, |ci| 6 Ci, gcd(ai, cj) = gcd(ci, cj) = 1 (4.15) |µ(a1a2a3)| = 1, gcd(ai, bj , bk) = 1. (4.16) Here, we recall that Z3∗ denotes the set of primitive vectors in Z 3 with all components non-zero. It will be convenient to set A = A1A2A3, B = B1B2B3, C = C1C2C3. THE MANIN CONJECTURE IN DIMENSION 2 51 Arguing exactly as in the previous section, it is not difficult to deduce from Lemmas 4.1 and 4.2 that M(A,B,C) ≪ε Amin{C,AεB1+ε}+A2/3B2/3C1/3 +A1/2+εB1/2+εC1/2 for any ε > 0, whence M(A,B,C) ≪ε A2/3B2/3C1/3 +A1+εB1/2+εC1/2. (4.17) By working a little harder, we would like to replace the terms Aε, Bε with something rather smaller. The main problem to be faced emerges in the application of Lemma 4.2, which gives M(A,B,C) ≪ |a1a2a3b1b2b3|1/2 2ω(a1a2a3b1b2b3). Rather than using the trivial bound 2ω(n) = Oε(n ε), as above, we can try to make use of the fact that 2ω(n) has average order ζ(2)−1 log n in order to get some saving. Following this line of thought it is fairly straightforward to show that AεBε can be replaced by (logA)3/2(logB)3 in (4.17). However this would still not be enough to deduce the best possible upper bound for NU (B) that we would like. Let us simplify matters by considering only the contribution S(A,B) = 2ω(a1a2a3b1b2b3), to the above estimate for M(A,B,C). Then S(A,B) has exact order of magnitude (logAi)(logBi), so how can we hope to do better than this? The crucial observation comes in noting that we are only interested in summing over values of a,b for which the corresponding conic (4.14) has a non-zero solution c ∈ Z3, with gcd(ci, cj) = 1. If we denote this finer quantity by S∗(A,B), then it is actually possible to show that S∗(A,B) ≪ AB. (4.18) This is established in [13, Lemma 1], and is simply a facet of the well-known fact that a random plane conic doesn’t have a rational point. This should be compared with the work of Serre [58]. Using the large sieve inequality, Serre has shown #{y ∈ Z3 : |y| 6 Y, (−y1y3,−y2y3)Q = 1} ≪ (log Y )3/2 where (a, b)Q = 1, if ax2 + by2 = z2 has a solution (x, y, z) 6= 0 in Q3, −1, otherwise, denotes the Hilbert symbol. Guo [32] has established an asymptotic for- mula for the corresponding quantity in which one counts only odd values of y1, y2, y3 such that the product y1y2y3 is square-free. 52 T.D. BROWNING Thus in addition to considering the density of integer solutions to diagonal quadratic equations, as in the previous section, we also need to consider how often such an equation has at least one non-trivial integer solution in order to derive sufficiently sharp bounds. The outcome of this investigation is the following result, which is established in [13, Lemma 2]. Lemma 4.5. For any ε > 0, we have M(A,B,C) ≪ε A2/3B2/3C1/3 + στAB1/2C1/2, where σ = 1 + min{A,B}ε min{BiBj}1/16 , τ = 1 + min{BiBj}1/16 It is clear that this constitutes a substantial sharpening over our earlier estimate (4.17) for M(A,B,C). Nonetheless this is still not enough on its own, and we will need an alternative estimate when B1, B2, B3 have particularly awkward sizes. The following result is rather easy to establish. Lemma 4.6. We have M(A,B,C) ≪ ABiBj(Ck + CiCjA−1k )(logAC) for any permutation {i, j, k} of the set {1, 2, 3}. Proof. For fixed integers a, b, q, let ρ(q; a, b) denote the number of solutions to the congruence at2 + b ≡ 0 mod q. We then have ρ(q; a, b) 6 |µ(d)| . (4.19) It will clearly suffice to establish Lemma 4.6 in the case (i, j, k) = (1, 2, 3), say. Now it follows from (4.14) that for given ai, b1, b2, c3, and each corre- sponding solution t of the congruence a1b1t 2 + a2b2 ≡ 0 (mod a3c23), we must have c1 ≡ tc2 mod a3c23 in any solution to be counted. This gives rise to an equation of the form h.w = 0, with h = (1,−t, a3c23) and w = (c1, c2, k). Upon recalling that gcd(c1, c2) = 1 from (4.15), an application of Lemma 4.1 therefore yields the bound ≪ ρ(a3c23; a1b2, a2b2) |a3c23| for the number of possible b3, c1, c2 given fixed choices of ai, b1, b2 and c3. It now follows from (4.19) that M(A,B,C) ≪ ai,b1,b2,c3 ρ(a3c 3; a1b2, a2b2) |a3c23| ai,b1,b2,c3 d|a3c3 |µ(d)| (−a1a2b1b2 |a3c23| ai,b1,b2,c3 τ(a3)τ(c3) + C1C2 ai,b1,b2,c3 τ(a3)τ(c3) |a3c23| THE MANIN CONJECTURE IN DIMENSION 2 53 A simple application of partial summation now reveals that M(A,B,C) ≪ AB1B2C3 +A1A2B1B2C1C2 (logAC)2, as required to complete the proof of Lemma 4.6. � We are now ready to combine Lemmas 4.5 and 4.6 to get a sharper upper bound for NU (B). Taking Lemma 4.3 as our starting point we need to bound the quantity N = Nv(S;U;Y) defined in (4.8), for fixed choices of v ∈ N and Si, Ui, Yi > 0. As previously we will need to extract common factors from s1, s2, s3, leading to the equality (4.11). Writing S i = Si/k and S′ = k−1S, as before, it is a simple matter to check that we have N ∗v (S′;U;Y) 6 M(U,S′,Y), with (a,b, c) = (u, s,y). Indeed we plainly have gcd(ui, yj) = gcd(yi, yj) = 1, |µ(u1u2u3)| = gcd(ui, sj , sk) = 1, and u, s,y ∈ Z4∗ , for any vectors counted by N ∗v (S′;U;Y), as required for M(U,S′,Y). It now follows from (4.11) and Lemma 4.5 that Nv(S;U;Y) ≪ε (U2/3S2/3Y 1/3 + k2/16στ US1/2Y 1/2 ≪ε U2/3S2/3Y 1/3 + στUS1/2Y 1/2, for any ε > 0, where σ = 1 + min{S,U}ε min{SiSj}1/16 , τ = 1 + min{SiSj}1/16 In order to obtain our final estimate for NU (B) we need to sum this bound over all positive integers v 6 B1/3, as in Lemma 4.4, and over all possible dyadic intervals for Si, Ui, Yi, subject to (4.10). Suppose for the moment that we want to sum over all possible dyadic intervals X 6 |x| < 2X, for which |x| 6 X . Then in deducing Lemma 4.4 we employed the basic bound O(logX ) for the number of possible choices for X. In the present investigation we will be more efficient and take advantage of the easily established estimates Xδ ≪δ 1, if δ < 0, X δ, if δ > 0, where the sum is over dyadic intervals for X 6 X . We will make frequent use of these bounds without further mention. Returning to our estimate for Nv(S;U;Y), we may now conclude from the bound Yi 6 B 1/2/(v2SiUUi) 1/2 in (4.10) that NU (B) ≪ε v6B1/3 Si,Ui,Yi U2/3S2/3Y 1/3 + στUS1/2Y 1/2 ≪ε B1/2 v6B1/3 Si,Ui v6B1/3 Si,Ui,Yi στUS1/2Y 1/2 ≪ε B(logB)6 + v6B1/3 Si,Ui,Yi στUS1/2Y 1/2. 54 T.D. BROWNING The first term on the right-hand side is clearly satisfactory, and it remains to deal with the second term, which we denote by R for convenience. We would like to show that R ≪ B(logB)6. Suppose without loss of generality that S1 6 S2 6 S3, so that in particular min{SiSj} = S1S2 in σ and τ. If there is a constant A > 0 such that S3 6 (S1S2) A, then it follows that σ 6 (S1S2) ε−1/16Sε3 6 (S1S2) (1+A)ε−1/16 ≪ 1, provided that ε is sufficiently small. Taking τ ≪ logB, we may then argue as above to conclude that there is a contribution of O(B(logB)6) to R from this case. Suppose now that there exists A′ > 0 such that U 6 (S1S2) Then we have σ ≪ 1 and τ ≪ logB, so that there is a contribution of O(B(logB)6) to R in this case too. Finally it remains to consider the contribution to NU (B) from Si, Ui, Yi such that S1S2 6 min{S3, U}δ, (4.20) for some small value of δ > 0, with S1 6 S2 6 S3. Let us denote this contribution N0, say. To estimate N0 we will return to the task of estimat- ing Nv(S;U;Y) for fixed v, Si, Ui, Yi, but this time apply Lemma 4.6 with (i, j, k) = (1, 2, 3). This gives Nv(S;U;Y) ≪ (logB)2 US1S2Y3 + U1U2S1S2Y1Y2 We must now sum over dyadic intervals for Si, Ui, Yi. Thus it follows from the bound Yi 6 B 1/2/(v2SiUUi) 1/2 in (4.10) that N0 ≪ (logB)2 v6B1/3 Si,Ui,Yi US1S2Y3 + U1U2S1S2Y1Y2 ≪ (logB)2 v6B1/3 Si,Ui Y1,Y2 B1/2U1/2S1S2 Si,Ui,Y3 B(S1S2U1U2) Since U2 ≪ B/(v3Y1Y2) in (4.10), and S1S2 ≪ Sδ3 by (4.20), we therefore deduce that the overall contribution from the first inner sum is ≪ (logB)2 v6B1/3 Si,Y1,Y2 U2,U3 B3/4S1S2 v7/4S S2,S3,Y1,Y2 U2,U3 B3/4(logB)2 1/2−δ Turning to the contribution from the second inner sum, we deduce from a second application of (4.20) that N0 ≪ B + (logB)2 Si,Ui,Y3 B(S1S2U1U2) ≪ B + (logB)2 S2,S3,Ui,Y3 U (1−δ)/2 ≪ B(logB)5. THE MANIN CONJECTURE IN DIMENSION 2 55 Once combined with our earlier work, this therefore concludes the proof of Theorem 1.2. Exercise 20. By mimicking the argument in [13, §5], establish the lower bound NU2(B) ≫ B(logB)6. Acknowledgements. The author is grateful to Roger Heath-Brown for a number of useful conversations relating to the contents of §2.3, and to Michael Harvey for spotting several typographical errors in an earlier draft. References [1] V.V. Batyrev and Y.I. Manin, Sur le nombre des points rationnels de hauteur borné des variétés algébriques. Math. Ann. 286 (1990), 27–43. [2] V.V. Batyrev and Y. Tschinkel, Tamagawa numbers of polarized algebraic varieties. Astérisque 251 (1998), 299–340. [3] V.V. Batyrev and Y. Tschinkel, Manin’s conjecture for toric varieties. J. Alg. Geom. 7 (1998), 15–53. [4] B.J. Birch, Forms in many variables. Proc. Roy. Soc. Ser. A 265 (1961), 245–263. [5] R. de la Bretèche, Sur le nombre de points de hauteur bornée d’une certaine surface cubique singulière. Astérisque 251 (1998), 51–77. [6] R. de la Bretèche, Compter des points d’une variété torique. J. Number Theory 87 (2001), 315–331. [7] R. de la Bretèche, Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5. Duke Math. J. 113 (2002), 421–464. [8] R. de la Bretèche and T.D. Browning, On Manin’s conjecture for singular del Pezzo surfaces of degree four, I. Michigan Math. J., to appear. [9] R. de la Bretèche and T.D. Browning, On Manin’s conjecture for singular del Pezzo surfaces of degree four, II. Math. Proc. Camb. Phil. Soc., to appear. [10] R. de la Bretèche and É. Fouvry, L’éclaté du plan projectif en quatre points dont deux conjugués. J. Reine Angew. Math. 576 (2004), 63–122. [11] R. de la Bretèche and P. Swinnerton-Dyer, Fonction zêta des hauteurs associée à une certaine surface cubique. Submitted, 2006. [12] R. de la Bretèche, T.D. Browning and U. Derenthal, On Manin’s conjecture for a certain singular cubic surface. Ann. Sci. École Norm. Sup., to appear. [13] T.D. Browning, The density of rational points on a certain singular cubic surface. J. Number Theory 119, (2006), 242–283. [14] T.D. Browning, An overview of Manin’s conjecture for del Pezzo surfaces. Proceedings of the Gauss–Dirichlet conference (Göttingen), to appear. [15] T.D. Browning and D.R. Heath-Brown, Counting rational points on hypersurfaces. J. Reine Angew. Math. 584 (2005), 83–115. [16] J.W. Bruce and C.T.C. Wall, On the classification of cubic surfaces. J. London Math. Soc. 19 (1979), 245–256. [17] J.W.S. Cassels, Introduction to the geometry of numbers. 2nd ed., Springer-Verlag, 1997. [18] A. Cayley, A memoir on cubic surfaces. Phil. Trans. Roy. Soc. 159 (1869), 231–326. [19] A. Chambert-Loir and Y. Tschinkel, On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148 (2002), 421–452. [20] J.-L. Colliot-Thélène and J.-J. Sansuc, Torseurs sous des groupes de type multipli- catif; applications à l’étude des points rationnels de certaines variétés algébriques. C. R. Acad. Sci. Paris Sér. A-B 282 (1976), 1113–1116. [21] J.-L. Colliot-Thélène and J.-J. Sansuc, La descente sur les variétés rationnelles. II. Duke Math. J. 54 (1987), 375–492. [22] J.-L. Colliot-Thélène, D. Kanevsky, et J.-J. Sansuc, Arithmétique des surfaces cu- biques diagonales, Diophantine approximation and transcendenc theory (Bonn, 1985), Lecture Notes in Math. 1290, Springer-Verlag, 1987, pp. 1–108. [23] D.F. Coray and M.A. Tsfasman, Arithmetic on singular Del Pezzo surfaces. Proc. London Math. Soc. 57 (1988), 25–87. 56 T.D. BROWNING [24] H. Davenport, Cubic forms in 16 variables. Proc. Roy. Soc. A 272 (1963), 285–303. [25] H. Davenport, Analytic Methods in Diophantine Equations and Diophantine Inequal- ities. 2nd ed., edited by T.D. Browning, Cambridge University Press, 2005. [26] U. Derenthal, Singular del Pezzo surfaces whose universal torsors are hypersurfaces. Submitted, 2006. [27] U. Derenthal and Y. Tschinkel, Universal torsors over Del Pezzo surfaces and rational points, Equidistribution in Number theory, An Introduction, 169–196, NATO Science Series II 237, Springer, 2007. [28] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), 349–366. [29] É. Fouvry, Sur la hauteur des points d’une certaine surface cubique singulière. Astérisque 251 (1998), 31–49. [30] A. Granville, On the number of solutions to the generalized Fermat equation. Num- ber theory (Halifax, NS, 1994), 197–207, CMS Conf. Proc. 15, Amer. Math. Soc., Providence, RI, 1995. [31] J. Franke, Y.I. Manin and Y. Tschinkel, Rational points of bounded height on Fano varieties. Invent. Math. 95 (1989), 421–435. [32] C.R. Guo, On solvability of ternary quadratic forms. Proc. London Math. Soc. 70 (1995), 241–263. [33] R. Hartshorne, Algebraic geometry. Springer-Verlag, 1977. [34] B. Hassett and Y. Tschinkel, Universal torsors and Cox rings. Arithmetic of higher- dimensional algebraic varieties (Palo Alto, CA, 2002), 149–173, Progr. Math. 226, Birkhäuser, 2004. [35] D.R. Heath-Brown, Diophantine approximation with square-free numbers. Math. Zeit. 187 (1984), 335–344. [36] D.R. Heath-Brown, The density of zeros of forms for which weak approximation fails. Math. Comp. 59 (1992), 613–623. [37] D.R. Heath-Brown, A new form of the circle method, and its application to quadratic forms. J. Reine Angew. Math. 481 (1996), 149–206. [38] D.R. Heath-Brown, The density of rational points on cubic surfaces. Acta Arith. 79 (1997), 17–30. [39] D.R. Heath-Brown, The circle method and diagonal cubic forms. R. Soc. Lond. Philos. Trans. Ser. A 356 (1998), 673–699. [40] D.R. Heath-Brown, Counting rational points on cubic surfaces. Astérisque 251 (1998), 13–29. [41] D.R. Heath-Brown, The density of rational points on curves and surfaces. Annals of Math. 155 (2002), 553–595. [42] D.R. Heath-Brown, The density of rational points on Cayley’s cubic surface. Pro- ceedings of the session in analytic number theory and Diophantine equations, Bonner Math. Schriften 360, 2003. [43] D.R. Heath-Brown and B.Z. Moroz, The density of rational points on the cubic surface 0 = X1X2X3. Math. Proc. Camb. Soc. 125 (1999), 385–395. [44] M. Hindry and J. Silverman, Diophantine geometry. Springer-Verlag, 2000. [45] W.V.D. Hodge and D. Pedoe, Methods of algebraic geometry. Vol. 2, Cambridge University Press, 1952. [46] K. Ireland and M. Rosen, A classical introduction to modern number theory. 2nd ed., Springer-Verlag, 1990. [47] H. Knörrer, Isolierte singularitäten von durchschnitten zweier quadriken. Bonner Mathematische Schriften 117. Universität Bonn Math. Institut, Bonn, 1980. [48] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. [49] Y.I. Manin, Cubic forms. 2nd ed., North-Holland Mathematical Library 4, North- Holland Publishing Co., 1986. [50] L.J. Mordell, A remark on indeterminate equations in several variables. J. London Math. Soc. 12 (1937), 127–129. [51] E. Peyre, Hauteurs et nombres de Tamagawa sur les variétés de Fano. Duke Math. J. 79 (1995), 101–218. THE MANIN CONJECTURE IN DIMENSION 2 57 [52] E. Peyre, Counting points on varieties using universal torsors. Arithmetic of higher- dimensional algebraic varieties (Palo Alto, CA, 2002), 61–81, Progr. Math. 226, Birkhäuser, 2004. [53] E. Peyre and Y. Tschinkel, Tamagawa numbers of diagonal cubic surfaces, numerical evidence. Math. Comp. 70 (2001), 367–387. [54] E. Peyre and Y. Tschinkel, Tamagawa numbers of diagonal cubic surfaces of higher rank. Rational points on algebraic varieties, 275–305, Progr. Math., 199, Birkhäuser, 2001. [55] P. Salberger, Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque 251 (1998), 91–258. [56] L. Schläfli, On the distribution of surfaces of the third order into species. Phil. Trans. Roy. Soc. 153 (1864), 193–247. [57] B. Segre, A note on arithmetical properties of cubic surfaces. J. London Math. Soc. 18 (1943), 24–31. [58] J.- P. Serre, Spécialisation des éléments de Br2(Q(T1, . . . , Tn)). C. R. Acad. Sci. Paris 311 (1990), 397–402. [59] J.B. Slater and P. Swinnerton-Dyer, Counting points on cubic surfaces. I. Astérisque 251 (1998), 1–12. [60] G. Tenenbaum, Introduction to analytic and probabilistic number theory. Translated from the 2nd French ed., Cambridge Studies in Advanced Mathematics 46, Cam- bridge University Press, 1995. [61] E.C. Titchmarsh, The theory of the Riemann zeta-function. 2nd ed., edited by D.R. Heath-Brown, Oxford University Press, 1986. [62] C.T.C. Wall, The first canonical stratum. J. London Math. Soc. 21 (1980), 419–433. School of Mathematics, University of Bristol, Bristol BS8 1TW E-mail address: [email protected] 1. Introduction 1.1. Notation 1.2. The Manin conjectures 1.3. Degree 3 surfaces 1.4. Degree 4 surfaces 1.5. Degree 5 surfaces 1.6. Universal torsors 2. Further heuristics 2.1. The lines on a cubic surface 2.2. Cubic characters and Jacobi sums 2.3. The Hardy–Littlewood circle method 3. The A1 del Pezzo surface of degree 6 3.1. Elementary considerations 3.2. The asymptotic formula 4. The D4 del Pezzo surface of degree 3 4.1. Elementary considerations 4.2. A crude upper bound 4.3. A better upper bound References

0704.1218

The impact of radio feedback from active galactic nuclei in cosmological simulations: Formation of disk galaxies

Mon. Not. R. Astron. Soc. 000, 1–22 (2007) Printed 24 October 2018 (MN LATEX style file v2.2) The impact of radio feedback from active galactic nuclei in cosmological simulations: Formation of disk galaxies Takashi Okamoto1⋆, Rodrigo S. Nemmen2, Richard G. Bower1 1Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham, DH1 3LE 2Instituto de Fı́sica, Universidade Federal do Rio Grande do Sul, Campus do Vale, Porto Alegre, RS, Brazil ABSTRACT In this paper, we present a new implementation of feedback due to active galactic nuclei (AGN) in cosmological simulations of galaxy formation. We assume that a fraction of jet energy, which is generated by an AGN, is transferred to the surrounding gas as thermal en- ergy. Combining a theoretical model of mass accretion onto black holes with a multiphase description of star-forming gas, we self-consistently follow evolution of both galaxies and their central black holes. The novelty in our model is that we consider two distinct accretion modes: standard radiatively efficient thin accretion disks and radiatively inefficient accretion flows which we will generically refer to as RIAFs; motivated by theoretical models for jet pro- duction in accretion disks, we assume that only the RIAF is responsible for the AGN feedback. The focus of this paper is to investigate the interplay between galaxies and their central black holes during the formation of a disc galaxy. We find that, after an initial episode of bursting star formation, the accretion rate onto the central black hole drops so that the accretion disk switches to a RIAF structure. At this point, the feedback from the AGN becomes efficient and slightly suppresses star formation in the galactic disk and almost completely halts star formation in the bulge. This suppression of the star formation regulates mass accretion onto the black hole and associated AGN feedback. As a result, the nucleus becomes a stochas- tically fuelled low-luminosity AGN (Seyfert galaxy) with recurrent short-lived episodes of activity after the star bursts. During the “on” events the AGN produces reasonably power- ful jets (radio-loud state) and is less luminous than the host galaxy, while in the “off” phase the nucleus is inactive and “radio-quiet”. Our model predicts several properties of the low- luminosity AGN including the bolometric luminosity, jet powers, the effect on kpc-scale of the radio jet and the AGN lifetime, which are in broad agreement with observations of Seyfert galaxies and their radio activity. We also find that the ratios between the central black hole mass and the mass of the host spheroid at z = 0 are ∼ 10−3 regardless of the strength of either supernova feedback or AGN feedback because the radiation drag model directly relates the star formation activity in the galactic centre and the mass accretion rate onto the central black hole. Key words: black hole physics – galaxies: active – galaxies: starburst – galaxies: Seyfert – galaxies: evolution – galaxies: formation 1 INTRODUCTION Recent observations have suggested a fundamental connec- tion between active galactic nuclei (AGN) and the formation of galaxies. Firstly, there is now a well established correla- tion between the properties of galaxies and the masses of the black holes (BH) at their centres. The mass of the cen- tral black holes tightly correlates with the mass of the host galactic bulges (with a median BH-to-bulge mass ratio of ≃ 0.001, e.g. Kormendy & Richstone 1995; Magorrian et al. 1998; Merritt & Ferrarese 2001; McLure & Dunlop 2002; ⋆ E-mail: [email protected] Marconi & Hunt 2003), as well as with the stellar velocity disper- sion of the bulges (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002). This discovery points to a fundamental connection between the growth of central BHs and the formation of stellar spheroids in galaxies. Secondly, high resolution X-ray observations of galaxy clusters have revealed large, radio-plasma cavities in intra-cluster medium. These are usually associated with episodic outbursts from a central radio galaxy and indicate that huge amounts of mechanical energy are being deposited into the intracluster medium by powerful AGN-driven jets (e.g. Allen et al. 2006; Bı̂rzan et al. 2004; Fabian et al. 2006; Rafferty et al. 2006; Taylor et al. 2006). Simulations of the impact of these AGN- driven radio cavities suggest that this powerful feedback provides c© 2007 RAS http://arxiv.org/abs/0704.1218v3 2 T. Okamoto, R. S. Nemmen and R. G. Bower sufficient energy to offset the cooling radiation from the cluster and potentially explain why so little cool (T < 3 keV) gas is seen in these systems (Quilis et al. 2001; Churazov et al. 2002; Dalla Vecchia et al. 2004; Omma et al. 2004; Sijacki & Springel 2006; Sijacki et al. 2007). Motivated by these discoveries, simple prescriptions for the feedback from AGN (aka. “radio-mode feedback”) have recently been incorporated into semi-analytic galaxy formation models (e.g. Granato et al. 2004; Monaco & Fontanot 2005; Cattaneo et al. 2005; Croton et al. 2006; De Lucia et al. 2006; Bower et al. 2006). Including radio-mode AGN feedback has resulted in dramatic im- provements in the models’ ability to match the sharp decline of the galaxy luminosity function and to explain the “down-sizing” seen in the evolution of the galaxy population. In particular, Bower et al. shows that, by assuming that AGN feedback operates only in quasi- hydrostatic halos where the cooling time is longer than the dynami- cal time (Dalla Vecchia 2005; Sijacki & Springel 2006), the galaxy luminosity functions in local and higher redshifts universe can be matched well. Croton et al. (2006) and Kitzbichler & White (2007) showed that similar results were obtained by modifying the Bondi- Hoyle-Lyttleton accretion rate formula (Hoyle & Lyttleton 1939; Bondi & Hoyle 1944; Bondi 1952) to account for the multiphase structure of the accreting gas in rapidly cooling halos. Recent simulations also have begun to track the impact of AGN feedback on the galaxy population (Kawata & Gibson 2005; Sijacki & Springel 2006; Di Matteo et al. 2005; Springel et al. 2005b; Sijacki et al. 2007; Di Matteo et al. 2007). One of the first hydrodynamic simulation of galaxy formation which invoked AGN feedback was performed by Kawata & Gibson (2005) by using cosmological simulations with smoothed particle hydrodynamics (SPH). They reproduced observed X-ray and optical properties of elliptical galaxies by injecting thermal energy into the centre of the main progenitor at z < 1, assuming that sufficient BH mass was present for the AGN to be active when a convergent gas inflow ex- ists at the centre of the main progenitor. A self-consistent model for AGN “quasar-mode” feedback associated with BH accretion in simulations of galaxy mergers was proposed by Di Matteo et al. (2005) and Springel et al. (2005b). They estimated accretion rate onto BHs by using a Bondi-Hoyle-Lyttleton parameterisation and injected some fraction of accreted rest mass energy into the sur- rounding interstellar medium (ISM) in the form of thermal energy. By using this model and performing a series of merger simulations, Hopkins et al. (2006) simulated evolution of quasar luminosity and predicted luminosity functions of quasars. In this paper, we introduce a new methodology to incorporate BH growth and AGN radio feedback self-consistently in cosmolog- ical simulations of galaxy formation. We are motivated to do this by the recent semi-analytic models that we have discussed above. These indicate that the distinction between “quasar” and “radio” modes is necessary to achieve good matches to the global proper- ties of galaxies. We distinguish these two modes depending on the accretion rate onto the central BHs assuming that the radio-mode exists only when the accretion rate onto the BH is low. We estimate the mass accretion rate onto the central BHs by combining a radiation drag model by Kawakatu & Umemura (2002) with our description of the ISM and star formation. A virtue of the radiation drag model is that it relates star formation activity in galactic centre, which can be resolved in our simulations, to mass accretion onto a central BHs. It implies that a fixed fraction of the star formation in bulges is converted into BH mass just as assumed in most semi-analytic models (e.g. Croton et al. 2006; Bower et al. 2006). Since our estimate of the accretion rate onto central BHs is based on radiation from stars, the mass growth of BHs in our sim- ulations is resolution independent so long as the star formation is modelled in a resolution independent fashion. In order to model the AGN radio-mode feedback, we distin- guish two fundamentally different modes of AGN accretion: radia- tively efficient geometrically thin accretion flows (standard disks; Shakura & Sunyaev 1973) and geometrically thick, radiatively in- efficient accretion flows (which we will generically refer to as RIAFs; Narayan et al. 1998; Narayan 2005; Nemmen et al. 2006; Yuan 2007). We assume only the latter are responsible for the radio- mode feedback through production of powerful jets (e.g. Rees et al. 1982; Meier 2001; Maccarone et al. 2003; Churazov et al. 2005). This assertion is supported by theoretical models for jet production since the jet power is intimately linked to the scale of the magnetic field loops in the vicinity of the last stable orbit around the BH (e.g. Livio et al. 1999; Meier 2001; Nemmen et al. 2007). Since RIAFs exist only when the accretion rate is much lower than the Eddington rate (Narayan et al. 1998), AGNs can produce strong feedback in objects which have low specific star formation rates because the ra- diation drag model directly connects the star formation rate around BHs and mass accretion rate onto the BHs. Given the limited nu- merical resolution inherent in our simulations, we adopt a subgrid model for the impact of the AGN feedback. We simply assume that a certain fraction of the available jet power thermally couples to nearby diffuse halo gas. Until today, most of simulations of galaxy formation with AGN feedback aimed to form elliptical galaxies because the AGN feedback is the most plausible mechanism that halts the star formation activity in elliptical galaxies (Kawata & Gibson 2005; Springel et al. 2005b; Di Matteo et al. 2005; Sijacki et al. 2007). Therefore roles of AGN feedback in disk galaxy formation are still largely unknown in spite that disk galaxies also harbour cen- tral supermassive BHs in their bulges. In this paper we hence fo- cus on the impact that feedback from an AGN jet might have on the formation of a large disk galaxy, and the resulting coevolution of the AGN and the host galaxy. The majority of AGN hosted by disk galaxies are known to be either Seyfert nuclei or the lowest luminosity AGN, the low-ionisation nuclear emission-line regions (LINERs) (Ho et al. 1997; see Ho 2004 for a review). Several tar- geted observations of individual objects (e.g., Wilson & Ulvestad 1983; Morganti, Oosterloo, & Tsvetanov 1998; Cecil et al. 2000; Whittle & Wilson 2004; Keel et al. 2006; Kharb et al. 2006; Middelberg et al. 2007) indicate that Seyfert galaxies commonly host compact radio jets spanning ∼ 10 − 100 pc which strongly affect the state of the ISM and contribute to its ionisation via shocks (e.g., Kukula et al. 1993, 1996; Falcke, Wilson, & Simpson 1998; Capetti et al. 1999; Bicknell et al. 1998; Mundell et al. 2003; Riffel et al. 2006). It is also common that kpc-scale radio structures are observed, which are thought as extensions of the inner radio jets (e.g., Wilson & Ulvestad 1983; Morganti, Oosterloo, & Tsvetanov 1998; Cecil et al. 2000; Whittle & Wilson 2004; Keel et al. 2006; Kharb et al. 2006; Middelberg et al. 2007), and sometimes blow out of the galactic disk with no preferential direction (Schmitt et al. 2001; Gallimore et al. 2006). Very Large Array surveys of sam- ples of Seyfert and LINER galaxies indicate that radio outflows with extent & 1 kpc are a common feature in Seyfert galax- ies, and presumably are driven by the AGN jets (Colbert et al. 1996; Gallimore et al. 2006). Gallimore et al. (2006) in particu- lar find that & 44% of the galaxies in their complete sample display kpc-scale radio outflows, and they cannot rule out that most Seyfert galaxies produce extended radio outflows. The ki- netic power carried by the Seyfert jets as estimated from obser- c© 2007 RAS, MNRAS 000, 1–22 AGN feedback in cosmological simulations 3 vations can be as high as 1043erg s−1 (e.g., Wilson & Ulvestad 1983; Morganti, Oosterloo, & Tsvetanov 1998; Kharb et al. 2006). If an appreciable fraction of this power reaches out of the nuclear region, then the jets may affect the environment of the AGN and be a source of feedback in Seyfert galaxies. The impact of feedback from AGN jets on the nuclear environment of the host disk galax- ies and the evolution of the nuclear activity during the formation of the disk galaxy are open problems, which we attack using our cos- mological simulations. In this paper, we show that distinguishing the two modes of AGN accretion yields results which are in broad agreement with observations of Seyferts. The paper is organised as follows. We first describe our micro- scopic descriptions of the ISM, star formation, and stellar feedback in Section 2 and BH growth and AGN feedback in Section 3. We present the details of our cosmological simulations in Section 4 and the results in Section 5. Finally, we summarise and discuss our re- sults in Section 6. Readers who are observation-oriented and are not interested in the technical details of the simulation may wish to skip directly to Section 6, where we compare our results with observations of nuclear activity in Seyfert galaxies. 2 THE MODEL OF STAR-FORMING GAS AND STELLAR FEEDBACK In order to study the effect of AGN feedback on galaxy for- mation, we construct a physically motivated, well-controlled model of the star-forming gas. Based on the simplest star for- mation and feedback recipes used in the first generation of hy- drodynamic simulations of galaxy formation (Navarro & White 1993; Steinmetz & Mueller 1994; Cen & Ostriker 1992; Katz et al. 1996), several authors have introduced ‘multiphase’ models for star-forming gas in which the ISM is treated as a number of distinct phases by formulating differential equations that model the inter- actions between the phases (Yepes et al. 1997; Springel & Hern- quist 2003 hereafter SH03; Samland & Gerhard 2003; OEFJ05; Scannapieco et al. 2006 but see Booth et al. 2007 where they ex- plicitly model ‘clouds’ by means of sticky particles in SPH sim- ulations). Our model is based on the one used in OEFJ05 but has been significantly modified. We describe the details in the follow- ing subsections. 2.1 Gas cooling We compute the non-equilibrium cooling/heating rate and ionisa- tion state of each gas particle by using the cooling functions in Theuns et al. (2002) which include cooling by H, He, and metals, in the presence of an evolving but uniform ultraviolet background (Haardt & Madau 1996) that is switched on at z ≃ 6. Inverse Compton cooling is also included. Metals are carried by particles and once assigned to a parti- cle, they remain with it. Nonetheless, effective mixing takes place because we use the smoothed metallicity (smoothed in the same way as other SPH quantities) when computing the cooling rate and defining the metallicity of a newly-formed star. 2.2 Cloud formation and the cloud model As in SH03 and OEFJ05 we treat an SPH particle as a hybrid par- ticle that consists of two distinct phases, i.e. hot ambient gas and cold clouds, once its gas density, ρ, exceeds a density threshold, We assume that the cold clouds form and grow through ther- mal instability, that is Λnet(ρh, uh, Z) uh − uc , (1) where Mc and Mh are mass in cold clouds and mass in the hot phase associated with the particle, respectively, Λnet is the cooling function for gas of metallicity Z, and uh and uc represent specific internal energy of the hot phase and cold clouds, respectively. This implies that we assume that the gas is thermally unstable when ρ > ρth. Cold clouds remain at a fixed temperature, Tc = 100K hence uc is a constant as well. Note, however, as far as Tc is much lower than 104 K our results are hardly affected by the choice of Tc. We allow cold clouds to have a mass spectrum φ(m) ≡ . (2) Observationally α is in a range between α = 1.5 to 1.9 (Solomon & Rivolo 1989; Fukui et al. 2001; Heyer et al. 2001). Since the effect of changing the value of α is completely degen- erate with changing other parameters we will introduce such as star formation efficiency and thermal conduction efficiency, we simply fix α = 1.7 and the mass range of clouds, 102–106M⊙, throughout this paper. Following Samland & Gerhard (2003), we assume spherical clouds which follow the mass-radius relation of Elmegreen (1989) = 190 4 , (3) where P4 = 104K cm3 is the pressure in the ambient hot phase which is equal to the effective pressure of the particle. 2.3 Star formation We assume that giant molecular clouds in mass range of 104M⊙ to 106M⊙ are eligible to form stars with a star formation timescale that is proportional to the dynamical time of each cloud. Therefore the star formation rate for a cloud of mass m is ṁ∗ = , (4) where m∗ is stellar mass, tsf = tdyn/c∗ is the star formation timescale for a cloud of mass m, c∗ is a dimensionless star for- mation efficiency parameter that must be less than unity, and tdyn is the dynamical timescale of the cloud. For the mass-radius rela- tion in equation (3), the dynamical timescale of a cloud of mass m is given as tdyn = 32Gρ(m) ≃ 0.32 P Myr, (5) where ρ(m) is the mean density of a cloud of mass m. It should be noted that now the star formation timescale is a function of ambient pressure and it could be very short in high- pressure medium. Therefore we naturally have ‘shock-induced’ starbursts that OEFJ05 introduced by changing the star formation efficiency by hand when the rate of change of the entropy by shock heating exceeds a threshold value. The total star formation rate for clouds of total mass of 1M⊙ with the mass spectrum φ(m) is ob- tained by integrating ṁ∗ over the range of masses that are eligible to form stars, Ṁ∗ = Z 106M⊙ 104M⊙ ṁ∗(m)φ(m)dm, (6) c© 2007 RAS, MNRAS 000, 1–22 4 T. Okamoto, R. S. Nemmen and R. G. Bower where the normalisation Z 106M⊙ 102M⊙ mφ(m)dm = 1 is applied. 2.4 Cloud evaporation The different phases of the ISM exchange mass by evapora- tion. As in Samland & Gerhard (2003), we use the model of Cowie & McKee (1977) for the cloud evaporation, that is, ṁEVP = = ηEVP 16πµκr ≃ 4.4× 10 M⊙ Myr , (7) where ηEVP, µ, κ, r, and kB are the dimensionless conduction ef- ficiency parameter, the mean molecular weight, the conductivity where we use the Spitzer value (Spitzer 1962), the radius of the cloud, and the Boltzmann constant. The total evaporation rate for clouds of total mass 1M⊙ then becomes ṀEVP = Z 106M⊙ 102M⊙ ṁEVP(m)φ(m)dm. (8) 2.5 Supernova feedback and chemical enrichment We consider each stellar particle as a single stellar population (SSP) having its own age, metallicity and initial mass function (IMF) φ∗(m). Throughout this paper we assume that the IMF is always the Salpeter IMF (Salpeter 1955) whose lower and upper mass lim- its are 0.1 and 100 M⊙, respectively. The recycled mass fraction, type II and type Ia supernova (SN) rates, and metal production rates for some species are calculated according to the age, metallicity, and IMF of the SSP. Each SN is assumed to give ηSN10 51 erg of energy to surrounding gas, where ηSN is the feedback efficiency parameter. We refer the readers to OEFJ05 and references therein for a more detailed description. 2.6 Numerical implementation We have implemented the physics described above into a publicly available PM-TreeSPH code GADGET2 (Springel 2005), a succes- sor of the TreeSPH code GADGET (Springel et al. 2001). We do not use the instantaneous recycling approximation (IRA) which is often assumed in studies of galaxy formation. In this section we de- scribe how the physical processes described above are implemented in our code. For ρ > ρth, an SPH particle is eligible to form cold clouds. Hence it is eligible to form stars. We compute the probability p∗ of a SPH particle spawning a new star particle of mass m∗ during a time-step ∆t as 1− exp , (9) where the average star formation timescale t∗ is obtained by using equation (6): t∗(P ) = Ṁ∗(P ) Myr. (10) We use m∗ = morig/3 as in OEFJ05, where morig denotes the original SPH particle mass. When an SPH particle spawns a star particle, the mass in cold clouds is reduced by m∗. We relax the IRA through a probabilistic treatment of SNe II and Ia. First, we assume that stars more massive than 8M⊙ explode as SNe II. The probability pII of a star particle having an event of SN II explosion during a time-step ∆t is given by pII = R t+∆t rII(t ′)dt′ R t(8M⊙) rII(t′)dt′ , (11) where t is the age of the SSP, rII(t) is the SN II rate for the SSP of age t and t(8M⊙) is the lifetime of the star of original mass 8M⊙. When a star particle satisfies the condition to have an event of SN II we distribute mass, metals, and feedback energy, which are expelled by the total SNe II between t = 0 and t(8M⊙) to the surrounding gas particles using the SPH kernel weighting. Since the SN Ia rate is much lower than the SN II rate, we allow star particles to have only one SN Ia explosion event for each SN Ia time-step ∆tIa, which is much longer than the time-steps for dynamical calculation. By doing this, we substantially reduce the computational expense that is needed mainly for the neighbour search to distribute energy, mass, and metals. As for SN II, we can compute the SN Ia probability during a time-step ∆t as pIa = R t+∆t rIa(t ′)dt′ R t0+∆tIa rIa(t′)dt′ , (12) where t0 is the age at which the previous SN Ia time-step finished and rIa(t) is the SN Ia rate for a SSP of age t. We adopt ∆tIa = 100 Myr in this paper. The probabilistic method described above statistically relaxes the IRA. Finally, we update the mass of the hot phase Mh and the spe- cific internal energy of the hot phase uh. The new mass of the hot phase Mh ′ is given by solving the thermal energy equation implic- itly using equation (1), = Mh +∆MEVP − Λnet(ρh ′, uh, Z) uh − uc ∆t, (13) where ∆MEVP is the mass which evaporates from cold clouds dur- ing the time-step, and the new density of the hot phase ρh ′ is given by the mass conservation, MSPH = Mh+Mc, and by the assump- tion that the volume occupied by a SPH particle remains the same during the time-step, i.e. = Vol c + where Volc is the volume occupied by the cold clouds computed using the mass-radius relation (eq. 3). In addition to the adiabatic heating/cooling and shock heating, non-hydrodynamic processes such as SN feedback and cloud evap- oration also change the specific energy of the hot phase. The new specific energy of the hot phase uh ′ due to the non-hydrodynamic processes is also calculated implicitly as = uh + ∆Q+ (uc − uh ′)∆MEVP , (14) where ∆Q is the thermal energy received by the SPH particle dur- ing the time-step. Note however that we model the SN feedback not as the thermal heating of the ISM but as the kinetic heating described in the following subsection as winds. This is motivated by the results from high-resolution 2D and 3D simulations of the ISM (Wada & Norman 2001; Wada 2001; Wada & Norman 2007) in c© 2007 RAS, MNRAS 000, 1–22 AGN feedback in cosmological simulations 5 which they showed that the ISM is supported by the turbulent mo- tion originated in the self-gravity of the gas and galactic rotation. They found that energy injection by SNe hardly changes the struc- ture of the ISM. Kravtsov (2003) also indicated that the energy feedback from SNe does not change the structure of the ISM but significantly reduces the amount of gas in the galaxy in their cos- mological simulation performed with an adaptive mesh refinement code. Lacking the heating by the ∆Q term, the ISM loses its pres- sure quickly and would be fragmented. We therefore introduce min- imal heating by imposing the minimum pressure for the star form- ing gas as a function of density, Pmin(ρ) ∝ ρ γeff for ρ > ρth, (15) by which we mimic the heating from self-gravity and galactic shear motion claimed by Wada & Norman (2001), Wada (2001), and Wada & Norman (2007). We adopt in this paper the value γeff = 1.4 for the effective adiabatic index, which is close to the value found by Wada & Norman (2007) for dense gas and is stable against gravitational instability (the critical value is 4/3). We nor- malise the minimum pressure as Pmin(ρth) = (γ−1)u4ρth, where u4 denotes the specific internal energy of the gas with temperature 104 K. Once pressure from the hot phase drops below Pmin, we set the temperature of the hot phase to Thot and recalculate the mass in cold clouds so that the ambient pressure becomes Pmin. We set Thot to 10 6 K in this paper but our results are hardly affected by the value of Thot as far as Thot >> 10 4 K. In quiescent disks, most of the star forming gas has Pmin by construction. However, for exam- ple during a violent merger, the pressure can be significantly higher than Pmin due to shock heating and hence the gas can have very short star formation timescale. 2.7 Winds As discussed in SH03, we explicitly assume that cold clouds and hot ambient medium remain tightly coupled in high-density re- gions. Hence, there is no obvious route for the high entropy gas to escape from the star forming regions. We thus extend our feedback model to account for galactic winds by SN feedback. Our model is based on SH03 and is modified to take the non-instantaneous SN explosions into account. First, we specify the velocity of the wind, vw , when it leaves the disk. Then, we assign the probability pw with which the gas particle is added to the wind as MSPHvw , (16) where MSPH is the mass of the gas particle and ∆Q is the feedback energy received during the time-step. By doing this, our model be- comes insensitive to the numerical resolution, time-step in the sim- ulations, and the number of neighbours we use to distribute the feedback energy. When a particle is added to the wind, we modify the velocity v of the particle according to = v + vwn. (17) For the unit vector n, we choose random orientation along the di- rection v × agrav , where agrav is the gravitational acceleration. Therefore wind particles are preferentially ejected along the rota- tion axis of a spinning object as the ‘axial wind’ in SH03. We also decouple a new wind particle from hydrodynamic interactions for a brief time such that the winds originate from a region close to the surface of the star forming region. The full hydrodynamic interac- tions are enabled again once the density of the particle has fallen to 0.1 ρth, or once a time of ∆t = 20 Myr has elapsed, whichever happens earlier. 2.8 Parameter setting and test simulations In this paper we use the following parameters for star formation and SN feedback as a fiducial model; the threshold density, ρth = 5 × 10−25 g cm−3, the dimensionless star formation efficiency, c∗ = 0.005, the dimensionless conductivity efficiency, ηEVP = 0.1, the SN feedback efficiency, ηSN = 1, and the wind velocity vw = 500 km s−1. The relatively low value of c∗ compared to other papers comes from the fact that we use the dynamical time of each cloud not the dynamical time for SPH particles. We choose the values of ρth and c∗ so that our model reproduce the observed relation between surface gas density and surface star formation rate density (Kennicutt 1998). The most important parameters in our model are the feedback efficiency and the wind velocity. The wind velocity we use is comparable to the strong wind model in Nagamine et al. (2004). For our choice of the parameters and the IMF, the wind production rate by SNe II is Ṁw/Ṁ∗ ≃ 3. There will be further contribution from SNe Ia at later times. In this subsection, we show the behaviour of our model in ide- alised simulations of disk formation from virialised gas in the ex- ternal NFW potential (Navarro et al. 1996). Here, Mvir is the virial mass of the system and we always assume that 10 per cent of the mass is in baryonic form. The initial angular momentum in terms of the spin parameter, λ = J |E|1/2/(GM vir ), is set to 0.07 with the assumption that the specific angular momenta j(r) of spherical shells are all aligned and their magnitude is given by j(r) ∝ In Fig. 1 we show the star formation rates in the simulations for Mvir = 10 12M⊙ with three different resolutions: Ngas = 10000, 50000, and 250000. The star formation rates quickly con- verge. The simulation with Ngas = 50000, which is a compara- ble resolution that we will use in cosmological simulations, has very similar star formation history to that in the simulation with Ngas = 250000. Hereafter we thus only show the simulations with Ngas = 50000. Now we show how star formation rates depend on the values of feedback parameters, ηSN and vw . We here run the simulations with two additional models: the slow wind model (ηSN = 1 and vw = 250 km s −1) and the weak feedback model (ηSN = 0.5 and vw = 250 km s −1). The wind speed in the slow wind model is half of that in the fiducial model, therefore it has an uncomfortably high mass loading factor ηw ≃ 12. Of course the mass loading in the weak feedback model becomes half. Fig. 2 shows the star formation rates in galaxies with these three wind models for halos of Mvir = 10 12M⊙ and 10 10M⊙. Since vw = 500 km s −1 exceeds the escape velocity of the halo withMvir = 10 12 M⊙ and vw = 250 km s −1 does not, the star for- mation is more strongly suppressed in the fiducial model in spite of the same feedback efficiency. We find that the star formation rates of the slow wind and weak feedback models have regularly time spaced peaks of activity. For the adopted NFW potential, the wind particles with the initial velocity, vw = 250 km s −1, will return to the disk in ∼ 0.13 Gyr. This time-scale might be responsible for the periodicity seen in the star formation rates. The star formation rate is highest in the weak feedback model for this halo as expected. c© 2007 RAS, MNRAS 000, 1–22 6 T. Okamoto, R. S. Nemmen and R. G. Bower Figure 1. Star formation rates in simulations of disks growing within ha- los of Mvir = 10 12M⊙, with different resolutions. The simulations with Ngas = 10000, 50000, and 250000 are indicated by the solid, dotted, and dashed lines, respectively. However situation changes for the halo of Mvir = 10 10 M⊙. Now the mass loading factor is more important than the wind velocity because both wind velocities exceeds the escape velocity of this halo. Hence the star formation rate is highest in the fiducial model and lowest in the slow wind model although all models have star formation rates ∼ 10−3 M⊙ yr −1 after 1 Gyr. Since we have to suppress star formation even in relatively large halos to form disk galaxies (OEFJ05), we employ the fiducial parameter set in our cos- mological simulations. In Fig. 3 we show the relations between the star formation rate per unit area and the surface gas density for the fiducial (top panel) and weak feedback (middle panel) models. We estimate the surface star formation rate densities by computing the surface den- sity of stars younger than 3 × 107 yr in cylindrical bins. Both models show reasonably good agreement with the target relation (Kennicutt 1998, ; the target relation we show is eq. (25) of SH03). The use of different wind parameters do not affect this relation in our simulations, while the ISM can have higher gas density in the weak feedback model. For the galaxy in Mvir = 10 10M⊙ halo, winds are so effective that there are only a handful of young stars in the galaxy. We hence only show, in the bottom panel of Fig. 3, the relation between the global surface gas density and the global surface star formation rate , which are averaged over the galaxy. The different points in this panel correspond to the relations at dif- ferent epochs. We find that the global relation in the 1010M⊙ halo is also in reasonable agreement with the target relation. 3 BLACK HOLE ACCRETION AND AGN FEEDBACK 3.1 Seed black holes In the cosmological simulations we run a friends-of-friends halo- finder (Davis et al. 1985) 200 times on the fly from z = 15 to 0. When we find a halo that contains more than Nth dark matter parti- cles and no BH particles, we put a seed BH on a density peak of the Figure 2. Star formation rates in simulations of disks growing within halos of Mvir = 10 12M⊙ (upper panel) and Mvir = 10 10M⊙ (lower panel). The solid, dotted, and dashed lines indicate the fiducial, slow wind, and weak feedback models, respectively. stellar distribution in the halo. We assume 260 M⊙ as the mass of the seed black hole. This mass is consistent with the theory of stel- lar evolution, which shows that the nuclear burning in very massive stars above 260 M⊙ is unable to halt gravitational collapse (e.g. Heger et al. 2003). Note that as far as the seed BH mass is suffi- ciently small, our results hardly change because we do not impose an upper limit on the mass accretion rate. Throughout this paper we use Nth = 128. The simulation employing Nth = 256 yields almost the same results except for the total number of BHs. The insensitivity on the choice of these parameters relies on our AGN feedback model that is only effective when the mass accretion rate onto the BH is much lower than the Eddington rate (see §3.3). 3.2 Mass accretion by the radiation drag One mechanism yielding the proportionality of the BH mass to its host bulge mass has been proposed by Umemura (2001) and Kawakatu & Umemura (2002). According to Kawakatu & Umemura, in the central regions of galaxies the drag due to stellar radiation may result in a loss of angular momentum of a clumpy ISM and gives rise to mass inflow toward the centre. The optical depth of a gas cloud is τ̄ = χdρcrc ≃ χdmc/r where ρc, mc, and rc are the density, mass, and a size of a cloud. c© 2007 RAS, MNRAS 000, 1–22 AGN feedback in cosmological simulations 7 Figure 3. Star formation rate per unit area versus gas surface density in an idealised simulation of disk formation in a halo with Mvir = 10 12M⊙ (top and middle panels). The surface star formation rate densities are computed using the surface density of stars younger than 3 × 107 yr in cylindrical bins. The symbols represent star formation rates at t = 0.5, 1, 2, 3 Gyr, indicated by plus signs, diamonds, triangles, and squares respectively. The top and middle panels respectively show the fiducial and weak feedback models. The galaxy in the halo with Mvir = 10 10M⊙ is shown in the bottom panel, and there both the surface star formation rate density and the surface gas density are averaged over the galaxy. Points represent the evolution of the relation from 0.25 Gyr to 3 Gyr with a time-step of 0.25 Gyr. The solid and dotted lines indicate the target relation and the cut-off surface density respectively. In principle we can use the cloud model we described in §2.2. We however assume that the clouds are all identical and randomly distributed in a star forming region as in Kawakatu & Umemura for simplicity. The mass extinction coefficient χd is given by χd = 300 cm 2g−1(ad/0.1µm −1)(ρs/g cm −3)(Z/0.3Z⊙), where ad is the grain radius, ρs is the density of solid material within the grain (e.g. Spitzer 1978), and Z is the metallicity of gas. We use ad = 0.1 µm, ρs = 1 g cm −3, and Z is directly obtained in the simulations as the mean metallicity of the gas in a star forming region. The total optical depth of a star forming region can then be approximated as τSFR(t) ≃ Mc(t) RSFR(t)2 , (18) where RSFR and Mc are the size of a star forming region and the total mass of the clouds in the star forming region. The mass accretion rate due to the radiation drag mechanism described above is well approximated by Ṁdrag = ηdrag LSFR(t) (1− e −τSFR(t)), (19) where LSFR is the total luminosity of stars in the star form- ing region. We calculate LSFR by summing up the bolometric luminosities of all stars, each of which is obtained as a func- tion of its age from a look-up table generated by PÉGASE2 (Fioc & Rocca-Volmerange 1997). We here ignore the metallic- ity dependence of the stellar luminosity and use a table for the solar metallicity because the dependence on metallicity is weak. Kawakatu & Umemura found that the efficiency ηdrag is maxi- mally 0.34 in the optically thick regime assuming a spherical star forming region with a uniform density. In principle ηdrag can be much larger or smaller depending on the geometry and density structure of the star forming region (Kawakatu private communica- tion). We thus change the efficiency and adopt ηdrag = 1 in this pa- per, considering more centrally concentrated density distribution in real star forming regions than assumed in Kawakatu & Umemura. Once we define the radius of a star forming region around a BH, RSFR, we obtain χd, Mc, and LSFR in equations (18) and (19) directly in simulations, and hence we can compute τSFR and Ṁdrag self-consistently; the effects of outflows are also implicitly incorpo- rated in the estimate of these variables because we calculate them self-consistently at each time-step of the simulation. In order to de- fine the size of the star forming region, RSFR we start from a sphere around a BH that contains 40 SPH particles, and then we search the radius of the sphere that maximises Ṁdrag by increasing the radius of the sphere. We employ this radius as the size of the star forming region. When the mean gas density of the initial sphere is lower than the threshold density for star formation, ρth, we consider that there is not enough gas around the BH to fuel accretion and we simply set Ṁdrag = 0. Kawakatu & Umemura (2004) argued that radiation from bulge stars contributes to the mass accretion to the central hole more efficiently than that from disc stars. We crudely mimic this effect by using spherically averaged values to define star forming region; by doing this we put less weight to the star forming gas that has a flattened distribution. We will show how star forma- tion in the disc and bulge correlate with the mass accretion rate, Ṁdrag, in our simulations later. Strictly speaking, one should distinguish Ṁdrag from the ac- cretion rate onto the BH, Ṁ , because the radiation drag is not likely to remove the angular momentum thoroughly; some residual an- gular momentum will terminate the radial contraction of the ac- creted gas (Sato et al. 2004). Granato et al. (2004) introduced the viscous timescale in which the mass accumulated at the galactic centre by the radiation drag is fed to the BH. We however assume Ṁ = Ṁdrag for simplicity. The time delay in mass fuelling intro- duced by the viscous timescale may become important when we compare the luminosity of AGNs and their host galaxies in detail. 3.3 AGN feedback Motivated by the success of recent semi-analytic models (Croton et al. 2006; Bower et al. 2006), we assume that AGNs di- rectly heat diffuse hot gas confined in dark halos through the pro- duction of jets. As the jet production mechanism, we employ the model proposed by Meier (2001) which is an hybrid version of the Blandford-Znajek (Blandford & Znajek 1977) and the Blandford- Payne (Blandford & Payne 1982) processes. In this hybrid model the jet power is generated by the magnetic field threading the ac- cretion flow inside and outside the the ergosphere of the BH. This model is able to draw upon the rotational energy of the accretion flow as well as the spinning black hole in order to drive jets, and its c© 2007 RAS, MNRAS 000, 1–22 8 T. Okamoto, R. S. Nemmen and R. G. Bower basic features are supported by recent relativistic numerical simu- lations of jet formation (e.g. Hawley & Krolik 2006b). We refer the readers to Meier (2001) and Nemmen et al. (2007) for more details. Meier (2001) presented models both for non-spinning (Schwarzschild) and for spinning (Kerr) BHs. Semi-analytic cos- mological simulations of the spin evolution of BHs through merg- ers and gas accretion (Volonteri et al. 2005) and estimates of the radiative efficiencies of global populations of quasars based on Soltan-type arguments (e.g. Soltan 1982; Yu & Tremaine 2002; Wang et al. 2006) suggest that most nearby massive BHs are rapidly rotating. Recently Nemmen et al. (2007) also reported that BHs in nearby elliptical galaxies are likely to have high spins j ≃ 0.7 − 1. We thus only consider spinning BHs in this pa- per. Meier (2001) parameterised the jet luminosity for standard thin discs as jet ≈ 10 erg s ”−0.1 × (1 + 1.1j + 0.29j ), (20) and for RIAFs as jet ≈ 10 erg s αRIAF × (0.55f + 1.5fj + j ), (21) where αSD and αRIAF are the viscosity parameters for standard thin discs and for RIAFs, respectively, m9 is the black hole mass in units of 109M⊙, ṁ ≡ Ṁ/ṀEdd is the accretion rate scaled to the Eddington rate assuming the 10% energy conversion effi- ciency (ṀEdd ≡ 22 m9 M⊙ yr −1), and f and g are the ratios of the actual angular velocity and the azimuthal magnetic field to those calculated by Narayan & Yi (1995) respectively. According to Meier (2001) we employ f = 1 and g = 2.3. Since the limited numerical resolution of our cosmological simulations prevents us from resolving the details of the propagation of jets and its impact on the surrounding medium, we simply assume that a fraction of the jet energy ∆EjetFB = ηAGNLjet∆t is delivered as thermal en- ergy to the neighbouring 40 diffuse gas particles that have density, ρ < 0.1ρth. In our simulations, the size of the region heated by the AGN feedback is typically ∼ 10 kpc. Unless otherwise stated, we use αSD = αRIAF = 0.1, j = 0.5, and ηAGN = 0.1. The adopted value of j corresponds to the intermediate case between Schwarzschild (no-spin) and maximally rotating BHs. As previously mentioned, we adopt two distinct regimes of accretion flows: standard thin discs (optically thick, geometrically thin, radiatively efficient) and RIAFs (optically thin, geometrically thick, radiatively inefficient). The parameter which controls the state of the accretion flow is ṁ, and the critical value of the accre- tion rate which sets the division between the two regimes is given by ṁcrit ≈ α 2 (e.g. Narayan et al. 1998). Since the RIAF solu- tion ceases to be valid above ṁcrit, for ṁ 6 ṁcrit a RIAF exists and for ṁ > ṁcrit a thin disc occurs. As a consequence of the adopted theoretical model of jet production, we have strong AGN feedback only for RIAFs and the maximum AGN feedback occurs at ṁ = ṁcrit. For the parameters we chose, we have the following jet production efficiency in RIAFs, jet ≈ 2.6× 10 for ṁ 6 ṁcrit, (22) which is much higher than that in standard discs, given by jet ≈ 8.1× 10 for ṁ > ṁcrit, (23) where in equation (23) we adopt m9 = ṁ = 1. The equations (22) and (21) correspond to the “radio-loud mode” in our simulations, and the equations (23) and (20) represent the “radio-quiet mode”. Note that, in principle, we could have even higher efficiencies of jet production, because the energy is drawn from the “spin en- ergy” stored in the BH. We explore the case of a higher efficiency corresponding to a higher BH spin later in §5.3. 4 COSMOLOGICAL SIMULATIONS OF DISC GALAXY FORMATION WITH AGN FEEDBACK To study the role of AGN feedback in disk galaxy formation, we use the initial conditions presented in OEFJ05, from which they produced an extended disk galaxy. The background cosmology is the so-called ΛCDM with the mean matter density Ω0 = 0.3, Hub- ble parameter, h ≡ H0/100 km s −1Mpc−1 = 0.7, cosmological constant term, ΩΛ ≡ Λ0/(3H 0 ) = 0.7, amplitude of mass fluc- tuations, σ8 = 0.9, and mean baryon density, Ωb = 0.04. The size of the periodic simulation box is 35.325h−1Mpc and the ini- tial redshift is z = 49. We put the high-resolution dark matter particles and gas particles in the region where a halo with mass Mvir = 1.2 × 10 12h−1M⊙ forms at z = 0. The region external to this is populated with high-mass dark matter particles (bound- ary particles), the function of which is to reproduce the appro- priate tidal fields. The circular velocity, spin parameter and col- lapse redshift of the selected halo are vc(rvir) = 155 km s λ ≡ J |E|1/2/(GM5/2) = 0.038 and zc ≃ 1.5, respectively, where zc is defined as the redshift at which the main progenitor had half the final halo mass. The masses of the SPH and high-resolution dark matter parti- cles are ∼ 2.6× 106 and ∼ 1.7× 107h−1M⊙, respectively. Thus the Nth = 128 implies that halos larger than ∼ 2.6× 10 9h−1M⊙ are allowed to have BHs. The gravitational softening length are kept fixed in comoving coordinates for z > 3; thereafter they are frozen in physical units at a value (of the equivalent Plummer softening) of ǫ = 0.5 and 1 kpc for the baryonic particles (SPH particles, stars, and BHs) and high-resolution dark matter particles, respectively. The gravitational force obeys the exact r−2 law at r > 2.8ǫ. 4.1 Black hole particles When we find a halo with Ndm > Nth that does not contain any BH particles and is not contaminated by the boundary particles, we turn the star particle that has the largest stellar density into a seed BH. We thus discriminate the BH mass from the mass of the BH particle (particle mass). The former is updated every time-step ac- cording to the mass accretion rate obtained from the radiation drag model. Once BH mass exceeds its particle mass, the BH particle increases its particle mass by absorbing gas particles n in the star- forming region around it with the probability: MBH −Mp , (24) where MBH and Mp are, respectively, the BH mass and the mass of the BH particle, NSFR is the number of gas particles in the star- forming region, and Mn is the mass of the gas particle n. BH particles can increase their BH masses and particle masses by BH mergers. However, how long it would take to harden a BH binary until finally gravitational wave emission becomes impor- tant is still unclear (Makino & Funato 2004; Escala et al. 2004). We simply assume that BHs can merge into a single BH if two or more c© 2007 RAS, MNRAS 000, 1–22 AGN feedback in cosmological simulations 9 BHs are within a gravitational softening radius and they are grav- itationally bound. Therefore the BH merger rates obtained in our simulations are likely to be the maximum possible rate. Techni- cally, we apply the friends-of-friends algorithm to find groups of BHs which may potentially merge originating new BHs. An interesting problem arises owing to the finite numerical resolution. In reality, supermassive BHs are much heavier than stars and dark matter. Therefore they are quickly brought to the galac- tic centres and stay at the bottom of the potential wells owing to the dynamical friction. In cosmological simulations, the mass of a BH particle is comparable to those of gas and stellar particles and lighter than the dark matter particles in the early stage of its evolu- tion. As a result, BH particles tend to oscillate around galactic cen- tres with the velocity dispersions of the stellar systems. To circum- vent this problem, we allow BH particles to behave as peak tracers at galactic centres. At each time-step, we compute local stellar den- sity fields within the gravitational softening radii of BH particles, ǫ. We then displace the BH particles by a small distance, ∆l, along the local density gradients. After several tests, we chose ∆l = MIN(0.01ǫ, 0.03 |v| ∆t), (25) where v is the velocity of a BH particle and ∆t is the time-step. This ∆l correction sufficiently suppresses the oscillation of the BH particles at galactic centres. We adopt a maximum time step for BHs, ∆tmax = 1 Myr, in order to follow mass accretion and associated feedback with a reasonable time resolution. Hence a time step for a BH particle is defined as ∆tBH = min(∆tmax,∆tdyn), where ∆tdyn is a time step determined by the dynamics. 4.2 Models We first compare three simulations in order to show the effects of AGN feedback. The first simulation does not have feedback from AGN. We refer this model as the ‘no-fb’ model. In the second model, we distribute 1% of jet energy to the ambient halo gas par- ticles defined by their densities, ρ < 0.1ρth in the form of thermal energy using SPH smoothing. Since only 1% of jet energy is avail- able for heating, we call this model the ‘weak-fb’ model. In the ‘reference’ model, we adopt the same AGN feedback scheme as in the weak-fb model but assume 10% of jet energy is available to heat the ambient halo gas. We will explore other models in which we change the implementation of AGN feedback and strength of SN feedback in §5.3. 5 RESULTS To give a visual impression of our simulations, we show some snapshots of the simulation with AGN-radio feedback (‘reference’ model) in Fig. 4. There are a number of merger events at high red- shift (z > 1). The disk formation starts at z ∼ 1, and finally a disk galaxy forms. A lot of BHs form in the small halos and they are brought to the main progenitor by merging satellites. The BH parti- cles nicely trace the centres of the stellar objects resulting from our correction applied to the positions of BH particles. There are also many BHs which are not associated to any stellar objects. They are scattered into the halo when small satellites, which harbour a small BH at their centres, are destroyed during mergers. Most of the drifting BHs hence have very small mass. In the following sections,we present the main results of the paper. In §5.1, we compare the results of simulations run with Figure 5. Projected stellar distributions within 50 h−1 kpc boxes centred on the galaxies at z = 0. The left, middle, and right panels show the ‘no-fb’, ‘weak-fb’, and ‘reference’ simulations, respectively. The viewing angles are chosen to be edge-on for upper panels and face-on for lower panels. The brightness indicates the projected stellar mass density, and the same scaling is used for each simulation. the standard feedback parameters (the ‘reference simulation’) with simulations using weaker AGN feedback (the ‘weak-fb’ model) and no AGN feedback at all (the ‘no-fb’ model). In §5.2 we look at the importance of the AGN feedback for the bolometric luminosity of the forming galaxy, while in §5.3, we explore the dependence of the results on the way in which the feedback is coupled to the surrounding gas and the impact of assuming a lower efficiency of the feedback from SNe. 5.1 Simulations with different AGN feedback efficiencies Now we compare galaxies formed in three simulations, i.e. the ‘no- fb’, ‘weak-fb’, and ‘reference’ simulations. We first show the stellar distributions within 50h−1 kpc boxes centred on the galaxies at z = 0 in Fig. 5. The edge-on and face-on projections are selected to be perpendicular and parallel to the angular momentum vector of the stellar component within the central 10h−1 kpc sphere. In all of the models, the galaxy has an extended stellar disk. The ‘no-fb’ and ‘weak-fb’ models are very similar. From the face-on view, it is found that the ‘reference’ galaxy has a disk with lower surface stellar density. Fig. 6 shows the I band surface brightness profiles for these galaxies. We also show the surface brightness profile of a disk galaxy obtained by OEFJ05 using the same initial condition. An in- teresting difference from the surface brightness profile of the galaxy by OEFJ05 can be seen at the centre. Our galaxies do not show the extreme central concentration. The origin of this difference is the different implementation of SN feedback. While we implement it as kinetic winds, they deposited feedback energy in the form of thermal energy. Therefore they had to calculate the pressure gradi- ents caused by the thermal feedback to blow the gas away. Unfortu- nately, the SPH does not allow steep gradients in physical quantities by definition unless one uses a sufficiently high resolution. All our three galaxies show similar central surface brightness. The main difference due to AGN feedback can be seen in the sur- face brightness of the disks. While ‘no-fb’ and ‘weak-fb’ galaxies show the almost same surface brightness profiles, the ‘reference’ model has a less extended disk. We do not try to determine the bulge-disk decomposition based on the surface brightness profiles c© 2007 RAS, MNRAS 000, 1–22 10 T. Okamoto, R. S. Nemmen and R. G. Bower Figure 4. Snapshots of the ‘reference’ model (x–y projections). The rows correspond to z = 6, 3, 2, 1, and 0 from top to bottom. The first and the second columns show gas and stellar distributions in the comoving 3 h−1 Mpc cubes centred on the main progenitor, respectively, where the brightness represents three dimensional density. The third and the fourth columns show gas and stellar distributions in the physical 50 h−1 kpc cubes, respectively. The circles in each panel indicate the position of the BH particles and their radii are proportional to the log of the BH masses. c© 2007 RAS, MNRAS 000, 1–22 AGN feedback in cosmological simulations 11 Figure 6. Surface brightness profiles in the I band for the ‘no-fb’ (pluses), ‘weak-fb’ (asterisks), ‘reference’ (diamonds) galaxies, and for the galaxy with the ‘shock-burst’ model of Okamoto et al. (2005; dotted-line). The solid lines are exponential fits for r < 15 kpc with the scale lengths 5.4, 5.6, and 4.0 kpc for the ‘no-fb’, ‘weak-fb’, and ‘reference’ galaxies, respec- tively. since they provide only partial information about the morphology of galaxies. Below, we will show the decomposition based on the dynamics of stars (Abadi et al. 2003). This is a more accurate way to characterise the relative importance of the bulge and disk com- ponents. To investigate effects of AGN feedback quantitatively, we present the star formation histories of the stars that lie within 25h−1 kpc from the galactic centres at z = 0 in Fig. 7. All galaxies have qualitatively similar star formation histories and have peak star for- mation rates at z ∼ 1.5. During the starburst (t = 2.5−6 Gyr), the ‘reference’ galaxy shows lower peak star formation rate compared with other galaxies. As we will show later, the AGN is not a RIAF during the starburst. Hence AGN feedback from a RIAF is not re- sponsible for this lower star formation rate. The feedback from standard disk (equation (23)) might suppress the starburst. However we should not overinterpret the differences during the starburst be- cause the system is strongly non-linear during this epoch and hence small changes can result in relatively large difference. In fact, the model which employs stronger AGN feedback has a higher star for- mation rate during the starburst than in the ‘reference’ model as we will show later in Fig. 15. After the starburst, the ‘reference’ galaxy has significantly lower star formation rate than other galaxies. This suppression of star formation is likely to be caused by AGN feedback from a RIAF. The lower star formation rate in the ‘reference’ galaxy explains its less extended disk in Fig. 6 because it is the star formation after z ∼ 1 that builds up the stellar disk (see Fig. 4). The ‘weak-fb’ galaxy has almost the same star formation rate as the ‘no-fb’ galaxy. Hence it can be said that, in order to give a visible impact on galaxy formation, at least ∼ 10% of jet energy has to be given to the halo Figure 7. The formation history (star formation rate as a function of time) of the stars that lie within 25h−1 kpc from the galactic centre at z = 0. The solid and dotted lines correspond to the galaxies in the simulation without and with BHs, respectively. gas. We will later explore the case in which the jet energy is not given to the diffuse halo gas but delivered to the surrounding ISM. Since the strength of AGN feedback in our model is closely re- lated to the central BH mass and the accretion rate onto the BH, we plot the evolution of the central BH of the main progenitor and the mass accretion rate in units of the Eddington rate in Fig. 8 in each model. Note that the mass accretion rate Ṁ is different from the mass growth rate, ṀBH; the latter includes the mass increase due to BH mergers. In all simulations, the central BHs rapidly evolve un- til t ∼ 5.5 Gyr at which the starburst ceases. The final BH masses reach ∼ 107M⊙. The stepwise jumps seen in the evolution of the BH mass indicate the mass growth by mergers. The consumption of the available gas for star formation ex- plains the decline of the accretion rate onto the central BH. At t ≃ 5.5 Gyr ṁ reaches the critical accretion rate ṁcrit, below which the accretion disk becomes a RIAF. Our model has the max- imum AGN feedback when the mass accretion rate is equal to the critical accretion rate. This epoch is indicated by the crossing of the solid line and the horizontal dashed line in the lower panel of Fig. 8. Once the accretion disk becomes a RIAF after z < 1 , in the ‘reference’ model, the accretion rate drops to zero rapidly and there is no mass accretion after the strongest production of AGN feedback at t ∼ 5.5 Gyr except for some episodic accretion events. This implies that there is almost no star formation activity at the centre of the main progenitor at z < 1 for this galaxy. On the other hand, lacking the strong AGN feedback, there are not such sudden decrease in accretion rates in the ‘no-fb’ and ‘weak-fb’ models. The gradual decline is caused by consumption of gas due to star formation and associated SN feedback(winds). Our implementation of AGN feedback cannot directly sup- c© 2007 RAS, MNRAS 000, 1–22 12 T. Okamoto, R. S. Nemmen and R. G. Bower Figure 8. Upper panel: the mass growth of the central BH of the main progenitor. Lower panel: the mass accretion rate onto the central BH of the main progenitor in units of the Eddington rate (ṁ = Ṁ/ṀEdd). The horizontal dashed line indicate the critical mass accretion rate ṁcrit below which the accretion flow becomes radiatively inefficient (RIAF). The ‘no-fb’, ‘weak-fb’, and ‘reference’ simulations are shown from left to right panels. press the star formation because it injects energy not into the star forming dense gas but into the diffuse halo gas (ρ < 0.1ρth). Thus it is the suppression of the cooling of the halo gas around the centre of the galaxy which is responsible for the halting of star formation around the BH in the ‘reference’ simulation. The impulsive feature seen in the accretion rate implies the existence of a self-regulated cycle that operates between gas cooling, star formation, and pro- duction of jets in the RIAF. The fact that the central surface bright- ness is almost the same in the three models in spite that the galaxy with AGN feedback hardly form stars at its centre after z ∼ 1 sug- gests that the vast majority of stars at the centre have already been formed at z ∼ 1 in all galaxies. We now return to the separation of the disk and bulge compo- nents. As noted earlier, it is best to separate the bulge and disk com- ponents using a dynamical decomposition (Abadi et al. 2003). For this, we first compute the angular momentum Jz of each star par- ticle parallel to the net angular momentum of stars within 10h−1 kpc, and the angular momentum of the corotating circular orbit, Jc(E), where E is the energy of each particle. The ratio Jz/Jc defines an orbital circularity. In Fig. 9, we show the probability distribution of this orbital circularity for stars within 25h−1 kpc from the centre of each galaxy. A disk component should have Jz/Jc(E) ≃ 1 and we see that such a component clearly dom- inates in each galaxy. Comparing to the results by OEFJ05, the suppression of the formation of spheroidal components seems to rely largely on the strong SN feedback by winds. Governato et al. (2007) also produced disk dominated galaxies in halos with quieter merger histories (in which the epoch of the last major merger is zlmm > 2) using a different feedback recipe from ours. To iden- tify the stars in the spheroid, we assume a non-rotating spheroid, i.e. that stars in the spheroid are symmetrically distributed around zero in Fig. 9, where all stars having Jz/Jc(E) 6 0 are identified as a half of the spheroidal component. Another half of the spheroid with Jz/Jc(E) > 0 are defined statistically so that the total angular momentum of the spheroid becomes zero. All remaining stars are identified as the disk component. As expected from the similarity in c© 2007 RAS, MNRAS 000, 1–22 AGN feedback in cosmological simulations 13 Figure 9. Mass-weighted probability distributions of the orbital circular- ity, Jz/Jc(E), of stars within 25h−1 kpc from the galactic centres. The solid, dotted, and dashed lines indicate the ‘no-fb’, ‘weak-fb’, and ‘refer- ence’ galaxies, respectively. the star formation histories, all galaxies have similar distributions in this plane while the disc component in the ‘reference’ model is slightly less significant because of the relatively strong AGN feed- back. In Table 1 we show the total and spheroid’s masses, total lu- minosity, and the disk-to-total ratios in mass, g band, and r band. Since colour of a stellar population is sensitive to its metallicity, we here include the metallicity dependence to compute the luminosity at each passband by using PÉGASE2 (Fioc & Rocca-Volmerange 1997). The masses of the spheroidal components directly reflect amplitude of starbursts and therefore the ‘no-fb’ galaxy has the most massive spheroidal component. Nevertheless, all galaxies have disk-dominated morphology. It is interesting to see when each component forms and how the formation history of each component is related to the accretion history of the central BH. In Fig. 10, we show star formation histo- ries of the disk and spheroidal components in each galaxy at z = 0. We also show the rescaled accretion rate onto the central BH of the main progenitor of each galaxy. Note that the star formation rates include star formation in all building blocks (all progenitors) but the mass accretion rates are only the rates onto the central BHs of the main progenitors. Thus at high redshift, the mass accretion rates presented in the figure considerably underestimate the net accretion rates. Since the galaxies do not undergo any significant mergers af- ter z ∼ 1, it is safe to compare the star formation rates and the mass accretion rates at low redshift. We find that there are good correla- tions between the star formation rates of the spheroid and the mass accretion rates onto the central BHs at low redshift, in particular between t ∼ 5.5 and 8 Gyr. After the starburst, the star forma- tion rates of the spheroids rapidly decreases in all simulations. The accretion rates onto the BHs also decrease at the same time. The contribution from disk stars to the mass accretion rate due to the radiation drag effect should be largest during the peak of star for- Figure 10. Star formation histories for the disk and spheroid stars within 25h−1 kpc from the centre of each galaxy at z = 0. The ‘no-fb’, ‘weak-fb’, and ‘reference’ galaxies are shown in the top, middle, and bottom panels, respectively. The star formation histories for the disk components are repre- sented by the solid lines and those for the spheroidal components are indi- cated by the dotted lines. We also show the mass accretion rates, which are multiplied by 1000, onto the central BHs in the main progenitors (dashed lines). Note that while star formation not only in the main progenitors but also in all other building blocks is counted in the star formation histories, the mass accretion rates only represent the rates onto the central BHs of the main progenitors. mation in the disc component (t ≃ 5 Gyr) because at this epoch the disc is gas rich and very compact. We have however confirmed that even at this epoch the contribution from disc stars is less than 20% and most of the time less than 5% in all models. Considering the fact that half of the spheroid stars are defined statistically, the correlation between the star formation rate and the mass accretion rate of the spheroid in each galaxy is significant. After t ∼ 8 Gyr there are many short lived star formation episodes in the spheroids, which do not correspond to gas accretion events. These star forma- tion events occur in small satellites before they become a part of the spheroidal components by minor mergers. Next we show in Fig. 11 the evolution of the relations be- tween mass of the central BHs of the main progenitors and the mass of the host spheroids at z = 3, 2, 1.5, 1, 0.5, 0.2, and 0. We here define the main progenitors as stars within 10% of the virial radii at each redshift. We have confirmed that the spheres with radii of 0.1rvir contain the stellar systems defined as main pro- genitors by the friends-of-friends algorithm in §5.2. We have also checked by eye that at selected redshifts the main progenitors are not undergoing major mergers and the spheres do not contain mas- sive satellites, both of which ruin the dynamical decomposition. c© 2007 RAS, MNRAS 000, 1–22 14 T. Okamoto, R. S. Nemmen and R. G. Bower Table 1. Total and spheroid’s mass, g and r band total luminosities, and disk-to-total mass and luminosity ratios for the galaxies at z = 0. Mtot/ M⊙ MSP/ M⊙ Mg Mr no-fb 6.01× 1010 1.98× 1010 -20.8 -21.3 0.67 0.83 0.79 weak-fb 5.98× 1010 1.44× 1010 -20.9 -21.3 0.76 0.88 0.85 reference 4.32× 1010 1.21× 1010 -20.6 -21.0 0.72 0.86 0.83 Figure 11. The relation between mass of the central BHs of the main pro- genitors and the stellar mass of the host spheroids at z = 3, 2, 1.5, 1, 0.5, 0.2, and 0. The relations for the ‘no-fb’, ‘weak-fb’, and ‘reference’ simula- tion are shown by the plus signs, diamonds, and triangles respectively. The solid line indicates MBH = 0.001MSP. We then compute the mass of the spheroidal component of each main progenitor based on the dynamical decomposition, noting that this method of decomposition is less vulnerable to insufficient res- olution compared with the decomposition based on surface density profiles. This feature is important at high redshift where the size of the galaxies are much smaller than that at z = 0. Fig. 11 shows that the MBH-MSP relations evolve similarly in all simulations be- cause the BHs and spheroids mainly increase their mass during the starbursts and our AGN feedback is not effective during this phase. Before the starburst, the BH mass is far below the local relation, MBH ∼ 10 −3MSP, and then it rapidly moves towards the local relation during the starburst (z = 2 ∼ 1). The delay in the mass evolution of the BHs compared to the spheroids is resulting from the fact that the mass accretion rate due to the radiation drag model depends on the metallicity of the ISM. The mass accretion by this model is not efficient until the ISM is metal enriched and the opti- cal depth becomes ∼ 1. After the starburst, the mass of BHs stays almost constant as seen in Fig. 8. The relation is imprinted in the ra- diation drag model because the model provides a good correlation between star formation in a spheroid and the mass accretion onto a BH. It is important to note that the radiation drag model predicts considerably lower mass ratio between central BHs and their host spheroids before the starburst. Figure 12. Colour-magnitude diagrams for the ‘no-fb’ (left), ‘weak- fb’ (middle), and ‘reference’ (right) galaxies. The galaxies at z = 3, 2, 1.5, 1, 0.5, 0.2 and 0 are indicated by the plus signs connected by the dotted line. The colours of their spheroidal components are also indi- cated by the diamonds. The upper and lower thin solid lines represent mean colours of the red and blue populations taken from Baldry et al. (2004). The u − r colour of simulated galaxies are converted to the SDSS colours by using (u − r)(SDSS) = (u − r)(AB) + 0.05 (Abazajian et al. 2003) to compare with the data by Baldry et al. (2004). In order to illustrate how the colours of our galaxies evolve, we show the evolutions of the main progenitor of each galaxy and its spheroid in a colour-magnitude diagram (Fig. 12). The mean colours of the red and blue populations of the SDSS galax- ies (Baldry et al. 2004) are indicated by solid lines. All galaxies show similar evolution on the colour-magnitude plane. This con- firms that the AGN feedback has only minor effects on evolution of disk galaxies. The colour of the galaxies and their spheroids at z = 0 are consistent with the observed colours of local blue and red galaxies, respectively. 5.2 Relative importance of the AGN In order to show the relative importance of the AGN in galaxy for- mation, we show the bolometric luminosities generated by stars in the main progenitors and the bolometric luminosities radiated by the AGNs of the main progenitors in the upper panels of Fig. 13 for our three simulated galaxies. A main progenitor is defined by applying the friends-of-friends algorithm for stellar particles with a linking length llink = 0.02 〈l〉, where 〈l〉 is the the mean separation of dark matter particles. To estimate the luminosity of an AGN, we assume that the efficiency of conversion of rest mass energy of the accretion flow into radiation (LAGN/(Ṁc 2)) is ηSD = 10% for a standard thin disk and for a RIAF we use the equation ηRIAF = 0.1 ṁcrit + δ + 10 where δ represents the fraction of the turbulent energy which heats the electrons (Quataert 2001). While the value of δ is quite uncer- tain, we employ δ = 0.1 as commonly assumed in spectral models of RIAFs (Nemmen et al. 2006). Note that the radiative efficiency for a thin disk can be a factor of a few higher than 0.1 in the case c© 2007 RAS, MNRAS 000, 1–22 AGN feedback in cosmological simulations 15 Figure 13. Upper panels: Bolometric luminosities radiated by the stars in the main progenitors (solid lines) and by the AGNs of the main progenitors (dotted lines). A 10% radiative efficiency is assumed for standard thin disks and the efficiency given by eq. (26) is assumed for RIAFs. Lower panel: Raw feedback luminosities ejected by SNe (II and Ia) in the main progenitors (solid lines) and by jets from the central BHs (AGNs) of the main progenitors (dotted lines). From left to right, the ‘no-fb’, ‘weak-fb’, and ‘reference’ simulations are shown. of a spinning BH, and may reach efficiencies as high as 42% for an extreme Kerr BH (Novikov & Thorne 1973). It is found that the main progenitors are always brighter than the AGNs for these galaxies as we expect for disk-dominated galax- ies. Since the radiation drag is more efficient for more compact star forming region, we expect that the significance of the AGN lumi- nosity would increase for galaxies which are undergoing gas-rich mergers. There is a notable difference among our models when the accretion flows become RIAFs after z ∼ 1. While all the AGNs have much lower luminosities than their host galaxies at low red- shift because of the low accretion rates and the low radiative ef- ficiency of the RIAF, only the AGN in the ‘reference’ galaxy has almost zero luminosity right after the accretion disc switches to a RIAF. In this simulation, the low accretion rate, due to the suppres- sion of star formation around the BH by AGN feedback, makes the AGN invisible most of the time. In the ‘no-fb’ and ‘weak-fb’ simu- lations, the AGNs still continuously emit light for a few giga years after the onset of the RIAFs. This result suggests that if 10% of the jet luminosity is used to heat the ambient halo gas AGNs become quite dim and can be hardly observed after the starburst regime. It should be noted that it is beyond the scope of this work to incorpo- rate in the simulations possible inclination effects in the appearance of the AGN, caused by dust obscuration inside the nucleus as in the obscuring dust torus model invoked in unified models of AGN (e.g. Antonucci 1993; Krolik 1999). In the lower panels of Fig. 13 we show the raw feedback lu- minosities ejected by the total SNe in the main progenitors and jet luminosities produced by the AGNs computed by using equations (20) and (21). Until the end of the starburst, the feedback energy is dominated by energy produced by SNe except for a few temporary c© 2007 RAS, MNRAS 000, 1–22 16 T. Okamoto, R. S. Nemmen and R. G. Bower RIAF modes seen in the ‘weak-fb’ simulation. The jet luminosi- ties suddenly exceed the SN luminosities when the accretion flow becomes radiatively inefficient. Since in the ‘no-fb’ and ‘weak-fb’ simulations, the efficiencies, ηAGN by which the AGN jet energy is given to the surrounding halo gas are small (ηAGN = 0 and 0.01 respectively), the effects of the AGN feedback are none or negligible in these simulations. In the ‘reference’ simulation, the efficiency, ηAGN = 0.1 is sufficiently high to suppress the star for- mation around the BH therefore quenching the mass accretion to the BH. This is an important feature of our AGN feedback. The im- pulsive events in Fig. 13 represent epochs where the AGN would be seen as radio-loud, and they constitute the radio-loud mode of the AGN. The duration in which the AGN is in the radio-loud mode is strongly dependent on the feedback efficiency, since the AGN feedback terminates the radio-loud mode through the suppression of gas cooling and hence star formation. Interestingly enough, in the ‘reference’ simulation, the AGN spends most of the time in the radio-quiet mode and only a fraction 8.8% of the time in the radio- loud mode. Contrarily, in the ‘no-fb’ and ‘weak-fb’ galaxies, the AGNs spend ∼ 25% of the time in the radio-loud mode. Note that we are likely to overestimate the star formation rate at the centre and hence AGN activity particularly in the quiescent star forming regime because of the angular momentum transfer due to the artifi- cial and numerical viscosities which can bring gas particles to the centre (Okamoto et al. 2003; Kaufmann et al. 2007). 5.3 Dependence on feedback parameters and implementations of feedback In §5.1, we compared simulations with different AGN feedback ef- ficiencies. In this sections, we compare simulations which all have strong AGN feedback, but which differ in other aspects. We com- pare three additional models. The models investigate the coupling of the AGN feedback, the role of BH spin and the effect of the wind speed generated by SN feedback. In the first model, AGN feedback energy is not given to the ambient halo gas (diffuse gas) but deposited to the nearest 40 ISM particles (dense gas). We use ηAGN = 1 so that all the jet en- ergy is used to heat the surrounding ISM because a model with ηAGN = 0.1 does not show any visible difference from the ‘no-fb’ model. We call this model the ‘ism-fb’ model. In the second model, we assume the higher spin, j = 0.9, and the maximum feedback efficiency, ηAGN = 1, as we may need such strong feedback in order to explain the X-ray properties of clusters of galaxies (e.g. Nemmen et al. 2007). As a result, the efficiency of AGN feedback in this model becomes 17 times higher than in the ‘reference’ model . We dub this model the ‘max-fb’ model. In the third model, we use the same AGN feedback as in the ‘reference’ model but employ a slow wind speed, 250 km s−1 for SN feedback in order to explore effects of the wind speed. We refer this model as the ‘slow-wind’ model. In Fig. 14, we show the edge-on and face-on views of the stel- lar distributions of the galaxies produced by the three additional models. The ‘ism-fb’ model is similar to the ‘reference’ model. As expected the disk component of the ‘max-fb’ galaxy is less significant than that of the ‘reference’ model since stronger AGN feedback suppresses the disk formation more strongly. The ‘slow- wind’ model exhibits completely different morphology from oth- ers. There is no sign of a disk component. Now we carry out quantitative comparison by showing the star formation histories of the galaxies and the accretion histories to the central BHs (Fig. 15). In spite of the fact that we employ an order of Figure 14. Same as Fig. 5 but for the ‘ism-fb’ (left), ‘max-fb’ (middle), and ‘slow-wind’ (left) models. magnitude higher AGN feedback efficiency in ‘ism-fb’ model, the star formation history of this galaxy is very similar to that of the ‘reference’ model. It suggests that delivering AGN feedback en- ergy to the diffuse hot gas causes the stronger feedback effect than depositing it into the surrounding dense ISM where cooling time is very short. The ‘max-fb’ galaxy shows a resembling star formation history to the ‘reference’ model (upper-middle panel) at high red- shift (t < 6 Gyr) where the AGN harbours a thin accretion disc. Once the accretion flow becomes a RIAF, strong AGN feedback suppresses the star formation dramatically and this galaxy have far less star formation after t ∼ 6 Gyr than the ‘reference’ galaxy. The ‘slow-wind’ galaxy has a completely different star formation his- tory from others. Owing to the slow wind speed, it has a higher star formation rate at high redshift including two burst events at t ∼ 2 and 3 Gyr. The AGN in this galaxy becomes underfed earlier than in other models (at t ∼ 4 Gyr) and the AGN feedback strongly suppresses the subsequent star formation. We also show the model which employs the same wind speed as the ‘slow-wind’ model but without AGN feedback (dashed line in the upper-right panel). By comparing these two slow-wind models, we can see that the AGN feedback is responsible for the suppression of the star formation at low redshift. The bulge-to-total mass ratio of the ‘slow-wind’ model is 0.9 and the u− r colour of this galaxy is consistent with the mean colour of the red population at z = 0. The evolution of the accretion rates in the ‘ism-fb’ and ‘max- fb’ models are similar to that in the ‘reference’ model particu- larly at high redshift (t < 6 Gyr) where the accretion disc is thin (ṁ > 10−2). In these two models, the accretion flows become RI- AFs after the starbursts as in the ‘reference’ model. After that the mass accretion onto the central BH in the ‘ism-fb’ model is more strongly suppressed than in the ‘reference’ model, while the ‘ism- fb’ model has a higher star formation rate. Since the AGN feedback energy is deposited into the surrounding star forming ISM particles in this model, it can terminate star formation around the BH more directly than in the ‘reference’ model where the AGN feedback en- ergy is distributed to the halo gas. In the max-fb model, there is no accretion event after t ≃ 6 Gyr. The strong AGN feedback in this model removes all the low angular momentum gas around the BH and prohibits subsequent star formation around the galactic centre. The ‘slow-wind’ model shows a different accretion history from other models. Since the starbursts occur earlier in this model ow- ing to the slow wind speed, the AGN also becomes underfed earlier (t ∼ 4 Gyr) than in other models. Contrary to other models, the c© 2007 RAS, MNRAS 000, 1–22 AGN feedback in cosmological simulations 17 Figure 15. Upper panel: star formation histories for the ‘ism-fb’ (left), ‘max-fb’ (middle), and ‘slow-wind’ (right) models. The star formation history for the reference model (dashed line in Fig. refSFH) is indicated by the dotted line. In the right panel, we also show the ‘slow-nofb’ model in which we use the same wind speed as in the ‘slow-wind’ model and do not include AGN feedback (dashed line) for comparison. Lower panel: same as the lower panel of Fig. 8 but for the ‘ism-fb’ (left), ‘max-fb’ (middle), and ‘slow-wind’ (right) models. mass accretion continues at t 6 4 Gyr. This is because the slow wind speed, 250 km s−1, does not allow gas particles to escape from galactic centre and the AGN feedback energy is not given to the star forming gas. The accretion rate might become more episodic if we deposit a significant fraction of the AGN feedback energy to the central star forming gas as in the ‘ism-fb’ model. In Fig. 16, we show the relations between mass of the central BHs of the main progenitors and the mass of the host spheroids at z = 3, 2, 1.5, 1, 0.5, 0.2 and 0. The relations for ‘ism-fb’ and ‘max- fb’ models evolve similarly to the relations for the models pre- sented in Fig. 11. The relation for the ‘slow-wind’ model reaches the local relation earlier (at z = 2) because this galaxy undergoes starbursts earlier than other galaxies. At low redshift, this model has slightly smaller BH mass to spheroid mass ratio than in other models. This is because infalling satellites increase the mass of the spheroid without changing the mass of the central BH very much when these infalling galaxies are disk dominated. Due to the slow wind speed, these infalling systems have more stars than in other models. Fig. 11 and Fig. 16 tell us an important feature of the ra- diation drag model, that is, the MBH −MSP relation after the last starburst does not depend on either the wind speed or efficiency of AGN feedback. In all models, the central BH masses are ∼ 0.1% of the mass of their host spheroids, MBH ≃ 10 −3MSP, when the accretion becomes radiatively inefficient. 6 COMPARISON WITH OBSERVATIONS OF AGN IN SPIRAL GALAXIES As we have shown in §5.2, for z . 1 the bolometric luminos- ity of the AGN is feeble compared to the host galaxy and the nu- cleus undergoes several recurring episodes of activity. During these episodes when the nucleus is active, its characteristic accretion rate is in the range ṁ ≈ 10−4 − 10−3, while the typical nuclear bolo- metric luminosities and jet powers are Lbol ≈ 10 40−1042 erg s−1 and Ljet ≈ 10 42 − 1043 erg s−1 respectively. Given these prop- erties, at z . 1 during which the galactic disk is formed, the AGN can be considered to be in a highly sub-Eddington, low- luminosity or Seyfert state which undergoes a stochastic cycle of activity. The Seyfert nucleus is sometimes turned on, but most of c© 2007 RAS, MNRAS 000, 1–22 18 T. Okamoto, R. S. Nemmen and R. G. Bower Figure 16. Same as Fig. 11 but for the ‘ism-fb’ (pluses), ‘max-fb’ (dia- monds), and ‘slow-wind’ (triangles) models. the time is “off” or inactive with ṁ ≈ 0, resembling the nucleus of a normal spiral galaxy. The “on” episodes are induced by the inward motion of clumps of cold gas into the centre of the galaxy, which provide a replenishment of new stars. The radiation drag or Poynting-Robertson effect caused by stellar radiation in the cen- tral star-forming region then dissipates the angular momentum of the gas and fuels the AGN. Our model suggests that reasonably powerful jet outbursts accompany each “on” episode of the low- luminosity AGN, suggesting that, in analogy to the observed states of AGNs, during the active phases the Seyfert nucleus would be seen as radio-loud, becoming radio-quiet during the “off” phase. In this section, we compare the properties of our simulated AGN with those of observed ones in disk galaxies. In the reference model, the range of values obtained for the AGN bolometric luminosity for z . 1 agrees well with the median observational values for LINERs (median Lbol ≈ 10 41 erg s−1) and Seyferts (median Lbol ≈ 10 42 erg s−1) inferred from the Palomar spectroscopic survey of nearby galaxies (Ho 2004). Since the Seyfert episodes of the central AGN coincide with jet out- burst events during which the AGN would be seen as radio-loud in our simulations, this suggests that observed Seyfert/LINER nu- clei would be mostly radio-loud objects rather than radio-quiet given their formation history, contrary to the generally held no- tion existent in the literature (e.g., Krolik 1999; Laor 2000). Ho & Ulvestad (2001) and Terashima & Wilson (2003) however investigated several low-luminosity AGN and their radio-loudness classification, obtaining that a large fraction of Seyfert/LINER nu- clei are radio-loud, a finding that corroborates our results. In par- ticular, Ho & Ulvestad (2001) obtained that & 60% of the sources in their sample are radio-loud. In our simulations, the numerical resolution is not sufficient to resolve the details of the propagation of the jet as it flows out of the AGN and how it transfer its power to the surrounding medium, for instance the spatial resolution is 0.5 kpc. As such, we assume that a fraction of the jet power is injected as thermal energy in the diffuse gas around the AGN on kpc-scales. The size of the region which is affected by the feedback from the AGN may be as large as ≈ 10−15 kpc, although the jet itself is not resolved in the simulation. Two important questions arise: (a) are the heating effects due to jets generated in Seyfert galaxies able to affect kpc-scales and (b) can they be a source of feedback? Observations provides appealing clues to answering these questions. Several observational studies of Seyferts suggest that Seyfert jets may indeed reach kpc-scales and beyond. The Very Large Array surveys of Colbert et al. (1996) and Gallimore et al. (2006) find that extended kpc-scale radio outflows most probably asso- ciated with jets are common in Seyfert galaxies, although they cannot rule out entirely the possibility that these outflows arise from starburst-driven winds in some sources. These radio outflows have random orientations with respect to the host galaxy, a result which is maintained for jets observed from sub-kpc to kpc scales (Kinney et al. 2000; Schmitt et al. 2001), and have distorted, un- collimated morphologies. Therefore the observed radio outflows could provide a broad effective surface for depositing the jet power in the surrounding gas, as envisioned in our simulations. Several targeted observations of individual Seyfert galaxies also demon- strate that kpc-scale radio jets are not a rare feature in disk galaxies (e.g., Wilson & Ulvestad 1983; Morganti, Oosterloo, & Tsvetanov 1998; Cecil et al. 2000; Whittle & Wilson 2004; Keel et al. 2006; Kharb et al. 2006; Middelberg et al. 2007). X-ray observations also reveal kpc-scale outflows of X- ray-emitting hot plasma in Seyfert galaxies (Colbert et al. 1998; Rupke, Veilleux, & Sanders 2005), cospatial with the large scale radio outflows. Two favoured interpretations are that these kpc- scale X-ray outflows are generated either by the AGN jets that en- train and heat gas as they flow out of the nucleus, or through heat- ing of the surrounding gas via thermalisation of the kinetic power of the jet, generating a wind. Either way, these processes bear re- semblance to the generation of cavities or bubbles of X-ray emit- ting plasma in the central elliptical galaxies of clusters of galaxies, which are inflated by the radio jets (e.g., Dalla Vecchia et al. 2004; Bı̂rzan et al. 2004; Fabian et al. 2006). These observations there- fore provide evidence for the possible importance of feedback due to the radio jets in Seyfert galaxies, which could heat the circum- nuclear plasma and generate X-ray emitting outflows. Observational estimates of the kinetic power carried by the jets in Seyferts using different methods indicate that it is in the range ∼ 1042 − 1043erg s−1 (e.g., Wilson & Ulvestad 1983; Morganti, Oosterloo, & Tsvetanov 1998; Bicknell et al. 1998; Capetti et al. 1999; Kharb et al. 2006). This observational range is in agreement with the values of the jet power which are generated during the radio-loud episodes for z . 1 in the simulation, Ljet ≈ 10 42 − 1043 erg s−1 (Fig. 13). It is worth not- ing that the jet power in our simulation is estimated using a physical model for the energetics of the jet, which depends on fundamental parameters of central engine, especially the accretion rate which is self-consistently calculated. The fraction of the jet power generated by the AGN which is actually available to heat the gas on kpc-scales is a matter of debate. In the ‘reference’ simulation we adopt that a fraction ηAGN = 0.1 of the jet power is deposited as thermal energy of the diffuse gas, although it is possible that this may be an overestimate of the value of ηAGN. For instance, based on ram pressure and terminal ve- locity considerations of the lobes of kpc-scale radio outflows in Seyferts, Gallimore et al. (2006) argues that the Seyfert jets lose almost all their power within the inner kpc, implying that ηAGN may be considerably smaller than 0.1. The other simulations we present encompass the uncertainty in the value of ηAGN. In this re- c© 2007 RAS, MNRAS 000, 1–22 AGN feedback in cosmological simulations 19 spect, the simulation adopting ηAGN = 0.01 does not show any significant difference from the simulation without AGN feedback (ηAGN = 0). On the other hand, when 100% of the jet energy is transferred to the surrounding star forming gas (‘ism-fb’ model), most of the feedback energy is rapidly radiated away, and the re- sults are very similar to the ‘reference’ model in which 10% of jet energy is given to the ambient halo gas. Thus, a better knowledge on the “microscopic” physics of jet-ISM interactions is required in order to improve the understanding of the effects of AGN feedback in Seyfert galaxies. Our simulations show that during the cosmological forma- tion of a typical spiral galaxy, a low-luminosity AGN (Seyfert or LINER) is formed which has a stochastic cycle of activity, with many short-lived recurrent accretion events accompanied by jet outbursts. This finding agrees with the conclusion of Sanders (1984) that the Seyfert activity is the result of short-lived stochas- tic accretion events. He arrived at this conclusion by consider- ing plausible mechanisms that may fuel the central black holes of Seyferts, which are stochastic in nature, and our simulations track the evolution of the AGN by implementing one of these mechanisms: the accretion of molecular clouds to the galactic cen- tre. Our findings therefore provide support to the assumption of Hopkins & Hernquist (2006), central to their work, that the central supermassive BHs of Seyferts stochastically increase their mass by accretion of cold gas. In the period z . 1 after the galactic disk is formed, the AGN is active only during a fraction ≈ 9% of the its life in the ‘reference’ model, ≈ 25% in the ‘weak-fb’ model and ≈ 1% in the ‘ism- fb’ model. This implies that the lifetime of Seyfert activity in the spiral galaxy is ≈ 109 yr for the ‘reference’ or ‘weak-fb’ model and ≈ 108 yr for the ‘ism-fb’ model. This is in agreement with statistical Seyfert lifetime ∼ 108−109 yr estimated by counting the fraction of spiral galaxies that are observed to be Seyferts (Woltjer 1968; Sanders 1984; Ho et al. 1997). Sanders (1984) estimated that each single episode of activity would not last longer than 106 yr based on the size of Seyfert jets, a prediction which is in agreement with the detailed study of Kharb et al. (2006). Unfortunately the time-step we used to output snapshot files is 16.8 Myr and therefore we may miss many of such short-lived events. Considering this, the ‘ism-fb’ model may be more comfortable than the ‘reference’ model because there is no long-lived accretion event in the ‘ism- fb’ model for z . 1 (lower-left panel of Fig. 15). It is also possible that the observed short duration of activity is not caused by the large-scale mass accretion to the galactic centres but by the physical processes in the accretion disk (e.g., instabilities) which we do not take into account in this paper. 7 SUMMARY AND CONCLUSIONS In this paper, we have introduced a new methodology that enables us to simultaneously model star formation, SN feedback, BH ac- cretion, and AGN feedback in cosmological simulations of galaxy formation. Our treatment of black hole accretion is based on a the- oretical model of the mass accretion into the galactic centre due to radiation drag. This connects the kpc-scale star formation activity in the galactic centre with the BH accretion rate. Furthermore, mo- tivated by the recent semi-analytic models of Croton et al. (2006) and Bower et al. (2006), we have distinguished two distinct accre- tion modes of BH fuelling, namely standard (optically thick, geo- metrically thin) Shakura-Sunyaev disks and (geometrically thick, optically thin) RIAFs. This allows us to distinguish radio loud and radio quiet AGNs depending on their mass accretion rate. In this model, we only consider AGN radio feedback in which we assume a fraction of the jet power is thermally coupled with the surround- ing diffuse gas, and assume RIAFs are responsible for production of powerful jets. As the first application of this model, we have per- formed cosmological simulations of the formation of a disk galaxy and its coevolution with the central AGN. Our results reveal a fundamental AGN-starburst connection during the evolution of the galaxy and the central AGN. During the starburst period (z & 1) there is a lot of gas available to be accreted and the AGN is fuelled at high accretion rates, as a consequence the accretion disk is in the standard/thin state and the AGN is relatively luminous and produces weak jets (radio-quiet state). The starburst phase ceases at z ∼ 1 because most of the gas at the galactic cen- tre is consumed by star formation or blown out by the associated SN feedback. Afterwards, the AGN fuelling rate drops and the ac- cretion disk then switches to a radiatively inefficient (RIAF) state. Due to a combination of small accretion rates and the low radiative efficiency of the RIAF, the AGN becomes a low-luminosity AGN or Seyfert nucleus which is almost invisible compared to the host galaxy. At this point even a little star formation activity around the central BH triggers intense jet production that regulates further gas cooling to the centre. Thus, in the quiescent star forming regime after z ∼ 1 the AGN has a stochastic fuelling, undergoing several short-lived recurrent episodes of activity. During these accretion episodes the AGN brightness increases and jet outbursts are gen- erated, which correspond to the radio-loud state. In-between these Seyfert episodes the nucleus is turned off and the galaxy is inactive. Below z ∼ 1 AGN feedback becomes important if 10% of jet energy is available to heat the ambient diffuse halo gas (or 100% is fed into the surrounding ISM). This powerful episodic AGN feed- back suppresses star formation in the disc and almost completely halts the star formation in the bulge. This effect is minor when the wind speed due to SN feedback is fast compared to the circular velocity of the host halo because the SN feedback alone can sup- press the star formation in the bulge reasonably well. When the wind speed is slow compared to the circular velocity of the host halo, the AGN feedback becomes more important. This leads to the formation of an old, red elliptical galaxy in the ‘slow-wind’ model. Many properties of the simulated low-luminosity AGN and its jets are in broad agreement with observations of Seyfert galax- ies and their jets: namely its typical nuclear bolometric luminosity and radio luminosity, the coincidence of the radio-loud state with the Seyfert episodes, the fact that the jet may reach kpc-scales and strongly affect the surrounding gas on these scales, and the pre- dicted AGN lifetime and duration of individual episodes of activ- ity. The stochastic pattern of nuclear activity that emerges from our simulation seems to confirm the prediction of Sanders (1984) for Seyfert galaxies. The ratios between the central BH mass and the mass of the host spheroid at z = 0 in our simulations are ∼ 10−3 regard- less of the strength of either SN or AGN feedback. This result is direct outcome of the radiation drag model which relates the star formation rate in the central star-forming region and the mass ac- cretion rate to the central BH. The ratio is in good agreement with the observed ratio (Kormendy & Richstone 1995; Magorrian et al. 1998; Merritt & Ferrarese 2001; McLure & Dunlop 2002), al- though the efficiency of the radiation drag we assumed to ob- tain this ratio is somewhat higher than the value suggested by Kawakatu & Umemura (2002). However, as we have discussed, the geometry and density structure of star forming regions can easily change the efficiency. We also note that recent semi-analytic mod- c© 2007 RAS, MNRAS 000, 1–22 20 T. Okamoto, R. S. Nemmen and R. G. Bower els (Baugh et al. 2005; Nagashima et al. 2005a,b) and cosmologi- cal simulations (OEFJ05) suggest that stars may be born with top- heavy IMFs (and thus stronger radiation fields and higher radiation drag efficiencies) in starbursts. The accretion rate by radiation drag can be much higher with this assumption because stellar popula- tions with top-heavy IMFs produce more radiation. The self-consistent treatment of BH accretion and associated AGN feedback in galaxy formation enables us to explore the in- terplay between galaxies and AGNs in cosmological simulations of galaxy formation. The self-regulation of mass accretion due to feedback from RIAFs produces galaxies whose AGN properties are well matched to the observations. The model is also qualitatively consistent with recent semi-analytic models that successfully re- produce many of the global properties of the galaxy population. The effects of our AGN feedback implementation on formation of groups of galaxies, where more powerful AGN feedback from larger BHs is expected, will be presented in a forthcoming paper. ACKNOWLEDGMENTS We greatly appreciate the detailed reading and shrewd comments of the anonymous referee that strengthened this paper. We thank Takayuki Saitoh, Nozomu Kawakatu, Yasuyuki Watabe, Carlos Frenk, Tom Theuns and Thaisa Storchi-Bergmann for useful dis- cussions. TO acknowledges financial support from the Japan Soci- ety for the Promotion of Science for Young Scientists (1089) under which part of this work was carried out. RSN acknowledges support from the Brazilian institutions CNPq and CAPES, and by the Eu- ropean Commissions ALFA-II programme through its funding of the Latin-american European Network for Astrophysics and Cos- mology (LENAC). RGB thanks PPARC for the support of a senior research fellowship. We are grateful to Volker Springel for making the GADGET2 code public. All simulations were performed on the Cosmology Machine at the Institute for Computational Cosmology in Durham University. Part of analyses were carried out on GRAPE system at the Center for Computational Astrophysics, CfCA, of the National Astronomical Observatory of Japan (Project-ID: g06b13). REFERENCES Abadi M. G., Navarro J. F., Steinmetz M., Eke V. R., 2003, ApJ, 597, 21 Abazajian K., et al., 2003, AJ, 126, 2081 Allen S. W., Dunn R. J. H., Fabian A. C., Taylor G. B., Reynolds C. S., 2006, MNRAS, 372, 21 Antonucci R., 1993, ARA&A, 31, 473 Baugh C. M., Lacey C. G., Frenk C. S., Granato G. L., Silva L., Bressan A., Benson A. J., Cole S., 2005, MNRAS, 356, 1191 Bicknell G. V., Dopita M. A., Tsvetanov Z. I., Sutherland R. S., 1998, ApJ, 495, 680 Bı̂rzan L., Rafferty D. A., McNamara B. R., Wise M. W., Nulsen P. E. J., 2004, ApJ, 607, 800 Blandford R. D., Payne D. G., 1982, MNRAS, 199, 883 Blandford R. D., Znajek R. L., 1977, MNRAS, 179, 433 Bondi H., 1952, MNRAS, 112, 195 Bondi H., Hoyle F., 1944, MNRAS, 104, 273 Booth C. M., Theuns T., Okamoto T., 2007, MNRAS accepted (astro-ph/0701877) Bower R. G., Benson A. J., Malbon R., Helly J. C., Frenk C. S., Baugh C. M., Cole S., Lacey C. G., 2006, MNRAS, 370, 645 Baldry I. K., Glazebrook K., Brinkmann J., Ivezić Ž., Lupton R. H., Nichol R. C., Szalay A. S., 2004, ApJ, 600, 681 Capetti A., Axon D. J., Macchetto F. D., Marconi A., Winge C., 1999, ApJ, 516, 187 Cattaneo A., Blaizot J., Devriendt J., Guiderdoni B., 2005, MN- RAS, 364, 407 Cecil G., et al., 2000, ApJ, 536, 675 Cen R., Ostriker J. P., 1992, ApJL, 399, L113 Churazov E., Sazonov S., Sunyaev R., Forman W., Jones C., Böhringer H., 2005, MNRAS, 363, L91 Churazov E., Sunyaev R., Forman W., Böhringer H., 2002, MN- RAS, 332, 729 Colbert E. J. M., Baum S. A., O’Dea C. P., Veilleux S., 1998, ApJ, 496, 786 Colbert E. J. M., Baum S. A., Gallimore J. F., O’Dea C. P., Chris- tensen J. A., 1996, ApJ, 467, 551 Cowie L. L., McKee C. F., 1977, ApJ, 211, 135 Croton D. J., Springel V., White S. D. M., De Lucia G., Frenk C. S., Gao L., Jenkins A., Kauffmann G., Navarro J. F., Yoshida N., 2006, MNRAS, 365, 11 Dalla Vecchia C., 2005, PhD. thesis Dalla Vecchia C., Bower R. G., Theuns T., Balogh M. L., Maz- zotta P., Frenk C. S., 2004, MNRAS, 355, 995 Davis M., Efstathiou G., Frenk C. S., White S. D. M., 1985, ApJ, 292, 371 De Lucia G., Springel V., White S. D. M., Croton D., Kauffmann G., 2006, MNRAS, 366, 499 Di Matteo T., Croft R. A. C., Springel V., Hernquist L., 2003, ApJ, 593, 56 Di Matteo T., Springel V., Hernquist L., 2005, Nature, 433, 604 Di Matteo, T., Colberg, J., Springel, V., Hernquist, L., Sijacki, D., 2007, MNRAS, submitted (arXiv:0705.2269) Elmegreen B. G., 1989, ApJ, 338, 178 Enoki M., Nagashima M., Gouda N., 2003, PASJ, 55, 133 Escala A., Larson R. B., Coppi P. S., Mardones D., 2004, ApJ, 607, 765 Fabian A. C., Sanders J. S., Taylor G. B., Allen S. W., Crawford C. S., Johnstone R. M., Iwasawa K., 2006, MNRAS, 366, 417 Falcke H., Wilson A. S., Simpson C., 1998, ApJ, 502, 199 Ferrarese L., Merritt D., 2000, ApJL, 539, L9 Fioc M., Rocca-Volmerange B., 1997, A&A, 326, 950 Fukui Y., Mizuno N., Yamaguchi R., Mizuno A., Onishi T., 2001, PASJ, 53, L41 Gallimore J. F., Axon D. J., O’Dea C. P., Baum S. A., Pedlar A., 2006, AJ, 132, 546 Gebhardt K., Bender R., Bower G., Dressler A., Faber S. M., Fil- ippenko A. V., Green R., Grillmair C., Ho L. C., Kormendy J., Lauer T. R., Magorrian J., Pinkney J., Richstone D., Tremaine S., 2000, ApJL, 539, L13 Governato F., Willman B., Mayer L., Brooks A., Stinson G., Valenzuela O., Wadsley J., Quinn T., 2007, MNRAS, 374, 1479 Granato G. L., De Zotti G., Silva L., Bressan A., Danese L., 2004, ApJ, 600, 580 Haardt F., Madau P., 1996, ApJ, 461, 20 Hawley J. F., Krolik J. H., 2006, ApJ, 641, 103 Heger A., Fryer C. L., Woosley S. E., Langer N., Hartmann D. H., 2003, ApJ, 591, 288 Heyer M. H., Carpenter J. M., Snell R. L., 2001, ApJ, 551, 852 Ho L. C., Fillipenko A. V., Sargent W. L. W. 1997, ApJ, 487, 568 Ho L. C. 2004, in Coevolution of Black Holes and Galaxies, ed. L. C. Ho (Cambridge: Cambridge Univ. Press), 293 Ho L. C., Ulvestad J. S. 2001, ApJ, 555, 650 c© 2007 RAS, MNRAS 000, 1–22 http://arxiv.org/abs/astro-ph/0701877 http://arxiv.org/abs/0705.2269 AGN feedback in cosmological simulations 21 Hopkins P. F., Hernquist L., 2006, ApJS, 166, 1 Hopkins P. F., Hernquist L., Cox T. J., Di Matteo T., Robertson B., Springel V., 2006, ApJS, 163, 1 Hoyle F., Lyttleton R. A., 1939, in Proceedings of the Cambridge Philisophical Society The effect of interstellar matter on climatic variation. p. 405 Katz N., Weinberg D. H., Hernquist L., 1996, ApJS, 105, 19 Kauffmann G., Haehnelt M., 2000, MNRAS, 311, 576 Kaufmann T., Mayer L., Wadsley J., Stadel J., Moore B., 2007, MNRAS, 375, 53 Kawakatu N., Saitoh T. R., Wada K., 2005, ApJ, 628, 129 Kawakatu N., Umemura M., 2002, MNRAS, 329, 572 Kawakatu N., Umemura M., 2004, ApJ, 601, L21 Kawata D., Gibson B. K., 2005, MNRAS, 358, L16 Keel W. C., White R. E., III, Owen F. N., Ledlow M. J., 2006, AJ, 132, 2233 Kennicutt Jr. R. C., 1998, ApJ, 498, 541 Kinney A. L., Schmitt H. R., Clarke C. J., Pringle J. E., Ulvestad J. S., Antonucci R. R. J., 2000, ApJ, 537, 152 Kharb P., O’Dea C. P., Baum S. A., Colbert E. J. M., Xu C., 2006, ApJ, 652, 177 Kitzbichler M. G., White S. D. M., 2007, MNRAS, pp 75–+ Kormendy J., Richstone D., 1995, ARA&A, 33, 581 Kravtsov A. V., 2003, ApJL, 590, L1 Krolik J. H., 1999, in Krolik J. H., ed., Active Galactic Nu- clei: From the Central Black Hole to the Galactic Environment. Princeton Univ. Press, Princeton, NJ, c1999 Kukula M. J., Ghosh T., Pedlar A., Schilizzi R. T., Miley G. K., de Bruyn A. G., Saikia D. J., 1993, MNRAS, 264, 893 Kukula M. J., Holloway A. J., Pedlar A., Meaburn J., Lopez J. A., Axon D. J., Schilizzi R. T., Baum S. A., 1996, MNRAS, 280, Laor A., 2000, ApJ, 543, L111 Livio M., Ogilvie G. I., Pringle J. E., 1999, ApJ, 512, 100 Maccarone T. J., Gallo E., Fender R., 2003, MNRAS, 345, L19 Magorrian J., Tremaine S., Richstone D., Bender R., Bower G., Dressler A., Faber S. M., Gebhardt K., Green R., Grillmair C., Kormendy J., Lauer T., 1998, AJ, 115, 2285 Makino J., Funato Y., 2004, ApJ, 602, 93 Malbon R. K., Baugh C. M., Frenk C. S., Lacey C. G., 2007, MN- RAS (in press) Marconi A., Hunt L. K., 2003, ApJL, 589, L21 McLure R. J., Dunlop J. S., 2002, MNRAS, 331, 795 Meier D. L., 2001, ApJL, 548, L9 Merritt D., Ferrarese L., 2001, ApJ, 547, 140 Middelberg E., Agudo I., Roy A. L., Krichbaum T. P., 2007, MN- RAS, 377, 731 Monaco P., Fontanot F., 2005, MNRAS, 359, 283 Morganti R., Oosterloo T., Tsvetanov Z., 1998, AJ, 115, 915 Mundell C. G., Wrobel J. M., Pedlar A., Gallimore J. F., 2003, ApJ, 583, 192 Nagamine K., Springel V., Hernquist L., 2004, MNRAS, 348, 421 Nagashima M., Lacey C. G., Baugh C. M., Frenk C. S., Cole S., 2005a, MNRAS, 358, 1247 Nagashima M., Lacey C. G., Okamoto T., Baugh C. M., Frenk C. S., Cole S., 2005b, MNRAS, 363, L31 Narayan R., 2005, Ap&SS, 300, 177 Narayan R., Mahadevan R., Quataert E., 1998, in Abramowicz M. A., Bjornsson G., Pringle J. E., eds, Theory of Black Hole Accretion Disks Advection-dominated accretion around black holess. p. 148 Narayan R., Yi I., 1995, ApJ, 444, 231 Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563 Navarro J. F., White S. D. M., 1993, MNRAS, 265, 271 Nemmen R. S., Bower R. G., Babul A., Storchi-Bergmann T., 2007, MNRAS, 377, 1652 Nemmen R. S., Storchi-Bergmann T., Yuan F., Eracleous M., Terashima Y., Wilson A. S., 2006, ApJ, 643, 652 Novikov I. D., Thorne K. S., 1973, in DeWitt C., DeWitt B. S., eds, Black Holes. New York: Gordon & Breach, p. 344 Okamoto T., Eke V. R., Frenk C. S., Jenkins A., 2005, MNRAS, 363, 1299 (OEFJ05) Okamoto T., Jenkins A., Eke V. R., Quilis V., Frenk C. S., 2003, MNRAS, 345, 429 Omma H., Binney J., Bryan G., Slyz A., 2004, MNRAS, 348, Quataert E., 2001, in Peterson B. M., Pogge R. W., Polidan R. S., eds, ASP Conf. Ser. 224: Probing the Physics of Active Galactic Nuclei Low Radiative-Efficiency Accretion Flows. p. 71 Quilis V., Bower R. G., Balogh M. L., 2001, MNRAS, 328, 1091 Rafferty D. A., McNamara B. R., Nulsen P. E. J., Wise M. W., 2006, ApJ, 652, 216 Rees M. J., Phinney E. S., Begelman M. C., Blandford R. D., 1982, Nature, 295, 17 Riffel R. A., Storchi-Bergmann T., Winge C., Barbosa F. K. B., 2006, MNRAS, 373, 2 Rupke D. S., Veilleux S., Sanders D. B., 2005, ApJ, 632, 751 Salpeter E. E., 1955, ApJ, 121, 161 Samland M., Gerhard O. E., 2003, A&A, 399, 961 Sanders R. H., 1984, A&A, 140, 52 Sato J., Umemura M., Sawada K., Matsuyama S., 2004, MNRAS, 354, 176 Scannapieco C., Tissera P. B., White S. D. M., Springel V., 2006, MNRAS, 371, 1125 Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 Schmitt H. R., Ulvestad J. S., Antonucci R. R. J., Kinney A. L., 2001, ApJS, 132, 199 Sijacki D., Springel V., 2006, MNRAS, 366, 397 Sijacki D., Springel V., di Matteo T., Hernquist L., 2007, MNRAS, 380, 877 Solomon P. M., Rivolo A. R., 1989, ApJ, 339, 919 Soltan A., 1982, MNRAS, 200, 115 Spitzer L. J., 1962, Physics of Fully Ionized Gases. New York: Interscience Spitzer L. J., 1978, Physical Processes in the Interstellar Medium. New York: Wiley-Interscience Springel V., 2005, MNRAS, 364, 1105 Springel V., Di Matteo T., Hernquist L., 2005a, ApJL, 620, L79 Springel V., Di Matteo T., Hernquist L., 2005b, NNRAS, 361, 776 Springel V., Hernquist L., 2003, MNRAS, 339, 289 (SH03) Springel V., Yoshida N., White S. D. M., 2001, NewA, 6, 79 Steinmetz M., Mueller E., 1994, A&A, 281, L97 Taylor G. B., Sanders J. S., Fabian A. C., Allen S. W., 2006, MN- RAS, 365, 705 Terashima Y., Wilson A. S., 2003, ApJ, 583, 145 Theuns T., Schaye J., Zaroubi S., Kim T.-S., Tzanavaris P., Car- swell B., 2002, ApJL, 567, L103 Tremaine S., Gebhardt K., Bender R., Bower G., Dressler A., Faber S. M., Filippenko A. V., Green R., Grillmair C., Ho L. C., Kormendy J., Lauer T. R., Magorrian J., Pinkney J., Richstone D., 2002, ApJ, 574, 740 Umemura M., 2001, ApJL, 560, L29 Volonteri M., Madau P., Quataert E., Rees M. J., 2005, ApJ, 620, c© 2007 RAS, MNRAS 000, 1–22 22 T. Okamoto, R. S. Nemmen and R. G. Bower Wada K., 2001, ApJL, 559, L41 Wada K., Norman C. A., 2007, ApJ, 660, 276 Wada K., Norman C. A., 2001, ApJ, 547, 172 Wang J.-M., Chen Y.-M., Ho L. C., McLure R. J., 2006, ApJL, 642, L111 Whittle M., Wilson A. S., 2004, AJ, 127, 606 Wilson A. S., Ulvestad J. S., 1983, ApJ, 275, 8 Woltjer L., 1968, AJ, 73, 914 Yepes G., Kates R., Khokhlov A., Klypin A., 1997, MNRAS, 284, Yu Q., Tremaine S., 2002, MNRAS, 335, 965 Yuan F., 2007, in ASP Conf. Ser. 373, The Central Engine of Ac- tive Galactic Nuclei, ed. L. C. Ho & J.-M. Wang (San Francisco: ASP), 95 c© 2007 RAS, MNRAS 000, 1–22 Introduction The model of star-forming gas and stellar feedback Gas cooling Cloud formation and the cloud model Star formation Cloud evaporation Supernova feedback and chemical enrichment Numerical implementation Winds Parameter setting and test simulations Black hole accretion and AGN feedback Seed black holes Mass accretion by the radiation drag AGN feedback Cosmological simulations of disc galaxy formation with AGN feedback Black hole particles Models Results Simulations with different AGN feedback efficiencies Relative importance of the AGN Dependence on feedback parameters and implementations of feedback Comparison with observations of AGN in spiral galaxies Summary and Conclusions